Methods and systems for beam control in quantum computers

Abstract

Systems, methods, and computer-readable media for a virtually imaged phased array (VIPA) device may include: an electro-optic material, wherein a path of a plurality of optical beams output from said VIPA device is modifiable via applying a voltage to said electro-optic material of said VIPA device.

Description

WSGR Docket No. 55436-733.601METHODS AND SYSTEMS FOR BEAM CONTROL IN QUANTUM COMPUTERS CROSS-REFERENCE

[0001] This application claims the benefit of U.S. Provisional Patent Application No. 63 / 698,499, entitled “METHODS AND SYSTEMS FOR VIRTUALLY IMAGED PHASED ARRAY CONTROL FOR NON-CLASSICAL COMPUTING,” filed on September 24, 2024, U.S. Provisional Patent Application No. 63 / 717,117, entitled “METHODS AND DEVICES FOR MULTIPLEXED SPOT PROJECTION,” filed on November 6, 2024, U.S. Provisional Patent Application No. 63 / 717,239, entitled “METHODS AND SYSTEMS FOR MULTITRANSDUCER AODS FOR INDEPENDENT GENERATION OF OPTICAL BEAMS IN TWO DIMENSIONS,” filed on November 6, 2024, and U.S. Provisional Patent Application No. 63 / 717,210, entitled “METHODS AND SYSTEMS FOR COMBINING MANY TWEEZERS BY WAVELENGTH MULTIPLEXING,” filed on November 6, 2024, each of which is incorporated by reference herein in its entirety.BACKGROUND

[0002] Quantum computers may make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers may be different from digital electronic computers based on transistors. For instance, whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits), which can be in superpositions of states.

[0003] In neutral-atom quantum computers or simulation devices, qubits may be encoded in optically trapped atoms. The qubit can be represented by a linear superposition of its two orthonormal basis states. The two orthonormal basis states are usually denoted as |0) = [ ] (the “zero state”) and | 1) = (the “one state”). The two orthonormal basis states,{| 0), 11)}, together called the computational basis, span the two-dimensional linear vector (Hilbert) space of the qubit. The basis states can also be combined to form product basis states, e.g., |00), 101), 110), | 11), each called a quantum register. For example, n qubits may be represented by a superposition state vector in 2ndimensional Hilbert space.SUMMARY

[0004] In an aspect, the present disclosure provides a device for beam steering, comprising a medium that is at least partially optically transparent, wherein said medium is bounded by a first surface that is partially internally reflective, and a second surface that is opposed to said first surface and is at least partially internally reflective, said medium forming a transmission guide,WSGR Docket No. 55436-733.601 between said first surface and said second surface, for an input optical beam that transmits, from said second surface, a plurality of output optical beams, and wherein said medium comprises an electro-optic material having an index of refraction that is responsive to a voltage difference, and a plurality of electrodes electrically coupled to said medium, wherein said plurality of electrodes are configured to generate an electric field, thereby changing said index of refraction of said electro-optic material of said medium to change a relative phase of said plurality of output optical beams.

[0005] In some embodiments, said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate. In some embodiments, said plurality of output optical beams crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits. In some embodiments, said plurality of qubits form at least a portion of a non-classical computer. In some embodiments, said first surface comprises a first portion that is substantially transparent and a second portion that is substantially fully internally reflective. In some embodiments, said second surface is substantially fully internally reflective. In some embodiments, said medium comprises a first face comprising said first surface, a second face comprising said second surface, a third face, and a fourth face. In some embodiments, said plurality of electrodes are electrically coupled to said medium via connecting to said third face and said fourth face. In some embodiments, said medium is a rectangular prism. In some embodiments, a set of virtual images of a light source said input optical beam are aligned at a constant interval. In some embodiments, said electric field is oriented substantially along an axis of optical polarization of said electrooptic material of said medium. In some embodiments, the device further comprises further comprising a second device for beam steering described herein, wherein said pair of devices is configured to steer a beam in two dimensions.

[0006] In another aspect, the present disclosure provides a virtually imaged phased array (VIP A) device comprising an electro-optic material, wherein a path of a plurality of optical beams output from said VIPA device is modifiable via applying an electric field to said electro-optic material of said VIPA device.

[0007] In some embodiments, wherein said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate. In some embodiments, said path of said plurality of optical beams output from said VIPA device crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits. In some embodiments, said plurality of qubits form at least a portion of a non-classical computer. In some embodiments, a set of virtual images of a light source of said VIPA device are aligned at a constant interval. In some embodiments, said electricWSGR Docket No. 55436-733.601 field is oriented substantially along an axis of optical polarization of said electro-optic material of said VIPA device. In some embodiments, the VIPA device further comprises a second VIPA device described herein, wherein said pair of VIPA devices is configured to steer said path of said plurality of optical beams in two dimensions.

[0008] In another aspect, the present disclosure provides a method for beam steering, comprising directing a optical beam into a virtually imaged phase array (VIPA) device that comprises an electro-optic material, and applying an electric field to said electro-optic material of said VIPA device to modify an index of refraction of said electro-optic material of said VIPA device, thereby controlling a path of a plurality of output optical beams from said VIPA device.

[0009] In some embodiments, said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate. In some embodiments, said path of said plurality of optical beams output from said VIPA device crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits. In some embodiments, said plurality of qubits for at least a portion of a non-classical computer. In some embodiments, the method further comprising a second VIPA device herein, and further comprising steering, using said pair of VIPA devices, said path of said plurality of optical beams in two dimensions. In some embodiments, the method further comprises substantially aligning a set of virtual images of a light source of said VIPA device. In some embodiments, said VIPA device emits a substantially collimated optical beam in one direction. In some embodiments, the method further comprises directing said substantially collimated optical beam at a wavelength dependent angle, wherein said angle is configured to produce angular dispersion of said substantially collimated optical beam. In some embodiments, said electric field is oriented substantially along an axis of optical polarization of said electrooptic material of said VIPA device.

[0010] In another aspect, the present disclosure provides a system for generating multiple optical beams, said system comprising an acousto-optic deflector (AOD) comprising an acousto-optic material, wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to generate an acoustic wave in said acousto-optic material of said AOD, wherein said acoustic wave is configured to generate a plurality of independent output optical beams from an input optical beam directed to the AOD.

[0011] In some embodiments, said acoustic wave comprises one or more discrete momentum components. In some embodiments, each discrete momentum component is associated with a drive vector. In some embodiments, a drive vector comprises a set of one or more drive signals. In some embodiments, one or more drive signals share a common frequency. In some embodiments, said one or more drive signals is configured to be applied to two or more acousticWSGR Docket No. 55436-733.601 transducers of said plurality of acoustic transducers, wherein said drive signals further comprise a phase relationship between two or more signals applied to said two or more acoustic transducers of said plurality of acoustic transducers. In some embodiments, said phase relationship comprises a spatial gradient. In some embodiments, each independent output optical beam corresponds to a drive vector. In some embodiments, an independent optical beam is deflected based at least in part on a frequency and phase gradient of a corresponding drive vector. In some embodiments, said plurality of transducers is operated in an acoustic near-field limit. In some embodiments, said plurality of transducers is operated in said acoustic far-field limit.

[0012] In another aspect, the present disclosure provides a method for generating multiple optical beams, the method comprising providing an acousto-optic deflector (AOD) comprising an acousto-optic material, wherein said AOD comprises a plurality of acoustic transducers, and generating, using said plurality of acoustic transducers, an acoustic wave in said acousto-optic material of said AOD, wherein said acoustic wave is configured to generate a plurality of independent output optical beams from an input optical beam directed to the AOD.

[0013] In some embodiments, said acoustic wave comprises one or more discrete momentum components. In some embodiments, each discrete momentum component is associated with a drive vector. In some embodiments, a drive vector comprises a set of one or more drive signals. In some embodiments, said one or more drive signals share a common frequency. In some embodiments, said one or more drive signals is configured to be applied to two or more acoustic transducers of said plurality of acoustic transducers, wherein said drive signals further comprise a phase relationship between two or more signals applied to said two or more acoustic transducers of said plurality of acoustic transducers. In some embodiments, said phase relationship comprises a spatial gradient. In some embodiments, each independent output optical beam corresponds to a drive vector. In some embodiments, the method further comprises deflecting an independent output optical beam based at least in part on a frequency and phase gradient of a corresponding drive vector. In some embodiments, said plurality of transducers is operated in an acoustic near- field limit. In some embodiments, said plurality of transducers is operated in said acoustic far- field limit.

[0014] In another aspect, the present disclosure provides a system for optical control, the system comprising an optical source configured to generate an optical beam having an optical path, and an acousto-optic deflector (AOD), wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to generate an acoustic wave in said AOD that is configured to modify said optical path of said optical beam at least in part while said optical beam passes through said AOD.WSGR Docket No. 55436-733.601

[0015] In some embodiments, said plurality of acoustic transducers is driven in a phase -coherent manner to generate multiple independent output beams.

[0016] In another aspect, the present disclosure provides a system for generating multiple optical beams, said system comprising an acousto-optic deflector (AOD), wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to be driven in a phase-coherent manner to generate multiple independent output beams.

[0017] In some embodiments, substantially the same frequency is applied to each acoustic transducer of said plurality of acoustic transducers. In some embodiments, a deflected optical azimuthal angle, perpendicular to said plurality of acoustic transducers, may be changed based at least in part on applied frequency. In some embodiments, one or more spatial phase gradients are applied to one or more frequencies applied to said plurality of acoustic transducers to vary a deflected elevation angle in said direction along said plurality of acoustic transducers. In some embodiments, said one or more spatial phase gradients are applied in a first dimension, and one or more frequencies are applied in a second dimension, thereby yielding two-dimensional control of a deflected output beam. In some embodiments, a sum of said one or more frequencies used to generate each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams. In some embodiments, said plurality of transducers is operated in said acoustic near-field limit. In some embodiments, said plurality of transducers is operated in said acoustic far-field limit.

[0018] In another aspect, the present disclosure provides a system for modulating a phase of an optical beam, said system comprising an acoustic medium configured to propagate an acoustic wave, wherein said acoustic wave is configured to transform said phase of said optical beam into a modulated phase at least in part while said optical beam passes through said acoustic medium.

[0019] In another aspect, the present disclosure provides an acousto-optic deflector (AOD), said AOD comprising an acousto-optic medium configured to propagate an acoustic wave, and a plurality of acoustic transducers configured to provide an acoustic wave, wherein said acoustic wave is configured to generate a plurality of optical beams from an optical beam incident on said acoustic medium.

[0020] In another aspect, the present disclosure provides a method for generating multiple optical beams, said method comprising providing an acousto-optic deflector (AOD) comprising a plurality of acoustic transducers, and driving said plurality of acoustic transducers in a phase- coherent manner, thereby generating multiple independent output beams.

[0021] In another aspect, the present disclosure provides a method of optical control, said method comprising generating an optical beam having an optical path, providing an acousto- optic deflector (AOD), and modifying said optical path of said optical beam at least in part whileWSGR Docket No. 55436-733.601 said optical beam passes through said AOD, wherein said modifying of said optical path is controlled by a plurality of acoustic transducers.

[0022] In some embodiments, said modifying of said optical path is controlled by a plurality of acoustic transducers comprises generating, using said plurality of acoustic transducers, an acoustic wave configured to modify said optical path of said optical beam at least in part while said optical beam passes through said AOD. In some embodiments, substantially the said same frequency is applied to each acoustic transducer of said plurality of acoustic transducers. In some embodiments, a deflected optical azimuthal angle of said optical path, perpendicular to said plurality of acoustic transducers, may be changed based on an applied frequency. In some embodiments, one or more spatial phase gradients are applied to one or more frequencies applied to said plurality of acoustic transducers to vary a deflected elevation angle in a direction along said plurality of acoustic transducers. In some embodiments, said one or more spatial phase gradients are applied in a first dimension, and one or more frequencies are applied in a second dimension, thereby yielding two-dimensional control of a deflected output beam. In some embodiments, a sum of said one or more frequencies used to generate each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams. In some embodiments, a sum of said one or more spatial phase gradients used to deflect each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams. In some embodiments, a sum of said one or more frequencies used to generate each individual optical beam, and a sum of one or more spatial phase gradients used to deflect each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams. In some embodiments, said plurality of transducers is operated in said acoustic near -field limit. In some embodiments, said plurality of transducers is operated in said acoustic far-field limit.

[0023] In another aspect, the present disclosure provides a control system for generating drive vector control, the system comprising at least one controller, wherein a controller of the at least one controller is configured to deconstruct an input control signal into an output control signal, wherein said output control signal of said controller of said at least one controller is configured to generate an acoustic wave in an acousto-optic medium.

[0024] In some embodiments, said at least one controller comprises two or more controllers. In some embodiments, said two or more controllers are configured to generate an acoustic wave in an acousto-optic medium, at least in part, by combining output control signals from each of the two or more controllers. In some embodiments, said controller of said at least one controller is configured to generate said acoustic wave in said acousto-optic medium, at least in part, by at least one acoustic transducer. In some embodiments, said at least one acoustic transducerWSGR Docket No. 55436-733.601 comprises two or more acoustic transducers. In some embodiments, said input control signal comprises one or more drive vectors, wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam. In some embodiments, said input control signal comprises a piecewise function. In some embodiments, said input control signal comprises an interpolated curve. In some embodiments, said input control signal comprises a B- spline curve. In some embodiments, said input control signal comprises a Bezier curve. In some embodiments, said input control signal comprises a parameterized curve and a set of offsets. In some embodiments, said set of offsets comprises a plurality of offset vectors. In some embodiments, a memory, wherein said plurality of offset vectors are stored in said memory. In some embodiments, said plurality of offset vectors further comprises a location in said memory, and wherein said input control signal further comprises locations in said memory corresponding to each of said plurality of offset vectors. In some embodiments, said plurality of offset vectors comprise one or more displacement vectors specified on a grid. In some embodiments, said grid comprises a regular grid. In some embodiments, said regular grid comprises a rectangular grid. In some embodiments, said rectangular grid comprises a square grid. In some embodiments, said grid comprises a non-regular grid. In some embodiments, said two or more controllers are configured to deconstruct said input control signal into distinct output control signals. In some embodiments, said distinct output control signals correspond to distinct regions of a target area. In some embodiments, said distinct regions of a target area comprise overlapping regions. In some embodiments, said distinct regions of a target area comprise non-overlapping regions. In some embodiments, said distinct regions of a target area comprise both overlapping regions and non-overlapping regions. In some embodiments, said distinct regions of a target area are rectangular regions. In some embodiments, said distinct output control signals are selected to reduce crossings between distinct region of said target area. In some embodiments, deconstructing said input control signal into said distinct output control signals comprises oversampling, wherein said oversampling is configured to provide smooth movement from a first region of said target area corresponding to a first control signal to a second region of said target area corresponding to a second control signal. In some embodiments, said input control signal further comprises one or more pre-calculated moves. In some embodiments, a move of said one or more pre-calculated moves comprises one or more drive vectors wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam. In some embodiments, said one or more pre-calculated moves comprises one or more individual movements of an optical beam. In some embodiments, said one or more pre-calculated moves comprises one or more movements of a group of optical beams. In some embodiments, said one or more pre-calculated moves is initiated, at least in part, by an external trigger. In someWSGR Docket No. 55436-733.601 embodiments, said external trigger comprises an external command signal from one or more users. In some embodiments, said external trigger comprises an autonomous trigger. In some embodiments, said autonomous trigger comprises a clock-based periodic trigger. In some embodiments, the control system is configured to hold a position of one or more optical beams absent said input control signal. In some embodiments, said input control signal further comprises one or more intensity commands, wherein an intensity command of said one or more intensity commands is configured to modulate an intensity of an optical beam. In some embodiments, an intensity command of said one or more intensity commands is configured to turn an optical beam on and off. In some embodiments, an intensity command of said one or more intensity commands is configured to provide time dependent control of an intensity of an optical beam. In some embodiments, said input control signal comprises a phase gradient control signal. In some embodiments, said phase gradient control signal further comprises a global phase. In some embodiments, said global phase is chosen randomly.

[0025] In another aspect, the present disclosure provides a method of generating drive vector control, the method comprising deconstructing, using at least one controller, an input control signal to produce an output control signal, and generating using said output control signal, an acoustic wave in an acousto-optic medium.

[0026] In some embodiments, said at least one controller comprises two or more controllers. In some embodiments, generating is done, at least in part, by combining output control signals from each of the two or more controllers. In some embodiments, generating) is performed, at least in part, by at least one acoustic transducer. In some embodiments, said at least one acoustic transducer comprises two or more acoustic transducers. In some embodiments, said input control signal comprises one or more drive vectors, wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam. In some embodiments, said input control signal comprises a piecewise function. In some embodiments, said input control signal comprises an interpolated curve. In some embodiments, said input control signal comprises a B-spline curve. In some embodiments, said input control signal comprises a Bezier curve. In some embodiments, said input control signal comprises a parameterized curve and a set of offsets. In some embodiments, said set of offsets comprises a plurality of offset vectors. In some embodiments, the method further comprises storing said plurality of offset vectors in a memory. In some embodiments, said plurality of offset vectors further comprises a location in said memory, and wherein said input control signal further comprises locations in said memory corresponding to each of said plurality of offset vectors. In some embodiments, said plurality of offset vectors comprise one or more displacement vectors specified on a grid. In some embodiments, said grid comprises a regular grid. In some embodiments, said regular gridWSGR Docket No. 55436-733.601 comprises a rectangular grid. In some embodiments, said rectangular grid comprises a square grid. In some embodiments, said grid comprises a non-regular grid. In some embodiments, the method further comprises deconstructing, at least in part using said two or more controllers, said input control signal into distinct output control signals. In some embodiments, said distinct output control signals correspond to distinct regions of a target area. In some embodiments, said distinct regions of a target area comprise overlapping regions. In some embodiments, said distinct regions of a target area comprise non-overlapping regions. In some embodiments, said distinct regions of a target area comprise both overlapping regions and non-overlapping regions. In some embodiments, said distinct regions of a target area are rectangular regions. In some embodiments, the method further comprises selecting said distinct output control signals to reduce crossings between distinct region of said target area. In some embodiments, deconstructing said input control signal into said distinct output control signals further comprises oversampling, wherein said oversampling is configured to provide smooth movement from a first region of said target area corresponding to a first control signal to a second region of said target area corresponding to a second control signal. In some embodiments, the method further comprises pre-calculating one or more moves of said input control signal. In some embodiments, pre-calculating said one or more moves comprises pre-calculating one or more drive vectors wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam. In some embodiments, pre-calculating said one or more moves comprises pre-calculating one or more individual movements of an optical beam. In some embodiments, pre-calculating one or more moves comprises pre-calculating one or more movements of a group of optical beams. In some embodiments, the method further comprises, initiating said pre-calculated one or more moves, at least in part, by an external trigger. In some embodiments, said external trigger comprises receiving an external command signal from one or more users. In some embodiments, said external trigger comprises an autonomous trigger. In some embodiments, aid autonomous trigger comprises a clock -based periodic trigger. In some embodiments, the method further comprises holding a position of one or more optical beams absent said input control signal. In some embodiments, the method further comprises controlling an intensity of one or more optical beams, at least in part, using said input control signal. In some embodiments, controlling said intensity of one or more optical beams further comprises turning at least one optical beam of said one or more optical beams on or off. In some embodiments, controlling said intensity of one or more optical beams further comprises time dependent control of an intensity of one or more optical beams. In some embodiments, controlling said intensity of one or more optical beams is done, at least in part, using a phase gradient control signal. In someWSGR Docket No. 55436-733.601 embodiments, said phase gradient control signal further comprises a global phase. In some embodiments, said global phase is chosen randomly.

[0027] In another aspect, the present disclosure provides a system for array generation, the system comprising one or more optical sources configured to generate one or more optical beams, one or more acousto-optic deflectors (AODs), and one or more spatial light modulators (SLMs), wherein said one or more AODs is illuminated by said one or more optical beams, and the one or more AODs are configured to illuminate the one or more SLMs with one or more transmitted optical beams.

[0028] In some embodiments, an SLM of said one or more SLMs comprises one or more SLM masks configured to produce a specific pattern from an emitted optical beam. In some embodiments, an SLM mask of said one or more SLM masks comprises a superimposed correction pattern. In some embodiments, the system further comprises one or more relay telescopes. In some embodiments, said one or more AODs comprises a pair of AODs arranged in a crossed AOD (xAOD) configuration. In some embodiments, the system further comprises a first relay telescope of said one or more relay telescopes, said first relay telescope located in an optical path of said one or more transmitted optical beams between a first AOD of said pair of AODs and a second AOD of said pair of AODs. In some embodiments, the system further comprises a second relay telescope of said one or more relay telescopes, said second relay telescope located in an optical path of said one or more transmitted optical beams between said second AOD of said pair of AODs and an SLM of said one or more SLMs. In some embodiments, said one or more relay telescopes comprise one or more of: a 4f relay telescope, a non 4f relay telescope, a two element relay telescope, a 2fl relay telescope, a 2f2 relay telescope, a two element 2fl + 2f2 relay telescope, or an unobscured two-mirror relay telescope.

[0029] In another aspect, the present disclosure provides a method for array generation, said method comprising generating one or more optical beams, illuminating one or more acousto- optic deflectors (AODs) with said one or more optical beams, said one or more AODs thereby producing one or more transmitted optical beams, and illuminating one or more spatial light modulators (SLMs) with said one or more transmitted optical beams.

[0030] In some embodiments, an SLM of said one or more SLMs comprises one or more SLM masks configured to produce a specific pattern from an emitted optical beam. In some embodiments, an SLM mask of said one or more SLM masks comprises a superimposed correction pattern. In some embodiments, said one or more AODs comprises a pair of AODs arranged in a crossed AOD (xAOD) configuration. In some embodiments, the method further comprises one or more relay telescopes. In some embodiments, the method further comprises projecting said one or more transmitted optical beams from said one or more AODs using a relayWSGR Docket No. 55436-733.601 telescope of said one or more relay telescopes. In some embodiments, the method further comprises projecting, using a first relay telescope of said one or more relay telescopes, said one or more transmitted optical beams from a first AOD of said pair of AODs onto a second AOD of said pair of AODs. In some embodiments, the method further comprises projecting, using a second relay telescope of said one or more relay telescopes, said one or more transmitted optical beams from said second AOD of said pair of AODs onto an SLM of said one or more SLMs. In some embodiments, said one or more relay telescopes comprise one or more of: a 4f relay telescope, a non 4f relay telescope, a two element relay telescope, a 2fl relay telescope, a 2f2 relay telescope, a two element 2fl + 2f2 relay telescope, or an unobscured two-mirror relay telescope. In some embodiments, the method further comprises generating a different pattern of spots of light at a different location. In some embodiments, the method further comprises, generating the same pattern of spots of light at a different location. In some embodiments, the method further comprises generating a different pattern of spots of light at the same location.

[0031] In another aspect, the present disclosure provides a method of optically addressing atoms, said method comprising obtaining a plurality of atoms in a plurality of spatially distinct optical trapping sites, and addressing at least a subset of said plurality of atoms with a plurality of optical tweezers, said plurality of optical tweezers collectively having multiplexed wavelengths combined on an optical grating.

[0032] In some embodiments, said optical grating corresponds, at least in part, to one or more real-space planes of said plurality of atoms. In some embodiments, said optical grating comprises a millimeter-scale optical grating. In some embodiments, said optical grating comprises a meterscale optical grating. In some embodiments, said optical grating corresponds, at least in part, to one or more Fourier space planes of said plurality of atoms. In some embodiments, said optical grating comprises a microlens array. In some embodiments, addressing is done, at least in part, using spectrally broadband light. In some embodiments, addressing is done, at least in part, using spatially incoherent light. In some embodiments, addressing is done, at least in part, using a lightfield directing array (LDA). In some embodiments, said LDA is a one-dimensional LDA. In some embodiments, said LDA is a two-dimensional LDA. In some embodiments, aid optical grating comprises a blazed grating. In some embodiments, said optical grating comprises an echelle grating. In some embodiments, addressing is done, at least in part, using a virtually imaged phase array (VIP A) optical grating pair. In some embodiments, said optical grating corresponds to an intermediate plane between a real-space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms.

[0033] In another aspect, the present disclosure provides a method for optically addressing atoms, said method comprising obtaining a plurality of atoms in a plurality of spatially distinctWSGR Docket No. 55436-733.601 optical trapping sites, and addressing at least a first subset of said plurality of atoms with a first plurality of optical tweezers, said first plurality of optical tweezers collectively having multiplexed wavelengths corresponding, at least in part, to one or more real-space planes of said plurality of atoms, or addressing at least a second subset of said plurality of atoms with a second plurality of optical tweezers, said second plurality of optical tweezers collectively having multiplexed wavelengths corresponding, at least in part, to one or more Fourier space planes of said plurality of atoms, or interpolating between said first plurality of optical tweezers and said second plurality of optical tweezers, at least in part, using an optical grating in an intermediate optical plane.

[0034] In some embodiments, said optical grating corresponds, at least in part, to one or more real-space planes of said plurality of atoms. In some embodiments, said optical grating comprises a millimeter-scale optical grating. In some embodiments, said optical grating comprises a meterscale optical grating. In some embodiments, said optical grating corresponds, at least in part, to one or more Fourier space planes of said plurality of atoms. In some embodiments, said optical grating comprises a microlens array. In some embodiments, addressing is done, at least in part, using spectrally broadband light. In some embodiments, addressing is done, at least in part, using spatially incoherent light. In some embodiments, addressing is done, at least in part, using a lightfield directing array (LDA). In some embodiments, said LDA is a one-dimensional LDA. In some embodiments, said LDA is a two-dimensional LDA. In some embodiments, aid optical grating comprises a blazed grating. In some embodiments, said optical grating comprises an echelle grating. In some embodiments, addressing is done, at least in part, using a virtually imaged phase array (VIP A) optical grating pair. In some embodiments, said optical grating comprises a millimeter scale optical grating. In some embodiments, said optical grating comprises a meter scale optical grating. In some embodiments, said optical grating comprises a microlens array (MLA). In some embodiments, said plurality of atoms corresponds to a plurality of qubits. In some embodiments, said optical grating corresponds to a real-space plane of said plurality of atoms. In some embodiments, said optical grating corresponds to an intermediate plane between a real-space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms. In some embodiments, wherein said optical grating corresponds to an intermediate plane between a real-space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms.

[0035] In another aspect, the present disclosure provides a system for optically addressing atoms, said system comprising a plurality of atoms in a plurality of spatially distinct optical trapping sites, an optical grating, and an optical source configured to supply a plurality of optical tweezers, said plurality of optical tweezers collectively having multiplexed wavelengths, whereinWSGR Docket No. 55436-733.601 said plurality of optical tweezers is directed onto said plurality of atoms, at least in part, using said optical grating.

[0036] In some embodiments, said optical source comprises spectrally broadband light. In some embodiments, said optical source comprises spatially incoherent light. In some embodiments, said optical source comprises a plurality of optical sources. In some embodiments, the system further comprises a lightfield directing array (LDA). In some embodiments, said LDA is a onedimensional LDA. In some embodiments, said LDA is a two-dimensional LDA. In some embodiments, said optical grating comprises a blazed grating. In some embodiments, said optical grating comprises an echelle grating. In some embodiments, said system further comprises a virtually imaged phase array (VIP A) optical grating pair. In some embodiments, said optical grating comprises a millimeter scale optical grating. In some embodiments, said optical grating comprises a meter scale optical grating. In some embodiments, said optical grating comprises a microlens array (MLA). In some embodiments, said plurality of atoms corresponds to a plurality of qubits. In some embodiments, said optical grating corresponds to a real -space plane of said plurality of atoms. In some embodiments, said optical grating corresponds to a Fourier-space plane of said plurality of atoms. In some embodiments, said optical grating corresponds to an intermediate plane between a real-space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms.

[0037] In some embodiments, the systems, methods or devices herein further comprises a plurality of atoms. In some embodiments, said plurality of atoms comprises a plurality of ions. In some embodiments, said plurality of atoms comprises a plurality of neutral atoms. In some embodiments, said plurality of atoms comprises a plurality of group II atoms. In some embodiments, said plurality of atoms comprises a plurality of qubits. In some embodiments, a deflected optical beam is used, at least in part, to control said plurality of atoms. In some embodiments, said control of said plurality of atoms comprises a moving an atom of said plurality of atoms. In some embodiments, said control of said plurality of qubits comprises a non- classical computation.

[0038] Another aspect of the present disclosure provides a system comprising one or more computer processors and computer memory coupled thereto. The computer memory comprises machine executable code that, upon execution by the one or more computer processors, implements any of the methods above or elsewhere herein.

[0039] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in this art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its several details are capable ofWSGR Docket No. 55436-733.601 modifications in various obvious respects, all without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.INCORPORATION BY REFERENCE

[0040] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and / or take precedence over any such contradictory material.BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The novel features of the disclosure are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present disclosure may be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the disclosure are utilized, and the accompanying drawings (also “Figure” and “FIG.” herein), of which:

[0042] FIG. 1 shows a computer control system that is programmed or otherwise configured to implement methods provided herein;

[0043] FIG. 2 shows an example of a system for performing a non-classical computation;

[0044] FIG. 3A shows an example of an optical trapping unit;

[0045] FIG. 3B shows an example of a plurality of optical trapping sites;

[0046] FIG. 3C shows an example of an optical trapping unit that is partially filled with atoms;

[0047] FIG. 3D shows an example of an optical trapping unit that is completely filled with atoms;

[0048] FIG. 4 shows an example of an electromagnetic delivery unit;

[0049] FIG. 5 shows an example of a state preparation unit;

[0050] FIG. 6 shows a flowchart for an example of a first method for performing a non-classical computation;

[0051] FIG. 7 shows a flowchart for an example of a second method for performing a non- classical computation;

[0052] FIG. 8 shows a flowchart for an example of a third method for performing a non-classical computation;

[0053] FIG. 9 shows an energy level structure for single-qubit and multi-qubit operations in strontium-87;WSGR Docket No. 55436-733.601

[0054] FIG. 10 shows an example method for error corrected non-classical computation;

[0055] FIG. 11 shows an example of a system for error corrected non-classical computing that is programmed or otherwise configured to implement methods provided herein;

[0056] FIG. 12 shows an example process for performing continuous, non-classical computation;

[0057] FIG. 13A shows example of a plurality of mirrors configured to provide a plurality of optical cavities;

[0058] FIG. 13B shows example of a spacer with a complicated set of cavities to be aligned, including example views and images of Rayleigh scattered light taken from each view;

[0059] FIG. 14 shows an example of an electro-optic virtually imaged phased array;

[0060] FIG. 15 shows an example of an optic cavity for a virtually imaged phased array;

[0061] FIG. 16 illustrates a combined crossed acousto-optic deflector and spatial light modulator system;

[0062] FIG. 17 illustrates a combined crossed acousto-optic deflector and spatial light modulator system;

[0063] FIG. 18 illustrates a combined acousto-optic deflector and spatial light modulator system;

[0064] FIG. 19 illustrates an example of a multi -transducer AOD;

[0065] FIG. 20 provides a non-limiting example of optical wavefronts;

[0066] FIG. 21 provides a non-limiting example of a polyphase synthesis bank;

[0067] FIG. 22 provides a non-limiting example of a column controller;

[0068] FIG. 23 provides a non-limiting example of a row controller;

[0069] FIG. 24 provides a first example of local amplification in a multi-channel AOD;

[0070] FIG. 25 provides a second example of local amplification in a multi-channel AOD;

[0071] FIG. 26 illustrates an example of a grating in real-space; and

[0072] FIG. 27 illustrates an example of a grating in Fourier space.DETAILED DESCRIPTION

[0073] While various embodiments of the disclosure have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the disclosure. It should be understood that various alternatives to the embodiments of the disclosure described herein may be employed.

[0074] As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. Any reference to “or” herein is intended to encompass “and / or” unless otherwise stated.WSGR Docket No. 55436-733.601

[0075] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “at least,” “greater than” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3.

[0076] Whenever the term “no more than,” “less than,” “less than or equal to,” or “at most” precedes the first numerical value in a series of two or more numerical values, the term “no more than,” “less than,” “less than or equal to,” or “at most” applies to each of the numerical values in that series of numerical values. For example, less than or equal to 3, 2, or 1 is equivalent to less than or equal to 3, less than or equal to 2, or less than or equal to 1.

[0077] Where values are described as ranges, it will be understood that such disclosure includes the disclosure of all possible sub-ranges within such ranges, as well as specific numerical values that fall within such ranges irrespective of whether a specific numerical value or specific subrange is expressly stated.

[0078] As used herein, like characters refer to like elements.

[0079] As used herein, the terms “non-classical computation,” “non-classical procedure,” “non- classical operation,” any “non-classical computer” generally refer to any method or system for performing computational procedures outside of the paradigm of classical computing. A non- classical computation, non-classical procedure, non-classical operation, or non-classical computer may comprise a quantum computation, quantum procedure, quantum operation, or quantum computer.

[0080] As used herein, the terms “quantum computation,” “quantum procedure,” “quantum operation,” and “quantum computer” generally refer to any method or system for performing computations using quantum mechanical operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) on a Hilbert space represented by a quantum device. As such, quantum and classical (or digital) computation may be similar in the following aspects: both computations may comprise sequences of instructions performed on input information to then provide an output. Various paradigms of quantum computation may break the quantum operations down into sequences of basic quantum operations that affect a subset of qubits of the quantum device simultaneously. The quantum operations may be selected based on, for instance, their locality or their ease of physical implementation. A quantum procedure or computation may then consist of a sequence of such instructions that in various applications may represent different quantum evolutions on the quantum device. For example, procedures to compute or simulate quantum chemistry may represent the quantum states and the annihilation and creation operators of electron spin-orbitalsWSGR Docket No. 55436-733.601 by using qubits (such as two-level quantum systems) and a universal quantum gate set (such as the Hadamard, controlled-not (CNOT), and rotations) through the so-called Jordan-Wigner transformation or Bravyi-Kitaev transformation.

[0081] Additional examples of quantum procedures or computations may include procedures for optimization such as quantum approximate optimization algorithm (QAOA) or quantum minimum finding. QAOA may comprise performing rotations of single qubits and entangling gates of multiple qubits. In quantum adiabatic computation, the instructions may carry stochastic or non- stochastic paths of evolution of an initial quantum system to a final one.

[0082] Quantum-inspired procedures may include simulated annealing, parallel tempering, master equation solver, Monte Carlo procedures and the like. Quantum -classical or hybrid algorithms or procedures may comprise such procedures as variational quantum eigensolver (VQE) and the variational and adiabatically navigated quantum eigensolver (VanQver).

[0083] A quantum computer may comprise one or more adiabatic quantum computers, quantum gate arrays, one-way quantum computers, topological quantum computers, quantum Turing machines, quantum annealers, Ising solvers, or gate models of quantum computing.

[0084] As used herein, the term “adiabatic” refers to any process performed on a quantum mechanical system in which the parameters of the Hamiltonian are changed slowly in comparison to the natural timescale of evolution of the system.

[0085] As used herein, the term “non-adiabatic” refers to any process performed quantum mechanical system in which the parameters of the Hamiltonian are changed quickly in comparison to the natural timescale of evolution of the system or on a similar timescale as the natural timescale of evolution of the system.Examples of Beam Steering

[0086] In one non-limiting example, systems, methods, computer-readable media, and techniques disclosed herein may use an electro-optic material, as described herein, in a virtually imaged phased array (VIP A) device to create a high speed beam deflector. In some cases, a VIPA may be used for adjusting path length of an optical beam directed therethrough. Additionally, the use of the electro-optic material in the VIPA may enable beam steering of an optical beam directed therethrough. Accordingly, the systems, the methods, the computer-readable media, and the techniques disclosed herein can provide a technical solution to a need for a high-speed beam deflector by combining electro-optic path adjustment with VIPA technology to build a deflector.

[0087] Rapid steering of laser beams may be broadly useful in quantum computing applications. For example, rapid steering of laser beams can be used to direct beams into different spatial modes for switching purposes, or can be used to create movable spots of light of an optical beam (e.g., laser light) in free space. For quantum computing applications with neutral or chargedWSGR Docket No. 55436-733.601 atoms, it can be useful to be able to rapidly direct laser light. As a non-limiting example, it can be useful to rapidly direct laser light onto different parts of an atomic register for atomic control.

[0088] Some non-limiting examples of technologies for rapid beam steering may include spatial light modulators (SLMs), acousto-optic deflectors (AODs), digital micromirror devices (DMDs), acousto-optic modulator arrays (AOM arrays), electro-optic deflectors (EODs), etc. In some cases, within a quantum computing applications, one or more of these technologies may be implemented, e.g., in conjunction with one another. In some cases, within quantum computing applications, one or more of these technologies may be implemented in conjunction with a VIP A. As a non-limiting example, SLMs can create arbitrary patterns of spots of light of an optical beam, but may be slow (e.g., millisecond-scale) to change patterns in some cases. This example feature of an SLM can make it slow to create a continuously moving spot. In another nonlimiting example, AODs may create ID or 2D arrays of spots of light of an optical beam, but may additionally possess speed limitations associated with the speed of sound in the deflector crystal. As an additional non -limiting example, the use of AODs may encounter challenges associated with intermodulation when creating multiple spots of light of an optical beam. In another non-limiting example, DMDs may create 2D patterns of spots of light of an optical beam by turning on and off mirrors. In a non-limiting example, using DMDs to optically address a plurality of atoms with spots of light of an optical beam may be used in a real-space image plane of the atoms (e.g., with limited ability to create moving spots of light) or in a frequency-space Fourier plane of the atoms (e.g., with lesser efficiency). In another non-limiting example, AOM arrays may provide arbitrary amplitude and phase control of pre-positioned spots of light of an optical beam, but may encounter limited ability to change the positions of these spots of light. In another non-limiting example, EODs may provide continuous scanning of a single spot of an optical beam. In some cases, however, the small size of an optical beam passing through the EOD, combined with the limited angular deflection that can be generated for achievable voltages, may limit the resolution of EODs (e.g., as the product of the beam size and angle, or the number of resolvable spots of light that can be generated). Notably, between these non-limiting example technologies, there are numerous examples of challenges in providing fast, continuous steering of one or more laser beams in a power-efficient manner.

[0089] In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may provide fast, continuous steering of one or more optical beams in a power-efficient manner by leveraging virtually imaged phased array (VIP A) technology. In some cases, a VIPA may refer to the 1903 Lummer-Gehrcke interferometer used for spectral analysis used early in the 20th century (or the like) further comprising, e.g., an Echelle grating for crossed spectral dispersion. In some cases, a Lummer-Gehrcke interferometer can compriseWSGR Docket No. 55436-733.601 an optical beam propagating near the total internal reflection angle to achieve multiple bounces of the optical beam in a parallel plate. In some cases, VIPA devices may use a high reflection coating on one side and a partial reflector on the other so that the optical beam may propagate closer to normal to the plate surfaces. In some cases, VIPA devices can comprise a variable path length inside their cavity for changing the optical distance between surfaces.Examples of Virtually Imaged Phased Array Devices

[0090] In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may provide fast, continuous steering of one or more optical beams in a power-efficient manner via leveraging VIPA technology. In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may modify a VIPA device, where the optical beam passes multiple times through a material which comprises electrooptic effects. In some cases, these electro-optic effects may be used to steer a laser beam in response to an applied voltage.

[0091] A non-limiting example of a VIPA may comprise an angular dispersive device that, like a prism or a diffraction grating, may split light into its spectral components. Such a non-limiting example of a VIPA may work substantially independently of polarization, in some cases. In contrast to prisms or regular diffraction gratings, non-limiting examples of VIP As may comprise a higher angular dispersion but may also comprise a smaller free spectral range. This characteristic of a non-limiting example of a VIPA may be similar to that of an Echelle grating, as both the example VIPA and an Echelle grating may use high diffraction orders. This high diffraction order may pose challenges in certain cases. For the potential challenges of high diffraction orders, a VIPA can be combined with one or more diffraction gratings. A VIPA may comprise a compact spectral disperser, e.g., further comprising high wavelength resolving power.

[0092] In some cases, the phased array of a VIPA may be interpreted as the optical analogue of a phased array antenna at radio frequencies. Unlike a diffraction grating, which can be interpreted as a real phased array, the phased array of a VIPA may be created in a virtual image. More specifically, the optical phased array of a VIPA may be virtually formed with multiple virtual images of a light source. This formation of the phase array from multiple virtual images of a light source can distinguish an example VIPA from an Echelle grating, where a similar phased array may be formed in the real space. In some cases, the virtual images of a light source in a VIPA may be aligned (e.g., automatically) at a constant interval, which may be used for optical interference. This operation of a VIPA can provide an advantage over an Echelle grating, in some cases. When the output light of the light source is observed, the VIPA may function as if light were emitted from a real phased array.WSGR Docket No. 55436-733.601

[0093] In some cases, a VIPA comprises a plate of optical material, described herein. For example, a normal to the plate may be slightly tilted with respect to the input optical beam. In a non-limiting example, one side (e.g., light input side) of the plate of optical material may be coated with a substantially (e.g., about 100%, about 95%, about 90%, etc.) reflective mirror and the other side (e.g., light output side) may be coated with a highly reflective but partially transmissive mirror. In the non-limiting example, the side with the substantially reflective mirror may have an anti -refl ection coated light entrance area, through which a light beam enters the glass plate. In some cases, the input optical beam may be line-focused to a line (e.g., focal line) on the partially transmissive mirror on the light output side. In some cases, a line-focusing lens may include a cylindrical lens (or other suitable lens). In some cases, the line-focusing lens may be used to line-focus the input optical beam to generate a light beam. In some cases, the light beam may diverge after the beam waist located at the line-focused position.

[0094] In some cases, optical materials described herein may comprise one or more of glass, borosilicate glass, fused silica glass, naturally occurring crystalline material, lab grown crystalline material, quartz, sapphire, germanium, polymers, acrylic, polycarbonate, semiconductors, silicon, gallium arsenide, or any suitable optical material, or a combination thereof.

[0095] In some cases, after the optical beam enters the plate of optical material through the light entrance area, light of the optical beam may be reflected at the partially transmissive mirror and the substantially reflective mirror. Thus, in some cases, the optical beam may travel back and forth between the partially transmissive mirror and the substantially reflective mirror.

[0096] In some cases, the plate of optical material may be tilted as a result of a slight rotation where the axis of rotation is the focal line. In some cases, this rotation or tilt may help to reduce light leaving the plate of optical material out of the light entrance area. Therefore, in order for the optical system to work as a VIPA, there may be a critical minimum angle of tilt that allows the light entering through the light entrance area to return to the substantially reflective mirror. In some cases, below this angle, the function of the VIPA may be severely impaired. In some cases, if the tilting angle were zero, the reflected light from the partially transmissive mirror may travel exactly in reverse and exit the plate of optical material through the light entrance area without being reflected by the substantially reflective mirror.

[0097] In some embodiments, a critical minimum angle of tilt comprises an angle of at least about 0.1°. In some embodiments, a critical minimum angle of tilt comprises an angle of about 0.1° to about 1° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 1° to about 5° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 5° to about 10° and incrementsWSGR Docket No. 55436-733.601 therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 10° to about 15° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 15° to about 20° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 20° to about 25° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 25° to about 30° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 30° to about 50° and increments therein. In some embodiments, a critical minimum angle of tilt comprises an angle of about 50° or more.

[0098] In some cases, when the light of an optical beam is reflected (e.g., each time) at the partially transmissive mirror, a small portion of the light power can pass through the mirror and travels away from the plate of optical material. For an optical beam passing through the mirror after multiple reflections, the position of the line-focus may be seen in the virtual image of the VIP A, e.g., when observed from the light output side. In some cases, this optical beam may travel as if the optical beam originated at a virtual light source located at the position of the line-focus and diverged from the virtual light source. In some cases, the positions of the virtual light sources for all the transmitted optical beams may align (e.g., automatically) along the normal to the glass plate with approximately constant spacing, that is, a number of virtual light sources may be superimposed to create an optical phased array. Due to the interference of the multiple optical beams from the virtual light sources, the phased array may emit a substantially collimated optical beam in one direction, which is at a wavelength dependent angle, and therefore, an angular dispersion is produced.

[0099] In some embodiments, a transmitted light power, herein, comprises a transmitted power of at least about 0.01%. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 0.01% to about 0.1% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 0.1% to about 1% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 1% to about 5% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 5% to about 10% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 10% to about 20% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 20% to about 50% and increments therein. In some embodiments, a transmitted light power, herein, comprises a transmitted power of about 50% or more.

[0100] FIG. 14 shows an example of an electro-optic virtually imaged phased array 1400. As illustrated, the VIPA 1400 includes a transparent (or semi-transparent) plate of optical materialWSGR Docket No. 55436-733.601 with two parallel surfaces 1401 and 1402. In the non-limiting example, an input optical beam passes through a first portion 1401A (a lower portion) of the first surface 1401, where the portion 1401A is largely transparent. The input optical beam then bounces between the second surface 1402, which is partially transmissive, and a second portion 1401B (an upper potion) of the first surface 1401 that is fully (or approximately -fully) reflective.

[0101] After each round-trip pass through the transparent or semi-transparent plate of optical material, the optical beam creates a portion of the output optical beam from the VIPA 1400. When combined, these portions may be considered to create one or more output optical beams, whose angles are determined by the phase relationship between the sub-beams. Any effect that creates a phase shift between subsequent round-trip passes will then shift the angle of the output beam. In a non-limiting example, this may be used to analyze the frequency spectrum of the incident light, as a change in the input frequency leads to a relative phase shift of output beam components.

[0102] The systems, the methods, the computer-readable media, and the techniques disclosed herein provide an electro-optic material, described herein, as the transparent or semi-transparent plate of optical material, or part of the transparent or semi-transparent plate of optical material of the VIPA 1400. In some cases, a voltage applied to the electro-optic material of the VIPA 1400 changes an index of refraction of the electro-optic material. For example, the electro-optic material may comprise lithium niobate. In some cases, the voltage applied to the electro-optic material may be about 0.1 kV to about 20 kV. In some cases, the voltage applied to the electrooptic material may be about 0.1 kV to about 0.5 kV, about 0.1 kV to about 1 kV, about 0.1 kV to about 2.5 kV, about 0.1 kV to about 5 kV, about 0.1 kV to about 10 kV, about 0.1 kV to about 15 kV, about 0.1 kV to about 20 kV, about 0.5 kV to about 1 kV, about 0.5 kV to about 2.5 kV, about 0.5 kV to about 5 kV, about 0.5 kV to about 10 kV, about 0.5 kV to about 15 kV, about 0.5 kV to about 20 kV, about 1 kV to about 2.5 kV, about 1 kV to about 5 kV, about 1 kV to about 10 kV, about 1 kV to about 15 kV, about 1 kV to about 20 kV, about 2.5 kV to about 5 kV, about 2.5 kV to about 10 kV, about 2.5 kV to about 15 kV, about 2.5 kV to about 20 kV, about 5 kV to about 10 kV, about 5 kV to about 15 kV, about 5 kV to about 20 kV, about 10 kV to about 15 kV, about 10 kV to about 20 kV, or about 15 kV to about 20 kV. In some cases, the voltage applied to the electro-optic material may be about 0.1 kV, about 0.5 kV, about 1 kV, about 2.5 kV, about 5 kV, about 10 kV, about 15 kV, or about 20 kV. In some cases, the voltage applied to the electro-optic material may be at least about 0.1 kV, about 0.5 kV, about 1 kV, about 2.5 kV, about 5 kV, about 10 kV, or about 15 kV. In some cases, the voltage applied to the electro-optic material may be at most about 0.5 kV, about 1 kV, about 2.5 kV, about 5 kV, about 10 kV, about 15 kV, or about 20 kV.WSGR Docket No. 55436-733.601

[0103] In some cases, when an optical beam is directed through the electro-optic material while a voltage is applied to the electro-optic material, a phase shift may be induced between output subbeams of the optical beam, and an angle of the output optical beam. In some cases, the voltage may be applied between surfaces of the electro-optic material that are parallel to the propagation of the optical beam (e.g., laser beam), though other directions may be possible as well.

[0104] In some cases, a voltage applied to the electro-optic material of the VIPA may be varied based on the electro-optic material used. In some cases, applying a voltage to the electro-optic material comprises applying an electromagnetic field along an axis of the electro-optic material. The change in index of refraction of an electro-optic material due to an applied voltage or electric field is referred to as the electro-optic effect. The electro-optic effect is a material-dependent property. As an example, some electro-optic materials can exhibit a substantially linear change in index of refraction due to an applied voltage. This phenomenon may be modeled using the Pockels Effect. In some cases, electro-optic materials exhibit more complex responses to applied voltage. As a further example, some electro-optic materials can exhibit a substantially quadratic change in index of refraction due to an applied voltage, modeled by the Kerr Effect. In practice, the systems, methods, the computer-readable media, and the techniques disclosed herein may employ any applied voltage suitable to induce the desired phase shift using a specific electrooptic material. As an example, for Lithium Niobate, an applied voltage of up to 10 kV may be used.

[0105] In some embodiments, a desired phase shift comprises a phase shift of about On to about 2n and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about On to about 0. In and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about O.ln to about n / 2 and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about n / 2 to about n and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about n to about 3n / 2 and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about 3n / 2 to about 1.9n and increments therein. In some embodiments, a desired phase shift comprises a phase shift of about 2n.

[0106] Advantageously, in a non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable directing one or more optical beams onto atoms or ions for a variety of applications. These applications may include, for example, coherent or incoherent transfer of an atom or ion from one internal state to another internal state. In another example, these applications may include applying energy shifts to atomic states. In another example, these applications may include inducing fluorescence from atoms or ions, such as for imaging to determine presence or state occupation. In another example, these applications may includeWSGR Docket No. 55436-733.601 controlling the position or the velocity of atoms or ions. Further, the electro-optic material in the VIPA 1400 may be used to direct light into one or more optical fibers, optical cavities, or other photonic structures. Further, the electro-optic material in the VIPA 1400 may be used for imaging, such as, in Light Detection and Ranging (LiDAR) applications, in some cases.

[0107] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable beam steering using a DC voltage applied to the electrooptic material of the VIPA 1400 combined with additional control by changing the frequency components present in the light controlled by the electro-optic material of the VIPA 1400.

[0108] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable applying the electric field along any axis, either by applying a voltage to an opaque, conductive material on or near portions of the surfaces of the electro-optic material through which the optical beam does not pass, or to transparent conductive materials on or near surfaces through which the optical beam does pass. For example, the opaque, conductive material may comprise a metal, or the like. For example, the non-opaque, conductive material may comprise indium tin oxide (ITO), or the like.

[0109] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable using two or more devices that are the same as or similar to the VIPA 1400 with electro-optic material. In some cases, these two or more devices may be combined in series to provide beam steering in two or more dimensions.

[0110] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable applying a voltage as a time-varying DC voltage, a static DC voltage, or an AC voltage. The AC voltage may be combined with a phase-modulated input beam to selectively modify certain frequency components.[OHl] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable configurations with the electro-optical material in a fixed or dynamic form. In a fixed form, the electro-optic material may be clamped, to not allow the electro-optic material to expand. In a dynamic form, the electro-optic material may be unclamped, preventing (or reducing) the electro-optic material from expanding.

[0112] Still further advantageously, in another non-limiting example, the use of the electro-optic material in the VIPA 1400 may enable tuning the index of refraction of the electro-optic material. This tuning may be, in some cases, combined or replaced with strain-based tuning. For example, strain-based tuning may create large, slow changes in beam angle.WSGR Docket No. 55436-733.601Dynamic Refocusing

[0113] In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may enable dynamic beam refocusing. Such dynamic beam focusing may include refocusing the optical beam with each pass.

[0114] In a non-limiting example, electro-optic analysis may include a modification to the VIPA 1400 of FIG. 14 that included an electro-optic material. For this analysis, assume the transverse dimensions of the VIPA is m and the crystal is t long. The field is given by E=V / (o. TheRayleigh range over which the beam expands by V2 is zr= — -. The index change is An = A.— reffE , where re^E = r33for LiNbCh polarized in z and field in z.

[0115] In some cases, a series of outputs may step by a fraction of a wavelength to satisfy Nyquist sampling and use an approximate to a smooth output wave. This may include driving the system at a single pass maximum phase retardation of IV / to produce 4 samples at the largest angular deflection. To produce N resolvable spots of light, 4N bounces may be needed. IV may be found using yielding Vn= m (n3reyyt) \Note that in some cases, the Nyquist criteria may not be necessary for filtering out a single order of a multi-order diffraction (e.g., taking an efficiency reduction).

[0116] In some cases, 4Nt < zrto maintain the diffracting beam inside the width of the crystal. This constraint may raise the width, m, over that of a single pass Pockels cell, and hence the required voltage. Therefore, the cavity may refocus (e.g., continuously, at an interval, in response to a specified condition, etc.) the beam while it walks to progressive positions or angles.

[0117] FIG. 15 shows a non-limiting example of an optical cavity 1500 for a virtually imaged phased array. As illustrated, the optic cavity 1500 may include electro-optic material shaped as a roof hat prism, 1550, on one side of a lens 1560. The lens 1560 may have a focal point displaced vertically from the center of a roof hat prism, 1550. Each subsequent beam pass may be displaced from the previous making the beam spiral around inside the cavity. In some cases, when the right-hand side mirror 1511 is a partial reflector, each subsequent beam pass may be emitted in successive angles. In some cases, when the left-hand side mirror, 1510, is a partial reflector, each subsequent beam pass may be emitted in successive positions. In some cases, this may yield a VIPA-like device with power to keep the beam refocused. In some cases, the angular output (e.g., on the right) may be recollimated with another lens.

[0118] As illustrated in FIG. 15, the non-limiting example of an optical cavity 1500 may include the lens 1560 with a focal point offset vertically from a corner cube 1550 (which may be in addition or in alternative to the roof hat prism 1550) by a displacement 1520. A partial reflector 1570 is at the lens focal point and normal to the optic axis. An incident optical beam 1531 mayWSGR Docket No. 55436-733.601 be focused by the lens 1560 onto the partial reflector 1570. In some cases, the partial reflector 1570, may transmit a portion of the light of the optical beam and reflects the rest back to the lens 1560. In some cases, upon second pass through the lens 1560, the reflected portion may be bent to be parallel with the incident optical beam 1531.

[0119] The corner cube 1550 may have reflective surfaces 1510 and 1511 that reflect this light of the optical beam symmetrically about the corner of the corner cube 1550 into the optical beam 1532. The light beam 1532 may be displaced from the optical beam 1531 by twice the displacement 1520. Subsequent passes of the optical beam may provide a series of optical beams, the first four of which are shown as beams 1531, 1532, 1533, and 1534. The corner cube 1550 can be made of an electro-optic material. In some cases, the z-axis of the crystal may be normal to the plane of FIG. 15 and electrodes may be arranged to apply a field in the z-axis. In some cases, the optical polarization may be oriented in the z-axis. In such cases, the electro-optic coefficient r (or similar) may be employed to change the refractive index in proportion to the applied electric field.

[0120] In some cases, the electro-optical material, described herein, may change its refractive index when an electric field is applied, via the electro-optic effect. In some cases, he electro- optical material, described herein, may be a non-centrosymmetric material. In some cases, the electro-optical material, described herein, may comprise one or more of lithium, niobium, oxygen, etc. For example, the electro-optical material may comprise lithium niobate (LiNbCh), an inorganic material that can be made into a film and bonded to a silicon layer. The electro- optical material may comprise one or more of lithium tantalate, potassium titanyl phosphate, P- barium borate, etc. In some cases, the electro-optic material, described herein, may comprise any suitable transparent or semi-transparent optical material. In some cases, the electro-optic material, described herein, may comprise any suitable transparent or semi-transparent optical material wherein the index of refraction may be changed, at least in part, via the electro-optic effect.Examples of Combined Modulator-Based Array Generation

[0121] In some cases, a combination of acousto-optic deflectors (AODs) and spatial light modulators (SLMs) can be used to generate spot patterns (e.g., arbitrary spot patterns) for use in the methods and systems of the present disclosure. The use of a combination of AODs and SLMs can provide high speed and reconfigurable spot arrays for use in quantum computing systems.

[0122] In some cases, AODs can deflect an input optical beam (e.g., laser beam) into one or more higher-order output beams by applying acoustic frequency modulation to a crystal aligned with the input optical beam to satisfy the Bragg condition of the crystal. In some cases, a single AOD can deflect an optical beam along one axis; to achieve 2D beam steering, two or moreWSGR Docket No. 55436-733.601AODs may be used together (e.g., with a second AOD rotated 90 degrees relative to the first along the optical axis). This configuration may be a crossed AOD (xAOD) configuration.

[0123] In some cases, in the xAOD configuration, the beam deflection by the first AOD can cause the optical beam to partially walk off the active area of the second AOD (e.g., the area of the AOD configured to impart beam deflection). Additionally, the location along the optical axis where the beam diffracts can differ for each axis, which can create alignment challenges for downstream optical systems, as each axis can have a different apparent focal point. In one nonlimiting example, to correct for these issues, one or more relay telescopes can be placed between the two AODs. In some cases, this may serve to reimage the exit aperture of the first AOD onto a conjugate plane where the entrance aperture of the second AOD is placed. In some cases, the one or more relay telescopes may comprise one or more of a 4f relay telescope, a non-4f relay telescope, a two element relay telescope, a 2fl relay telescope, a 2f2 relay telescope, a two element 2fl + 2f2 relay telescope, an unobscured two-mirror relay telescope, or any other suitable optical device for reimaging an optical beam.

[0124] In some cases, crossed AODs can produce multiple spots of light by modulating the acousto-optic material, described herein, in one or both of the AODs at multiple frequencies. In some cases, however, this method may be limited because each AOD may only steer along a single axis. Furthermore, multiple spots light generated by the first AOD can each be split into multiple spots of light by the second AOD, which can, in some cases, limit the xAODs to producing spot patterns that are the outer product of the patterns generated by each AOD.

[0125] In some cases, SLMs may comprise a plurality of pixels configured to apply a pixelated phase or amplitude mask to an input optical beam. In some cases, SLM mask can be computed to produce a specific pattern in the far field or through an optical system, by, for example, using iterative algorithms (e.g., the Gerchberg-Saxton algorithm, the mixed region amplitude freedom (MRAF) algorithm, or other suitable algorithms). Although this non-limiting example of an iterative algorithm can generate the optimal mask pattern for arbitrary spot patterns, it may be, in some cases, computationally intensive.

[0126] In some cases, SLMs can offer the benefit of creating arbitrary spot patterns, which may be limited by their resolution. In some cases, SLMs can also compensate for aberrations in downstream optical systems by superimposing correction patterns onto the mask. However, SLMs may be limited by speed, with update frequencies of about 60 Hz to about 1 kHz in some cases. As a non-limiting example, this speed limitation combined with the computational demands of pattern calculations, can prevent SLMs from rapidly moving spots of light in arbitrary patterns or along arbitrary trajectories.WSGR Docket No. 55436-733.601

[0127] Combining a pair of crossed AODs with one or more SLMs can, in some cases, mitigate the limitations of both technologies and enhance corresponding strengths. A non-limiting example of an xAOD / SLM system may be found in FIG. 16. In the non-limiting example, a pair of xAODs 1610 with a first relay telescope 1620 between them may be followed by a second relay telescope 1630, projecting the beam onto an SLM 1640. The SLM 1640 may then produce an arbitrary pattern of spots of light from an incident optical beam, while the xAODs 1610 steer the optical beam’s angle of incidence onto the SLM 1640. In some cases, this combination can allow the SLM 1640 to generate patterns of spots of light not limited to the outer product of the xAOD 1610 spot positions. In some cases, the AODs 1610, by quickly adjusting the incident beam angle, can enable rapid movement of these spots of light, surpassing the speed limitations of the SLM 1640 alone. In some cases, the AODs 1610 can also perform intensity control by smoothly modulating the power delivered to the spots of light.

[0128] In some cases, this configuration can also allow the SLM 1640 to correct for aberrations in the optical system, which the AODs 1610 alone may not be able to address. The combination of xAODs 1610 and SLMs 1640 can be extended with one or both AODs 1610 producing multiple optical beams, 1710 and 1720 as shown in FIG. 17, each striking the one or more SLMs 1640 at different angles. In some cases, each incident optical beam 1710 and 1720 can generate the same pattern of spots of light at different locations, depending on the incident angle. This approach can produce groups of spots of light whose pattern matches the outer product of the beams generated by the AODs 1610, with the individual spots of light within the group determined by the SLM 1640. In some cases, each incident optical beam 1710 and 1720 can generate a different pattern of spots of light at different locations. In some cases, each incident optical beam 1710 and 1720 can generate a different pattern of spots of light at the same location. In some cases, a single AOD 1810 can be combined with one or more relay telescopes 1820 and one or more SLMs 1830 for a system 1800 that uses a single dimension of movement as shown in FIG. 18Examples of Multi-Transducer Acousto-Optic Deflectors

[0129] This disclosure hereby incorporates by reference for all purposes: Dennis R. Pape, “Multichannel Bragg cells: design, performance, applications,” Opt. Eng. 31(10) (1 October 1992) https: / / doi.org / 10.1117 / 12.58871; Kelvin Wagner and Demetri Psaltis, “A space integrating acousto-optic matrix-matrix multiplier” Optics Communications, Volume 52, Issue 3, 1 December 1984, Pages 173-177; and Robert T. Weverka and Kelvin H. Wagner, “Wide- angular aperture acousto-optic Bragg cell”. Proceedings of the SPIE, Volume 1562, Devices for Optical Processing; (1991).WSGR Docket No. 55436-733.601

[0130] In many cases, rapid steering of laser beams is broadly useful in non-classical computing. As a non-limiting example, rapid steering may be used to direct optical beams (e.g., laser light) into different spatial modes for switching purposes. As a further non-limiting example, rapid steering may be used to create movable spots of light (e.g., laser light) in free space. As a further non-limiting example, for quantum computing applications with neutral or charged atoms, it may be useful to rapidly direct an optical onto different parts of an atomic register for atomic control.

[0131] Various technologies for rapid beam steering may be used with the systems, the methods, the computer-readable media, and the techniques disclosed herein.

[0132] For example, spatial light modulators (SLMs) may create arbitrary patterns of spots of light, but may be slow (e.g., on a millisecond-scale) to change patterns. Accordingly, it may be slow to create a continuously moving spot of light, for example.

[0133] In another example, digital micromirror devices (DMDs) can create 2D patterns by turning on and off mirrors. In some cases, DMDs can be used in the real-space image plane (e.g., with limited ability to create moving spots of light) or in the frequency-space Fourier plane (e.g., with poor efficiency).

[0134] In another example, acousto-optic modulator arrays (AOM arrays) may provide arbitrary amplitude and phase control of pre-positioned spots of light. In some cases, AOM arrays may have limited ability to change the positions of the spots of light.

[0135] In another example, electro-optic deflectors (EODs) can provide fast continuous scanning of a single spot of light. However, the small size of the optical beam going through the deflector, combined, e.g., with the limited angular deflection that can be generated for achievable voltages, may limit the resolution of these devices. In some cases, the limited resolution may be the product of the optical beam size and angle, or the number of resolvable spots of light that can be generated.

[0136] In another example, acousto-optic deflectors (AODs) can create ID-arrays and 2D-arrays of spots of light that can be changed relatively quickly. In some cases, there may speed limitations associated with the speed of sound in the deflector crystal. In some cases, 2D-arrays of spots of light can be formed by sequentially passing light through two AODs with diffraction in orthogonal directions. In this configuration, the spots of light may not be fully independent, as the same deflections may be applied by the second modulator to all deflections applied by the first modulator.

[0137] As a non-limiting example, the systems, the methods, the computer-readable media, and the techniques of the present disclosure provide a type of acousto-optic deflector that enables the generation of multiple independent beams in two dimensions.WSGR Docket No. 55436-733.601

[0138] As a non-limiting example, the systems, the methods, the computer-readable media, and the techniques of the present disclosure provide an acousto-optic deflector with multiple acoustic transducers that are driven in a phase -coherent manner to generate multiple independent output beams.

[0139] FIG. 19 illustrates a non-limiting example of a multi -transducer AOD 1900, in accordance with some embodiments herein. In some cases, an AOD comprises an acousto-optic material, in which acoustic waves are generated by one or more transducers. Laser light passing through the medium acquires a phase modulation as a result of the variation in index of refraction associated with the acoustic wave. A sinusoidal signal gives a periodic modulation which leads to the diffraction of part of the incoming light at an angle associated with the period of the acoustic wave. By applying a linear combination of multiple sinusoidal waves, multiple diffracted angles can be generated, which are independent in the limit of low diffraction efficiency.

[0140] In some cases, for a perfectly isotropic acousto-optic material, the associated angle of diffraction of the incoming laser light may be changed proportional to a frequency applied to the acousto-optic material. In many cases, however, the acousto-optic material is anisotropic. In such cases, the change in associated angle of diffraction of the incoming laser may not exhibit a proportional relationship to applied frequency due to crystal orientation and cross -coupling effects. In each case, however, a material -dependent mapping may be provided between change in associated angle of diffraction and applied frequency. In some cases, this mapping may be precalculated. In some cases, this mapping may be calculated in-situ. In some cases, this mapping is orientation-dependent.

[0141] In some cases, multiple transducers may be affixed to the acousto-optic material and may be used in a “phased array” configuration to generate acoustic waves that propagate with tunable angle and periodicity. When a laser interacts with the combination of acoustic waves created by the multiple transducers, a portion of it is deflected with the azimuthal and elevation angle associated with the composite wave generated by the transducers.

[0142] For example, to generate a single deflected spot of light, the same (or similar) frequency may be applied to each transducer. The deflected optical azimuthal angle, perpendicular to the transducer array, may be changed based on the applied frequency. By applying a phase gradient to the drive tones applied to the transducers, the deflected elevation angle in the direction along the transducer array may be varied. Spatial variation of this phase gradient may allow for two- dimensional control of the deflected spot. A frequency and spatial phase gradient for deflecting one optical beam to produce a spot of light may be thought of as a drive vector.

[0143] To generate multiple independent spots of light, the sum of the drive vectors that may be used to generate each individual spot can be applied to the transducer array. In such a case, theWSGR Docket No. 55436-733.601 sum of the drive vectors for each individual spot of light may be summed to produce an acoustic wave for deflecting all spots of light.

[0144] The device can be operated in either the acoustic near-field limit of the transducer element (where diffraction and overlap of the acoustic waves is minimal) or in the far -field limit of the acoustic transducer element (where the diffraction and overlap of acoustic waves from multiple transducer elements is significant). Operation in the far field allows the AOD elements to be designed with smaller heights, allowing for higher diffraction efficiency per unit power applied to the transducers.

[0145] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with non-classical computations systems and methods disclosed herein in various ways.

[0146] For example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used to direct optical beams (e.g., laser) onto atoms or ions. Directing optical beams onto atoms or ions may be used for: coherent or incoherent transfer of the atom or ion from one internal state to another; applying energy shifts to atomic states; inducing fluorescence from atoms or ions, for example for imaging to determine presence or state occupation; controlling the position or velocity of the atoms or ions; etc.

[0147] In another example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used to direct light into one or more optical fibers, optical cavities, or other photonic structures.

[0148] In another example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used for imaging, for example in LiDAR applications.

[0149] In another example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used in a resonant optical cavity for two-dimensional deflection. In a resonant cavity excited by an external optical source, the optical field built up is a single spatial mode. The increased intensity inside the cavity allows use of a deflector that deflects only a small portion of the incident optical power (e.g., about 1%, 5%, 10%, 15%, or about 20%), and still get a large diffracting power as compared to the external optical source.Examples of Wavefront Control in AODs

[0150] In some cases, the wavefront of each outgoing spot of light may be arbitrarily controlled along the axis of the electrodes. As a non-limiting example, the phase of the control signal sent to each electrode may be directly converted to the optical phase at that position in the AO crystal. As described above, a linear phase (e.g., in which cp is approximately proportional to y) sets the angle of the outgoing beam, which ultimately determines the position of the spot in the atom plane. In the language of optical wavefronts, this may be referred to as “tilt.” In some cases, thisWSGR Docket No. 55436-733.601 idea can be extended further: a parabolic phase (e.g., in which cp is approximately proportional to y2) can be written as a sum of - (y2+ x2) + - (2— x2) (a combination of “defocus” and “astigmatism”), a cubic phase is a sum of “tilt,” “coma,” “trefoil,” etc. In the expression, - (y2+ x2) + - (y2— x2), the first term represents symmetric curvature along the two axes 4 4(which corresponds to a uniform change in focus for both axes), the second term corresponds to the differential focus, and the sum corresponds to the focus in the y-axis alone.

[0151] A non-limiting example of optical wavefronts is provided in FIG. 20, in accordance with some embodiments. FIG. 20 illustrates a curved wavefront 2010 and a curved and tilted wavefront 2020.

[0152] One non-limiting example application occurs when a spot is moved along the frequency direction. Without being bound by any particular theory, through a separate mechanism, this can create an optical phase proportional to x2, which can be written as a sum of defocus and astigmatism. By adding a y2term, this is converted to pure defocus with no astigmatism. This can be beneficial for certain applications, including moving atoms, where astigmatism may be more detrimental than defocus.Examples of Additional AOD Configurations

[0153] In some cases, an AOD need not operate using acoustic waves in the bulk of a material. Without being bound by any particular theory, a reflective substrate with surface acoustic wave transmitters may be used. In some cases, the substrate may be made of a bulk electrostrictive (e.g., piezoelectric) material, such as Lithium Niobate, Zinc Oxide, Lead Zirconate Titanate, or Aluminum Nitride, or may be a composite substrate configured for surface acoustic wave applications such as Lithium Niobate on Insulator technology. In some cases, a reflective coating may be atop the substrate. For example, the reflective coating may include evaporated metal, a dielectric stack, or combination thereof. In some cases, the material with the substrate or the reflective coating may include an intermediate layer (or layers) for planarization or compatibility of the reflective surface with the substrate configured to surface acoustic waves.

[0154] In some cases, patterning and driving these surface acoustic wave transmitters may generate ripples on the reflective surface of the substrate to create a Bragg grating in the far field of the transmitters with tunable orientation and wavelength. Such a device may have a benefit of decoupling substrate material choice from optical properties, as the coating of the material may be chosen arbitrarily (e.g., as long as the choice does not substantially negatively affect the surface wave properties of the material). Additionally, a reflective multichannel AOD (MAOD) may be a planar device, making heat removal easier. Further, such a device may be easier to fit into the capabilities of more standard semiconductor and MEMS processes, easingWSGR Docket No. 55436-733.601 manufacturability. In some cases, this may apply to any AOD, not just the MAOD. Back reflections of the acoustic wave off the crystal end may return to the region of the optical beam and cause problems. Using a crystal which is at least as long as (e.g., longer than) the desired optical beam size allows the acoustic wave to attenuate naturally through the crystal, so the back reflection may be weakened (e.g., substantially eliminated). In some cases, the crystal is at least about: 1%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 100%, etc. longer than the desired optical beam size. In some cases, the crystal is at least about: 100%, 250%, 500%, 1000%, 2500%, 5000%, 10000%, 25000%, 50000%, 1000000%, etc. longer than the desired optical beam size. In some cases, the back reflection is diminished by at least about 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 96%, 97%, 98%, 99%, or at least about 100%.Examples of Multi-Channel AOD Control Schemes

[0155] Control of a multichannel AOD may involve a large amount of data, in some cases. For example, for a multichannel AOD with N channels, each being fed by a data converter of bandwidth ( ), where each sample has a bit depth (M), this involves 2 x B x M bits per second. For an example device with a bandwidth of 200 MHz, 16-bit samples, and 256 channels, this is about 1.5 terabits per second of data (roughly equivalent to the data contained in all emails sent globally). Presenting an additional challenge, in some cases, this data must be generated in response to control data with low latency. The systems, the methods, the computer-readable media, and the techniques disclosed herein provide a collection of schemes to divide this challenge up, such that portions of the challenge may be addressed in parallel, using many functional units, and then recombined.

[0156] In some cases, a key to controlling the MAOD is that while the total data rate out of the control system is extremely high, the data rate into the control system may be an order of magnitude smaller. To control one spot, the data needed to encode spot movement is the drive vector for the spot of light. The drive vector comprises frequency information and phase gradient information which gives arbitrary control over deflection of the spot of light in two dimensions. In some cases, the two dimensions of control over the spot of light may exist in a cartesian (flat space) plane. In some cases, the two dimensions of control over a spot of light may exist in a plane diffeomorphic to a cartesian plane. In some cases, the two dimensions of control over a spot of light may exist on the surface of an ellipsoid embedded in cartesian space. In some cases, the two dimensions of control over a spot of light are a tensor relationship.

[0157] In order to control a plurality of spots of light, the input control signal needed will be the sum of the drive vectors for the individual spots of light. The data needed to encode the input control signal may comprise one or more of a list of spot locations, a piecewise function, an interpolated curve, a b-spline curve, or a Bezier curve. In some cases, the data needed to encodeWSGR Docket No. 55436-733.601 the input control signal may comprise an interpolated curve and a set of offsets. In some cases, the set of offsets may comprise a set of offset vectors, where an offset vector comprises one or more displacement vectors on a grid. As an example, this grid may be a regular grid, such as a rectangular or cartesian grid, or an irregular grid. In some cases, a move in a neutral atom quantum computer involves a movement from an atom’s current position into a lane, such as, from the first point in the lane to the end point in the lane, and then to the atom’s final destination. In some cases, if each atom position is encoded as a tuple (x, y, f) with each coordinate being a 32-bit integer, then the move coordinates take up 48 bytes, plus a spot identifier number which may be 32-bits, totaling about 52 bytes per move. A modem error correcting code may comprise approximately 400 million moves per second, resulting in a total data bandwidth of 20 gigabytes per second. This data rate can be further reduced by considering that it may be common to move large groups of atoms together. Suppose for instance that a subspace of identifiers are responsible for groups of atoms. In this case, every atom in this group can be moved together with one single move command. If, for example, each group contains A atoms, this scales the bandwidth on average by a factor of 7 / A.

[0158] The systems, the methods, the computer-readable media, and the techniques disclosed herein may subdivide the problem of spot movement into (e.g., mostly) independent control units. Suppose in some cases, that one such control unit is responsible for a subdivision of the optical field of the multichannel AOD. In such cases, all movement commands may be distributed to all control units, but the control units may be able to determine which moves in this command stream that the control units are responsible for. For single-spot moves encoded with absolute paths, a control unit can determine the control unit’s responsibility by calculating the control unit’s overlap with the path specified in the move command. It may be possible to execute multi-spot moves by a controller enumerating out the single-spot moves and then executing these, but several optimizations are also possible. In some cases, a control unit may be only concerned with moves whose trajectory overlaps the subdivision the control unit is responsible for. The control unit may therefore prune movement commands (e.g., all movement commands) that the control unit need not concern itself with. Hence, this stage command may be referred to as pruning.

[0159] In some cases, for moves that stay within a subdivision, this decreases the data rate processed by a particular control unit by the total number of independent control units. In the case of moves that cross control unit fields, this same data rate may be expanded by the average number of fields crossed. In some cases, this may break the control problem up such that control units will be responsible for a regular grid of rectangular fields. Given that for the application of neutral atom movement, in some cases, movements (e.g., most movements) may be tall andWSGR Docket No. 55436-733.601 narrow. Sizing fields to also be tall and narrow may help to minimize the number of crossings. For example, the shape of the fields may be tailored to the shape of the most common moves to reduce the number of crossings.

[0160] In some cases, subdividing the field may comprise splitting the field along the frequency, or vertical axis of the AOD, such that a control unit may be responsible for a tall, rectangular subdivision of the optical field. In some cases, the final data stream to the DAC may comprise driving a particular electrode, for an AOD of bandwidth , with a minimum data rate of 2 X B samples per second. The logic fabric may be run at this frequency, or run at some fraction 1 / N of this speed to generate N samples per clock cycle. This may be a challenge as the signal delivered to each electrode may be a function of the position of spots (e.g., every spot) in the field.

[0161] Instead, in some cases, suppose N column control units, each one effectively driving a MAOD of bandwidth B / N. The benefit here may be that each control unit generates 2B / N samples per second for each virtual electrode, and each control unit may be responsible for 1 / N of the spots of light, assuming equally-distributed spots of light. In some cases, each one of these control units may utilize a strategy disclosed herein, where each column control unit is provided with identical commands, but prunes all movement commands which do not start, end, or pass through the assigned field.

[0162] In some cases, to efficiently recombine each data stream, the systems, the methods, the computer-readable media, and the techniques disclosed herein may use a polyphase synthesis filter bank. Such a filter may accept AT signals of sample rate , where the ithsignal is effectively frequency shifted by a bandwidth ixB and summed into the outgoing data stream, generating an output signal of bandwidth M B. This process may be functionally equivalent to a digital Frequency Division Multiplexer (FDM), taking multiple parallel, low-bandwidth streams and “stacking” the streams side-by-side in the frequency domain to form a single, composite high- bandwidth signal.

[0163] One example approach to this technique comprises up-sampling each of the AT input signals by a factor of AT (e.g., by inserting M — 1 zeros between each sample) and then passing each of the now high-rate signals through a unique, bandpass filter to place in the corresponding assigned frequency slot. In some cases, the bandpass filter may be a sharp bandpass filter. In some cases, the bandpass filter may be approximately flat over the usable bandwidth. This technique may be computationally heavy because the complex filtering operations may be performed at the high output sample rate (M B). One advantage provided by the instant disclosure is the implementation of the polyphase synthesis bank, which may avoid this heavy computational load by restructuring the problem.WSGR Docket No. 55436-733.601

[0164] A non-limiting example of a polyphase synthesis bank is provided in FIG. 21, in accordance with some embodiments. In some cases, a polyphase synthesis bank may efficiently reconstruct a single wideband signal from M narrowband sub-band inputs. The computational leverage may be derived from an Inverse Fast Fourier Transform (IFFT), which acts as a highly efficient modulator bank. Notably, inverse and forward Fourier transforms may be computationally efficient to implement in an FPGA, ASIC, GPU, etc. In some cases, the IFFT may simultaneously (or nearly-simultaneously) place each input signal into the input signal’s correct frequency slot in a single, unified operation. In some cases, this may be generalized to time-multiplexing or “serialized” operation, in which the FFT delivers a fraction of the total output samples per clock cycle. Further, in these cases, the filter bank may be configured to reduce the minimum clock speed (there may be advantages in time-multiplexing the operation of certain subsets of the logic). This multiplexing operation may create a time-area trade-off, allowing faster runtime, but shrinking in area.

[0165] In some cases, after performing the transform, a subsequent bank of simple polyphase component filters may then complete the time-domain signal reconstruction. This structure may help avoid the immense computational cost of a naive implementation using AT distinct up- samplers and bandpass filters.

[0166] In some cases, FIG. 21 provides (on the left side) an array of synthesizers, each accepting movement commands pruned for their respective field. In some cases, the array of synthesizers supplies a data stream corresponding to the electrode signal to a virtual AOD of width corresponding to the field. On the right half of FIG. 21 is provided an array of electrode drivers, each with input filter banks taking the virtual electrode inputs for all bands for a virtual electrode. Restated, a column control unit is assigned to one frequency bin and outputs data for many electrodes, and an electrode control unit accepts data for one electrode and uses data for many frequency bins. The connectivity between column control units and the electrode control unit filter bank inputs constitutes that of a matrix transpose. Some possible ways to implement this may include one or more of a network (e.g., Benes, Butterfly, Omega, etc.), using a dual port, or n-buffered memory bank with strided access, a Barrel shifter, or potentially by grouping signals on copper wire or fiber interconnects and physically connecting the cables to implement some level of this transpose network.

[0167] In some cases, a polyphase synthesizer bank may be implemented. In one non-limiting example, this may include using a critically sampled polyphase synthesizer bank, as is used in audio codecs, such as, MP3 and AAC, where the total cumulative sample rate across all inputs is equal to the sample rate of the single output. Such a filter bank may comprise a unitary operation. However, it may also be possible to construct a polyphase synthesizer where the cumulativeWSGR Docket No. 55436-733.601 input sample rate is larger than the output sample rate. This is the case where the multiple parallel signals are not stacked in the frequency domain but are made to overlap. This may be referred to as an oversampled polyphase filter bank. This may be analogous to the multiple narrow virtual AODs having overlapping output fields. Such oversampling may limit phase and amplitude distortion near the filter edges. Additionally, this configuration may provide a means for a seamless handoff when a moving spot travels across the boundary between two controllers. In some cases the two controllers may correspond to adjacent regions or subdivided regions. In some cases, the system achieves this by crossfading the control signal: the first controller fades its signal out while the second controller simultaneously fades its signal in. This process helps to ensure the moving spot follows a smooth, continuous path with no (or approximately no) phase discontinuity, by controlled fading and avoiding phase distortion near the filter’s band edge. In some cases, when a spot is moving, the constituent control signals may be potentially shifted or broadened. This may simplify the control system to add margin in the filter -bank input to allow for shifting or broadening without cooperation between neighboring controllers.

[0168] The systems, the methods, the computer-readable media, and the techniques disclosed herein may create a distinct acoustic plane wave for each light spot, enabling precise control of the wave’s k-vector (e.g., with respect to magnitude and direction). To achieve this, signals may be generated for each input of the synthesis filter bank. To provide a non-limiting physical analogy for the sake of clarity, these signals may be thought of as “electrodes.” While not physical components, they serve the same function, in some cases: each is a distinct channel through which a control signal may be applied to a virtual AOD.

[0169] To generate a single spot of light, a drive vector of the same frequency may be provided to all of these “electrodes,” but with a phase gradient applied across them. In some cases, the phase gradient may be approximately uniform. In some cases, the phase gradient may correspond to (or be configured for) any phase relationship between drive vector components. For example, this may include a linear relation. In another example, this may include a non-linear phase gradient, e.g., with compensatory strategies. This constant frequency may form the single spot of light, while the linear change in phase across the channels may steer the single spot of light.

[0170] A non-limiting example of a column controller is provided in FIG. 22, in accordance with some embodiments. In implementing the column controller, a dedicated Numerically Controlled Oscillator (NCO), a precise digital signal generator, may be assigned to each light spot. When the system receives a movement command, the system may dynamically adjust the frequency of the NCO corresponding to that spot. This tuning process may generate the signal for controlling the spot’s horizontal deflection (its x-coordinate) as the spot moves along the commanded path.WSGR Docket No. 55436-733.601Notably, the phase of an NCO may start with a random value, e.g., to help reduce signal crest factor.

[0171] In addition to the NCOs, this architecture may use a grid of NxM phase shifters, where N is the number of virtual AOD electrodes, and A / is the number of spots of light. The signal from a single spot of light’s NCO may be broadcast to an entire column of N phase shifters. Each phase shifter in that column may apply a specific phase shift to that spot’s signal for its corresponding electrode. The final drive signal for each electrode may be created by summing the outputs of all AT phase shifters in its row. This way, each electrode may receive a combined signal that includes a precisely phase-adjusted contribution from every signal spot. The column of phase shifters for a particular spot may be collectively modulated to apply the proper phase gradient to achieve the vertical deflection commanded at the current time in a move profile.

[0172] In some cases, the hardware used by each column controller (e.g., the number of oscillators and phase shifters) may be determined by the maximum number of spots of light expected within its own field. This prevents the controller's complexity from growing with the total number of spots of light in the system, helping ensure the design remains efficient and scalable.

[0173] A non-limiting example for implementing a row controller is provided in FIG. 23, in accordance with some embodiments. In this additional example, instead of using a network of phase shifters, the electrode signals may be the result of the output of a Fourier Transform. In some cases, unlike in audio synthesis where the Fourier synthesis is implemented in space, the Fourier transform here may be along the electrode axis of signals. Notably, the Fourier transform may provide an output per electrode at the rate needed to satisfy the downstream polyphase synthesizer. For a physical AOD of bandwidth , channelized into N virtual AODs that overlapby a fraction O, the FFT may provide, on each virtual electrode output, at least — - — samples per second. This may be readily satisfied by FFT cores on FPGAs.

[0174] In some cases, the Fourier transform may execute a transformation to acoustic k-space on the output, where on the input each row addresses one row of resolved spots of light. Therefore, to generate a spot on a row A at a horizontal displacement of frequency / , a signal,A x sin( x t + ) may be placed on the FFT input terminal R, thus efficiently addressing the optical output field. However, challenges may arise when wanting to implement smooth vertical motion, as the FFT discretizes the output field with a granularity of resolved spots of light. Smooth spot movements with the described arrangement may run into the issue of Gibbs phenomena, where intermediate deflections in k-space require a signal that distributes across the entire FFT input. One way to get around this problem may be to oversample the FFT. Supposing the goal is to address N electrodes, oversampling by a factor K, by using an FFT of width N KWSGR Docket No. 55436-733.601 allows vertical movement with a granularity of a resolved spot divided by K. Choosing K becomes a compromise between the discretized vertical spot resolution and the hardware resources required to implement an FFT of width N K.

[0175] In some cases, an additional strategy to implement smooth vertical movement may include implementing a non-uniform FFT (NUFFT). Such a transform may be implemented by connecting a signal A X sin( X t + ) to several neighboring inputs, with each input given by an interpolation kernel, such as a Slepian or Kaiser-Bessel kernel, multiplied by the time varying signal. This may allow for smooth (e.g., completely smooth) interpolation. The number of inputs needed for a particular oscillator may be determined by the kernel truncation, where a more truncated kernel results in a spot in the output field which spreads due to Heisenberg uncertainty. In some cases, a combination of oversampling and a degree of kernel truncation may be chosen to result in an acceptable spot in the output field with acceptable hardware resource usage.

[0176] The systems, the methods, the computer-readable media, and the techniques disclosed herein may provide NCO connectivity to the FFT input. Such connectivity may be trivial when this scheme is implemented in software. In hardware, a handful of schemes may be possible. For example, a bank of NCOs may be arbitrarily routed using a crossbar switch or a Benes network. Alternatively, NCO outputs may be written into RAM which may be read into the FFT in a dualport, n-buffered memory bank, etc.Examples of Local Amplification in Multi-Channel AODs

[0177] A non-limiting example for local amplification in multi-channel AODs may be provided in FIG. 24, in accordance with some embodiments. In the non-limiting example, each transducer on the MAOD may constitute a load that may be poorly matched to the feed line. The impedance of these transducers, while feasible to control, may be driven by the acoustic requirements of the device, such as acoustic impedance matching to the AO crystal, and may be of high impedance. When imposing a voltage across a load from a long transmission line of impedance different to that of the load, outgoing waves from the generator may reflect off the load and power transfer may be poor. Additionally, any deviation of the generator impedance from the transmission line impedance may result in standing waves in the transmission line that lead to frequency dependent amplitude fluctuations in the voltage across the transducer. To alleviate this, it may be standard to include a resonant circuit electrically close to the electrode which, from the perspective of the generator and the transmission line, converts the impedance of the load into an impedance, which may be close to that of the transmission line, optimizing power transfer and minimizing reflected power. Such impedance matching circuits may use lumped inductive and capacitive elements which may be physically large, especially relative to the electrode pitches intended to target. The impedance matching circuit may correspond to a large labor cost in the manufacturing of the AOWSGR Docket No. 55436-733.601 device. As the electrical impedance of the AO transducer may be difficult to predict ahead of time, being sensitive to various manufacturing variations, the impedance matching circuit may be manually assembled and tuned out of bulk electrical components such as capacitors and hand wound inductors. Scaling this process to high volumes or densities presents a labor bottleneck.

[0178] In some cases, while electrical cross-talk between nearest-neighbor electrodes may be of little impact, imperfections in grounding may result in signals of one electrode leaking to all electrodes through a phenomenon called ground bounce. Advanced grounding schemes, such as using fully coaxial transmission lines may be possible, but for high channel counts, the coaxial bundles may become large and unruly. Such large coaxial bundles may also constitute significant mechanical coupling between the carefully vibration-decoupled optical elements and the control electronics, which may not be configured to minimize vibration. An alternative to single-ended signal transmission, may be differential signaling, where the control signal may be encoded as a voltage difference between two electrical conductors. For example, a possible multichannel interconnect may include a ribbon cable, where control signals are carried by pairs of conductors. In some cases, when carrying high frequency single-ended signals over a ribbon cable, signal and ground connections may be alternated in a GROUND-[SIGNAL1]-GROUND-[SIGNAL2]- GROUND pattern.

[0179] An additional embodiment for local amplification in multi-channel AODs is provided in FIG. 25, in accordance with some embodiments. In this non-limiting example, implementing the local amplification and differential decoding, amplifiers may be placed spatially close to transducers. Such a local amplifier may be integrated on a bump-bonded substrate, or may be placed close to a short electrical fan-out structure and wire bonded using a fine pitch wire bonder.

[0180] In an example AO device with an RF frequency of about 100 MHz passing through a surface microstrip PCB made of FR-4 laminate, the signal speed may be 30-50% of the speed of light. If the length of the transmission line is limited to 5% of the wavelength of that signal, then the amplifier may be placed 7.5 cm from the transducer without suffering significantly from signal reflections of power loss due to impedance mismatch. In other words, the FR-4 has an epsilon ranging from about 4 to about 5. Then, in the case that of using a “5% of the wavelength" rule, that means that about 7.5 cm is acceptable. Note that the rule may correspond to an RF design rule that about X / 20 to X / 10 is an acceptable range for staying within the lumped regime.

[0181] In some cases, if many AO channels are to be driven, such as for a phased array AOD, many RF amplifier channels may be packaged together and located near the AO transducers. Amplifiers may also have some power consumption, and the thermal load from that power consumption may be managed.WSGR Docket No. 55436-733.601

[0182] In some cases, a large portion of the quantum instruction stream includes quantum error correction (QEC) schemes. The atom positions and associated movements may be tiled across a number of repetitions of QEC blocks. In some cases, the output of a smaller AOD, covering one “tile” of spots of light, may be replicated with optics after the AOD across many tiles. This replication can be used to reduce computational controls system requirements or for the concept to function with a smaller AOD.

[0183] In some cases, a large portion of the quantum instruction stream includes quantum error correction (QEC) schemes. The atom positions and associated movements may be known ahead of time and may be predictable. In some cases, a hardware solution or software solution may store pre-computed movement patterns to “replay” across the AOD. This can apply to storing individual atoms movements and drive tones or to the aggregate control signal moving a large number of atoms. This may help reduce the computational burden of generating the control signal in the control system. In some cases, these pre-computed movements may be stored local to the controllers. In some cases, these pre-computed movements may be stored local to other hardware. In some cases, a possible location is the input to the column controller just after pruning takes place. In some cases, the possible location may be on the output of the column controller, with a subsystem for summing this generated RF into the data stream. In some cases, for systems without the disclosed control system, the location may be anywhere between the movement commands and the DACs where commands trigger replay of these movements or the movements are triggered by a periodic timer.Examples of Wavelength Multiplexing

[0184] One challenge in quantum computing is how to create an array of independent tweezers for resorting qubits (e.g., atoms) efficiently, and for performing the highly parallelized moves that are an important enabling technology for high-threshold error-correcting codes. Regardless of what technology is used to generate N independent tweezers (e.g., AOD, MAOD, galvos, LDA, etc.), if all N tweezers are the same wavelength, then the tweezers may be combined in a way that: (A) wastes all but 1 / N of the power; (B) limits each tweezer field of view to 1 / N of the total objective FoV; or (C) limits each tweezer to 1 / N of the total objective NA (or some combination thereof).

[0185] In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein employ wavelength multiplexing to address this above challenge, and present two different architectures that efficiently circumvent the use for a network of dichroics while still generating and combining N independently controlled tweezers.

[0186] At a high level, a first architecture may put the gratings in image planes of the atoms, while a second architecture may put the gratings in Fourier planes of the atoms. The firstWSGR Docket No. 55436-733.601 architecture may use an enormous (~1 m x ~1 mm scale) grating, while the second architecture may use a microlens array (MLA) with substantially more resolving power per lens than a cavityarray MLA (which may work better with a fly’s eye array than a standard microlens array). Accordingly, this forms a new trade space for understanding control of many tweezers - it may even be possible to interpolate between the two architectures by putting the gratings in intermediate optical planes.

[0187] With respect to each of the two architectures, in some cases, working with spectrally broadband light for each tweezer has some advantages in terms of suppressing speckle. This may not be as effective as fully spatially incoherent light, but at least it means the tweezers may not interfere with one another and back-reflections will be strongly suppressed.

[0188] With respect to each of the two architectures, in some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may use a ID lightfield directing array (LDA) with -100 mirrors in it. If each mirror is about 1 mm, that, itself is about 10 cm across. In some cases, the mirrors may be about 10 pm, 100 pm, 1 mm, 1 cm, 10 cm, etc.

[0189] The systems, the methods, the computer-readable media, and the techniques disclosed herein may diffract the light in 2D to enable use of a more typical 2D LDA. One example includes the use of an Echelle grating (e.g., blazed grating at high order) to provide periodic diffraction angle with wavelength. Another example combines a grating and a VIPA to enable 2D resolution, similar to as explained on page 64 of: Willits, J. T. (2012). Dynamic Line-by-line Pulse Shaping. University of Colorado, Thesis, which is incorporated by reference herein for all purposes. Using this combined grating and VIPA to enable 2D resolution leaves the challenge of achieving imaging with a VIPA, for which various techniques may address.

[0190] With respect to each of the two architectures, in some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein use a high power, broadband light source. However, in other cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may combine low power light-sources of different wavelengths, e.g., instead of using the input grating.

[0191] With respect to each of the two architectures, in some cases, chromatic aberration of the objective may move some tweezers out of focus; this may be correctable downstream with additional optics.

[0192] With respect to each of the two architectures, in some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may use high-efficiency, small period gratings, such as blazed gratings, or echelle gratings.WSGR Docket No. 55436-733.601Gratings in Real-Space

[0193] FIG. 26 illustrates an example of a grating in real-space 2600. The grating 2600 may receive a spectrally broadband input beam with total spectral width SA and focuses the input beam down onto a grating of size w, which may be a virtual image plane of the atoms. The grating 2600 may disperse the light into N independent beams which, in the Fourier plane of the subsequent lens, illuminate an array of N independently controlled 2D deflecting mirrors in a lightfield directing array (LDA). Examples of an LDA that may be used with the systems, the methods, the computer-readable media, and the techniques disclosed herein may be included in Panas, R. (2015). Micro-Reflector Array for High-Speed Directed-Light-Field Projection, which is incorporated by reference herein for all purposes. Each of these mirrors in the LDA may be viewed as a full-NA Fourier plane of the atom imaging system, albeit each for a different wavelength range. Tilting these mirrors may move the tweezers in the virtual image plane that follows the subsequent lens. Because they come from displaced regions of Fourier space, tweezers in the same spatial location will come in at different angles, but this will be corrected in by the enormous grating in this image plane. In other words, just before the second grating, may be tweezers of different colors occupying different angular ranges; just after the second grating may be tweezers of different colors occupying the same angular range, corresponding to the full aperture of the objective Fourier plane.

[0194] In some cases, note that in order to disperse a beam with spectral width SA into N independent beams uses a grating of size w =xd, where d is the grating spacing.Choosing the smallest allowable grating spacing d = A, this means that, for A = ~485 nm, N = ~100, and SA = ~10 nm, there may be an input grating of size w = ~1.1 mm. This may mean that the output side will use a grating that is larger by a factor of the ratio R of the tweezer size to the FoV. For a 500 nm tweezer and a 500 pm field of view, A=~1000 spots of light, and this may mean the output grating has a size of about 1.1 m.

[0195] In some cases, the grating 2600 may use an enormous, meter-scale grating - it may not need be perfectly periodic over the full grating size, but only over the scale of a single tweezer, meaning on the scale of about a few millimeters.

[0196] In some cases, because the gratings are in image planes, the grating may have the potential downside of being susceptible to dust and other imperfections, which may map directly onto beam-distortions on the atoms. Therefore, techniques may be implemented to shield the gratings, replace the gratings, or clean the gratings.

[0197] In some cases, to reduce crosstalk, discrete frequencies (e.g., not a continuous spectrum) may be used. For example, this may mean a comb with ~0.1 nm tooth spacing, which may be extremely wide. At 485 nm this may be -lOOGHz. Frequency combs may not have such largeWSGR Docket No. 55436-733.601 spacings but a moire filter (or, e.g., a single filter cavity with about 101: 100 FSR ratio compared to the comb cavity) of a GHz comb, prior to amplification, may provide such large spacings, albeit at a large initial power cost that may rely on multiple amplification stages. A fiber amplifier may work well, as the tooth spacing is larger than the SBS bandwidth, but may introduce dispersion. For example, a Thorlabs diode 488p60 (or the like), which provides 60 mW with comb teeth separated by ~0.1 nm over ~2 nm, may be used with an LDA and pair of lens arrays. In another example, a whispering gallery mode frequency comb may help to get stable -100GHz mode spacing, as disclosed in Savchenkov, A. A., Matsko, A. B., Ilchenko, V. S., Solomatine, I., Seidel, D., & Maleki, L. (2008). Tunable optical frequency comb with a crystalline whispering gallery mode resonator. Physical Review Letters, 101(9). https: / / doi.org / 10.1103 / physrevlett.101.093902, which is incorporated by reference herein for all purposes.Gratings in Fourier Space

[0198] FIG. 27 illustrates an example of a grating 2700 in Fourier space. The grating 2700 puts the gratings in Fourier space: the input grating+ spherical lens+MLA creates a 4f imaging system that images a wavelength range onto each LDA mirror, which is then re-imaged onto the output grating via second 4f imaging system created from the second MLA and second spherical lens.

[0199] In some cases, the grating 2700 has the advantage that each grating has a lower size of w which, for realistic parameters discussed above, is about 1.1 mm. With thegrating 1100, techniques may be implemented to ensure that tilting the LDA mirrors does not deflect the beams past the edges of the microlenses. In practice, this likely means that fully broadband light will incur some loss in the MLA-based approach (e.g., filling all of the input microlenses). In some cases, techniques may be implemented to ensure that the microlenses are not excessively aberrated at the large angles of incidence induced by the tilting of the LDA.

[0200] In some cases, the grating 2700 may face the challenge that each lens in the output MLA should be able to resolve, in angle, a number of independent beams given by the ratio of the FoV to the tweezer size, R (defined in the previous section). Each microlens may have a diffractionlimited angular resolution (e.g., when the spot fills its image plane) of 60 « and can producea range of angles across a FoV set by its size of A0 = , here, d is the diameter of the lens, and f is its focal length. As such, the number of points that the lens can produce in ID is N1D= — = o u d2n X — Plugging in example values (Thorlabs MLA300-14AR) d = 300 [im, f = 15 mm, yields for = 485 nm that N1D« 42, well short of the R=~1000 spots of light (for a 500 nm spot in a 500 pm field of view). Choosing the Thorlabs MLA150-7AR instead, with d = 150 pm, f =WSGR Docket No. 55436-733.6015.2 mm, yields N1D« 28. Another option, the Edmund #21-152, with d = 300 m, f =4.8 mm, yields N1D« 121.

[0201] In some cases, the grating 2700 faces the challenge of yielding more spots of light. First, note that a plano-convex lens of focal length / has radius of curvature ROC = (n — 1) , where n is the refractive index of the lens, and also note that a lens of diameter d and radius of curvature d21 d2TROC has a center thickness T « - 8 ROC = — 8( -n-l -) — f . Accordingly, Niun= 8fn — 1) J x - Because these MLA may be etched and hence may be quite shallow, it may be a challenge to yield a lot of points. By contrast, a fly’s eye array, such as Edmunds 84-132, can have d = 300 gm, f = 0.6 mm, resulting in N1D« 971. This may be far improved, provided, for example, there are not limitations by aberrations on the lens.

[0202] In some cases, these gratings of the grating 2700 are in Fourier planes and may be less susceptible to dust and other imperfections. This approach of the grating 2700 may use relatively large angles of incidence on the lenses of the output lens array, and, as such, may potentially suffer from lens aberration that is difficult to control without aspheric microlenses. This is an advantage of the grating 2600 of FIG. 26. For the grating 2700 of FIG. 27, really large input / output gratings may allow using small frequency spacing. For example, a l m grating may enable 1 GHz frequency spacing, which could be generated with a standard frequency comb. In another example, a 10 cm grating may enable 10 GHz spacing, which is possible for opticals with intracavity EOMs, as well as EOM-based comb generators.Examples of Systems for Performing a Non-Classical Computation

[0203] FIG. 2 shows an example of a system 200 for performing a non-classical computation. The non-classical computation may comprise a quantum computation. The quantum computation may comprise a gate-model quantum computation.

[0204] The system 200 may comprise one or more trapping units 210. The trapping units may comprise one or more optical trapping units. The optical trapping units may comprise any optical trapping unit described herein, such as an optical trapping unit described herein with respect to FIG. 3A. The optical trapping units may be configured to generate a plurality of optical trapping sites. The optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites. For instance, the optical trapping units may be configured to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. The optical trapping units may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000,WSGR Docket No. 55436-733.601400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer optical trapping sites. The optical trapping units may be configured to trap a number of optical trapping sites that is within a range defined by any two of the preceding values.

[0205] The optical trapping units may be configured to trap a plurality of atoms. For instance, the optical trapping units may be configured to trap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. The optical trapping units may be configured to trap at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The optical trapping units may be configured to trap a number of atoms that is within a range defined by any two of the preceding values.

[0206] Each optical trapping site of the optical trapping units may be configured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. Each optical trapping site may be configured to trap at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each optical trapping site may be configured to trap a number of atoms that is within a range defined by any two of the preceding values. Each optical trapping site may be configured to trap a single atom.

[0207] One or more atoms of the plurality of atoms may comprise qubits, as described herein (for instance, with respect to FIG. 4). Two or more atoms may be quantum mechanically entangled.Two or more atoms may be quantum mechanically entangled with a coherence lifetime of at least about 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, 6 ms, 7 ms, 8 ms, 9 ms, 10 ms, 20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms, 300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2 s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or more. Two or more atoms may be quantum mechanically entangled with a coherence lifetime of at most about 10 s, 9 s, 8 s, 7 s, 6 s, 5 s, 4 s, 3 s, 2 s, 1 s, 900 ms, 800 ms, 700 ms, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100 ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms, 8 ms, 7 ms, 6 ms, 5 ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, or less. Two or more atoms may be quantum mechanicallyWSGR Docket No. 55436-733.601 entangled with a coherence lifetime that is within a range defined by any two of the preceding values. One or more atoms may comprise neutral atoms. One or more atoms may comprise uncharged atoms.

[0208] One or more atoms may comprise alkali atoms. One or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium -45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium - 145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium- 175 atoms, or lutetium- 176 atoms.

[0209] The plurality of atoms may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. TheWSGR Docket No. 55436-733.601 plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium -6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium- 86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium- 156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium- 164 atoms, erbium - 166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium- 172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium- 176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-WSGR Docket No. 55436-733.60184 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium- 130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-WSGR Docket No. 55436-733.601171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance that is within a range defined by any two of the preceding values.

[0210] The system 200 may comprise one or more first electromagnetic delivery units 220. The first electromagnetic delivery units may comprise any electromagnetic delivery unit described herein, such as an electromagnetic delivery unit described herein with respect to FIG. 4. The first electromagnetic delivery units may be configured to apply first electromagnetic energy to one or more atoms of the plurality of atoms. Applying the first electromagnetic energy may induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state that is different from the first atomic state.

[0211] The first atomic state may comprise a first single-qubit state. The second atomic state may comprise a second single-qubit state. The first atomic state or second atomic state may be elevated in energy with respect to a ground atomic state of the atoms. The first atomic state or second atomic state may be equal in energy with respect to the ground atomic state of the atoms.

[0212] The first atomic state may comprise a first hyperfine electronic state and the second atomic state may comprise a second hyperfine electronic state that is different from the first hyperfine electronic state. For instance, the first and second atomic states may comprise first and second hyperfine states on a multiplet manifold, such as a triplet manifold. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi or3P2 manifold. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi or3P2 manifold of any atom described herein, such as a strontium-873Pi manifold or a strontium-873P2 manifold.

[0213] In some cases, the first and second atomic states are first and second hyperfine states of a first electronic state. Optical excitation may be applied between a first electronic state and a second electronic state. The optical excitation may excite the first hyperfine state or the second hyperfine state to the second electronic state. A single-qubit transition may comprise a two- photon transition between two hyperfine states within the first electronic state using a second electronic state as an intermediate state. To drive a single-qubit transition, a pair of frequencies, each detuned from a single-photon transition to the intermediate state, may be applied to drive a two-photon transition. In some cases, the first and second hyperfine states are hyperfine states of the ground electronic state. The ground electronic state may not decay by spontaneous or stimulated emission to a lower electronic state. The hyperfine states may comprise nuclear spin states.

[0214] In some cases, the hyperfine states comprise nuclear spin states of a strontium-87 'So manifold and the qubit transition drives one or both of two nuclear spin states of strontium-87xSoWSGR Docket No. 55436-733.601 to a state detuned from or within the3P2 or3Pi manifold. In some cases, the one-qubit transition is a two photon Raman transition between nuclear spin states of strontium-87 'So via a state detuned from or within the3P2 or3Pi manifold. In some cases, the nuclear spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift may be driven off resonance with any, all, or a combination of a single-qubit transition, a two-qubit transition, a shelving transition, an imaging transition, etc.

[0215] In some cases, the hyperfine states comprise nuclear spin states of ytterbium.

[0216] The first atomic state may comprise a first nuclear spin state and the second atomic state may comprise a second nuclear spin state that is different from the first nuclear spin state. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a quadrupolar nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin- 9 / 2 nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of any atom described herein, such as first and second spin states of strontium-87.

[0217] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1 / 2 (such as a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 7 / 2 spin state, may also drive mN = 7 / 2 to mN = 5 / 2, mN = 5 / 2 to mN = 3 / 2, mN = 3 / 2 to mN = 1 / 2, mN = 1 / 2 to mN = -1 / 2, mN = -1 / 2 to mN = -3 / 2, mN = -3 / 2 to mN = -5 / 2, mN = -5 / 2 to mN = -7 / 2, and mN = -7 / 2 to mN = -9 / 2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 5 / 2 spin state, may also drive mN = 7 / 2 to mN = 3 / 2, mN = 5 / 2 to mN = 1 / 2, mN = 3 / 2 to mN = -1 / 2, mN = 1 / 2 to mN = -3 / 2, mN = -1 / 2 to mN = -5 / 2, mN = -3 / 2 to mN = -7 / 2, and mN = -5 / 2 to mN = -9 / 2. Such a transition may thus not be selective for inducing transitions between particular spin states on the nuclear spin manifold.

[0218] It may be desirable to instead implement selective transitions between particular first and second spins states on the nuclear spin manifold. This may be accomplished by providing light from a light source that provides an AC Stark shift and pushes neighboring nuclear spin states out of resonance with a transition between the desired transition between the first and second nuclear spin states. For instance, if a transition from first and second nuclear spin states having mN = -9 / 2 and mN = -7 / 2 is desired, the light may provide an AC Stark shift to the mN = -5 / 2 spin state, thereby greatly reducing transitions between the mN = -7 / 2 and mN = -5 / 2 states. Similarly, if aWSGR Docket No. 55436-733.601 transition from first and second nuclear spin states having mN = -9 / 2 and mN = -5 / 2 is desired, the light may provide an AC Stark shift to the mN = -1 / 2 spin state, thereby greatly reducing transitions between the mN = -5 / 2 and mN = -1 / 2 states. This may effectively create a two-level subsystem within the nuclear spin manifold that is decoupled from the remainder of the nuclear spin manifold, greatly simplifying the dynamics of the qubit systems. It may be advantageous to use nuclear spin states near the edge of the nuclear spin manifold (e.g., mN = -9 / 2 and mN = -7 / 2, mN = 7 / 2 and mN = 9 / 2, mN = -9 / 2 and mN = -5 / 2, or mN = 5 / 2 and mN = 9 / 2 for a spin-9 / 2 nucleus) such that only one AC Stark shift is required. Alternatively, nuclear spin states farther from the edge of the nuclear spin manifold (e.g., mN = -5 / 2 and mN = -3 / 2 or mN = -5 / 2 and mN = -1 / 2) may be used and two AC Stark shifts may be implemented (e.g., at mN = -7 / 2 and mN = -1 / 2 or mN = -9 / 2 and mN = 3 / 2).

[0219] Stark shifting of the nuclear spin manifold may shift neighboring nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. Stark shifting may decrease leakage from the first and second nuclear spin state to other states in the nuclear spin manifold. Starks shifts may be achievable up to 100s of kHz for less than 10 mW beam powers. Upper state frequency selectivity may decrease scattering from imperfect polarization control. Separation of different angular momentum states in the3Pi manifold may be many gigahertz from the single and two- qubit gate light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for two-qubit transitions (e.g., how quickly the transition can be driven) may be faster than the decoherence rate. Scattering from the intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve fidelity of two-qubit transitions.

[0220] Qubits based on nuclear spin states in the electronic ground state may allow exploitation of long-lived metastable excited electronic states (such as a3Po state in strontium-87) for qubit storage. Atoms may be selectively transferred into such a state to reduce cross-talk or to improve gate or detection fidelity. Such a storage or shelving process may be atom -selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the ' So state in strontium-87 to the3Po or3P2 state in strontium-87.

[0221] The clock transition (also a “shelving transition” or a “storage transition” herein) may be qubit-state selective. The upper state of the clock transition may have a very long natural lifetime, e.g., greater than 1 second. The linewidth of the clock transition may be much narrower than the qubit energy spacing. This may allow direct spectral resolution. Population may be transferred from one of the qubit states into the clock state. This may allow individual qubit states to be read out separately, by first transferring population from one qubit state into the clockWSGR Docket No. 55436-733.601 state, performing imaging on the qubits, then transferring the population back into the ground state from the clock state and imaging again. In some cases, a magic wavelength transition is used to drive the clock transition.

[0222] The clock light for shelving can be atom -selective or not atom-selective. In some cases, the clock transition is globally applied (e.g., not atom selective). A globally applied clock transition may include directing the light without passing through a microscope objective or structuring the light. In some cases, the clock transition is atom-selective. Clock transition which are atom-selective may potentially allow us to improve gate fidelities by minimizing cross-talk. For example, to reduce cross talk in an atom, the atom may be shelved in the clock state where it may not be affected by the light. This may reduce cross-talk between neighboring qubits undergoing transitions. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc.

[0223] The system 200 may comprise one or more readout units 230. The readout units may comprise one or more readout optical units. The readout optical units may be configured to perform one or more measurements of the one or more superposition states to obtain the non- classical computation. The readout optical units may comprise one or more optical detectors. The detectors may comprise one or more photomultiplier tubes (PMTs), photodiodes, avalanche diodes, single-photon avalanche diodes, single-photon avalanche diode arrays, phototransistors, reverse-biased light emitting diodes (LEDs), charge coupled devices (CCDs), or complementary metal oxide semiconductor (CMOS) cameras. The optical detectors may comprise one or more fluorescence detectors. The readout optical unit may comprise one or more objectives, such as one or more objective having a numerical aperture (NA) of at least about 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, or more. The objective may have an NA of at most about 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, or less. The objective may have an NA that is within a range defined by any two of the preceding values.

[0224] The one or more readout optical units 230 may make measurements, such as projective measurements, by applying light resonant with an imaging transition. The imaging transition may cause fluorescence. An imaging transition may comprise a transition between the ' So state in strontium-87 to thexPi state in strontium-87. ThexPi state in strontium-87 may fluoresce. The lower state of the qubit transition may comprise two nuclear spin states in the ' So manifold. The one or more states may be resonant with the imaging transition. A measurement may comprise two excitations. In a first excitation, one of the two lower states may be excited to the shelving state (e.g.,3Po state in strontium-87). In a second excitation, the imaging transition may beWSGR Docket No. 55436-733.601 excited. The first transition may reduce cross-talk between neighboring atoms during computation. Fluorescence generated from the imaging transition may be collected on one or more readout optical units 230.

[0225] The imaging units may be used to determine if one or more atoms were lost from the trap. The imaging units may be used to observe the arrangement of atoms in the trap.

[0226] The system 200 may comprise one or more vacuum units 240. The one or more vacuum units may comprise one or more vacuum pumps. The vacuum units may comprise one or more roughing vacuum pumps, such as one or more rotary pumps, rotary vane pumps, rotary piston pumps, diaphragm pumps, piston pumps, reciprocating piston pumps, scroll pumps, or screw pumps. The one or more roughing vacuum pumps may comprise one or more wet (for instance, oil-sealed) or dry roughing vacuum pumps. The vacuum units may comprise one or more high- vacuum pumps, such as one or more cryosorption pumps, diffusion pumps, turbomolecular pumps, molecular drag pumps, turbo-drag hybrid pumps, cryogenic pumps, ions pumps, or getter pumps.

[0227] The vacuum units may comprise any combination of vacuum pumps described herein. For instance, the vacuum units may comprise one or more roughing pumps (such as a scroll pump) configured to provide a first stage of rough vacuum pumping. The roughing vacuum pumps may be configured to pump gases out of the system 200 to achieve a low vacuum pressure condition. For instance, the roughing pumps may be configured to pump gases out of the system 200 to achieve a low vacuum pressure of at most about 103Pascals (Pa). The vacuum units may further comprise one or more high-vacuum pumps (such as one or more ion pumps, getter pumps, or both) configured to provide a second stage of high vacuum pumping or ultra -high vacuum pumping. The high-vacuum pumps may be configured to pump gases out of the system 200 to achieve a high vacuum pressure of at most about 10'3Pa or an ultra-high vacuum pressure of at most about 10'6Pa once the system 200 has reached the low vacuum pressure condition provided by the one or more roughing pumps.

[0228] The vacuum units may be configured to maintain the system 200 at a pressure of at most about IO’6Pa, 9 x IO’7Pa, 8 x IO’7Pa, 7 x IO’7Pa, 6 x IO’7Pa, 5 x IO’7Pa, 4 x IO’7Pa, 3 x IO’7Pa, 2 x IO’7Pa, IO’7Pa, 9 x IO’8Pa, 8 x IO’8Pa, 7 x IO’8Pa, 6 x IO’8Pa, 5 x IO’8Pa, 4 x IO’8Pa, 3 x IO’8Pa, 2 x IO’8Pa, IO’8Pa, 9 x IO’9Pa, 8 x IO’9Pa, 7 x IO’9Pa, 6 x IO’9Pa, 5 x IO’9Pa, 4 x 10’9Pa, 3 x IO’9Pa, 2 x IO’9Pa, IO’9Pa, 9 x IO’10Pa, 8 x IO’10Pa, 7 x IO’10Pa, 6 x IO’10Pa, 5 x IO’10Pa, 4 x IO’10Pa, 3 x IO’10Pa, 2 x IO’10Pa, IO’10Pa, 9 x 10’11Pa, 8 x 10’11Pa, 7 x 10’11Pa, 6 xlO’11Pa, 5 x 10’11Pa, 4 x 10’11Pa, 3 x 10’11Pa, 2 x 10’11Pa, 10’11Pa, 9 x IO’12Pa, 8 x IO’12Pa, 7 x IO’12Pa, 6 x IO’12Pa, 5 x 10'12Pa, 4 x 10'12Pa, 3 x 10'12Pa, 2 x 10'12Pa, 10'12Pa, or lower. The vacuum units may be configured to maintain the system 200 at a pressure of at least about 10'12WSGR Docket No. 55436-733.601Pa, 2 x IO’12Pa, 3 x IO’12Pa, 4 x IO’12Pa, 5 x IO’12Pa, 6 x IO’12Pa, 7 x IO’12Pa, 8 x IO’12Pa, 9 x IO’12Pa, IO’11Pa, 2 x IO’11Pa, 3 x IO’11Pa, 4 x IO’11Pa, 5 x IO’11Pa, 6 x IO’11Pa, 7 x IO’11Pa, 8 x IO’11Pa, 9 x IO’11Pa, IO’10Pa, 2 x IO’10Pa, 3 x IO’10Pa, 4 x IO’10Pa, 5 x IO’10Pa, 6 x IO’10Pa,7 x IO’10Pa, 8 x IO’10Pa, 9 x IO’10Pa, IO’9Pa, 2 x IO’9Pa, 3 x IO’9Pa, 4 x IO’9Pa, 5 x IO’9Pa, 6 x IO’9Pa, 7 x IO’9Pa, 8 x IO’9Pa, 9 x IO’9Pa, IO’8Pa, 2 x IO’8Pa, 3 x IO’8Pa, 4 x IO’8Pa, 5 x 10’8Pa, 6 x IO’8Pa, 7 x IO’8Pa, 8 x IO’8Pa, 9 x IO’8Pa, IO’7Pa, 2 x IO’7Pa, 3 x IO’7Pa, 4 x IO’7Pa, 5 x 10'7Pa, 6 x 10'7Pa, 7 x 10'7Pa, 8 x 10'7Pa, 9 x 10'7Pa, 10'6Pa, or higher. The vacuum units may be configured to maintain the system 200 at a pressure that is within a range defined by any two of the preceding values.

[0229] The system 200 may comprise one or more state preparation units 250. The state preparation units may comprise any state preparation unit described herein, such as a state preparation unit described herein with respect to FIG. 5. The state preparation units may be configured to prepare a state of the plurality of atoms.

[0230] The system 200 may comprise one or more atom reservoirs 260. The atom reservoirs may be configured to supply one or more replacement atoms to replace one or more atoms at one or more optical trapping sites upon loss of the atoms from the optical trapping sites. The atom reservoirs may be spatially separated from the optical trapping units. For instance, the atom reservoirs may be located at a distance from the optical trapping units.

[0231] Alternatively or in addition, the atom reservoirs may comprise a portion of the optical trapping sites of the optical trapping units. A first subset of the optical trapping sites may be utilized for performing quantum computations and may be referred to as a set of computationally-active optical trapping sites, while a second subset of the optical trapping sites may serve as an atom reservoir. For instance, the first subset of optical trapping sites may comprise an interior array of optical trapping sites, while the second subset of optical trapping sites comprises an exterior array of optical trapping sites surrounding the interior array. The interior array may comprise a rectangular, square, rectangular prism, or cubic array of optical trapping sites.

[0232] The system 200 may comprise one or more atom movement units 270. The atom movement units may be configured to move the one or more replacement atoms from the one or more atoms reservoirs to the one or more optical trapping sites. For instance, the one or more atom movement units may comprise one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs).

[0233] The system 200 may comprise one or more entanglement units 280. The entanglement units may be configured to quantum mechanically entangle at least a first atom of the plurality of atoms with at least a second atom of the plurality of atoms. The first or second atom may be in aWSGR Docket No. 55436-733.601 superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first or second atom may not be in a superposition state at the time of quantum mechanical entanglement. The first atom and the second atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The entanglement units may be configured to quantum mechanically entangle any number of atoms described herein.

[0234] The entanglement units may also be configured to quantum mechanically entangle at least a subset of the atoms with at least another atom to form one or more multi-qubit units. The multiqubit units may comprise two-qubit units, three-qubit units, four-qubit units, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For instance, a two-qubit unit may comprise a first atom quantum mechanically entangled with a second atom, a three-qubit unit may comprise a first atom quantum mechanically entangled with a second and third atom, a four-qubit unit may comprise a first atom quantum mechanically entangled with a second, third, and fourth atom, and so forth. The first, second, third, or fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first, second, third, or fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first, second, third, and fourth atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions.

[0235] The entanglement units may comprise one or more Rydberg units. The Rydberg units may be configured to electronically excite the at least first atom to a Rydberg state or to a superposition of a Rydberg state and a lower-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg units may be configured to induce one or more quantum mechanical entanglements between the Rydberg atoms or dressed Rydberg atoms and the at least second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (pm), 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance from the Rydberg atoms or dressed Rydberg atoms that is within a range defined by any two of the preceding values. The Rydberg units may be configured to allow the Rydberg atoms or dressed Rydberg atoms to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg units may be configured to induce the Rydberg atoms or dressed Rydberg atoms to relax to a lower-WSGR Docket No. 55436-733.601 energy atomic state. The Rydberg units may be configured to drive the Rydberg atoms or dressed Rydberg atoms to a lower-energy atomic state. For instance, the Rydberg units may be configured to apply electromagnetic radiation (such as RF radiation or optical radiation) to drive the Rydberg atoms or dressed Rydberg atoms to a lower-energy atomic state. The Rydberg units may be configured to induce any number of quantum mechanical entanglements between any number of atoms of the plurality of atoms.

[0236] The Rydberg units may comprise one or more light sources (such as any light source described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths may be selected to correspond to a wavelength that forms the Rydberg atoms or dressed Rydberg atoms. For instance, the light may comprise one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. The light may comprise one or more wavelengths of at most about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 300 nm to 400 nm.

[0237] The Rydberg units may be configured to induce a two-photon transition to generate an entanglement. The Rydberg units may be configured to induce a two-photon transition to generate an entanglement between two atoms. The Rydberg units may be configured to selectively induce a two-photon transition to selectively generate an entanglement between two atoms. For instance, the Rydberg units may be configured to direct electromagnetic energy (such as optical energy) to particular optical trapping sites to selectively induce a two-photon transition to selectively generate the entanglement between the two atoms. The two atoms may be trapped in nearby optical trapping sites. For instance, the two atoms may be trapped in adjacent optical trapping sites. The two-photon transition may be induced using first and second light from first and second light sources, respectively. The first and second light sources may each comprise any light source described herein (such as any laser described herein). The first light source may be the same or similar to a light source used to perform a single-qubit operation described herein. Alternatively, different light sources may be used to perform a single-qubit operation and to induce a two-photon transition to generate an entanglement. The first light source may emit light comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from 650 nm to 700 nm). The second light source may emit light comprising one or more wavelengths in the ultraviolet region of the optical spectrum (e.g.,WSGR Docket No. 55436-733.601 within a range from 200 nm to 400 nm or from 300 nm to 350 nm). The first and second light sources may emit light having substantially equal and opposite spatially-dependent frequency shifts.

[0238] The Rydberg atoms or dressed Rydberg atoms may comprise a Rydberg state that may have sufficiently strong interatomic interactions with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) to enable the implementation of multi -qubit operations. The Rydberg states may comprise a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or more. The Rydberg states may comprise a principal quantum number of at most about 100, 90, 80, 70, 60, 50, or less. The Rydberg states may comprise a principal quantum number that is within a range defined by any two of the preceding values. The Rydberg states may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift atomic energy levels of the atoms.

[0239] State selective excitation of atoms to Rydberg levels may enable the implementation of multi -qubit operations. The multi -qubit operations may comprise two-qubit operations, three- qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or more. Two-photon transitions may be used to excite atoms from a ground state (such as a ' So ground state) to a Rydberg state (such as an n3Si state, wherein n is a principal quantum number described herein). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transitions may be implemented using first and second laser sources, as described herein. The first laser source may emit pi -polarized light, which may not change the projection of atomic angular momentum along a magnetic field. The second laser may emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. The first and second qubit levels may be excited to Rydberg level using this polarization. However, the Rydberg levels may be more sensitive to magnetic fields than the ground state so that large splittings (for instance, on the order of 100s of MHz) may be readily obtained. This spectral selectivity may allow state selective excitation to Rydberg levels.

[0240] Multi-qubit operations (such as two-qubit operations, three-qubit operations, four-qubit operations, and so forth) may rely on energy shifts of levels due to van der Waals interactions described herein. Such shifts may either prevent the excitation of one atom conditional on the state of the other or change the coherent dynamics of excitation of the two-atom system to enact a two-qubit operation. In some cases, “dressed states” may be generated under continuous driving to enact two-qubit operations without requiring full excitation to a Rydberg level (for instance, as described in www.arxiv.org / abs / 1605.05207, which is incorporated herein by reference in its entirety for all purposes).WSGR Docket No. 55436-733.601

[0241] The system 200 may comprise one or more second electromagnetic delivery units (not shown in FIG. 2). The second electromagnetic delivery units may comprise any electromagnetic delivery unit described herein, such as an electromagnetic delivery unit described herein with respect to FIG. 4. The first and second electromagnetic delivery units may be the same. The first and second electromagnetic delivery units may be different. The second electromagnetic delivery units may be configured to apply second electromagnetic energy to the one or more multi -qubit units. The second electromagnetic energy may comprise one or more pulse sequences. The first electromagnetic energy may precede, be simultaneous with, or follow the second electromagnetic energy.

[0242] The pulse sequences may comprise any number of pulses. For instance, the pulse sequences may comprise at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, or more pulses. The pulse sequences may comprise at most about 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 pulses. The pulse sequences may comprise a number of pulses that is within a range defined by any two of the preceding values. Each pulse of the pulse sequence may comprise any pulse shape, such as any pulse shape described herein.

[0243] The pulse sequences may be configured to decrease the duration of time required to implement multi -qubit operations, as described herein (for instance, with respect to Example 3). For instance, the pulse sequences may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, or more. The pulse sequences may comprise a duration of at most about 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. The pulse sequences may comprise a duration that is within a range defined by any two of the preceding values.

[0244] The pulse sequences may be configured to increase the fidelity of multi -qubit operations, as described herein. For instance, the pulse sequences may enable multi-qubit operations with a fidelity of at least about 0.5, 0.6, 0.7, 0.8, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 0.991, 0.992, 0.993, 0.994, 0.995, 0.996, 0.997, 0.998, 0.999, 0.9991, 0.9992, 0.9993, 0.9994, 0.9995, 0.9996, 0.9997, 0.9998, 0.9999, 0.99991, 0.99992, 0.99993, 0.99994, 0.99995, 0.99996, 0.99997, 0.99998, 0.99999, 0.999991, 0.999992, 0.999993, 0.999994, 0.999995, 0.999996, 0.999997, 0.999998, 0.999999, or more. The pulse sequences may enable multi-qubit operations with a fidelity of at most about 0.999999, 0.999998, 0.999997, 0.999996, 0.999995, 0.999994,WSGR Docket No. 55436-733.6010.999993, 0.999992, 0.999991, 0.99999, 0.99998, 0.99997, 0.99996, 0.99995, 0.99994, 0.99993, 0.99992, 0.99991, 0.9999, 0.9998, 0.9997, 0.9996, 0.9995, 0.9994, 0.9993, 0.9992, 0.9991, 0.999, 0.998, 0.997, 0.996, 0.995, 0.994, 0.993, 0.992, 0.991, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.9, 0.8, 0.7, 0.6, 0.5, or less. The pulse sequences may enable multi-qubit operations with a fidelity that is within a range defined by any two of the preceding values.

[0245] The pulse sequences may enable the implementation of multi-qubit operations on non- adiabatic timescales while maintaining effectively adiabatic dynamics. For instance, the pulse sequences may comprise one or more of shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, counterdiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences. For instance, the pulse sequences may be similar to those described in M. V. Berry, “Transitionless Quantum Driving,” Journal of Physics A: Mathematical and Theoretical 42(36), 365303 (2009), www.doi.org / 10.1088 / 1751-8113 / 42 / 36 / 365303; Y.-Y. Jau et al., “Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction,” Nature Physics 12(1), 71-74 (2016); T. Keating et al., “Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing,” Physical Review A 91, 012337 (2015); A. Mitra et al., “Robust Mblmer-Sbrenson Gate for Neutral Atoms Using Rapid Adiabatic Rydberg Dressing,” www.arxiv.org / abs / 1911.04045 (2019); or L.S. Theis et al., “Counteracting Systems of Diabaticities Using DRAG Controls: The Status after 10 Years,” Europhysics Letters 123(6), 60001 (2018), each of which is incorporated herein by reference in its entirety for all purposes.

[0246] The pulse sequences may further comprise one or more optimal control pulse sequences. The optimal control pulse sequences may be derived from one or more procedures, including gradient ascent pulse engineering (GRAPE) methods, Krotov’s method, chopped basis methods, chopped random basis (CRAB) methods, Nelder-Mead methods, gradient optimization using parametrization (GROUP) methods, genetic algorithm methods, and gradient optimization of analytic controls (GOAT) methods. For instance, the pulse sequences may be similar to those described in N. Khaneja et al., “Optimal Control of Coupled Spin Dynamics: Design of NMR Pulse Sequences by Gradient Ascent Algorithms,” Journal of Magnetic Resonance 172(2), 296- 305 (2005); or J.T. Merrill et al., “Progress in Compensating Pulse Sequences for Quantum Computation,” Advances in Chemical Physics 154, 241-294 (2014), each of which is incorporated by reference in its entirety for all purposes.Examples of Cloud Computing

[0247] The system 200 may be operatively coupled to a digital computer described herein (such as a digital computer described herein with respect to FIG. 1) over a network described hereinWSGR Docket No. 55436-733.601(such as a network described herein with respect to FIG. 1). The network may comprise a cloud computing network.Examples of Optical Trapping Units

[0248] FIG. 3A shows an example of an optical trapping unit 210. The optical trapping unit may be configured to generate a plurality 211 of spatially distinct optical trapping sites, as described herein. For instance, as shown in FIG. 3B, the optical trapping unit may be configured to generate a first optical trapping site 211a, second optical trapping site 211b, third optical trapping site 211c, fourth optical trapping site 21 Id, fifth optical trapping site 21 le, sixth optical trapping site 21 If, seventh optical trapping site 211g, eighth optical trapping site 21 Ih, and ninth optical trapping site 21 li, as depicted in FIG. 3A. The plurality of spatially distinct optical trapping sites may be configured to trap a plurality of atoms, such as first atom 212a, second atom 212b, third atom 212c, and fourth atom 212d, as depicted in FIG. 3A. As depicted in FIG. 3B, each optical trapping site may be configured to trap a single atom. As depicted in FIG. 3B, some of the optical trapping sites may be empty (i.e., not trap an atom).

[0249] As shown in FIG. 3B, the plurality of optical trapping sites may comprise a two- dimensional (2D) array. The 2D array may be perpendicular to the optical axis of optical components of the optical trapping unit depicted in FIG. 3A. Alternatively, the plurality of optical trapping sites may comprise a one-dimensional (ID) array or a three-dimensional (3D) array.

[0250] Although depicted as comprising nine optical trapping sites filled by four atoms in FIG. 3B, the optical trapping unit 210 may be configured to generate any number of spatially distinct optical trapping sites described herein and may be configured to trap any number of atoms described herein.

[0251] Each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 pm, 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site maybe spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the preceding values.

[0252] The optical trapping sites may comprise one or more optical tweezers. Optical tweezers may comprise one or more focused laser beams to provide an attractive or repulsive force to hold or move the one or more atoms. The beam waist of the focused laser beams may comprise a strong electric field gradient. The atoms may be attracted or repelled along the electric fieldWSGR Docket No. 55436-733.601 gradient to the center of the laser beam, which may contain the strongest electric field. The optical trapping sites may comprise one or more optical lattice sites of one or more optical lattices. The optical trapping sites may comprise one or more optical lattice sites of one or more one-dimensional (ID) optical lattices, two-dimensional (2D) optical lattices, or three-dimensional (3D) optical lattices. For instance, the optical trapping sites may comprise one or more optical lattice sites of a 2D optical lattice, as depicted in FIG. 3B.

[0253] The optical lattices may be generated by interfering counter -propagating light (such as counter-propagating laser light) to generate a standing wave pattern having a periodic succession of intensity minima and maxima along a particular direction. A ID optical lattice may be generated by interfering a single pair of counter-propagating light beams. A 2D optical lattice may be generated by interfering two pairs of counter-propagating light beams. A 3D optical lattice may be generated by interfering three pairs of counter-propagating lights beams. The light beams may be generated by different light sources or by the same light source. Therefore, an optical lattice may be generated by at least about 1, 2, 3, 4, 5, 6, or more light sources or at most about 6, 5, 4, 3, 2, or 1 light sources.

[0254] Returning to the description of FIG. 3A, the optical trapping unit may comprise one or more light sources configured to emit light to generate the plurality of optical trapping sites as described herein. For instance, the optical trapping unit may comprise a single light source 213, as depicted in FIG. 3A. Though depicted as comprising a single light source in FIG. 3A, the optical trapping unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources. The light sources may comprise one or more lasers. The lasers may be configured to operate at a resolution limit of the lasers. For example, the lasers can be configured to provide diffraction limited spot sizes for optical trapping.

[0255] The lasers may comprise one or more continuous wave lasers. The lasers may comprise one or more pulsed lasers. The lasers may comprise one or more gas lasers, such as one or more helium-neon (HeNe) lasers, argon (Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N2) lasers, carbon dioxide (CO2) lasers, carbon monoxide (CO) lasers, transversely excited atmospheric (TEA) lasers, or excimer lasers. For instance, the lasers may comprise one or more argon dimer (Ar2) excimer lasers, krypton dimer (KT2) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe2) excimer lasers, argon fluoride (ArF) excimer lasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF) excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride (XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The laser may comprise one or more dye lasers.WSGR Docket No. 55436-733.601

[0256] The lasers may comprise one or more metal-vapor lasers, such as one or more heliumcadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg) metal-vapor lasers, heliumselenium (HeSe) metal-vapor lasers, helium-silver (HeAg) metal-vapor lasers, strontium (Sr) metal-vapor lasers, neon-copper (NeCu) metal-vapor lasers, copper (Cu) metal-vapor lasers, gold (Au) metal-vapor lasers, manganese (Mn) metal-vapor laser, or manganese chloride (MnCh) metal-vapor lasers.

[0257] The lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal -doped crystal lasers, or metal -doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd: YAG) lasers, neodymium / chromium doped yttrium aluminum garnet (Nd / Cr: YAG) lasers, erbium-doped yttrium aluminum garnet (Er: YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd: YLF) lasers, neodymium- doped yttrium orthovanadate (NIFYVOf) lasers, neodymium-doped yttrium calcium oxoborate (Nd:YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm:YAG) lasers, ytterbium -doped ytrrium aluminum garnet (Yb:YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho: YAG) lasers, chromium-doped zinc selenide (Cr:ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) lasers, erbium-doped glass (Erglass) lasers, erbium-ytterbium-codoped glass (Er / Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers.

[0258] The lasers may comprise one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGalnP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.

[0259] The lasers may emit continuous wave laser light. The lasers may emit pulsed laser light. The lasers may have a pulse length of at least about 1 femtoseconds (fs), 2 fs, 3 fs, 4 fs, 5 fs, 6 fs, 7 fs, 8 fs, 9 fs, 10 fs, 20 fs, 30 fs, 40 fs, 50 fs, 60 fs, 70 fs, 80 fs, 90 fs, 100 fs, 200 fs, 300 fs, 400 fs, 500 fs, 600 fs, 700 fs, 800 fs, 900 fs, 1 picosecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 nanosecond (ns), 2 ns, 3 ns, 4 ns, 5 ns, 6 ns, 7 ns, 8 ns, 9 ns, 10 ns, 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1,000 ns, or more. The lasers may have a pulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3 ns, 2WSGR Docket No. 55436-733.601 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800 fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80 fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6 fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or less. The lasers may have a pulse length that is within a range defined by any two of the preceding values.

[0260] The lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values.

[0261] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (pj), 2 pj, 3 pj, 4 pj, 5 pj, 6 pj, 7 pj, 8 pj, 9 pj, 10 pj, 20 pj, 30 pj, 40 pj, 50 pj, 60 pj, 70 pj, 80 pj, 90 pj, 100 pj, 200 pj, 300 pj, 400 pj, 500 pj, 600 pj, 700 pj, 800 pj, 900 pj, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 pj, 800 pj, 700 pj, 600 pj, 500 pj, 400 pj, 300 pj, 200 pj, 100 pj, 90 pj, 80 pj, 70 pj, 60 pj, 50 pj, 40 pj, 30 pj, 20 pj, 10 pj, 9 pj, 8 pj, 7 pj, 6 pj, 5 pj, 4 pj, 3 pj, 2 pj, 1 pj, 900WSGR Docket No. 55436-733.601 nJ, 800 nJ, 700 nJ, 600 nJ, 500 nJ, 400 nJ, 300 nJ, 200 nJ, 100 nJ, 90 nJ, 80 nJ, 70 nJ, 60 nJ, 50 nJ, 40 nJ, 30 nJ, 20 nJ, 10 nJ, 9 nJ, 8 nJ, 7 nJ, 6 nJ, 5 nJ, 4 nJ, 3 nJ, 2 nJ, 1 nJ, or less. The lasers may emit light having a pulse energy that is within a range defined by any two of the preceding values.

[0262] The lasers may emit light having an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values.

[0263] The lasers may emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The lasers may emit light comprising one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm,360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm,470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm,580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm,690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm,800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm,910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm, 1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm, 1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm, 1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm, 1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm, 1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370WSGR Docket No. 55436-733.601 nm, 1,380 nm, 1,390 nm, 1,400 nm, or more. The lasers may emit light comprising one or more wavelengths of at most about 1,400 nm, 1,390 nm, 1,380 nm, 1,370 n, 1,360 nm, 1,350 nm, 1,340 nm, 1,330 nm, 1,320 nm, 1,310 nm, 1,300 nm, 1,290 nm, 1,280 nm, 1,270 n, 1,260 nm,1,250 nm, 1,240 nm, 1,230 nm, 1,220 nm, 1,210 nm, 1,200 nm, 1,190 nm, 1,180 nm, 1,170 n,1,160 nm, 1,150 nm, 1,140 nm, 1,130 nm, 1,120 nm, 1,110 nm, 1,100 nm, 1,090 nm, 1,080 nm,1,070 n, 1,060 nm, 1,050 nm, 1,040 nm, 1,030 nm, 1,020 nm, 1,010 nm, 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values.

[0264] The lasers may emit light having a bandwidth of at least about 1 x 10'15nm, 2 x 10'15nm, 3 x 10’15nm, 4 x 10'15nm, 5 x 10'15nm, 6 x 10'15nm, 7 x 10'15nm, 8 x 10'15nm, 9 x 10'15nm, 1 x 10'14nm, 2 x 10'14nm, 3 x 10'14nm, 4 x 10'14nm, 5 x 10'14nm, 6 x 10'14nm, 7 x 10'14nm, 8 x 10'14nm, 9 x 10'14nm, 1 x 10'13nm, 2 x 10'13nm, 3 x 10'13nm, 4 x 10'13nm, 5 x 10'13nm, 6 x10'13nm, 7 x 10'13nm, 8 x 10'13nm, 9 x 10'13nm, 1 x 10'12nm, 2 x 10'12nm, 3 x 10'12nm, 4 xIO’12nm, 5 x 10'12nm, 6 x 10'12nm, 7 x 10'12nm, 8 x 10'12nm, 9 x 10'12nm, 1 x 10'11nm, 2 x10'11nm, 3 x 10'11nm, 4 x 10'11nm, 5 x 10'11nm, 6 x 10'11nm, 7 x 10'11nm, 8 x 10'11nm, 9 x10'11nm, 1 x 10'10nm, 2 x 10'10nm, 3 x 10'10nm, 4 x 10'10nm, 5 x 10'10nm, 6 x 10'10nm, 7 x10'10nm, 8 x 10'10nm, 9 x 10'10nm, 1 x 10'9nm, 2 x 10'9nm, 3 x 10'9nm, 4 x 10'9nm, 5 x 10'9nm, 6 x 10'9nm, 7 x 10'9nm, 8 x 10'9nm, 9 x 10'9nm, 1 x 10'8nm, 2 x 10'8nm, 3 x 10'8nm, 4 x 10'8nm, 5 x 10'8nm, 6 x 10'8nm, 7 x 10'8nm, 8 x 10'8nm, 9 x 10'8nm, 1 x 10'7nm, 2 x 10'7nm, 3 x 10'7nm, 4 x 10'7nm, 5 x 10'7nm, 6 x 10'7nm, 7 x 10'7nm, 8 x 10'7nm, 9 x 10'7nm, 1 x 10'6nm, 2 x 10'6nm, 3 x 10'6nm, 4 x 10'6nm, 5 x 10'6nm, 6 x 10'6nm, 7 x 10'6nm, 8 x 10'6nm, 9 x 10'6nm, 1 x 10'5nm, 2 x 10'5nm, 3 x 10'5nm, 4 x 10'5nm, 5 x 10'5nm, 6 x 10'5nm, 7 x 10'5nm, 8 x 10'5nm, 9 x 10'5nm, 1 x 10'4nm, 2 x 10'4nm, 3 x 10'4nm, 4 x 10'4nm, 5 x 10'4nm, 6 x 10'4nm, 7 x 10'4nm, 8 x 10'4nm, 9 x 10'4nm, 1 x 10'3nm, or more. The lasers may emit light having a bandwidth of at most aboutl x 10'3nm, 9 x 10'4nm, 8 x 10'4nm, 7 x 10'4nm, 6 x 10'4nm, 5 x 10'4nm, 4 x 10'4nm, 3 x 10'4nm, 2 x 10'4nm, 1 x 10'4nm, 9 x 10'5nm, 8 x 10'5nm, 7 x 10'5nm, 6 x 10'5nm, 5 x 10'5nm, 4 x 10'5nm, 3 x 10'5nm, 2 x 10'5nm, 1 x 10'5nm, 9 x 10'6nm, 8 x 10'6nm, 7 x 10'6nm, 6 x 10'6nm, 5 x 10'6nm, 4 x 10'6nm, 3 x 10'6nm, 2 x 10'6nm, 1 x 10'6nm, 9 xWSGR Docket No. 55436-733.60110'7nm, 8 x 10'7nm, 7 x 10'7nm, 6 x 10'7nm, 5 x 10'7nm, 4 x 10'7nm, 3 x 10'7nm, 2 x 10'7nm, 1 x 10'7nm, 9 x 10'8nm, 8 x 10'8nm, 7 x 10'8nm, 6 x 10'8nm, 5 x 10'8nm, 4 x 10'8nm, 3 x 10'8nm, 2 x 10'8nm, 1 x 10'8nm, 9 x 10'9nm, 8 x 10'9nm, 7 x 10'9nm, 6 x 10'9nm, 5 x 10'9nm, 4 x 10'9nm, 3 x 10'9nm, 2 x 10'9nm, 1 x 10'9nm, 9 x IO'10nm, 8 x IO'10nm, 7 x IO'10nm, 6 x IO'10nm, 5 x IO'10nm, 4 x IO'10nm, 3 x IO'10nm, 2 x IO'10nm, 1 x IO'10nm, 9 x IO'11nm, 8 x IO'11nm, 7 x IO'11nm, 6 x IO'11nm, 5 x IO'11nm, 4 x IO'11nm, 3 x IO'11nm, 2 x IO'11nm, 1 x IO'11nm, 9 x IO’12nm, 8 x 10'12nm, 7 x 10'12nm, 6 x 10'12nm, 5 x 10'12nm, 4 x 10'12nm, 3 x 10'12nm, 2 x IO’12nm, 1 x 10'12nm, 9 x 10'13nm, 8 x 10'13nm, 7 x 10'13nm, 6 x 10'13nm, 5 x 10'13nm, 4 x 10'13nm, 3 x 10'13nm, 2 x 10'13nm, 1 x 10'13nm, 9 x 10'14nm, 8 x 10'14nm, 7 x 10'14nm, 6 x 10'14nm, 5 x 10'14nm, 4 x 10'14nm, 3 x 10'14nm, 2 x 10'14nm, 1 x 10'14nm, 9 x IO'15nm, 8 x IO’15nm, 7 x IO'15nm, 6 x IO'15nm, 5 x IO'15nm, 4 x IO'15nm, 3 x IO'15nm, 2 x IO'15nm, 1 x IO’15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.

[0265] The light sources may be configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms. A magic wavelength corresponding to an atom may comprise any wavelength of light that gives rise to equal or nearly equal polarizabilities of the first and second atomic states. The magic wavelengths for a transition between the first and second atomic states may be determined by calculating the wavelengthdependent polarizabilities of the first and second atomic states and finding crossing points. Light tuned to such a magic wavelength may give rise to equal or nearly equal differential light shifts in the first and second atomic states, regardless of the intensity of the light emitted by the light sources. This may effectively decouple the first and second atomic states from motion of the atoms. The magic wavelengths may utilize one or more scalar or tensor light shifts. The scalar or tensor light shifts may depend on magnetic sublevels within the first and second atomic states.

[0266] For instance, group III atoms and metastable states of alkaline earth or alkaline earth-like atoms may possess relatively large tensor shifts whose angle relative to an applied magnetic field may be tuned to cause a situation in which scalar and tensor shifts balance and give a zero or near zero differential light shift between the first and second atomic states. The angle 0 may be tuned by selecting the polarization of the emitted light. For instance, when the emitted light is linearly polarized, the total polarizability a may be written as a sum of the scalar component ascaiar and the tensor component atenSor'.

[0267] By choosing 0 appropriately, the polarizability of the first and second atomic states may be chosen to be equal or nearly equal, corresponding to a zero or near zero differential light shift and the motion of the atoms may be decoupled.WSGR Docket No. 55436-733.601

[0268] The light sources may be configured to direct light to one or more optical modulators (OMs) configured to generate the plurality of optical trapping sites. Ffor instance, the optical trapping unit may comprise an OM 214 configured to generate the plurality of optical trapping sites. Although depicted as comprising one OM in FIG. 3A, the optical trapping unit may comprise any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMs may comprise one or more digital micromirror devices (DMDs). The OMs may comprise one or more liquid crystal devices, such as one or more liquid crystal on silicon (LCoS) devices. The OMs may comprise one or more spatial light modulators (SLMs). The OMs may comprise one or more acousto-optic deflectors (AODs) or acousto-optic modulators (AOMs). The OMs may comprise one or more electro-optic deflectors (EODs) or electro-optic modulators (EOMs).

[0269] The OM may be optically coupled to one or more optical element to generate a regular array of optical trapping sites. For instance, the OM may be optically coupled to optical element 219, as shown in FIG. 3A. The optical elements may comprise lenses or microscope objectives configured to re-direct light from the OMs to form a regular rectangular grid of optical trapping sites.

[0270] For instance, as shown in FIG. 3A, the OM may comprise an SLM, DMD, or LCoS device. The SLM, DMD, or LCoS device may be imaged onto the back focal plane of the microscope objectives. This may allow for the generation of an arbitrary configuration of optical trapping sites in two or three dimensions.

[0271] Alternatively or in addition, the OMs may comprise first and second AODs. The active regions of the first and second AODs may be imaged onto the back focal plane of the microscope objectives. The output of the first AOD may be optically coupled to the input of the second AOD. In this manner, the second AOD may make a copy of the optical output of the first AOD. This may allow for the generation of optical trapping sites in two or three dimensions.

[0272] Alternatively or in addition, the OMs may comprise static optical elements, such as one or more microlens arrays or holographic optical elements. The static optical elements may be imaged onto the back focal plane of the microscope objectives. This may allow for the generation of an arbitrary configuration of optical trapping sites in two or three dimensions.

[0273] The optical trapping unit may comprise one or more imaging units configured to obtain one or more images of a spatial configuration of the plurality of atoms trapped within the optical trapping sites. For instance, the optical trapping unit may comprise imaging unit 215. Although depicted as comprising a single imaging unit in FIG. 3A, the optical trapping unit may comprise any number of imaging units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more imaging units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 imaging units. The imaging units mayWSGR Docket No. 55436-733.601 comprise one or more lens or objectives. The imaging units may comprise one or more PMTs, photodiodes, avalanche photodiodes, phototransistors, reverse-biased LEDs, CCDs, or CMOS cameras. The imaging unit may comprise one or more fluorescence detectors. The images may comprise one or more fluorescence images, single-atom fluorescence images, absorption images, single-atom absorption images, phase contrast images, or single-atom phase contrast images.

[0274] The optical trapping unit may comprise one or more spatial configuration artificial intelligence (Al) units configured to perform one or more Al operations to determine the spatial configuration of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial configuration Al unit 216. Although depicted as comprising a single spatial configuration Al unit in FIG. 3A, the optical trapping unit may comprise any number of spatial configuration Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0275] The optical trapping unit may comprise one or more atom rearrangement units configured to impart an altered spatial arrangement of the plurality of atoms trapped with the optical trapping sites based on the one or more images obtained by the imaging unit. For instance, the optical trapping unit may comprise atom rearrangement unit 217. Although depicted as comprising a single atom rearrangement unit in FIG. 3A, the optical trapping unit may comprise any number of atom rearrangement units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atom rearrangement units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units.

[0276] The optical trapping unit may comprise one or more spatial arrangement artificial intelligence (Al) units configured to perform one or more Al operations to determine the altered spatial arrangement of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial arrangement Al unit 218. Although depicted as comprising a single spatial arrangement Al unit in FIG. 3A, the optical trapping unit may comprise any number of spatial arrangement Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial arrangement Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial arrangement Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0277] In some cases, the spatial configuration Al units and the spatial arrangement Al units may be integrated into an integrated Al unit. The optical trapping unit may comprise any number of integrated Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated Al units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated Al units.WSGR Docket No. 55436-733.601

[0278] The atom rearrangement unit may be configured to alter the spatial arrangement in order to obtain an increase in a filling factor of the plurality of optical trapping sites. A filling factor may be defined as a ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total number of computationally active optical trapping sites available in the optical trapping unit or in a portion of the optical trapping unit. For instance, initial loading of atoms within the computationally active optical trapping sites may give rise to a filling factor of less than 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, the atom rearrangement unit may attain a filling factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atom rearrangement unit may attain a filling factor of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atom rearrangement unit may attain a filling factor that is within a range defined by any two of the preceding values.

[0279] By way of example, FIG. 3C shows an example of an optical trapping unit that is partially filled with atoms. As depicted in FIG. 3C, initial loading of atoms within the optical trapping sites may give rise to a filling factor of 44.4% (4 atoms filling 9 available optical trapping sites). By moving atoms from different regions of the optical trapping unit (not shown in FIG. 3C) to unoccupied optical trapping sites or by moving atoms from an atom reservoir described herein, a much higher filling factor may be obtained, as shown in FIG. 3D.

[0280] FIG. 3D shows an example of an optical trapping unit that is completely filled with atoms. As depicted in FIG. 3D, fifth atom 212e, sixth atom 212f, seventh atom 212g, eighth atom 212h, and ninth atom 212i may be moved to fill unoccupied optical trapping sites. The fifth, sixth, seventh, eighth, and ninth atoms may be moved from different regions of the optical trapping unit (not shown in FIG. 3C) or by moving atoms from an atom reservoir described herein. Thus, the filling factor may be substantially improved following rearrangement of atoms within the optical trapping sites. For instance, a filling factor of up to 100% (such 9 atoms filling 9 available optical trapping sites, as shown in FIG. 3D) may be attained.

[0281] Atom rearrangement may be performed by (i) acquiring an image of the optical trapping unit, identifying filled and unfilled optical trapping sites, (ii) determining a set of moves to bringWSGR Docket No. 55436-733.601 atoms from filled optical trapping sites to unfilled optical trapping sites, and (iii) moving the atoms from filled optical trapping sites to unfilled optical trapping sites. Operations (i), (ii), and (iii) may be performed iteratively until a large filling factor is achieved. Operation (iii) may comprise translating the moves identified in operation (ii) to waveforms that may be sent to an arbitrary waveform generator (AWG) and using the AWG to drive AODs to move the atoms. The set of moves may be determined using the Hungarian algorithm described in W. Lee et al, “Defect-Free Atomic Array Formation Using Hungarian Rearrangement Algorithm,” Physical Review A 95, 053424 (2017), which is incorporated herein by reference in its entirety for all purposes.Examples of Electromagnetic Delivery Units

[0282] FIG. 4 shows an example of an electromagnetic delivery unit 220. The electromagnetic delivery unit may be configured to apply electromagnetic energy to one or more atoms of the plurality of atoms, as described herein. The electromagnetic delivery unit may comprise one or more light sources, such as any light source described herein. The electromagnetic energy may comprise optical energy. The optical energy may comprise any repetition rate, pulse energy, average power, wavelength, or bandwidth described herein.

[0283] The electromagnetic delivery unit may comprise one or more microwave or radiofrequency (RF) energy sources, such as one or more magnetrons, klystrons, traveling-wave tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMP ATT) diodes, or masers. The electromagnetic energy may comprise microwave energy or RF energy. The RF energy may comprise one or more wavelengths of at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km,8 km, 9 km, 10 km, or more. The RF energy may comprise one or more wavelengths of at most about 10 km, 9 km, 8 km, 7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m, 500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m, 30 m, 20 m, 10 m,9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm, 800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm, 80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or less. The RF energy may comprise one or more wavelengths that are within a range defined by any two of the preceding values.WSGR Docket No. 55436-733.601

[0284] The RF energy may comprise an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 Watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The RF energy may comprise an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or less. The RF energy may comprise an average power that is within a range defined by any two of the preceding values.

[0285] The electromagnetic delivery unit may comprise one or more light sources, such as any light source described herein. For instance, the electromagnetic delivery unit may comprise light source 221. Although depicted as comprising a single light source in FIG. 4, the electromagnetic delivery unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources.

[0286] The light sources may be configured to direct light to one or more OMs configured to selectively apply the electromagnetic energy to one or more atoms of the plurality of atoms. For instance, the electromagnetic delivery unit may comprise OM 222. Although depicted as comprising a single OM in FIG. 4, the electromagnetic delivery unit may comprise any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMs may comprise one or more SLMs, AODs, or AOMs. The OMs may comprise one or more DMDs. The OMs may comprise one or more liquid crystal devices, such as one or more LCoS devices.

[0287] The electromagnetic delivery unit may comprise one or more electromagnetic energy artificial intelligence (Al) units configured to perform one or more Al operations to selectively apply the electromagnetic energy to the atoms. For instance, the electromagnetic delivery unit may comprise Al unit 223. Although depicted as comprising a single Al unit in FIG. 4, the electromagnetic delivery unit may comprise any number of Al units, such as at least about 1, 2,WSGR Docket No. 55436-733.6013, 4, 5, 6, 7, 8, 9, 10, or more Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0288] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) on the qubits described herein. The electromagnetic delivery unit may be configured to apply one or more two-qubit operations (such as one or more two-qubit gate operations) on the two-qubit units described herein. Each singlequbit or two-qubit operation may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, or more. Each single-qubit or two- qubit operation may comprise a duration of at most about 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. Each single-qubit or two-qubit operation may comprise a duration that is within a range defined by any two of the preceding values. The single-qubit or two-qubit operations may be applied with a repetition frequency of at least 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or more. The single-qubit or two-qubit operations may be applied with a repetition frequency of at most 1,000 kHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The single-qubit or two-qubit operations may be applied with a repetition frequency that is within a range defined by any two of the preceding values.

[0289] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations by inducing one or more Raman transitions between a first qubit state and a second qubit state described herein. The Raman transitions may be detuned from a3Po or3Pi line described herein. For instance, the Raman transitions may be detuned by at least about 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1 GHz, or more. The Raman transitions may be detuned by at most about 1 GHz, 900 MHz, 800WSGR Docket No. 55436-733.601MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The Raman transitions may be detuned by a value that is within a range defined by any two of the preceding values.

[0290] Raman transitions may be induced on individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle or a frequency shift to a light beam based on an applied radio-frequency (RF) signal. The SLM or AOD may be combined with an optical conditioning system that images the SLM or AOD active region onto the back focal plane of a microscope objective. The microscope objective may perform a spatial Fourier transform on the optical field at the position of the SLM or AOD. As such, angle (which may be proportional to RF frequency) may be converted into position. For example, applying a comb of radio frequencies to an AOD may generate a linear array of spots of light at a focal plane of the objective, with each spot having a finite extent determined by the characteristics of the optical conditioning system (such as the point spread function of the optical conditioning system).

[0291] To perform a Raman transition on a single atom with a single SLM or AOD, a pair of frequencies may be applied to the SLM or AOD simultaneously. The two frequencies of the pair may have a frequency difference that matches or nearly matches the splitting energy between the first and second qubit states. For instance, the frequency difference may differ from the splitting energy by at most about 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The frequency difference may differ from the splitting energy by at least about 1 Hz, 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, or more. The frequency difference may differ from the splitting energy by about 0 Hz. The frequency difference may differ from the splitting energy by a value that is within a range defined by any two of the preceding values. The optical system may be configuredWSGR Docket No. 55436-733.601 such that the position spacing corresponding to the frequency difference is not resolved and such that light at both of the two frequencies interacts with a single atom.

[0292] The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at least about 10 nm, 50 nm, 75 nm, 100 nm, 125 nm, 150 nm, 175 nm, 200 nm, 225 nm, 250 nm, 275 nm, 300 nm, 325 nm, 350 nm, 375 nm, 400 nm, 425 nm, 450 nm, 475 nm, 500 nm, 525 nm, 550 nm, 575 nm, 600 nm, 625 nm, 650 nm, 675 nm, 700 nm, 725 nm, 750 nm, 775 nm, 800 nm, 825 nm, 850 nm, 875 nm, 900 nm, 925 nm, 950 nm, 975 nm, 1 micrometer (pm), 1.5 pm, 2 pm, 2.5 pm 3 pm, 3.5 pm, 4 pm, 4.5 pm, 5 pm, 5.5 pm, 6 pm, 6.5 pm, 7 pm, 7.5 pm, 8 pm, 8.5 pm, 9 pm, 9.5 pm, 10 pm, or more. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at most about 10 pm, 9.5 pm, 9 pm, 8.5 pm, 8 pm, 7.5 pm, 7 pm, 6.5 pm, 6 pm, 5.5 pm, 5 pm, 4.5 pm, 4 pm, 3.5 pm, 3 pm, 2.5 pm, 2 pm, 1.5 pm, 1 pm, 975 nm, 950 nm, 925 nm, 900 nm, 875 nm, 850 nm, 825 nm, 800 nm, 775 nm, 750 nm, 725 nm, 700 nm, 675 nm, 650 nm, 625 nm, 600 nm, 575 nm, 550 nm, 525 nm, 500 nm, 475 nm, 450 nm, 425 nm, 400 nm, 375 nm, 350 nm, 325 nm, 300 nm, 275 nm, 250 nm, 225 nm, 200 nm, 175 nm, 150 nm, 125 nm, 100 nm, 75 nm, 25 nm, 10 nm, or less.The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension as defined by any two of the proceeding values. For example, the beam can have a characteristic dimension of about 1.5 micrometers to about 2.5 micrometers. Examples of characteristic dimensions include, but are not limited to, a Gaussian beam waist, the full width at half maximum (FWHM) of the beam size, the beam diameter, the 1 / e2width, the D4G width, the D86 width, and the like. For example, the beam may have a Gaussian beam waist of at least about 1.5 micrometers.

[0293] The characteristic dimension of the beam may be bounded at the low end by the size of the atomic wavepacket of an optical trapping site. For example, the beam can be formed such that the intensity variation of the beam over the trapping site is sufficiently small as to be substantially homogeneous over the trapping site. In this example, the beam homogeneity can improve the fidelity of a qubit in the trapping site. The characteristic dimension of the beam may be bounded at the high end by the spacing between trapping sites. For example, a beam can be formed such that it is small enough that the effect of the beam on a neighboring trapping site / atom is negligible. In this example, the effect may be negligible if the effect can be minimized by techniques such as, for example, composite pulse engineering. The characteristic dimension may be different from a maximum achievable resolution of the system. For example, a system can have a maximum resolution of 700 nm, but the system may be operated at 1.5 micrometers. In this example, the value of the characteristic dimension may be selected to optimize the performance of the system in view of the considerations described elsewhere herein.WSGR Docket No. 55436-733.601The characteristic dimension may be invariant for different maximally achievable resolutions. For example, a system with a maximum resolution of 500 nm and a system with a maximum resolution of 2 micrometers may both be configured to operate at a characteristic dimension of 2 micrometers. In this example, 2 micrometers may be the optimal resolution based on the size of the trapping sites.Examples of Integrated Optical Trapping Units and Electromagnetic Delivery Units

[0294] The optical trapping units and electromagnetic delivery units described herein may be integrated into a single optical system. A microscope objective may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit described herein and to deliver light for trapping atoms generated by an optical trapping unit described herein. Alternatively or in addition, different objectives may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit and to deliver light from trapping atoms generated by an optical trapping unit.

[0295] A single SLM or AOD may allow the implementation of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or in addition, two separate SLMs or AODs may be configured to each handle light with orthogonal polarizations. The light with orthogonal polarizations may be overlapped before the microscope objective. In such a scheme, each photon used in a two-photon transition described herein may be passed to the objective by a separate SLM or AOD, which may allow for increased polarization control. Qubit operations may be performed on a two-dimensional arrangement of atoms by bringing light from a first SLM or AOD into a second SLM or AOD that is oriented substantially orthogonally to the first SLM or AOD via an optical relay. Alternatively or in addition, qubit operations may be performed on a two-dimensional arrangement of atoms by using a onedimensional array of SLMs or AODs.

[0296] The stability of qubit gate fidelity may be improved by maintaining overlap of light from the various light sources described herein (such as light sources associated with the optical trapping units or electromagnetic delivery units described herein). Such overlap may be maintained by an optical subsystem that measures the direction of light emitted by the various light sources, allowing closed-loop control of the direction of light emission. The optical subsystem may comprise a pickoff mirror located before the microscope objective. The pickoff mirror may be configured to direct a small amount of light to a lens, which may focus a collimated beam and convert angular deviation into position deviation. A position-sensitive optical detector, such as a lateral -effect position sensor or quadrant photodiode, may convert the position deviation into an electronic signal and information about the deviation may be fed into a compensation optic, such as an active mirror.WSGR Docket No. 55436-733.601

[0297] The stability of qubit gate manipulation may be improved by controlling the intensity of light from the various light sources described herein (such as light sources associated with the optical trapping units or electromagnetic delivery units described herein). Such intensity control may be maintained by an optical subsystem that measures the intensity of light emitted by the various light sources, allowing closed-loop control of the intensity. Each light source may be coupled to an intensity actuator, such as an intensity servo control. The actuator may comprise an acousto-optic modulator (AOM) or electro-optic modulator (EOM). The intensity may be measured using an optical detector, such as a photodiode or any other optical detector described herein. Information about the intensity may be integrated into a feedback loop to stabilize the intensity.Examples of State Preparation Units

[0298] FIG. 5 shows an example of a state preparation unit 250. The state preparation unit may be configured to prepare a state of the plurality of atoms, as described herein. The state preparation unit may be coupled to the optical trapping unit and may direct atoms that have been prepared by the state preparation unit to the optical trapping unit. The state preparation unit may be configured to cool the plurality of atoms. The state preparation unit may be configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites.

[0299] The state preparation unit may comprise one or more Zeeman slowers. For instance, the state preparation unit may comprise a Zeeman slower 251. Although depicted as comprising a single Zeeman slower in FIG. 5, the state preparation may comprise any number of Zeeman slowers, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Zeeman slowers or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Zeeman slowers. The Zeeman slowers may be configured to cool one or more atoms of the plurality of atoms from a first velocity or distribution of velocities (such an emission velocity from an of an atom source, room temperature, liquid nitrogen temperature, or any other temperature) to a second velocity that is lower than the first velocity or distribution of velocities.

[0300] The first velocity or distribution of velocities may be associated with a temperature of at least about 50 Kelvin (K), 60 K, 70 K, 80 K, 90 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1,000 K, or more. The first velocity or distribution of velocities may be associated with a temperature of at most about 1,000 K, 900 K, 800 K, 700 K, 600 K, 500 K, 400 K, 300 K, 200 K, 100 K, 90 K, 80 K, 70 K, 60 K, 50 K, or less. The first velocity or distribution of velocities may be associated with a temperature that is within a range defined by any two of the preceding values. The second velocity may be at least about 1 meter per second (m / s), 2 m / s, 3 m / s, 4 m / s, 5 m / s, 6 m / s, 7 m / s, 8 m / s, 9 m / s, 10 m / s, or more. The second velocity may be atWSGR Docket No. 55436-733.601 most about 10 m / s, 9 m / s, 8 m / s, 7 m / s, 6 m / s, 5 m / s, 4 m / s, 3 m / s, 2 m / s, 1 m / s, or less. The second velocity may be within a range defined by any two of the preceding values. The Zeeman slowers may comprise ID Zeeman slowers.

[0301] The state preparation unit may comprise a first magneto -optical trap (MOT) 252. The first MOT may be configured to cool the atoms to a first temperature. The first temperature may be at most about 10 millikelvin (mK), 9 mK, 8 mK, 7 mK, 6 mK, 5 mK, 4 mK, 3 mK, 2 mK, 1 mK, 0.9 mK, 0.8 mK, 0.7 mK, 0.6 mK, 0.5 mK, 0.4 mK, 0.3 mK, 0.2 mK, 0.1 mK, or less. The first temperature may be at least about 0.1 mK, 0.2 mK, 0.3 mK, 0.4 mK, 0.5 mK, 0.6 mK, 0.7 mK, 0.8 mK, 0.9 mK, 1 mK, 2 mK, 3 mK, 4 mK, 5 mK, 6 mK, 7 mK, 8 mK, 9 mK, 10 mK, or more. The first temperature may be within a range defined by any two of the preceding values. The first MOT may comprise a ID, 2D, or 3D MOT.

[0302] The first MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm,840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm,730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm,620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm,510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm,400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0303] The state preparation unit may comprise a second MOT 253. The second MOT may be configured to cool the atoms from the first temperature to a second temperature that is lower than the first temperature. The second temperature may be at most about 100 microkelvin (pK), 90 pK, 80 pK, 70 pK, 60 pK, 50 pK, 40 pK, 30 pK, 20 pK, 10 pK, 9 pK, 8 pK, 7 pK, 6 pK, 5 pK, 4 pK, 3 pK, 2 pK, 1 pK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300WSGR Docket No. 55436-733.601 nK, 200 nK, 100 nK, or less. The second temperature may be at least about 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 pK, 2 pK, 3 pK, 4 pK, 5 pK, 6 pK, 7 pK, 8 pK, 9 pK, 10 pK, 20 pK, 30 pK, 40 pK, 50 pK, 60 pK, 70 pK, 80 pK, 90 pK, 100 pK, or more. The second temperature may be within a range defined by any two of the preceding values. The second MOT may comprise a ID, 2D, or 3D MOT.

[0304] The second MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm,840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm,730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm,620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm,510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm,400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0305] Although depicted as comprising two MOTs in FIG. 5, the state preparation unit may comprise any number of MOTs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more MOTs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 MOTs.

[0306] The state preparation unit may comprise one or more sideband cooling units or Sisyphus cooling units (such as a sideband cooling unit described in www.arxiv.org / abs / 1810.06626 or a Sisyphus cooling unit described in www.arxiv.org / abs / 1811.06014, each of which is incorporated herein by reference in its entirety for all purposes). For instance, the state preparation unit may comprise sideband cooling unit or Sisyphus cooling unit 254. Although depicted as comprising a single sideband cooling unit or Sisyphus cooling unit in FIG. 5, the state preparation may comprise any number of sideband cooling units or Sisyphus cooling units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more sideband cooling units or Sisyphus cooling units, or at mostWSGR Docket No. 55436-733.601 about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband cooling units or Sisyphus cooling units. The sideband cooling units or Sisyphus cooling units may be configured to use sideband cooling to cool the atoms from the second temperature to a third temperature that is lower than the second temperature. The third temperature may be at most about 10 pK, 9 pK, 8 pK, 7 pK, 6 pK, 5 pK, 4 pK, 3 pK, 2 pK, 1 pK, 900 nK, 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, 90 nK, 80 nK, 70 nK, 60 nK, 50 nK, 40 nK, 30 nK, 20 nK, 10 nK, or less. The third temperature may be at most about 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80 nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 pK, 2 pK, 3 pK, 4 pK, 5 pK, 6 pK, 7 pK, 8 pK, 9 pK, 10 pK, or more. The third temperature may be within a range defined by any two of the preceding values.

[0307] The sideband cooling units or Sisyphus cooling units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm,560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm,670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm,780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm,890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm,1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm,880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm,770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm,660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm,550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm,440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0308] The state preparation unit may comprise one or more optical pumping units. For instance, the state preparation unit may comprise optical pumping unit 255. Although depicted as comprising a single optical pumping unit in FIG. 5, the state preparation may comprise any number of optical pumping units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more optical pumping units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 optical pumping units. The opticalWSGR Docket No. 55436-733.601 pumping units may be configured to emit light to optically pump the atoms from an equilibrium distribution of atomic states to a non-equilibrium atomic state. For instance, the optical pumping units may be configured to emit light to optically pump the atoms from an equilibrium distribution of atomic states to a single pure atomic state. The optical pumping units may be configured to emit light to optically pump the atoms to a ground atomic state or to any other atomic state. The optical pumping units may be configured to optically pump the atoms between any two atomic states. The optical pumping units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm,580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm,690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm,800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm,910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0309] The state preparation unit may comprise one or more coherent driving units. For instance, the state preparation unit may comprise coherent driving unit 256. Although depicted as comprising a coherent driving unit in FIG. 5, the state preparation may comprise any number of coherent driving units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more coherent driving units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 coherent driving units. The coherent driving units may be configured to coherently drive the atoms from the non-equilibrium state to the first or second atomic states described herein. Thus, the atoms may be optically pumped to an atomic state that is convenient to access (for instance, based on availability of light sources that emit particular wavelengths or based on other factors) and then coherently driven to atomic states described herein that are useful for performing quantum computations. The coherent drivingWSGR Docket No. 55436-733.601 units may be configured to induce a single photon transition between the non-equilibrium state and the first or second atomic state. The coherent driving units may be configured to induce a two-photon transition between the non-equilibrium state and the first or second atomic state. The two-photon transition may be induced using light from two light sources described herein (such as two lasers described herein).

[0310] The coherent driving units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm,860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm,750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm,640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm,530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm,420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0311] The coherent driving units may be configured to induce an RF transition between the nonequilibrium state and the first or second atomic state. The coherent driving units may comprise one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. For instance, the coherent driving units may comprise one or more RF sources (such as any RF source described herein) configured to emit RF radiation. The RF radiation may comprise one or more wavelengths of at least about 10 centimeters (cm), 20 cm, 30 cm, 40 cm, 50 cm, 60 cm, 70 cm, 80 cm, 90 cm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, or more. The RF radiation may comprise one or more wavelengths of at most about 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 90 cm, 80 cm, 70 cm, 60 cm, 50 cm, 40 cm, 30 cm, 20 cm, 10 cm, or less. The RF radiation may comprise one or more wavelengths that are within a range defined by any two of the preceding values. Alternatively orWSGR Docket No. 55436-733.601 in addition, the coherent driving units may comprise one or more light sources (such as any light sources described herein) configured to induce a two-photon transition corresponding to the RF transition.Examples of Controllers

[0312] The optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units may include one or more circuits or controllers (such as one or more electronic circuits or controllers) that is connected (for instance, by one or more electronic connections) to the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units. The circuits or controllers may be configured to control the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units.Examples of Non-Classical Computers

[0313] In some cases, the present disclosure provides a non-classical computer comprising: a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprise at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, which non-classical operation includes a superposition between at least the first qubit state and the second qubit state; one or more entanglement units configured to quantum mechanically entangle at least a subset of the plurality of qubits in the superposition with at least another qubit of the plurality of qubits; and one or more readout optical units configured to perform one or more measurements of the one or more qubits, thereby obtaining a non-classical computation.WSGR Docket No. 55436-733.601

[0314] In some cases, the present disclosure provides a non-classical computer comprising a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites.Examples of Methods for Performing a Non-Classical Computation

[0315] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms comprising greater than 60 atoms; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state that is different from the first atomic state; (c) quantum mechanically entangling at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more optical measurements of the one or more superposition state to obtain the non-classical computation.

[0316] FIG. 6 shows a flowchart for an example of a first method 600 for performing a non- classical computation.

[0317] In a first operation 610, the method 600 may comprise generating a plurality of spatially distinct optical trapping sites. The plurality of optical trapping sites may be configured to trap a plurality of atoms. The plurality of atoms may comprise greater than 60 atoms. The optical trapping sites may comprise any optical trapping sites described herein. The atoms may comprise any atoms described herein.

[0318] In a second operation 620, the method 600 may comprise applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state that is different from the first atomic state. The electromagnetic energy may comprise any electromagnetic energy described herein. The first atomic state may comprise any first atomic state described herein. The second atomic state may comprise any second atomic state described herein.

[0319] In a third operation 630, the method 600 may comprise quantum mechanically entangling at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality of atoms. The atoms may be quantum mechanically entangled in any manner described herein (for instance, as described herein with respect to FIG. 2).

[0320] In a fourth operation 640, the method 600 may comprise performing one or more optical measurements of the one or more superposition state to obtain the non-classical computation. The optical measurements may comprise any optical measurements described herein.WSGR Docket No. 55436-733.601

[0321] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) providing a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprise at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; (b) applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, which non-classical operation includes a superposition between at least the first qubit state and the second qubit state; (c) quantum mechanically entangling at least a subset of the plurality of qubits in the superposition with at least another qubit of the plurality of qubits; and (d) performing one or more optical measurements of the one or more qubits, thereby obtaining said the-classical computation.

[0322] FIG. 7 shows a flowchart for an example of a second method 700 for performing a non- classical computation.

[0323] In a first operation 710, the method 700 may comprise providing a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprise at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state. The optical trapping sites may comprise any optical trapping sites described herein. The qubits may comprise any qubits described herein. The atoms may comprise any atoms described herein. The first qubit state may comprise any first qubit state described herein. The second qubit state may comprise any second qubit state described herein. The first atomic state may comprise any first atomic state described herein. The second atomic state may comprise any second atomic state described herein.

[0324] In a second operation 720, the method 700 may comprise applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, which non-classical operation includes a superposition between at least the first qubit state and the second qubit state. The electromagnetic energy may comprise any electromagnetic energy described herein.

[0325] In a third operation 730, the method 700 may comprise quantum mechanically entangling at least a subset of the plurality of qubits in the superposition with at least another qubit of the plurality of qubits. The qubits may be quantum mechanically entangled in any manner described herein (for instance, as described herein with respect to FIG. 2).WSGR Docket No. 55436-733.601

[0326] In a fourth operation 740, the method 700 may comprise performing one or more optical measurements of the one or more qubits, thereby obtaining the non-classical computation. The optical measurements may comprise any optical measurements described herein.

[0327] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) providing a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, and (b) using at least a subset of the plurality of qubits to perform the non-classical computation.

[0328] FIG. 8 shows a flowchart for an example of a third method 800 for performing a non- classical computation.

[0329] In a first operation 810, the method 800 may comprise providing a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites. The qubits may comprise any qubits described herein. The atoms may comprise any atoms described herein. The optical trapping sites may comprise any optical trapping sites described herein.

[0330] In a second operation 820, the method 800 may comprise using at least a subset of the plurality of qubits to perform a non-classical computation.Examples of Parallel Addressing of Multi-Qubit Units

[0331] Direct excitation of strontium-87 from the ground state to Rydberg levels may require a laser with a wavelength of approximately 218 nm. Alternatively, the Rydberg excitation operation can be performed using two-photon excitation combining 689 nm and 319 nm light, each detuned from the intermediate3Pi state. The approximately 7 kHz width of the3Pi state provides an effective balance between the two-photon effective Rabi rate and scattering via spontaneous decay from the3Pi. FIG. 14 shows an energy level structure for single-qubit and multi -qubit operations in strontium-87.

[0332] The optical system for single-qubit operations is also designed to work well for multiqubit gates. One of the single-qubit beams is used as one leg of the two-photon excitation scheme that drives transitions to the Rydberg electronic manifold. To satisfy the spatially-dependent frequency and phase matching condition, AODs are also used for the UV light. Importantly, the optical systems are matched so that the frequency shift of the UV light from one site to another is identical to that of the 689 nm light. The consequence of this constraint is that the performance of state-of-the-art UV AODs dictate the accessible field of view (FOV) for multi -qubit operations. Further, because one of the single-qubit beams is being used for multi-qubit operations (and the two single-qubit beams are matched), the FOV for single-qubit operations will be the same. A figure of merit for UV AODs is the product of the active aperture and the RF bandwidth of theWSGR Docket No. 55436-733.601 device. For a fixed beam size in the back focal plane of the objective, increasing either of these quantities results in a larger scan angle of the beams, and thus a larger FOV in the plane of the qubit array. An FOV of approximately 100 pm x 100 pm was achieved, which is sufficient to address an array of approximately 1,000 atoms with a trapping site spacing of 3 pm.Examples of Methods for Error Corrected Quantum Computation

[0333] The systems, the methods, the computer-readable media, and the techniques disclosed herein may relate to qubit loss during error correction. Within qubit loss during error correction, there may be at least two general pieces. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be directed to detecting qubit loss without destroying the data stored on the qubits. Instead of appearing as a gate measurement error, if a qubit is lost, the data that would otherwise be there is absent rather than incorrect. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be directed to identifying when an error is caused by a missing qubit. The systems, the methods, the computer-readable media, and the techniques disclosed herein may also be directed to modifying the decoder to handle loss events. For example, the error correcting code may be directed to updating the calculation to address for error. In some cases, knowing about the error may be needed in order to implement error correcting code. However, in other cases, the error correcting code may be modified to account for missing data without explicit knowledge that qubit is lost.

[0334] The systems, the methods, the computer-readable media, and the techniques disclosed herein may not modify the topology of the underlying surface code. The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon methods of detecting atom loss by compressing the underlying protocol. The systems, the methods, the computer-readable media, and the techniques disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate. The systems, the methods, the computer-readable media, and the techniques disclosed herein may address uninduced erasure errors in addition to or alternatively to gate induced erasure errors.

[0335] The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon other procedures at least because the systems, the methods, the computer-readable media, and the techniques disclosed herein may not comprise or require operations that modify the topology of the underlying surface code. In some cases, the underlying surface code may be unchanged. Instead, a matching graph passed to a decoder algorithm may be updated to account for a predicted probability distribution of a lost qubit. Because the matching graph passed to the decoder is updated, the underlying decoder may also be unchanged. Because the decoder is unchanged, the error correcting code may also not be changed. Accordingly, the systems, the methods, the computer-readable media, and the techniques disclosed herein may beWSGR Docket No. 55436-733.601 used without changing the underlying surface code. Similarly, because changes to the underlying decoder and surface code are not needed, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with a wide variety of decoders and surface codes.

[0336] In some cases, tracking syndrome measurements may be used to detect loss events (e.g., defects). See, e.g., Siegel, A. et al., Adaptive Surface Code for Quantum Error Correction in the Presence of Temporary or Permanent Defects, arXiv:2211.08468vl [quant-ph] 15 Nov 2022, available at https: / / arxiv.org / pdf / 2211.08468.pdf, which is incorporated by reference herein in its entirety.

[0337] In some cases, gate induced erasure errors may be addressed by error correction. See e.g., Wu, Y., et al., Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays, Nat. Comms. Vol 13, P. 4657 (2022), which is incorporated by reference herein in its entirety. In the above, the atoms may still be present. Rather than addressing missing atoms or qubits, the above referenced application may turn gate errors into erasure errors.

[0338] In some cases, noise structure in the hardware may be used for error correction. See e.g., Shay, K., et al., High threshold codes for neutral atom qubits with biased erasure errors, arXiv:2302.03063vl [quant-ph] 6 Feb 2023), which is incorporated by reference herein in its entirety. Similar to Wu, the atoms may still be present. Rather than addressing missing atoms or qubits, the above referenced application may turn gate errors into erasure errors.Error Correction

[0339] Quantum error correction is a procedure for encoding quantum information in a distributed manner across many quantum systems in such a way that the information to be stored is protected from localized errors on the constituent systems, provided that these errors are sufficiently sparse. In some cases, the information to be stored and the constituent systems are both two-level quantum systems, or qubits. The information to be protected is encoded across many physical qubits, forming one or more logical qubits.

[0340] The process of detecting and correcting errors on the logical qubits amounts to measuring parities of a predetermined set of operators that act on the physical qubits and using the measured parities to diagnose and correct errors. In a simple example, a parity measurement checks the equality of two qubits to return a true or false answer, which can be used to determine whether a correction needs to occur. Additional measurements can be made for a system greater than two qubits. Since the physical qubits cannot be measured directly without collapsing the state of the logical system, these parities are measured using ancillary qubits (i.e., ancilla). Thus, there may be two types of physical qubits: data qubits on which the logical information is stored and ancilla qubits which are used to extract the desired parity checks.WSGR Docket No. 55436-733.601

[0341] In practice, an error correction cycle consists of a sequence of gates to transfer parity values onto the ancilla qubits followed by measurement of the ancilla qubits. This process may be referred to as syndrome extraction. Errors can occur at any point during the syndrome extraction process, including during readout of the ancilla qubits.

[0342] One method of error correction (Shor style) uses repeated rounds of syndrome extraction to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied. The decoding problem is commonly represented as a weighted graph or hypergraph. In this setting, each node in the graph corresponds to a collection of syndrome measurements. Such a collection of syndrome measurements is called a detector. Edges or hyperedges in the graph correspond to errors, and the weight of an edge corresponds to the likelihood of that error occurring. The occurrence of a given error may be expected cause the parity of all associated detectors to flip. The decoding problem can then be stated as follows: given a set of detectors (nodes) whose parities differ from those expected in the absence of error, determine the most likely set of physical errors (edges) that could cause the observed detections.Error Correction with Atom Loss

[0343] Quantum computers based on trapped atoms may be subject to errors generated by loss of qubit. In trapped atom quantum computers, a qubit may comprise atom in an array. That atom may be a neutral atom or an ion. Error correcting code may employ repeated implementations of the circuit implementing the quantum computation. As the circuit is implemented and reimplemented statistics may be generated on what errors occurred. However, error correcting code implemented on systems with qubit loss may differ from other systems. For example, a non-qubit loss error may be a gate error. Similarly, loss of coherence may be expressed as a gate error. In a gate error or an error that is similar to a gate error, there is a comparatively smaller set of possible error values because the result of a gate error is like a measurement of the system. In some cases, the atom loss rate may be similar or larger than the gate error rate, thus it may be helpful to provide improved methods of correcting for atom loss.

[0344] FIG. 10 is a flowchart of an example method 1000 for error corrected quantum computation. In some cases, an error correcting code which accounts for qubit loss may comprise identifying that a qubit has been lost (1010); replacing the qubit (1020); reimplementing the qubit into the circuit which may be in the wrong state when it is replaced 1030; and flagging measurements taken while the qubit was missing as untrustworthy (1040).

[0345] Referring to FIG. 10, at an operation 1010 of a method 1000 of error correction with atom loss, atom loss can be detected. For example, atom loss may be detected at the end of each syndrome extraction cycle. Methods of detecting atom loss are described herein. At an operationWSGR Docket No. 55436-733.6011020 of a method 1000 of error correction with atom loss, once a qubit is identified as lost it may be replaced with a new qubit. The new qubit may be, at least initially, in a random state.

[0346] At an operation 1030 of a method 1000 of error correction with atom loss, the qubit may be reimplemented into the circuit. In some cases, operation 1030 comprises use of a decoder algorithm. The decoder algorithm may take in a graph and determine a set of edges. In some cases, operation 1030 comprises prior implementing the decoder algorithm, updating a matching graph passed to the decoder algorithm based on a predicted probability distribution of a lost qubit replaced in operation 1020. Methods of updating the decoder algorithm are described herein.

[0347] In systems not subject to atom loss, the errors may be discrete Pauli errors on physical qubits. But when qubits are stored on atoms, the atoms — and therefore the qubits they contain — can be lost. The effect on syndrome extraction in the presence of loss depends on hardware details. For neutral atoms using Rydberg gates, the effect of atom loss manifests as a noninteraction instead of a two-qubit gate. Practically, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom.

[0348] In an example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may comprise a case in which a two-qubit interaction between a qubit and a lost qubit has an effect of a Pauli operation or identity operation on the qubit, as described in International Application PCT / US2024 / 018180, which is incorporated by reference in its entirety herein in its entirety for all purposes. In such a case, a first atom A and second atom B may be neighboring qubits. The qubits may be trapped ion qubits. The qubits may be trapped atom qubits. In some cases, one qubit acquires a phase conditioned on state-selective excitation of the other. In some cases, the state selective excitation is from a state |1> to a state |r>.

[0349] In some cases, a state |r> is a Rydberg state. In some cases, a state |r> is a Rydberg state of a neutral atom qubit. If at A is excited to a Rydberg state, then Atom B (if present) experiences a shift due to the Rydberg interaction. In some cases, an optical excitation may be tuned to a frequency difference between the |1> state and the Rydberg state. If Atom A is in state |1>, then Atom A is at least transiently driven to the Rydberg state and Atom B (if present) experiences a shift due to the Rydberg interaction. If Atom B is not present and Atom A is in state |1>, there is no Shift to Atom B. If Atom A is in state |0> and Atom B is present, then the energy gap is too large, and nothing happens to Atom A or Atom B. If Atom A is in state |0> and Atom B is not present, then the energy gap is still too large, and nothing happens to Atom A or Atom B (which isn’t present). Accordingly, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom. While this example describes a case where the two-qubit interaction with a lost atom affects the Identity operation, the systems, the methods, the computer-readable media, and the techniques disclosed herein also work when theWSGR Docket No. 55436-733.601 two-qubit interaction with a lost atom affects a Pauli operation. A Pauli operation may comprise a Pauli-X gate, a Pauli-Y gate, or a Pauli-Z gate. For example, the Pauli-X gate is a single-qubit rotation the pi radians around the X-axis. For example, the Pauli-Y gate is a single-qubit rotation the pi radians around the Y-axis. For example, the Pauli-Z gate is a single-qubit rotation the pi radians around the Z-axis. A rotation about an axis of two pi radians is an Identity operation.

[0350] The above works similarly if Atom A is also a lost qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit. For example, when the qubit is a lost qubit, a two-qubit gate between two lost qubits similarly affects the identity. Because the two-qubit operation between an atom a lost atom affects the identity. The protocol does not propagate errors (to first order) forward in time. For example, if the two-qubit operation is imperfect. The operation may propagate forward higher order errors in time.Examples of Identifying Qubit Loss

[0351] In some cases, the present disclosure provides at least two methods of identifying qubit loss; however, various methods of identifying qubit loss may be integrated into the systems, the methods, the computer-readable media, and the techniques disclosed herein. An example method of implementing an error correcting code which accounts for atom loss may comprise implementing a plurality of SWAP gates. Another example method of implementing an error correcting code which accounts for atom loss may comprise a modified knock-knock protocol. The systems, the methods, the computer-readable media, and the techniques disclosed herein for the implementation of a plurality of SWAP gates and modified knock-knock protocols may be the same as or similar to those described in International Application PCT / US2024 / 018180, which is incorporated by reference herein in its entirety for all purposes.Examples of Qubit Replacement

[0352] The systems, the methods, the computer-readable media, and the techniques disclosed herein may replace qubits into a quantum circuit after a vacancy has been identified. In some cases, the qubit is an atomic qubit. In some cases, the qubit is atom trapped in a spatially distinct optical trapping site. Examples presented herein may be recite qubits comprising neutral atoms; however, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be combined with various types of qubits.

[0353] In some examples, present techniques may be combined with methods for probabilistic, deterministic, or near-deterministic loading of optical or other traps, such as those disclosed herein. In some cases, atoms within the science region may or may not be rearranged as the science array is replenished. In some cases, atoms can be transferred between sites by optical tweezers. In some cases, atoms can be transferred between sites by optical lattices. In some cases,WSGR Docket No. 55436-733.601 atoms can be transferred between sites by tunneling / hopping between sites. In some cases, atoms can be transferred between sites by autonomous stabilization techniques.

[0354] In some cases, atom replacement is performed using one or both of a moving optical trap (e.g., moving optical lattice) or one or more optical tweezers. In some cases, an optical tweezer may be used to move a single atom (e.g., pick and place) or a subset of atoms between arrays or within an array. In some cases, a moving optical trap can be used to translate or compress an array. A moving optical trap (e.g., moving optical lattice) may implement a tone to sweep atoms from one location to another. The atom movement units may be configured to move the one or more replacement atoms from the one or more atoms reservoirs to the one or more optical trapping sites. For instance, the one or more atom movement units may comprise one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs).

[0355] The atom rearrangement unit may be configured to alter the spatial arrangement in order to obtain an increase in a filling factor of the plurality of optical trapping sites. A filling factor may be defined as a ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total number of computationally active optical trapping sites available in the optical trapping unit or in a portion of the optical trapping unit. For instance, initial loading of atoms within the computationally active optical trapping sites may give rise to a filling factor of less than 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, the atom rearrangement unit may attain a filling factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atom rearrangement unit may attain a filling factor of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atom rearrangement unit may attain a filling factor that is within a range defined by any two of the preceding values.Examples of Modifying Decoding Algorithms

[0356] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used in connection with error correction methodologies for quantum computing systems. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correcting code. A decoder may decodeWSGR Docket No. 55436-733.601 which errors occurred on which qubits. Once identified, these errors can be tracked and the information used to correct any subsequent measurement outcomes using the classical control software. The methods of updating the decoder described herein may not depend on the type of atom, the type of qubit, the type of error correction code, or the specific decoder used in the error correcting code. In some cases, an error correcting code may be of the class of stabilizer codes. If the two qubit-gate operation affects the Identity or a Pauli operation, then the matching graph passed to the decoder may be updated as described herein.

[0357] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various error correcting codes. An error correcting code may be a Shor style code. For example, in a Shor style code, repeated rounds of syndrome extraction may be implemented to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied.

[0358] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various stabilizer codes. A stabilizer code may be an error correcting code which uses stabilizers. A stabilizer code may be a class of error correcting code. The class of stabilizer codes may include toric codes, surface codes, etc. By repeatedly measuring a quantum system using a complete set of commuting stabilizers, the system may be forced into a simultaneous and unique eigenstate of all the stabilizers. One can measure the stabilizers without perturbing the system; when the measurement outcomes change, this corresponds to one or more qubit errors, and the quantum state is projected by the measurements onto a different stabilizer eigenstate.

[0359] An error correcting code may comprise a topological code. The class of topological codes may overlap with the class of stabilizer codes. A topological code may comprise a surface code, a color code, a toric code, etc. A topological code may also be referred to as a homological code. A topological code may comprise an array or lattice of qubits arranged on a surface (or higher dimensional structure). The systems, the methods, the computer-readable media, and the techniques disclosed herein may not change the underlying topology of a topological code.

[0360] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various surface codes. A surface code may be implemented as a stabilizer code. For example, in the surface code literature, surface codes may comprise two types of qubits data qubits and measurement qubits (e.g., ancilla qubits). The data qubits may contain the information carried by the quantum circuit, whose error is to be corrected. The measurement qubits may be used to stabilize and manipulate the quantum state of the data qubit. In a surface code, the measurement qubits may comprise two types: measure -Z qubits andWSGR Docket No. 55436-733.601 measure-X qubits. These two types of qubits may be called Z syndrome qubits and X-syndrome qubits respectively. The measure Z -qubits may measure the Z stabilizer. The measure X-qubits may measure the X stabilizer. In some cases, a surface code may be implemented with a decoder. In some cases, a surface code can address errors that occur during a surface code cycle as long as the errors that occur during each surface code cycle can be identified.

[0361] The systems, the methods, the computer-readable media, and the techniques disclosed herein may employ surface codes. Surface codes disclosed herein may include, for example, variations upon the minimum-weight perfect matching algorithm to decode the surface code. However, many surface codes may be applicable to the systems, the methods, the computer- readable media, and the techniques disclosed herein. A description of surface codes is provided for example at Fowler, A. G., et al., Surface codes: Towards Practical Large-scale Quantum Computation, arXiv: 1208.0928 [quant-ph] 4 Aug 2012, available at https: / / arxiv.org / pdf / 1208.0928.pdf, which is incorporated by reference herein in its entirety.

[0362] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various color codes. A color code may be implemented as a stabilizer code. For example, a color code may comprise a Steane code, etc. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various Shor style codes, for example, a Bacon-shor code. A Shor style code may be implemented as a stabilizer code. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various qLDPC codes, for example, hypergraph product codes. A qLDPC code may be implemented as a stabilizer code.

[0363] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various decoders. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correction code. A decoder may decode which errors occurred on which qubits. Once identified, these errors can be tracked and the information used to correct subsequent measurement outcomes using the classical control software. Decoder algorithms may include, for example, minimum-weight perfect matching, union find, tensor network decoder, belief propagation with ordered statistics decoder, maximum likelihood decoder, and look up table decoders. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be integrated with variations on the minimum -weight perfect matching such as sparse bloom and fusion blossom. A decoder may take in a matching graph. The systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to the decoder to account for a lost qubit.WSGR Docket No. 55436-733.601

[0364] In some cases, qubit loss may involve modification to surface code techniques that do not experience qubit loss errors. For example, error correcting code which does not account for qubit loss errors may keep track of a particular qubit changing from 1 to 0 or 0 to 1 unexpectedly. If a qubit has been lost, there is no change in state; instead, there is no value to measure.

[0365] Modifying the decoding algorithm may be a sub-operation of an operation for reimplementing the qubit into the circuit. A qubit reimplementing operation may comprise an embodiment, variation, or example of operation 1030 of the method 1000. In some cases, modifying the decoding algorithm may be performed subsequent to or during a reimplementing operation such as operation 1030 of a method 1000.

[0366] To augment the decoding algorithm, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update existing decoders to incorporate the change in error type. In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to a decoder. In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to a minimum-weight perfect matching decoder algorithm or any other decoder algorithm which takes in a matching graph.

[0367] In some examples, to update the matching graph, each node in the graph corresponds to a change-of-value of a particular stabilizer. Certain nodes are connected by edges corresponding to possible physical errors. These edges are weighted based on the likelihood of that particular error occurring. When an atom is lost and then replaced, the loss may be treated like a gate error that occurs with a probability of 50%. The procedure may change slightly if data vs. ancilla qubits are lost.

[0368] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be implement approaches for reimplementing a qubit, such as, as described in International Application PCT / US2024 / 018180, which is incorporated by reference herein in its entirety for all purposes.Examples of Measurement Operation

[0369] At an operation 1040 of a method 1000 of error correction with atom loss, measurements taken while the qubit was missing may be flagged as untrustworthy. In some cases, operation 1040 comprises flagging measurement taken during a window of time that includes a time when the qubit was missing as untrustworthy. For example, a window of time may comprise a round of syndrome measurements. It may not be necessary to know exactly which measurements were taken while the atom was lost, only what set of measurements were taken during a window of time that includes a lost atom. A flagging operation may comprise, prior to flagging a measurement, performing a measurement of one more qubits. The measurement may result in anWSGR Docket No. 55436-733.601 emission of a photon. In some cases, the measurement may be state selective. For example, a measurement may selectively probe either a |0> state or a |1> state. After a round of measurement, it may be possible to know whether a measured atom is lost. In some cases, measurement of ancilla atoms during an error correction protocol may indicate whether an ancilla atom is lost. An identification operation in 1010 (e.g., swap gates, a knock-knock protocol, etc.) may be performed in order to determine if a qubit is missing within a set of qubits including a data qubit without measurement of the data qubit.

[0370] In the systems, the methods, the computer-readable media, and the techniques disclosed herein, the fact of a missing qubit may not be immediately heralded. For example, it may become apparent that qubit is lost after completing a round of syndrome extraction, rather than immediately upon taking a measurement implicating a lost qubit. Once the round of syndrome extraction is complete, it may become apparent that there was a qubit loss, in order to proceed with the calculation in may be beneficial to flag a series of measurements taken during a window of time that includes a time when the qubit was missing. Each of these measurements may be flagged as untrustworthy. In some cases, the series of time which includes the flagged qubit may not be limited to the time in which the qubit is definitively lost. The series of time which includes the flagged qubit may include at least the time with the qubit was lost.

[0371] Advantageously, a qubit loss may be identified before measurement of the data qubits (and after a round of syndrome extraction). Since the data qubits have not yet been measured, it may be possible to continue on with a quantum circuit after replacing a lost qubit. The circuit may be adapted to retake or restart portions of the calculation implicating the lost qubit. As a consequence, the systems, the methods, the computer-readable media, and the techniques disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate.

[0372] For example, the error correcting code may be directed to updating the calculation to address for error. In some cases, knowing about the error may be needed in order to implement error correcting code. However, in other cases, the error correcting code may be modified to account for missing data without explicit knowledge that qubit is lost.Examples of Systems for Error Corrected Quantum Computing

[0373] FIG. 11 shows a system for error corrected quantum computing that is programmed or otherwise configured to implement methods provided herein. A system for error corrected computing may comprise an error correcting code. The present disclosure provides systems for error corrected quantum computing. The system may comprise an error correction code. An implementation of the error correcting code may comprise a decoder. The decoder may be configured to receive a matching graph and to determine a set of edges, and the matching graphWSGR Docket No. 55436-733.601 received by the decoder may be updated based on a predicted probability distribution of a lost qubit. In some cases, the error correction code comprises an operation in which a two-qubit interaction between a qubit and a lost qubit has an effect of a Pauli operation or an identity operation on the qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit.

[0374] In some cases, a system for error corrected quantum computing may comprise a non- classical computing system 1150. The non-classical computing system may be quantum computing system. The non-classical computing system may be a trapped atom quantum computing system. The trapped atom quantum computing system may comprise: an atom movement unit, an atom rearrangement unit, an optical trapping unit, an imaging unit, an optical pumping unit, an entanglement unit, a Rydberg unit, a non-classical computation unit, an electromagnetic deliver unit, or any combination thereof.

[0375] In some cases, the non-classical computing system may comprise a plurality of qubits.

[0376] In some cases, the non-classical computing system may comprise one or more electromagnetic delivery units. The electromagnetic delivery units may be configured to produce electromagnetic excitations to perform various operations, such as for example, atom movement, atom rearrangement, optical trapping, imaging, and various operations on atoms that may comprises portions of a nonclassical computation. Portions of a non-classical computation on trapped atoms may comprise optical pumping, entanglement operations, Rydberg operations, gate operations (e.g., one qubit operations, two qubit operations, etc.). In some cases, an atom movement unit may comprise an atom rearrangement unit. The components of a non-classical computing system are discussed herein above with respect to the operations they implement.

[0377] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises trapped atom qubits. In some cases, the trapped atom qubits comprise neutral atom qubits. In some cases, the neutral atom qubits comprise a Group II element or a Group Il-like element. In some cases, the Group II element or a Group II- like element comprises Ytterbium, Rubidium, Cesium, or Strontium. In some cases, the plurality of qubits comprises qubit states comprising nuclear spin states on the ' So manifold. In some cases, the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

[0378] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises a plurality of qubits, wherein the plurality of qubits comprises atomic qubits, and wherein an atom replacement operation is implemented using optical tweezers.WSGR Docket No. 55436-733.601

[0379] In some cases, the non-classical computing system may be configured to interact with a processor 1101. The processor may be classical processing system. The processor may be digital processing system.

[0380] In some cases, the system further comprises a processor configured to implement an error correcting code. In some cases, the processor is further configured to provide instructions to a non-classical computing system, wherein the non-classical computing system is configured to implement the instructions to: (i) identify that a qubit has been lost; (ii) replace the qubit; and (iii) reimplement the qubit into the circuit. In some cases, the processor is further configured to (iv) flag measurements taken while the qubit was missing as untrustworthy. In some cases, (i) comprise...

Claims

1. WSGR Docket No. 55436-733.601CLAIMSWHAT IS CLAIMED IS:

1. A device for beam steering, comprising a medium that is at least partially optically transparent, wherein said medium is bounded by a first surface that is partially internally reflective, and a second surface that is opposed to said first surface and is at least partially internally reflective, said medium forming a transmission guide, between said first surface and said second surface, for an input optical beam that transmits, from said second surface, a plurality of output optical beams, and wherein said medium comprises an electro-optic material having an index of refraction that is responsive to a voltage difference; and a plurality of electrodes electrically coupled to said medium, wherein said plurality of electrodes are configured to generate an electric field, thereby changing said index of refraction of said electro-optic material of said medium to change a relative phase of said plurality of output optical beams.

2. The device of claim 1, wherein said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate.

3. The device of claims 1 or 2, wherein said plurality of output optical beams crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits.

4. The device of claim 3, wherein said plurality of qubits form at least a portion of a non-classical computer.

5. The device of any one of the preceding claims, wherein said first surface comprises a first portion that is substantially transparent and a second portion that is substantially fully internally reflective.

6. The device of any one of the preceding claims, wherein said second surface is substantially fully internally reflective.

7. The device of any one of the preceding claims, wherein said medium comprises a first face comprising said first surface, a second face comprising said second surface, a third face, and a fourth face.

8. The device of any claim 7, wherein said plurality of electrodes are electrically coupled to said medium via connecting to said third face and said fourth face.

9. The device of either claim 7 or 8, wherein said medium is a rectangular prism.

10. The device of any one of the preceding claims, wherein a set of virtual images of a light source said input optical beam are aligned at a constant interval.WSGR Docket No. 55436-733.60111. The device of any one of the preceding claims, wherein said electric field is oriented substantially along an axis of optical polarization of said electro-optic material of said medium.

12. The device according to any one of the preceding claims, further comprising a second device for beam steering according to claim 1, wherein said pair of devices is configured to steer a beam in two dimensions.

13. A virtually imaged phased array (VIP A) device comprising an electro-optic material, wherein a path of a plurality of optical beams output from said VIPA device is modifiable via applying an electric field to said electro-optic material of said VIPA device.

14. The VIPA device of claim 13, wherein said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate.

15. The VIPA device of claim 14, wherein said path of said plurality of optical beams output from said VIPA device crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits.

16. The VIPA device of claim 15, wherein said plurality of qubits form at least a portion of a non-classical computer.

17. The VIPA device of any one of the preceding claims, wherein a set of virtual images of a light source of said VIPA device are aligned at a constant interval.

18. The VIPA device of any one of the preceding claims, wherein said electric field is oriented substantially along an axis of optical polarization of said electro-optic material of said VIPA device.

19. The VIPA device according to any one of the preceding claims, further comprising a second VIPA device according to claim 13, wherein said pair of VIPA devices is configured to steer said path of said plurality of optical beams in two dimensions.

20. A method for beam steering, comprising:(a) directing an optical beam into a virtually imaged phased array (VIPA) device that comprises an electro-optic material; and(b) applying an electric field to said electro-optic material of said VIPA device to modify an index of refraction of said electro-optic material of said VIPA device, thereby controlling a path of a plurality of output optical beams from said VIPA device.

21. The method of claim 20, wherein said electro-optic material comprises one or more of lithium niobate, lithium tantalate, potassium titanyl phosphate, or P-barium borate.

22. The method of claims 20 or 21, wherein said path of said plurality of optical beams output from said VIPA device crosses at least part of a plurality of optical trapping sites configured to trap a plurality of atoms that correspond to a plurality of qubits.WSGR Docket No. 55436-733.60123. The method of claim 22, wherein said plurality of qubits for at least a portion of a non-classical computer.

24. The method of any one of claims 20 to 23, further comprising a second VIPA device according to claim 20, and further comprising steering, using said pair of VIPA devices, said path of said plurality of optical beams in two dimensions.

25. The method of any one of claims 20 to 24, further comprising substantially aligning a set of virtual images of a light source of said VIPA device.

26. The method of any one of claims 20 to 25, wherein said VIPA device emits a substantially collimated optical beam in one direction.

27. The method of claim 26, further comprising directing said substantially collimated optical beam at a wavelength dependent angle, wherein said angle is configured to produce angular dispersion of said substantially collimated optical beam.

28. The method of any one of claims 20 to 27, wherein said electric field is oriented substantially along an axis of optical polarization of said electro-optic material of said VIPA device.

29. A system for generating multiple optical beams, said system comprising: an acousto-optic deflector (AOD) comprising an acousto-optic material, wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to generate an acoustic wave in said acousto-optic material of said AOD, wherein said acoustic wave is configured to generate a plurality of independent output optical beams from an input optical beam directed to the AOD30. The system of claim 29, wherein said acoustic wave comprises one or more discrete momentum components.

31. The system of claim 30, wherein each discrete momentum component is associated with a drive vector.

32. The system of claim 31, wherein a drive vector comprises a set of one or more drive signals.

33. The system of claim 32, wherein said one or more drive signals share a common frequency.

34. The system of claim 32 or 33, wherein said one or more drive signals is configured to be applied to two or more acoustic transducers of said plurality of acoustic transducers, wherein said drive signals further comprise a phase relationship between two or more signals applied to said two or more acoustic transducers of said plurality of acoustic transducers.WSGR Docket No. 55436-733.60135. The system of claim 34, wherein said phase relationship comprises a spatial gradient.

36. The system of any one of claims 31 to 35, wherein each independent output optical beam corresponds to a drive vector.

37. The system of claim 36, wherein an independent output optical beam is deflected based at least in part on a frequency and phase gradient of a corresponding drive vector.

38. The system of any one of claims 29 to 37, wherein said plurality of transducers is operated in an acoustic near-field limit.

39. The system of any one of claims 29 to 38, wherein said plurality of transducers is operated in said acoustic far-field limit.

40. A method for generating multiple optical beams, said method comprising: providing an acousto-optic deflector (AOD) comprising an acousto-optic material, wherein said AOD comprises a plurality of acoustic transducers; and generating, using said plurality of acoustic transducers, an acoustic wave in said acousto-optic material of said AOD, wherein said acoustic wave is configured to generate a plurality of independent output optical beams from an input optical beam directed to the AOD41. The method of claim 40, wherein said acoustic wave comprises one or more discrete momentum components.

42. The method of claim 41, wherein each discrete momentum component is associated with a drive vector.

43. The method of claim 42, wherein a drive vector comprises a set of one or more drive signals.

44. The method of claim 43, wherein said one or more drive signals share a common frequency.

45. The method of claim 43 or 44, wherein said one or more drive signals is configured to be applied to two or more acoustic transducers of said plurality of acoustic transducers, wherein said drive signals further comprise a phase relationship between two or more signals applied to said two or more acoustic transducers of said plurality of acoustic transducers.

46. The method of claim 45, wherein said phase relationship comprises a spatial gradient.

47. The method of any one of claims 42 to 46, wherein each independent output optical beam corresponds to a drive vector.WSGR Docket No. 55436-733.60148. The method of claim 47, further comprising deflecting an independent output optical beam based at least in part on a frequency and phase gradient of a corresponding drive vector.

49. The method of any one of claims 40 to 48, wherein said plurality of transducers is operated in an acoustic near-field limit.

50. The method of any one of claims 40 to 49, wherein said plurality of transducers is operated in said acoustic far-field limit.

51. A system for optical control, said system comprising: an optical source configured to generate an optical beam having an optical path; and an acousto-optic deflector (AOD), wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to generate an acoustic wave in said AOD that is configured to modify said optical path of said optical beam at least in part while said optical beam passes through said AOD.

52. The system of claim 51, wherein said plurality of acoustic transducers is driven in a phase-coherent manner to generate multiple independent output beams.

53. A system for generating multiple optical beams, said system comprising: an acousto-optic deflector (AOD), wherein said AOD comprises a plurality of acoustic transducers, wherein said plurality of acoustic transducers is configured to be driven in a phase- coherent manner to generate multiple independent output beams.

54. The system of claims 52 or 53, wherein substantially the same frequency is applied to each acoustic transducer of said plurality of acoustic transducers.

55. The system of claim 54, wherein a deflected optical azimuthal angle, perpendicular to said plurality of acoustic transducers, may be changed based at least in part on applied frequency.

56. The system of any one of claims 51 to 55, wherein one or more spatial phase gradients are applied to one or more frequencies applied to said plurality of acoustic transducers to vary a deflected elevation angle in said direction along said plurality of acoustic transducers.

57. The system of claim 56, wherein said one or more spatial phase gradients are applied in a first dimension, and one or more frequencies are applied in a second dimension, thereby yielding two-dimensional control of a deflected output beam.

58. The system of claims 56 or 57, wherein a sum of said one or more frequencies used to generate each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams.WSGR Docket No. 55436-733.60159. The system of any one of claims 56 to 58, wherein a sum of said one or more spatial phase gradients used to deflect each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams.

60. The system of any one of claims 56 to 59, wherein a sum of said one or more frequencies used to generate each individual optical beam, and a sum of said one or more spatial phase gradients used to deflect each optical beam are applied to said plurality of transducers to generate multiple independent optical beams.

61. The system of any one of claims 51 to 60, wherein said plurality of transducers is operated in said acoustic near-field limit.

62. The system of any one of claims 51 to 61, wherein said plurality of transducers is operated in said acoustic far-field limit.

63. A system for modulating a phase of an optical beam, said system comprising: an acoustic medium configured to propagate an acoustic wave, wherein said acoustic wave is configured to transform said phase of said optical beam into a modulated phase at least in part while said optical beam passes through said acoustic medium.

64. An acousto-optic deflector (AOD), said AOD comprising: an acousto-optic medium configured to propagate an acoustic wave; and a plurality of acoustic transducers configured to provide an acoustic wave, wherein said acoustic wave is configured to generate a plurality of optical beams from an optical beam incident on said acoustic medium.

65. A method for generating multiple optical beams, said method comprising: providing an acousto-optic deflector (AOD) comprising a plurality of acoustic transducers; and driving said plurality of acoustic transducers in a phase-coherent manner, thereby generating multiple independent output beams.

66. A method of optical control, said method comprising:(a) generating an optical beam having an optical path;(b) providing an acousto-optic deflector (AOD); and(c) modifying said optical path of said optical beam at least in part while said optical beam passes through said AOD, wherein said modifying of said optical path is controlled by a plurality of acoustic transducers.

67. The method of claim 66, wherein said modifying of said optical path is controlled by a plurality of acoustic transducers in (c) comprises generating, using said plurality of acoustic transducers, an acoustic wave configured to modify said optical path of said optical beam at least in part while said optical beam passes through said AOD.WSGR Docket No. 55436-733.60168. The method of any one of claims 66 or 67, wherein substantially the said same frequency is applied to each acoustic transducer of said plurality of acoustic transducers.

69. The method of any one of claims 66 to 68, wherein a deflected optical azimuthal angle of said optical path, perpendicular to said plurality of acoustic transducers, may be changed based on an applied frequency.

70. The method of any one of claims 66 to 69, wherein one or more spatial phase gradients are applied to one or more frequencies applied to said plurality of acoustic transducers to vary a deflected elevation angle in a direction along said plurality of acoustic transducers.

71. The method of claim 70, wherein said one or more spatial phase gradients are applied in a first dimension, and one or more frequencies are applied in a second dimension, thereby yielding two-dimensional control of a deflected output beam.

72. The method of claims 70 or 71, wherein a sum of said one or more frequencies used to generate each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams.

73. The method of any one of claims 70 to 72, wherein a sum of said one or more spatial phase gradients used to deflect each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams.

74. The method of any one of claims 70 to 73, wherein a sum of said one or more frequencies used to generate each individual optical beam, and a sum of one or more spatial phase gradients used to deflect each individual optical beam are applied to said plurality of acoustic transducers to generate multiple independent optical beams.

75. The method of any one of claims 66 to 74, wherein said plurality of transducers is operated in said acoustic near-field limit.

76. The method of any one of claims 66 to 75, wherein said plurality of transducers is operated in said acoustic far-field limit.

77. A control system for generating drive vector control, the system comprising: at least one controller, wherein a controller of the at least one controller is configured to deconstruct an input control signal into an output control signal, wherein said output control signal of said controller of said at least one controller is configured to generate an acoustic wave in an acousto-optic medium.

78. The control system of claim 77, wherein said at least one controller comprises two or more controllers.WSGR Docket No. 55436-733.60179. The control system of claim 78, wherein said two or more controllers are configured to generate an acoustic wave in an acousto-optic medium, at least in part, by combining output control signals from each of the two or more controllers.

80. The control system of claim 77, wherein said controller of said at least one controller is configured to generate said acoustic wave in said acousto-optic medium, at least in part, by at least one acoustic transducer.

81. The control system of claim 80, wherein said at least one acoustic transducer comprises two or more acoustic transducers.

82. The control system of claim 77, wherein said input control signal comprises one or more drive vectors, wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam.

83. The control system of claim 82, wherein said input control signal comprises a piecewise function.

84. The control system of claim 82, wherein said input control signal comprises an interpolated curve.

85. The control system of claim 82, wherein said input control signal comprises a B- spline curve.

86. The control system of claim 82, wherein said input control signal comprises a Bezier curve.

87. The control system of claim 82, wherein said input control signal comprises a parameterized curve and a set of offsets.

88. The control system of claim 87, wherein said set of offsets comprises a plurality of offset vectors.

89. The control system of claim 88, further comprising a memory, wherein said plurality of offset vectors are stored in said memory.

90. The control system of claim 89, wherein said plurality of offset vectors further comprises a location in said memory, and wherein said input control signal further comprises locations in said memory corresponding to each of said plurality of offset vectors.

91. The control system of any one of claims 88 to 90, wherein said plurality of offset vectors comprise one or more displacement vectors specified on a grid.

92. The control system of claim 91, wherein said grid comprises a regular grid.

93. The control system of claim 92, wherein said regular grid comprises a rectangular grid.

94. The control system of claim 93, wherein said rectangular grid comprises a square grid.WSGR Docket No. 55436-733.60195. The control system of claim 91, wherein said grid comprises a non -regular grid.

96. The control system of any one of claims 78 to 95, wherein said two or more controllers are configured to deconstruct said input control signal into distinct output control signals.

97. The control system of claim 96, wherein said distinct output control signals correspond to distinct regions of a target area.

98. The control system of claim 97, wherein said distinct regions of a target area comprise overlapping regions.

99. The control system of claim 97, wherein said distinct regions of a target area comprise non-overlapping regions.

100. The control system of claim 97, wherein said distinct regions of a target area comprise both overlapping regions and non-overlapping regions.

101. The control system of any one of claims 97 to 100, wherein said distinct regions of a target area are rectangular regions.

102. The control system of claims any one of claims 97 to 101, wherein said distinct output control signals are selected to reduce crossings between distinct region of said target area.

103. The control system of any one of claims 97 to 102, wherein deconstructing said input control signal into said distinct output control signals comprises oversampling, wherein said oversampling is configured to provide smooth movement from a first region of said target area corresponding to a first control signal to a second region of said target area corresponding to a second control signal.

104. The control system of any one of claims 77 to 103, wherein said input control signal further comprises one or more pre-calculated moves.

105. The control system of claim 104, wherein a move of said one or more precalculated moves comprises one or more drive vectors wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam.

106. The control system of claim 104 or 105, wherein said one or more pre -calculated moves comprises one or more individual movements of an optical beam.

107. The control system of any one of claims 104 to 106, wherein said one or more pre-calculated moves comprises one or more movements of a group of optical beams.

108. The control system of any one of claims 104 to 107, wherein said one or more pre-calculated moves is initiated, at least in part, by an external trigger.

109. The control system of claim 108, wherein said external trigger comprises an external command signal from one or more users.WSGR Docket No. 55436-733.601110. The control system of claim 108 or 109, wherein said external trigger comprises an autonomous trigger.

111. The control system of claim 110, wherein said autonomous trigger comprises a clock-based periodic trigger.

112. The control system of any one of claims 77 to 111, wherein the control system is configured to hold a position of one or more optical beams absent said input control signal.

113. The control system of any one of claims 77 to 112, wherein said input control signal further comprises one or more intensity commands, wherein an intensity command of said one or more intensity commands is configured to modulate an intensity of an optical beam.

114. The control system of claim 113, wherein an intensity command of said one or more intensity commands is configured to turn an optical beam on and off.

115. The control system of claim 113 or 114, wherein an intensity command of said one or more intensity commands is configured to provide time dependent control of an intensity of an optical beam.

116. The control system of any one of claims 77 to 115, wherein said input control signal comprises a phase gradient control signal.

117. The control system of claim 116, wherein said phase gradient control signal further comprises a global phase.

118. The control system of claim 117, wherein said global phase is chosen randomly.

119. A method of generating drive vector control, the method comprising:(a) deconstructing, using at least one controller, an input control signal to produce an output control signal; and(b) generating, using said output control signal, an acoustic wave in an acoustooptic medium.

120. The method of claim 119, wherein said at least one controller comprises two or more controllers.

121. The method of claim 120, wherein generating in (b) is done, at least in part, by combining output control signals from each of the two or more controllers.

122. The method of claim 119, wherein generating in (b) is performed, at least in part, by at least one acoustic transducer.

123. The method of claim 122, wherein said at least one acoustic transducer comprises two or more acoustic transducers.

124. The control system of claim 119, wherein said input control signal comprises one or more drive vectors, wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam.WSGR Docket No. 55436-733.601125. The method of claim 124, wherein said input control signal comprises a piecewise function.

126. The method of claim 124, wherein said input control signal comprises an interpolated curve.

127. The method of claim 124, wherein said input control signal comprises a B-spline curve.

128. The method of claim 124, wherein said input control signal comprises a Bezier curve.

129. The method of claim 124, wherein said input control signal comprises a parameterized curve and a set of offsets.

130. The method of claim 129, wherein said set of offsets comprises a plurality of offset vectors.

131. The method of claim 130, further comprising storing said plurality of offset vectors in a memory.

132. The method of claim 131, wherein said plurality of offset vectors further comprises a location in said memory, and wherein said input control signal further comprises locations in said memory corresponding to each of said plurality of offset vectors.

133. The method of any one of claims 130 to 132, wherein said plurality of offset vectors comprise one or more displacement vectors specified on a grid.

134. The method of claim 133, wherein said grid comprises a regular grid.

135. The method of claim 134, wherein said regular grid comprises a rectangular grid.

136. The method of claim 135, wherein said rectangular grid comprises a square grid.

137. The method of claim 133, wherein said grid comprises a non-regular grid.

138. The method of any one of claims 120 to 137, further comprising deconstructing, at least in part using said two or more controllers, said input control signal into distinct output control signals.

139. The method of claim 138, wherein said distinct output control signals correspond to distinct regions of a target area.

140. The method of claim 139, wherein said distinct regions of a target area comprise overlapping regions.

141. The method of claim 139, wherein said distinct regions of a target area comprise non-overlapping regions.

142. The method of claim 139, wherein said distinct regions of a target area comprise both overlapping regions and non-overlapping regions.WSGR Docket No. 55436-733.601143. The method of any one of claims 139 to 142, wherein said distinct regions of a target area are rectangular regions.

144. The method of claims any one of claims 139 to 143, further comprising selecting said distinct output control signals to reduce crossings between distinct region of said target area.

145. The method of any one of claims 139 to 144, wherein deconstructing said input control signal into said distinct output control signals further comprises oversampling, wherein said oversampling is configured to provide smooth movement from a first region of said target area corresponding to a first control signal to a second region of said target area corresponding to a second control signal.

146. The method of any one of claims 119 to 145, further comprising pre-calculating one or more moves of said input control signal.

147. The method of claim 146, wherein pre-calculating said one or more moves comprises pre-calculating one or more drive vectors wherein each drive vector comprises a phase gradient and a frequency corresponding to an independent optical beam.

148. The method of claim 146 or 147, wherein pre-calculating said one or more moves comprises pre-calculating one or more individual movements of an optical beam.

149. The method of any one of claims 146 to 148, wherein pre-calculating one or more moves comprises pre-calculating one or more movements of a group of optical beams.

150. The method of any one of claims 146 to 149, further comprising initiating said pre-calculated one or more moves, at least in part, by an external trigger.

151. The method of claim 150, wherein said external trigger comprises receiving an external command signal from one or more users.

152. The method of claim 150 or 151, wherein said external trigger comprises an autonomous trigger.

153. The method of claim 152, wherein said autonomous trigger comprises a clockbased periodic trigger.

154. The method of any one of claims 119 to 153, further comprising holding a position of one or more optical beams absent said input control signal.

155. The method of any one of claims 119 to 154, further comprising controlling an intensity of one or more optical beams, at least in part, using said input control signal.

156. The method of claim 155, wherein controlling said intensity of one or more optical beams further comprises turning at least one optical beam of said one or more optical beams on or off.WSGR Docket No. 55436-733.601157. The method of claim 155 or 156, wherein controlling said intensity of one or more optical beams further comprises time dependent control of an intensity of one or more optical beams.

158. The method of any one of claims 155 to 157, wherein controlling said intensity of one or more optical beams is done, at least in part, using a phase gradient control signal.

159. The method of claim 158, wherein said phase gradient control signal further comprises a global phase.

160. The method of claim 159, wherein said global phase is chosen randomly.

161. The system, method, or device of any one of the preceding claims, further comprising a plurality of atoms.

162. The system, method, or device of claim 161, wherein said plurality of atoms comprises a plurality of ions.

163. The system, method, or device of claim 161, wherein said plurality of atoms comprises a plurality of neutral atoms.

164. The system, method, or device of claim 161, wherein said plurality of atoms comprises a plurality of group II atoms.

165. The system, method, or device of any one of claims 161 to 164, wherein said plurality of atoms comprises a plurality of qubits.

166. The system, method, or device of any one of the preceding claims, wherein a deflected optical beam is used, at least in part, to control said plurality of atoms.

167. The system, method, or device of claim 166, wherein said control of said plurality of atoms comprises a moving an atom of said plurality of atoms.

168. The system, method, or device of claim 166 or 167, wherein said control of said plurality of qubits comprises a non-classical computation.

169. A system for array generation, comprising: one or more optical sources configured to generate one or more optical beams; one or more acousto-optic deflectors (AODs); and one or more spatial light modulators (SLMs), wherein said one or more AODs is illuminated by said one or more optical beams, and the one or more AODs are configured to illuminate the one or more SLMs with one or more transmitted optical beams.

170. The system of claim 169, wherein an SLM of said one or more SLMs comprises one or more SLM masks configured to produce a specific pattern from an emitted optical beam.

171. The system of claim 170, wherein an SLM mask of said one or more SLM masks comprises a superimposed correction pattern.WSGR Docket No. 55436-733.601172. The system of any one of claims 169 to 171, further comprising one or more relay telescopes.

173. The system of any one of claims 169 to 172, wherein said one or more AODs comprises a pair of AODs arranged in a crossed AOD (xAOD) configuration.

174. The system of any one of claims 169 to 173, further comprising one or more relay telescopes.

175. The system of claim 174, further comprising a first relay telescope of said one or more relay telescopes, said first relay telescope located in an optical path of said one or more transmitted optical beams between a first AOD of said pair of AODs and a second AOD of said pair of AODs.

176. The system of claim 175, further comprising a second relay telescope of said one or more relay telescopes, said second relay telescope located in an optical path of said one or more transmitted optical beams between said second AOD of said pair of AODs and an SLM of said one or more SLMs.

177. The system of any one of claims 174 to 176, wherein said one or more relay telescopes comprise one or more of: a 4f relay telescope, a non 4f relay telescope, a two element relay telescope, a 2fl relay telescope, a 2f2 relay telescope, a two element 2fl + 2f2 relay telescope, or an unobscured two-mirror relay telescope.

178. A method for array generation, said method comprising:(a) generating one or more optical beams;(b) illuminating one or more acousto-optic deflectors (AODs) with said one or more optical beams, said one or more AODs thereby producing one or more transmitted optical beams; and(c) illuminating one or more spatial light modulators (SLMs) with said one or more transmitted optical beams.

179. The method of claim 178, wherein an SLM of said one or more SLMs comprises one or more SLM masks configured to produce a specific pattern from an emitted optical beam.

180. The method of claim 179, wherein an SLM mask of said one or more SLM masks comprises a superimposed correction pattern.

181. The method of any one of claims 178 to 180, wherein said one or more AODs comprises a pair of AODs arranged in a crossed AOD (xAOD) configuration.

182. The method of any one of claims 178 to 181, further comprising one or more relay telescopes.WSGR Docket No. 55436-733.601183. The method of claim 182, further comprising projecting said one or more transmitted optical beams from said one or more AODs using a relay telescope of said one or more relay telescopes.

184. The method of claim 182 or 183, further comprising projecting, using a first relay telescope of said one or more relay telescopes, said one or more transmitted optical beams from a first AOD of said pair of AODs onto a second AOD of said pair of AODs.

185. The method of claim 184, further comprising projecting, using a second relay telescope of said one or more relay telescopes, said one or more transmitted optical beams from said second AOD of said pair of AODs onto an SLM of said one or more SLMs.

186. The method of any one of claims 182 to 185, wherein said one or more relay telescopes comprise one or more of: a 4f relay telescope, a non 4f relay telescope, a two element relay telescope, a 2fl relay telescope, a 2f2 relay telescope, a two element 2fl + 2f2 relay telescope, or an unobscured two-mirror relay telescope.

187. The method of any one of claims 178 to 186, further comprising (d) generating a different pattern of spots of light at a different location.

188. The method of any one of claims 178 to 187, further comprising (e) generating the same pattern of spots of light at a different location.

189. The method of any one of claims 178 to 188, further comprising (f) generating a different pattern of spots of light at the same location.

190. A method of optically addressing atoms, said method comprising:(a) obtaining a plurality of atoms in a plurality of spatially distinct optical trapping sites; and(b) addressing at least a subset of said plurality of atoms with a plurality of optical tweezers, said plurality of optical tweezers collectively having multiplexed wavelengths combined on an optical grating.

191. The method of claim 190, wherein said optical grating corresponds, at least in part, to one or more real-space planes of said plurality of atoms.

192. The method of claim 190 or 191, wherein said optical grating comprises a millimeter-scale optical grating.

193. The method of any one of claims 190 to 192, wherein said optical grating comprises a meter-scale optical grating.

194. The method of claim 190, wherein said optical grating corresponds, at least in part, to one or more Fourier space planes of said plurality of atoms.

195. The method of any one of claims 190 to 194, wherein said optical grating comprises a microlens array.WSGR Docket No. 55436-733.601196. The method of any one of claims 190 to 195, wherein addressing in (b) is done, at least in part, using spectrally broadband light.

197. The method of any one of claims 190 to 196, wherein addressing in (b) is done, at least in part, using spatially incoherent light.

198. The method of any one of claims 190 to 197, wherein addressing in (b) is done, at least in part, using a lightfield directing array (LDA).

199. The method of claim 198, wherein said LDA is a one-dimensional LDA.

200. The method of claim 198, wherein said LDA is a two-dimensional LDA.

201. The method of any one claims 190 to 200, wherein said optical grating comprises a blazed grating.

202. The method of any one of claims 190 to 201, wherein said optical grating comprises an echelle grating.

203. The method of any one of claims 190 to 202, wherein addressing in (b) is done, at least in part, using a virtually imaged phase array (VIP A) optical grating pair.

204. The method of any one of claims 190 to 203, wherein said optical grating corresponds to an intermediate plane between a real -space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms.

205. A method for optically addressing atoms, said method comprising:(a) obtaining a plurality of atoms in a plurality of spatially distinct optical trapping sites; and(b) addressing at least a first subset of said plurality of atoms with a first plurality of optical tweezers, said first plurality of optical tweezers collectively having multiplexed wavelengths corresponding, at least in part, to one or more real-space planes of said plurality of atoms; or(c) addressing at least a second subset of said plurality of atoms with a second plurality of optical tweezers, said second plurality of optical tweezers collectively having multiplexed wavelengths corresponding, at least in part, to one or more Fourier space planes of said plurality of atoms; or(d) interpolating between said first plurality of optical tweezers and said second plurality of optical tweezers, at least in part, using an optical grating in an intermediate optical plane.

206. The method of claim 205, wherein said optical grating comprises a millimeter scale optical grating.

207. The method of claim 205, wherein said optical grating comprises a meter scale optical grating.WSGR Docket No. 55436-733.601208. The method of claim 207, wherein said optical grating comprises a microlens array (ML A).

209. The method of any one of claims 205 to 208, wherein addressing in (b) and (c) is done, at least in part, using spectrally broadband light.

210. The method of any one of claims 205 to 209, wherein addressing in (b) and (c) is done, at least in part, using spatially incoherent light.

211. The method of any one of claims 205 to 210, wherein addressing in (b) and (c) is done, at least in part, using a lightfield directing array (LDA).

212. The method of claim 211, wherein said LDA is a one-dimensional LDA.

213. The method of claim 211, wherein said LDA is a two-dimensional LDA.

214. The method of any one of claims 205 to 213, wherein addressing in (b) and (c) is done, at least in part, using a blazed grating.

215. The method of any one of claims 205 to 214, wherein addressing in (b) and (c) is done, at least in part, using an echelle grating.

216. The method of any one of claims 205 to 215, wherein addressing in (b) and (c) is done, at least in part, using a virtually imaged phase array (VIP A) optical grating pair.

217. The method of any one of claims 205 to 216, wherein said plurality of atoms corresponds to a plurality of qubits.

218. A system for optically addressing atoms, said system comprising: a plurality of atoms in a plurality of spatially distinct optical trapping sites; an optical grating; and an optical source configured to supply a plurality of optical tweezers, said plurality of optical tweezers collectively having multiplexed wavelengths, wherein said plurality of optical tweezers is directed onto said plurality of atoms, at least in part, using said optical grating.

219. The system of claim 218, wherein said optical source comprises spectrally broadband light.

220. The system of claim 218, wherein said optical source comprises spatially incoherent light.

221. The system of claim 218, wherein said optical source comprises a plurality of optical sources.

222. The system of any one of claims 218 to 221, further comprising a lightfield directing array (LDA).

223. The system of claim 222, wherein said LDA is a one-dimensional LDA.

224. The system of claim 222, wherein said LDA is a two-dimensional LDA.WSGR Docket No. 55436-733.601225. The system of any one of claims 218 to 224, wherein said optical grating comprises a blazed grating.

226. The system of any one of claims 218 to 225, wherein said optical grating comprises an echelle grating.

227. The system of any one of claims 218 to 226, further comprising a virtually imaged phase array (VIP A) optical grating pair.

228. The system of any one of claims 218 to 227, wherein said optical grating comprises a millimeter scale optical grating.

229. The system of any one of claims 218 to 228, wherein said optical grating comprises a meter scale optical grating.

230. The system of any one of claims 218 to 229, wherein said optical grating comprises a microlens array (MLA).

231. The system of any one of claims 218 to 230, wherein said plurality of atoms corresponds to a plurality of qubits.

232. The system of any one of claims 219 to 231, wherein said optical grating corresponds to a real-space plane of said plurality of atoms.

233. The system of any one of claims 219 to 232, wherein said optical grating corresponds to a Fourier-space plane of said plurality of atoms.

234. The system of any one of claims 219 to 233, wherein said optical grating corresponds to an intermediate plane between a real -space plane of said plurality of atoms and a Fourier-space plane of said plurality of atoms.

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