Systems and methods for iterative assembly of atom arrays

Abstract

Systems, methods, and computer-readable media for implementing non-classical computing may comprise generating an array of atoms comprising greater than 150 atoms and a fill factor of greater than 95% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers.

Description

WSGR Docket No. 55436-726.601SYSTEMS AND METHODS FOR ITERATIVE ASSEMBLY OF ATOM ARRAYS CROSS-REFERENCE

[0001] This application claims the benefit of U.S. Provisional Application No.63 / 623,738, filed January 22, 2024, which application is incorporated herein by reference. BACKGROUND

[0002] Quantum computers typically 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 itstwo orthonormal basis states. The two orthonormal basis states are usually denoted as |0^ =[1] (the “zero state”) and |1^ =(the “one state”). The two orthonormal basis states, 0{|0^, |1^}, 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^, |01^, |10^, |11^, each called a quantum register. Generally, n qubits are represented by a superposition state vector in 2ndimensional Hilbert space. SUMMARY

[0004] In an aspect, the present application provides methods and systems for the generation of a first array of atoms, wherein said first array of atoms is loaded from a second array, and wherein the second array is loaded from a reservoir while a plurality of atoms are in the first array.

[0005] In one aspect, a method, comprises: (a) generating an array of atoms comprising greater than 150 atoms and a fill factor of greater than 95% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater thanWSGR Docket No. 55436-726.601500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites.

[0006] In another aspect, a method comprises: (a) generating an array of atoms comprising greater than 150 sites and a fill factor of greater than 95% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0007] In another aspect, a method comprises:: (a) generating an array of atoms comprising greater than 500 sites and a fill factor of greater than 50% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 60% occupancy. In some embodiments, the fill factor is greater than 90% occupancy. In some embodiments, the fill factor is greater than 95% occupancy. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0008] In another aspect, a method comprises: (a) filling an array of atoms comprising greater than 1000 atoms a plurality loading cycles, wherein a fill fraction of the array of atoms is greater than 0.95. In some embodiments, the fill fraction is greater than 0.98. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the plurality of loadingWSGR Docket No. 55436-726.601cycles comprises greater than 5 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0009] In another aspect, a method comprises: (a) filling an array of atoms comprising greater than 1000 atoms, wherein a fill fraction of the array of atoms is greater than 0.95; and (b) maintaining the fill fraction over a plurality of loading cycles. In some embodiments, the fill fraction is greater than 0.98. In some embodiments, the array of atoms comprises greater than 5,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the plurality of loading cycles comprises greater than 5 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0010] In another aspect, a method comprises: (a) generating a first array of atoms comprising greater than 150 atoms and a fill factor of greater than 95%, wherein the first array of atoms is loaded from at least a second array, and wherein the second array is loaded from a reservoir while a plurality of atoms are in the first array. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the method further comprises: maintaining the fill factor over a plurality of loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5 loading cycles. The method of claim 70, wherein the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0011] In another aspect, a method for loading atoms into an optical trap comprises: (a) trapping atoms from an incident atomic beam into a magneto-optical trap (MOT) using a core-shellWSGR Docket No. 55436-726.601configuration, wherein the core-shell configuration comprises a shell at a first wavelength surround a core with a second wavelength; and (b) transferring atoms from the MOT to an optical lattice, wherein the transferring is performed with optical tweezers. In some embodiments, a shell of the MOT comprises a first light near resonance with the1S0 to1P1 transition and a core of the MOT comprises a second light near resonance with the1S0 to3P1 transition. In some embodiments, the shell comprises 399 nm light near resonance with the1S0 to1P1 transition and the core comprises 556 nm light near resonance with the1S0 to3P1 transition. In some embodiments, a shell of the MOT is configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some embodiments, the atoms are Doppler cooled to about 10 µK. In some embodiments, the method further comprises: generating a cavity-enhanced optical lattice, wherein the cavity enhanced optical lattice comprises a 3-D optical lattice. In some embodiments, the cavity enhanced optical lattice is formed from a standing wave optical lattice and a running wave optical lattice. In some embodiments, the method further comprises: generating a core-shell magnetooptical trap (MOT). In some embodiments, the core-shell MOT comprises a ring at a first wavelength surround a central region with a second wavelength. In some embodiments, a shell of the MOT comprises a first light near resonance with the1S0 to1P1 transition and a core of the MOT comprises a second light near resonance with the1S0 to3P1 transition. In some embodiments, the shell comprises 399 nm light near resonance with the1S0 to1P1 transition and the core comprises 556 nm light near resonance with the1S0 to3P1 transition. In some embodiments, a shell of the MOT is configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some embodiments, the atoms are Doppler cooled to about 10 µK. In some embodiments, the method further comprises: implementing a transport operation, wherein the transport operation is operable to move one or more atoms from a MOT to an array of atoms. In some embodiments, the method further comprises: implementing a tweezer operation, wherein the tweezer operation comprises moving one or more atoms within an array of atom from a first site to a second site. In some embodiments, the first site and the second site are in a reservoir array. In some embodiments, the first site and the second site are in a target array. In some embodiments, the first site is in a reservoir array and the second site is in a target array. In some embodiments, the method further comprises:imaging one or more atoms in an array of atoms. In some embodiments, the method further comprises: Doppler cooling one or more atoms in an array of atoms. In some embodiments, the method further comprises: Raman sideband cooling one or more atoms in an array of atoms. In some embodiments, the method further comprises: moving the atoms from the MOT to a reservoir array via a moving optical lattice.WSGR Docket No. 55436-726.601

[0012] In another aspect, one or more non-transitory computer-readable media comprising machine-executable code comprise one or more instructions that, when executed, implement a method on a non-classical computer, wherein the non-classical computer is configured to execute the one or more instructions, the method comprising the method of any preceding aspect or embodiment.

[0013] In another aspect, a non-classical computing system is based on neutral atom qubits, wherein the non-classical computing system is operable to load atoms during a computing circuit, wherein the atoms loaded during the computing circuit are used in the computing circuit.

[0014] In another aspect, a non-classical computing system comprises: an array of atoms comprising greater than 150 atoms and a fill factor of greater than 95% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater than 500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites.

[0015] In another aspect, a non-classical computing system comprises: an array of atoms comprising greater than 150 sites and a fill factor of greater than 95% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0016] In another aspect, a non-classical computing system comprises: an array of atoms comprising greater than 500 sites and a fill factor of greater than 50% occupancy, wherein the plurality of atoms are trapped by a cavity-enhanced optical lattice and one or more optical tweezers. In some embodiments, the fill factor is greater than 60% occupancy. In some embodiments, the fill factor is greater than 90% occupancy. In some embodiments, the fill factorWSGR Docket No. 55436-726.601is greater than 95% occupancy. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0017] In another aspect, a non-classical computing system comprises: an array of atoms comprising greater than 1000 atoms a plurality loading cycles, wherein a fill fraction of the array of atoms is greater than 0.95. In some embodiments, the fill fraction is greater than 0.98. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the plurality of loading cycles comprises greater than 5 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0018] In another aspect, a non-classical computing system comprises: an array of atoms comprising greater than 1000 atoms, wherein a fill fraction of the array of atoms is greater than 0.95, wherein the fill fraction is configured to remain substantially constant over a plurality of loading cycles. In some embodiments, the fill fraction is greater than 0.98. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the plurality of loading cycles comprises greater than 5 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0019] In another aspect, a non-classical computing system comprises: a first array of atoms comprising greater than 150 atoms and a fill factor of greater than 95%, wherein the first arrayWSGR Docket No. 55436-726.601of atoms is loaded from at least a second array, and wherein the second array is loaded from a reservoir while a plurality of atoms are in the first array. In some embodiments, the fill factor is greater than 98% occupancy. In some embodiments, the array of atoms comprises greater than 500 sites. In some embodiments, the array of atoms comprises greater than 1,000 sites. In some embodiments, the array of atoms comprises greater than 10,000 sites. In some embodiments, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some embodiments, the array of atoms comprises greater than 500 atoms. In some embodiments, the array of atoms comprises greater than 1,000 atoms. In some embodiments, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some embodiments, the system further comprises: an optical unit configured to maintain the fill factor over a plurality of loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 500 loading cycles. In some embodiments, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0020] In another aspect, a non-classical computing system comprises: a magneto-optical trap (MOT), wherein the MOT is configured to trap atoms from an incident atomic beam into the MOT using a core-shell configuration, wherein the core-shell configuration comprises a ring at a first wavelength surrounding a central region with a second wavelength; an optical lattice; and optical tweezers, wherein the optical tweezers are configured to transfer atoms from the MOT to the optical lattice. In some embodiments, a shell of the MOT comprises a first light nearresonance with the 1S0 to 1P1 transition and a core of the MOT comprises a second light nearresonance with the1S0 to3P1 transition. In some embodiments, the shell comprises 399 nm light near resonance with the1S0 to1P1 transition and the core comprises 556 nm light near resonance with the1S0 to3P1 transition. In some embodiments, a shell of the MOT is configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some embodiments, the atoms are Doppler cooled to about 10 µK. In some embodiments, the optical unit is further configured to generate a cavity-enhanced optical lattice, wherein the cavity enhanced optical lattice comprises a 3-D optical lattice. In some embodiments, the cavity enhanced optical lattice is formed from a standing wave optical lattice and a running wave optical lattice. In some embodiments, the optical unit is further configured to generate a core-shell magnetooptical trap (MOT). In some embodiments, the core-shell MOT comprises a ring at a first wavelength surround a central region with a second wavelength. In some embodiments, a shell of the MOT comprises a first light near resonance with the1S0 to1P1 transition and a core of the MOT comprises a second light near resonance with the1S0 to3P1WSGR Docket No. 55436-726.601transition. In some embodiments, the shell comprises 399 nm light near resonance with the1S0 to1P1transition and the core comprises 556 nm light near resonance with the1S0to3P1transition. In some embodiments, a shell of the MOT is configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some embodiments, the atoms are Doppler cooled to about 10 µK. In some embodiments, the system further comprises: a transport unit configured to implement a transport operation, wherein the transport operation is operable to move one or more atoms from a MOT to an array of atoms. In some embodiments, the transport unit is further configured to implement a tweezer operation, wherein the tweezer operation comprises moving one or more atoms within an array of atom from a first site to a second site. In some embodiments, the first site and the second site are in a reservoir array. In some embodiments, the first site and the second site are in a target array. In some embodiments, the first site is in a reservoir array and the second site is in a target array. In some embodiments, the system further comprises: an imaging unit configured to image one or more atoms in an array of atoms. In some embodiments, the system further comprises: a cooling unit configured to Doppler cool one or more atoms in an array of atoms. In some embodiments, the cooling unit is further configured to Raman sideband cool one or more atoms in an array of atoms.

[0021] 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.

[0022] 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 of 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

[0023] 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 referenceWSGR Docket No. 55436-726.601contradict 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

[0024] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention may be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also “Figure” and “FIG.” herein), of which:

[0025] FIG.1 shows a computer control system that is programmed or otherwise configured to implement methods provided herein;

[0026] FIG.2 shows an example of a system for performinlg a non-classical computation;

[0027] FIG.3A shows an example of an optical trapping unit;

[0028] FIG.3B shows an example of a plurality of optical trapping sites;

[0029] FIG.3C shows an example of an optical trapping unit that is partially filled with atoms;

[0030] FIG.3D shows an example of an optical trapping unit that is completely filled with atoms;

[0031] FIG.4 shows an example of an electromagnetic delivery unit;

[0032] FIG.5 shows an example of a state preparation unit;

[0033] FIG.6 shows a flowchart for an example of a first method for performing a non-classical computation;

[0034] FIG.7 shows a flowchart for an example of a second method for performing a non- classical computation;

[0035] FIG.8 shows a flowchart for an example of a third method for performing a non- classical computation;

[0036] FIG.9 shows an energy level structure for single-qubit and multi-qubit operations in Strontium-87;

[0037] FIG.10 shows an example method for error corrected non-classical computation;

[0038] 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;

[0039] FIG.12 shows an example process for performing continuous, non-classical computation;

[0040] FIG.13A shows example of a plurality of mirrors configured to provide a plurality of optical cavities;WSGR Docket No. 55436-726.601

[0041] 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;

[0042] FIG.14 shows a diagram of a repeated loading sequence;

[0043] FIG.15A illustrates a timing diagram for a continuous loading sequence;

[0044] FIG.15B illustrates a number of atoms and fill fraction as a function of loading iteration number during initial loading (left panel) and while maintaining a filled array (right panel);

[0045] FIG.16A illustrates intersection of two cavity modes providing three-dimensional confinement for atoms within the field of view of a high-numerical-aperture imaging system for cavity-enhanced optical lattices for rapid low-loss imaging;

[0046] FIG.16B illustrates observed counts during a 7 ms long image of single atoms within the optical lattice, showing well-resolved peaks for occupied and unoccupied sites for cavity- enhanced optical lattices for rapid low-loss imaging;

[0047] FIG.16C illustrates lattice homogeneity across the target array as characterized by light shifts on the1S0 mf = ½ to3P1 F= 3 / 2, mf = -½ transition for cavity-enhanced optical lattices for rapid low-loss imaging;

[0048] FIG.16D illustrates light shift contributions from each cavity averaged over rows for cavity-enhanced optical lattices for rapid low-loss imaging;

[0049] FIG.16E illustrates light shift contributions from each cavity averaged over columns for cavity-enhanced optical lattices for rapid low-loss imaging;

[0050] FIG.17A illustrates a diagram of relevant directions and polarizations for Raman sideband cooling (RSC);

[0051] FIG.17B illustrates a level diagram for RSC cooling for Raman sideband cooling (RSC);

[0052] FIG.17C illustrates sideband spectra in the x, y plane for Raman sideband cooling (RSC);

[0053] FIG.17D illustrates sideband spectra in the z direction for Raman sideband cooling (RSC);

[0054] FIG.17E illustrates heating in the lattice, as measured from the sideband imbalance in the x, y plane for Raman sideband cooling (RSC);

[0055] FIG.18A illustrates loss per pair of handoffs for a characterization of per-cycle loss mechanisms;

[0056] FIG.18B illustrates another loss per pair of handoffs for a characterization of per-cycle loss mechanisms;

[0057] FIG.18C a final vacancy fraction versus rearrangement tweezer power, normalized to the default tweezer depth of about 150 µK for a characterization of per-cycle loss mechanisms;WSGR Docket No. 55436-726.601

[0058] FIG.19 illustrates alignment of the tweezer arrays to the cavity lattice performed by monitoring atomic survival after repeated handoffs, while scanning the position offset of the tweezer array;

[0059] FIG.20A illustrates a method for implementing non-classical computing;

[0060] FIG.20B illustrates another method for implementing non-classical computing;

[0061] FIG.20C illustrates another method for implementing non-classical computing;

[0062] FIG.20D illustrates another method for implementing non-classical computing;

[0063] FIG.20E illustrates another method for implementing non-classical computing;

[0064] FIG.20F illustrates another method for implementing non-classical computing; and

[0065] FIG.20G illustrates another method for implementing non-classical computing. DETAILED DESCRIPTION

[0066] While various embodiments of the invention 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 invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed.

[0067] Assembling and maintaining large arrays of atoms in optical tweezers may be useful for continued scaling of neutral-atom-based quantum computers and simulators. Disclosed herein are methods and systems for assembly of atomic arrays, based on a synergistic combination of optical tweezers and cavity-enhanced optical lattices. Disclosed herein are methods and systems for the incremental filling of a target array from a repetitively filled reservoir. In this protocol, the tweezers provide microscopic rearrangement of atoms, while the cavity-enhanced lattices enable the creation of large numbers of deep optical potentials that allow for rapid low-loss imaging of atoms. In some cases, deterministic filling (99% per-site occupancy) of 1225-site arrays is demonstrated. Because the reservoir may be repeatedly filled with fresh atoms, the array can be maintained in a filled state indefinitely. Methods and systems disclosed herein are compatible with mid-circuit reloading, which may be useful for running large-scale error- corrected quantum computations whose durations exceed the lifetime of a single atom in the system.

[0068] Individually controlled neutral atoms provide a promising platform for quantum information processing and simulation. Expanding the size and control of these platforms represent key challenges in the ongoing push to access regimes beyond the capabilities of classical simulation.WSGR Docket No. 55436-726.601

[0069] Tweezer arrays and optical lattices have emerged as useful technologies for optically trapping and manipulating cold atoms, each with complementary capabilities. Optical lattices provide a well-defined potential landscape with features on the sub-wavelength scale, which can enable tight confinement for imaging and a clean optical potential for simulations involving itinerant atoms.

[0070] In some cases, the optical traps may be formed by tightly focused light (tweezers) or by standing-wave lattices, or by imaged masks or gratings. Optical trapping may additionally include various methods where atoms are cooled with optical illumination, e.g., a laser, and a spatially varying magnetic field to create a trap. Such optical traps may be called magneto- optical traps (MOTs).

[0071] In some cases, the array is two dimensional. In some cases, the array is three dimensional. In some cases, the plurality of spatially distinct optical traps comprises a 1D, 2D, or 3D optical trap. In some cases, the arrays may be linear, two-dimensional, three-dimensional, or may involve synthetic dimensions. A synthetic dimension may include, for example, dimensions consisting of internal atomic states or motional states. The plurality of spatially distinct optical traps may comprise single or multiple reservoir regions. In some cases, the arrays may be of regular or irregular or quasi-regular geometry.

[0072] In some cases, the array is two-dimensional. For example, an array of two-dimensional optical traps can be formed. The two-dimensional array may include a rectangular, square, rectangular prism, or cubic array of optical trapping sites. In some cases, the method further comprises determining, based at least in part on determining that a number of spatially distinct optical trapping sites of the array of spatially distinct optical trapping sites is missing a qubit. For example, 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 µm, 2 µm, 3 µm, 4 µm, 5 µm, 6 µm, 7 µm, 8 µm, 9 µm, 10 µm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about: 10 µm, 9 µm, 8 µm, 7 µm, 6 µm, 5 µm, 4 µm, 3 µm, 2 µm, 1 µm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site may be spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the preceding values. In some cases, the array is three-dimensional. For example, an array of three-dimensional optical traps can be formed. Optical Tweezers

[0073] In some cases, the plurality of spatially distinct optical traps comprises optical tweezers. The optical trapping sites may comprise one or more optical tweezers. Optical tweezers mayWSGR Docket No. 55436-726.601comprise 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 field 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 tweezer sites of one or more optical arrays of tweezers. The optical trapping sites may comprise one or more optical tweezer sites of one or more one- dimensional (1D) optical arrays of tweezers, two-dimensional (2D) optical arrays of tweezers, or three-dimensional (3D) optical arrays of tweezers. In some cases, the methods and systems described herein may be applied similarly to optical lattices. Optical tweezers may be useful for manipulating atoms or arrays of atoms. Trap Electromagnetic Energy

[0074] In some cases, method and systems disclosed herein may be configured to form a plurality of optical trapping sites using a trap electromagnetic energy (e.g., a “trap excitation” herein). The trap excitation may be generated by a trapping optical source.

[0075] The trap excitation may comprise an optical excitation, such as in a magneto-optical trap, an optical tweezer, etc. In some cases, the trap excitation is delivered by one or more optical trapping systems as disclosed herein. In some cases, each optical trapping system comprises its own trap excitation (e.g., trap wavelength, trap power, trap focus, number of spots, etc.). In some cases, a single trap excitation may be split into multiple arrays in order to form a plurality of arrays of traps with similar characteristics.

[0076] In some cases, the optical traps described herein may comprise visible light. In some cases, the optical traps described herein may comprise ultraviolet light. In some cases, the optical traps described herein may comprise infrared light. In some cases, the optical traps described herein may comprise a wavelength of about 20 nm to about 1,000 nm. In some cases, the optical traps described herein may comprise a wavelength of about 100 nm to about 20 nm, about 100 nm to about 300 nm, about 100 nm to about 400 nm, about 100 nm to about 500 nm, about 100 nm to about 600 nm, about 100 nm to about 700 nm, about 100 nm to about 800 nm, about 100 nm to about 900 nm, about 100 nm to about 1,000 nm, about 20 nm to about 300 nm, about 20 nm to about 400 nm, about 20 nm to about 500 nm, about 20 nm to about 600 nm, about 20 nm to about 700 nm, about 20 nm to about 800 nm, about 20 nm to about 900 nm, about 20 nm to about 1,000 nm, about 300 nm to about 400 nm, about 300 nm to about 500 nm, about 300 nm to about 600 nm, about 300 nm to about 700 nm, about 300 nm to about 800 nm, about 300 nm to about 900 nm, about 300 nm to about 1,000 nm, about 400 nm to about 500 nm, about 400 nm to about 600 nm, about 400 nm to about 700 nm, about 400 nm to about 800 nm, about 400 nm to about 900 nm, about 400 nm to about 1,000 nm, about 500 nm to aboutWSGR Docket No. 55436-726.601600 nm, about 500 nm to about 700 nm, about 500 nm to about 800 nm, about 500 nm to about 900 nm, about 500 nm to about 1,000 nm, about 600 nm to about 700 nm, about 600 nm to about 800 nm, about 600 nm to about 900 nm, about 600 nm to about 1,000 nm, about 700 nm to about 800 nm, about 700 nm to about 900 nm, about 700 nm to about 1,000 nm, about 800 nm to about 900 nm, about 800 nm to about 1,000 nm, or about 900 nm to about 1,000 nm. In some cases, the optical traps described herein may comprise a wavelength of about 100 nm, about 20 nm, about 300 nm, about 400 nm, about 500 nm, about 600 nm, about 700 nm, about 800 nm, about 900 nm, or about 1,000 nm. In some cases, the optical traps described herein may comprise a wavelength of at least about 100 nm, about 20 nm, about 300 nm, about 400 nm, about 500 nm, about 600 nm, about 700 nm, about 800 nm, or about 900 nm. In some cases, the optical traps described herein may comprise a wavelength of at most about 20 nm, about 300 nm, about 400 nm, about 500 nm, about 600 nm, about 700 nm, about 800 nm, about 900 nm, or about 1,000 nm.

[0077] In some cases, the merits of optical lattices can be further enhanced through the use of optical buildup cavities. Optical cavities enhance the lattice depth by enabling the constructive interference of many retro-reflections of laser light, in turn enabling the creation of arrays of many deep traps. In some cases, the merits of optical lattices can be further enhanced through the use of optical buildup cavities, such as shown in FIGs.13A and 13B, and described in International Application PCT / US2023 / 022780, which is incorporated herein for all purposes.

[0078] In some cases, the present disclosure provides a method. The method may comprise generating a first array of atoms comprising greater than about 150, about 500, about 1000, etc. atoms and a fill factor of greater than about 50%, about 95%, about 99% occupancy, etc. wherein the first array of atoms is loaded from a second array, and wherein the second array is loaded from a reservoir while a plurality of atoms are in the first array.

[0079] In some cases, the fill factor may be about 50% to about 99%. In some cases, the fill factor may be about 50% to about 55%, about 50% to about 60%, about 50% to about 65%, about 50% to about 70%, about 50% to about 75%, about 50% to about 80%, about 50% to about 85%, about 50% to about 90%, about 50% to about 95%, about 50% to about 98%, about 50% to about 99%, about 55% to about 60%, about 55% to about 65%, about 55% to about 70%, about 55% to about 75%, about 55% to about 80%, about 55% to about 85%, about 55% to about 90%, about 55% to about 95%, about 55% to about 98%, about 55% to about 99%, about 60% to about 65%, about 60% to about 70%, about 60% to about 75%, about 60% to about 80%, about 60% to about 85%, about 60% to about 90%, about 60% to about 95%, about 60% to about 98%, about 60% to about 99%, about 65% to about 70%, about 65% to about 75%, about 65% to about 80%, about 65% to about 85%, about 65% to about 90%, about 65%WSGR Docket No. 55436-726.601to about 95%, about 65% to about 98%, about 65% to about 99%, about 70% to about 75%, about 70% to about 80%, about 70% to about 85%, about 70% to about 90%, about 70% to about 95%, about 70% to about 98%, about 70% to about 99%, about 75% to about 80%, about 75% to about 85%, about 75% to about 90%, about 75% to about 95%, about 75% to about 98%, about 75% to about 99%, about 80% to about 85%, about 80% to about 90%, about 80% to about 95%, about 80% to about 98%, about 80% to about 99%, about 85% to about 90%, about 85% to about 95%, about 85% to about 98%, about 85% to about 99%, about 90% to about 95%, about 90% to about 98%, about 90% to about 99%, about 95% to about 98%, about 95% to about 99%, or about 98% to about 99%. In some cases, the fill factor may be about 50%, about 55%, about 60%, about 65%, about 70%, about 75%, about 80%, about 85%, about 90%, about 95%, about 98%, or about 99%. In some cases, the fill factor may be at least about 50%, about 55%, about 60%, about 65%, about 70%, about 75%, about 80%, about 85%, about 90%, about 95%, or about 98%. In some cases, the fill factor may be at most about 55%, about 60%, about 65%, about 70%, about 75%, about 80%, about 85%, about 90%, about 95%, about 98%, or about 99%.

[0080] In some cases, the first array of atoms may comprise about 100 atoms to about 100,000 atoms. In some cases, the first array of atoms may comprise about 100 atoms to about 250 atoms, about 100 atoms to about 500 atoms, about 100 atoms to about 750 atoms, about 100 atoms to about 1,000 atoms, about 100 atoms to about 2,500 atoms, about 100 atoms to about 5,000 atoms, about 100 atoms to about 7,500 atoms, about 100 atoms to about 10,000 atoms, about 100 atoms to about 25,000 atoms, about 100 atoms to about 50,000 atoms, about 100 atoms to about 100,000 atoms, about 250 atoms to about 500 atoms, about 250 atoms to about 750 atoms, about 250 atoms to about 1,000 atoms, about 250 atoms to about 2,500 atoms, about 250 atoms to about 5,000 atoms, about 250 atoms to about 7,500 atoms, about 250 atoms to about 10,000 atoms, about 250 atoms to about 25,000 atoms, about 250 atoms to about 50,000 atoms, about 250 atoms to about 100,000 atoms, about 500 atoms to about 750 atoms, about 500 atoms to about 1,000 atoms, about 500 atoms to about 2,500 atoms, about 500 atoms to about 5,000 atoms, about 500 atoms to about 7,500 atoms, about 500 atoms to about 10,000 atoms, about 500 atoms to about 25,000 atoms, about 500 atoms to about 50,000 atoms, about 500 atoms to about 100,000 atoms, about 750 atoms to about 1,000 atoms, about 750 atoms to about 2,500 atoms, about 750 atoms to about 5,000 atoms, about 750 atoms to about 7,500 atoms, about 750 atoms to about 10,000 atoms, about 750 atoms to about 25,000 atoms, about 750 atoms to about 50,000 atoms, about 750 atoms to about 100,000 atoms, about 1,000 atoms to about 2,500 atoms, about 1,000 atoms to about 5,000 atoms, about 1,000 atoms to about 7,500 atoms, about 1,000 atoms to about 10,000 atoms, about 1,000 atoms to about 25,000 atoms,WSGR Docket No. 55436-726.601about 1,000 atoms to about 50,000 atoms, about 1,000 atoms to about 100,000 atoms, about 2,500 atoms to about 5,000 atoms, about 2,500 atoms to about 7,500 atoms, about 2,500 atoms to about 10,000 atoms, about 2,500 atoms to about 25,000 atoms, about 2,500 atoms to about 50,000 atoms, about 2,500 atoms to about 100,000 atoms, about 5,000 atoms to about 7,500 atoms, about 5,000 atoms to about 10,000 atoms, about 5,000 atoms to about 25,000 atoms, about 5,000 atoms to about 50,000 atoms, about 5,000 atoms to about 100,000 atoms, about 7,500 atoms to about 10,000 atoms, about 7,500 atoms to about 25,000 atoms, about 7,500 atoms to about 50,000 atoms, about 7,500 atoms to about 100,000 atoms, about 10,000 atoms to about 25,000 atoms, about 10,000 atoms to about 50,000 atoms, about 10,000 atoms to about 100,000 atoms, about 25,000 atoms to about 50,000 atoms, about 25,000 atoms to about 100,000 atoms, or about 50,000 atoms to about 100,000 atoms. In some cases, the first array of atoms may comprise about 100 atoms, about 250 atoms, about 500 atoms, about 750 atoms, about 1,000 atoms, about 2,500 atoms, about 5,000 atoms, about 7,500 atoms, about 10,000 atoms, about 25,000 atoms, about 50,000 atoms, or about 100,000 atoms. In some cases, the first array of atoms may comprise at least about 100 atoms, about 250 atoms, about 500 atoms, about 750 atoms, about 1,000 atoms, about 2,500 atoms, about 5,000 atoms, about 7,500 atoms, about 10,000 atoms, about 25,000 atoms, or about 50,000 atoms. In some cases, the first array of atoms may comprise at most about 250 atoms, about 500 atoms, about 750 atoms, about 1,000 atoms, about 2,500 atoms, about 5,000 atoms, about 7,500 atoms, about 10,000 atoms, about 25,000 atoms, about 50,000 atoms, or about 100,000 atoms.

[0081] In some cases, the first array of atoms may comprise about 100 sites to about 100,000 sites. In some cases, the first array of atoms may comprise about 100 sites to about 250 sites, about 100 sites to about 500 sites, about 100 sites to about 750 sites, about 100 sites to about 1,000 sites, about 100 sites to about 2,500 sites, about 100 sites to about 5,000 sites, about 100 sites to about 7,500 sites, about 100 sites to about 10,000 sites, about 100 sites to about 25,000 sites, about 100 sites to about 50,000 sites, about 100 sites to about 100,000 sites, about 250 sites to about 500 sites, about 250 sites to about 750 sites, about 250 sites to about 1,000 sites, about 250 sites to about 2,500 sites, about 250 sites to about 5,000 sites, about 250 sites to about 7,500 sites, about 250 sites to about 10,000 sites, about 250 sites to about 25,000 sites, about 250 sites to about 50,000 sites, about 250 sites to about 100,000 sites, about 500 sites to about 750 sites, about 500 sites to about 1,000 sites, about 500 sites to about 2,500 sites, about 500 sites to about 5,000 sites, about 500 sites to about 7,500 sites, about 500 sites to about 10,000 sites, about 500 sites to about 25,000 sites, about 500 sites to about 50,000 sites, about 500 sites to about 100,000 sites, about 750 sites to about 1,000 sites, about 750 sites to about 2,500 sites, about 750 sites to about 5,000 sites, about 750 sites to about 7,500 sites, about 750 sites to aboutWSGR Docket No. 55436-726.60110,000 sites, about 750 sites to about 25,000 sites, about 750 sites to about 50,000 sites, about 750 sites to about 100,000 sites, about 1,000 sites to about 2,500 sites, about 1,000 sites to about 5,000 sites, about 1,000 sites to about 7,500 sites, about 1,000 sites to about 10,000 sites, about 1,000 sites to about 25,000 sites, about 1,000 sites to about 50,000 sites, about 1,000 sites to about 100,000 sites, about 2,500 sites to about 5,000 sites, about 2,500 sites to about 7,500 sites, about 2,500 sites to about 10,000 sites, about 2,500 sites to about 25,000 sites, about 2,500 sites to about 50,000 sites, about 2,500 sites to about 100,000 sites, about 5,000 sites to about 7,500 sites, about 5,000 sites to about 10,000 sites, about 5,000 sites to about 25,000 sites, about 5,000 sites to about 50,000 sites, about 5,000 sites to about 100,000 sites, about 7,500 sites to about 10,000 sites, about 7,500 sites to about 25,000 sites, about 7,500 sites to about 50,000 sites, about 7,500 sites to about 100,000 sites, about 10,000 sites to about 25,000 sites, about 10,000 sites to about 50,000 sites, about 10,000 sites to about 100,000 sites, about 25,000 sites to about 50,000 sites, about 25,000 sites to about 100,000 sites, or about 50,000 sites to about 100,000 sites. In some cases, the first array of atoms may comprise about 100 sites, about 250 sites, about 500 sites, about 750 sites, about 1,000 sites, about 2,500 sites, about 5,000 sites, about 7,500 sites, about 10,000 sites, about 25,000 sites, about 50,000 sites, or about 100,000 sites. In some cases, the first array of atoms may comprise at least about 100 sites, about 250 sites, about 500 sites, about 750 sites, about 1,000 sites, about 2,500 sites, about 5,000 sites, about 7,500 sites, about 10,000 sites, about 25,000 sites, or about 50,000 sites. In some cases, the first array of atoms may comprise at most about 250 sites, about 500 sites, about 750 sites, about 1,000 sites, about 2,500 sites, about 5,000 sites, about 7,500 sites, about 10,000 sites, about 25,000 sites, about 50,000 sites, or about 100,000 sites.

[0082] In some cases, the present disclosure provides a method for loading one or more atoms into one or more optical traps, described herein. The method may comprise trapping said one or more atoms from an incident atomic beam into one or more magneto-optical traps (MOTs). Said one or more MOTs may further comprise a core-shell configuration, wherein the core-shell configuration comprises a ring at a first wavelength surrounding a central region with a second wavelength; and transferring atoms from one or more core-shell MOTs to one or more optical lattices, wherein said transferring is performed with one or more optical tweezers.

[0083] In some cases, said first wavelength comprising the ring comprises visible light. In some cases, said first wavelength comprising the ring comprises ultraviolet light. In some cases, said first wavelength comprising the ring comprises infrared light. In some cases, said first wavelength comprising the ring comprises about 100 nm light to about 1,000 nm light. In some cases, said first wavelength comprising the ring comprises about 100 nm light to about 200 nm light, about 100 nm light to about 300 nm light, about 100 nm light to about 400 nm light, aboutWSGR Docket No. 55436-726.601100 nm light to about 500 nm light, about 100 nm light to about 600 nm light, about 100 nm light to about 700 nm light, about 100 nm light to about 800 nm light, about 100 nm light to about 900 nm light, about 100 nm light to about 1,000 nm light, about 200 nm light to about 300 nm light, about 200 nm light to about 400 nm light, about 200 nm light to about 500 nm light, about 200 nm light to about 600 nm light, about 200 nm light to about 700 nm light, about 200 nm light to about 800 nm light, about 200 nm light to about 900 nm light, about 200 nm light to about 1,000 nm light, about 300 nm light to about 400 nm light, about 300 nm light to about 500 nm light, about 300 nm light to about 600 nm light, about 300 nm light to about 700 nm light, about 300 nm light to about 800 nm light, about 300 nm light to about 900 nm light, about 300 nm light to about 1,000 nm light, about 400 nm light to about 500 nm light, about 400 nm light to about 600 nm light, about 400 nm light to about 700 nm light, about 400 nm light to about 800 nm light, about 400 nm light to about 900 nm light, about 400 nm light to about 1,000 nm light, about 500 nm light to about 600 nm light, about 500 nm light to about 700 nm light, about 500 nm light to about 800 nm light, about 500 nm light to about 900 nm light, about 500 nm light to about 1,000 nm light, about 600 nm light to about 700 nm light, about 600 nm light to about 800 nm light, about 600 nm light to about 900 nm light, about 600 nm light to about 1,000 nm light, about 700 nm light to about 800 nm light, about 700 nm light to about 900 nm light, about 700 nm light to about 1,000 nm light, about 800 nm light to about 900 nm light, about 800 nm light to about 1,000 nm light, or about 900 nm light to about 1,000 nm light. In some cases, said first wavelength comprising the ring comprises about 100 nm light, about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, about 900 nm light, or about 1,000 nm light. In some cases, said first wavelength comprising the ring comprises at least about 100 nm light, about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, or about 900 nm light. In some cases, said first wavelength comprising the ring comprises at most about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, about 900 nm light, or about 1,000 nm light.

[0084] In some cases, said second wavelength comprising the central region comprises visible light. In some cases, said second wavelength comprising the central region comprises ultraviolet light. In some cases, said second wavelength comprising the central region comprises infrared light. In some cases, said second wavelength comprising the central region comprises about 100 nm light to about 1,000 nm light. In some cases, said second wavelength comprising the central region comprises about 100 nm light to about 200 nm light, about 100 nm light to about 300 nm light, about 100 nm light to about 400 nm light, about 100 nm light to about 500 nm light, aboutWSGR Docket No. 55436-726.601100 nm light to about 600 nm light, about 100 nm light to about 700 nm light, about 100 nm light to about 800 nm light, about 100 nm light to about 900 nm light, about 100 nm light to about 1,000 nm light, about 200 nm light to about 300 nm light, about 200 nm light to about 400 nm light, about 200 nm light to about 500 nm light, about 200 nm light to about 600 nm light, about 200 nm light to about 700 nm light, about 200 nm light to about 800 nm light, about 200 nm light to about 900 nm light, about 200 nm light to about 1,000 nm light, about 300 nm light to about 400 nm light, about 300 nm light to about 500 nm light, about 300 nm light to about 600 nm light, about 300 nm light to about 700 nm light, about 300 nm light to about 800 nm light, about 300 nm light to about 900 nm light, about 300 nm light to about 1,000 nm light, about 400 nm light to about 500 nm light, about 400 nm light to about 600 nm light, about 400 nm light to about 700 nm light, about 400 nm light to about 800 nm light, about 400 nm light to about 900 nm light, about 400 nm light to about 1,000 nm light, about 500 nm light to about 600 nm light, about 500 nm light to about 700 nm light, about 500 nm light to about 800 nm light, about 500 nm light to about 900 nm light, about 500 nm light to about 1,000 nm light, about 600 nm light to about 700 nm light, about 600 nm light to about 800 nm light, about 600 nm light to about 900 nm light, about 600 nm light to about 1,000 nm light, about 700 nm light to about 800 nm light, about 700 nm light to about 900 nm light, about 700 nm light to about 1,000 nm light, about 800 nm light to about 900 nm light, about 800 nm light to about 1,000 nm light, or about 900 nm light to about 1,000 nm light. In some cases, said second wavelength comprising the central region comprises about 100 nm light, about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, about 900 nm light, or about 1,000 nm light. In some cases, said second wavelength comprising the central region comprises at least about 100 nm light, about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, or about 900 nm light. In some cases, said second wavelength comprising the central region comprises at most about 200 nm light, about 300 nm light, about 400 nm light, about 500 nm light, about 600 nm light, about 700 nm light, about 800 nm light, about 900 nm light, or about 1,000 nm light.

[0085] In some cases, said first wavelength comprising the ring comprises 399 nm light near resonance with the broad linewidth 1S0 to 1P1 transition and said second wavelength comprising the central region comprises 556 nm light near resonance with the narrow linewidth 1S0 to 3P1 transition. In some cases, the ring is configured to capture atoms from the incident atomic beam and wherein the central region is configured to Doppler cool the atoms. In some cases, the atoms are Doppler cooled to several 10 µK.WSGR Docket No. 55436-726.601

[0086] In some cases, the present disclosure provides a non-classical computing system, wherein the non-classical computing system is based on neutral atom qubits, wherein the system is operable to load atoms during a computing circuit, wherein the atoms loaded during the circuit are used in the circuit. Examples of Loading Sequences

[0087] FIG.14 shows a diagram of a repeated loading sequence. One or more first “reservoir” optical tweezer arrays 1410 are repeatedly filled with one or more atoms transported from one or more spatially separated magneto-optical traps (MOTs), and ultimately transferred into one or more second “target” tweezer arrays 1420 using one or more third “rearrangement” tweezer arrays. In some cases, said one or more atoms are171Yb atoms. In some cases, said one or more atoms are a plurality of atoms, as described herein. The rearrangement moves required for the loading sequence are determined from low-loss images obtained by transferring atoms from the one or more first reservoir tweezers and one or more second target tweezers into one or more cavity-enhanced optical lattices and performing site-resolved fluorescence detection. The one or more cavity-enhanced optical lattices allow for the scalable generation of large numbers of deep traps. In some cases, images are shown prior to the 20th cycle of rearrangement 1430, and after the 70th cycle 1440, with the final reservoir reloading step omitted. In another example, this repeated loading sequence allows the loading of over 1200 atoms into 1225 target sites.

[0088] In some cases, methods and systems disclosed herein combine the capabilities of one or more tweezers and one or more cavity-enhanced optical lattices to demonstrate an iterative approach to creating large arrays of individually-controlled atoms. Systems and method disclosed herein may be combined with system and methods for continuous operation using a separate reservoir array, such as in FIG.12, and disclosed in International Application No. PCT / US2023 / 075948, which is incorporated by reference herein for all purposes.

[0089] In some cases, tweezer rearrangement may be performed by stochastically loading up to a single atom into each trap within an array, imaging the atoms to determine trap occupancy, and then rearranging atoms within the array to create a deterministically occupied sub-array. In some cases, the number of atoms contained in the final array with this approach is no greater than the number initially loaded. Further, because the initial loading is stochastic, the number of sites in the array must generally be substantially larger than the desired final sub-array (though under certain conditions, near-deterministic loading may be achieved). In some cases, repeated loading of a “buffer” array from an optical dipole trap “reservoir” demonstrates that one can decouple the filling of a multi-site target array from a single loading of a cold reservoir. Disclosed herein are methods and systems which extend this example to repeated loading of one or more firstWSGR Docket No. 55436-726.601reservoir tweezer arrays, from which one or more deterministically filled second target tweezer arrays may be formed (e.g., with a filling factor, described herein).

[0090] Systems and methods for a repeated loading sequence described herein may be, in some cases, 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 one or more second target tweezer arrays it or in a portion of the one or more second target tweezer arrays. For instance, initial loading of atoms within the computationally active one or more second target tweezer arrays may give rise to a filling factor of less than about: 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 systems and methods described herein, the one or more second target tweezer arrays 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 one or more second target tweezer arrays 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 one or more second target tweezer arrays may attain a filling factor that is within a range defined by any two of the preceding values. Optical Trapping Sites

[0091] In some cases, optical trapping systems described herein may be configured to generate a plurality of optical trapping sites. The optical trapping systems may be configured to generate a plurality of spatially distinct optical trapping sites. Each optical trapping system may comprise any number of sites disclosed herein. Each optical trapping system may comprise any number of trapped atoms disclosed herein.

[0092] For example, each optical trapping system 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. Each optical trapping system may be configured to generate at most about: 1,000,000, 900,000, 800,000, 700,000, 600,000,WSGR Docket No. 55436-726.601500,000, 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 optical trapping sites. The optical trapping system(s) may be configured to trap a number of optical trapping sites that is within a range defined by any two of the preceding values.

[0093] In some cases, each optical trapping system may be configured to trap a plurality of atoms. For example, each optical trapping system may be configured to trap a total number of atoms in the plurality of optical trapping sites of 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. For example, the optical trapping system(s) may be configured to trap a total number of atoms in the plurality of optical trapping sites of 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 system(s) may be configured to trap a number of atoms that is within a range defined by any two of the preceding values.

[0094] In some cases, said deterministically filled target array comprises over 1200171Yb atoms in 1225 sites, with a fill factor of about 99% occupancy. This is made possible at least in part by combining optical tweezer arrays with a cavity-enhanced optical lattice to provide both microscopic control and the large number of deep traps required for rapid, high-fidelity, low-loss imaging of large numbers of atoms.

[0095] In some cases, a single cycle of loading may proceed as follows (see e.g., FIG.14): atoms are collected and cooled in one or more magneto-optical traps (MOTs) at a first distance, d1, below a “science region” where one or more cavity-enhanced optical lattices is created, as well as one or more first reservoir tweezer arrays and one or more second target tweezer arrays, projected through a high-numerical-aperture imaging. The collected and cooled atoms in the one or more MOTs are then transported into the science region, where they are loaded into the set of one or more first tweezers that form the one or more first reservoir tweezer arrays. The one or more first reservoir arrays sit adjacent to the target one or more second target tweezer arrays, which on all loading cycles but the first is already partially occupied with one or more atoms loaded on previous cycles. The atoms from both the one or more first reservoir tweezer arrays and one or more second target tweezer arrays in the science region are then transferred into the trapping potential of the one or more cavity-enhanced optical lattices, where site-resolvedWSGR Docket No. 55436-726.601nondestructive fluorescence imaging is performed to determine which sites of each of the one or more first reservoir tweezer arrays and one or more second target tweezer arrays are occupied. Subsequently, the atoms are transferred back from the cavity lattice to the one or more first reservoir tweezer arrays and one or more second target tweezer arrays, and a separate one or more third “rearrangement” tweezers is used to move atoms from the one or more first reservoir tweezer arrays into empty one or more second target tweezer arrays. This may comprise one loading cycle. Over the course of multiple cycles, the number of atoms in the one or more second target tweezer arrays increases until the one or more second target tweezer arrays is filled. At this point, further operations may be performed on the atoms, and subsequent loading cycles are applied to maintain a filled array.

[0096] In some cases, the one or more first reservoir tweezer arrays are reloaded on each loading cycle from one or more fresh magneto-optical traps (MOTs); thus, it may be possible to continue loading the one or more target tweezer arrays indefinitely. In the near-term, this may allow for relatively high data-rates for quantum simulation and computation with large system sizes and could also be of benefit for optical clocks by providing high statistical precision with low dead-time. The ability to reload new atoms while maintaining both the presence and coherence of existing atoms may be useful for performing error-corrected quantum computations, where execution of an algorithm may take much longer than the lifetime of any given atom in the system. When combined with site-selective hiding and mid-circuit rearrangement techniques, systems and methods disclosed herein may facilitate mid-circuit reloading of atoms into the array. Site selective hiding techniques are disclosed in International Application No. WO2022 / 251435, which is incorporated by reference herein in its entirety. Mid- circuit rearrangement techniques are disclosed at “Mid-Circuit Qubit Measurement and Rearrangement in a 171Yb Atomic Array”, arXiv:2305.19119, https: / / doi.org / 10.48550 / arXiv.2305.19119, which is incorporated by reference herein in its entirety.

[0097] In some embodiments, the distance d1 is about 0.1 cm to about 100 cm. In some cases, the distance d1 is about 0.1 cm to about 1 cm, about 0.1 cm to about 5 cm, about 0.1 cm to about 10 cm, about 0.1 cm to about 15 cm, about 0.1 cm to about 20 cm, about 0.1 cm to about 25 cm, about 0.1 cm to about 30 cm, about 0.1 cm to about 50 cm, about 0.1 cm to about 100 cm, about 1 cm to about 5 cm, about 1 cm to about 10 cm, about 1 cm to about 15 cm, about 1 cm to about 20 cm, about 1 cm to about 25 cm, about 1 cm to about 30 cm, about 1 cm to about 50 cm, about 1 cm to about 100 cm, about 5 cm to about 10 cm, about 5 cm to about 15 cm, about 5 cm to about 20 cm, about 5 cm to about 25 cm, about 5 cm to about 30 cm, about 5 cm to about 50 cm, about 5 cm to about 100 cm, about 10 cm to about 15 cm, about 10 cm to about 20 cm, about 10WSGR Docket No. 55436-726.601cm to about 25 cm, about 10 cm to about 30 cm, about 10 cm to about 50 cm, about 10 cm to about 100 cm, about 15 cm to about 20 cm, about 15 cm to about 25 cm, about 15 cm to about 30 cm, about 15 cm to about 50 cm, about 15 cm to about 100 cm, about 20 cm to about 25 cm, about 20 cm to about 30 cm, about 20 cm to about 50 cm, about 20 cm to about 100 cm, about 25 cm to about 30 cm, about 25 cm to about 50 cm, about 25 cm to about 100 cm, about 30 cm to about 50 cm, about 30 cm to about 100 cm, or about 50 cm to about 100 cm. In some cases, the distance d1 is about 0.1 cm, about 1 cm, about 5 cm, about 10 cm, about 15 cm, about 20 cm, about 25 cm, about 30 cm, about 50 cm, or about 100 cm. In some cases, the distance d1 is at least about 0.1 cm, about 1 cm, about 5 cm, about 10 cm, about 15 cm, about 20 cm, about 25 cm, about 30 cm, or about 50 cm. In some cases, the distance d1 is at most about 1 cm, about 5 cm, about 10 cm, about 15 cm, about 20 cm, about 25 cm, about 30 cm, about 50 cm, or about 100 cm.

[0098] FIG.15A shows an example of a timing diagram 1500A for a continuous loading sequence. The sequence comprises an initialization period, with time scale shown below. The sequence further comprises a series of Repeated Blocks 1510A to fill and maintain the filling of one or more first target tweezer arrays. A Repeated Block may comprise an imaging sequence 1520A, described herein. A Repeated Block may further comprise a Variable-Time Rearrangement sequence 1530A. This Variable-Time Rearrangement sequence allocates a variable time duration for rearrangement of atoms that depends on the number and length of moves to be performed, as well as the time required to compute the moves. The following acronyms are used in FIG.15A: MOT: magneto-optical trap, LAC: light-assisted collisions, OP: optical pumping, and RSC: Raman sideband cooling. Disclosed herein are systems and methods for the trapping of atoms using magneto-optical traps (MOTs). Systems and methods described herein may be integrated with systems and methods for generating a substantially collimated hollow core beam from a light source, such as those described in patent application 18 / 978,501, which is incorporated by reference herein in its entirety.

[0099] FIG.15B shows an example of a number of atoms and fill fraction as a function of loading iteration number during initial loading 1510B and while maintaining a filled array 1520B Black points represent atom number inferred from images preceding the rearrangement step. Grey points 1530B represent atom number inferred from additional images inserted after rearrangement, representing atoms available for computation.

[0100] Systems and methods disclosed herein may provide a continuous loading protocol. Experimental data regarding its performance for loading large arrays is also presented.

[0101] In some cases, a loading cycle begins with collecting atoms from a pre-cooled atomic beam into one or more magneto-optical traps (MOTs) using a “core-shell” configuration. In saidWSGR Docket No. 55436-726.601one or more MOTs, each beam comprises a ring of 399 nm light near resonance with the broad linewidth1S0to1P1transition,1P1MOT Light in timing diagram 1500A, surrounding a central region with only 556 nm light near resonance with the narrow linewidth1S0 to3P1 transition,3P1 MOT Light in timing diagram 1500A. The shell, formed by the ring of1P1 MOT Light in timing diagram 1500A, is primarily responsible for capturing atoms from the incident beam, while the low Doppler temperature of the3P1 MOT Light in timing diagram 1500A used in the core may enable cooling to several 10 µK. Because atoms do not scatter light from the broad transition except during initial capture, in some embodiments, the core-shell design poses less risk of generating scattered photons that may be absorbed by atoms in the target array. In some these embodiments, the total scattering rate on the narrow transition may be much lower, and the large magnetic field present in the science region may dramatically suppresses reabsorption of photons via the narrow-line transition. In an example operating condition, the 399 nm photon scattering rate may be about 2 orders of magnitude lower than the broad-line stage of a sequential MOT.

[0102] From the one or more core-shell MOTs, atoms may be loaded into one or more vertically oriented standing-wave transport lattices formed by counterpropagating 532 nm laser beams focused at the location of the atoms. Loading is achieved by overlapping the one or more vertically oriented standing-wave transport lattices with the center of the one or more MOTs, followed by reducing the detuning and intensity of the 556 nm light in the central region of the one or more MOTs. Subsequently, the light of the one or more MOTs may then be extinguished, and the atoms may be transported vertically by synchronously applying a frequency offset to the upward-going and downward-going lattice beams, and translating their foci. Example techniques for long range transport of atoms are disclosed in International Application PCT / US2023 / 026595, which is incorporated by reference herein in its entirety for all purposes. This brings the atoms the first distance, d1,vertically into the vicinity of the target array. In some embodiments, the atoms are then transferred to said one or more first reservoir tweezer arrays formed by 483 nm tweezers (another magic wavelength for the1S0 to3P1 transition). To avoid disturbing atoms already loaded into the target array, this transfer takes place at a location displaced a second distance, d2, horizontally from said one or more second target tweezer arrays. No dissipation is required to transfer atoms from the vertically oriented standing-wave transport lattice to said one or more first reservoir tweezer arrays. Loading may be achieved by simply increasing the intensity of the 482 nm light forming said one or more first reservoir tweezer arrays while the one or more vertically oriented standing-wave transport lattices are overlapped and then decreasing the intensity of the 532 nm light forming said one or more vertically oriented standing-wave transport lattices. Once loaded, said one or more first reservoirWSGR Docket No. 55436-726.601tweezer arrays may be translated to be directly adjacent to said one or more first target arrays by changing the angle of one or more galvo mirrors of said one or more first reservoir tweezer arrays that is imaged onto the entrance pupil of the objective.

[0103] In some cases, the distance d2 is about 50 μm to about 500 μm. In some cases, the distance d2 is about 50 μm to about 100 μm, about 50 μm to about 120 μm, about 50 μm to about 150 μm, about 50 μm to about 160 μm, about 50 μm to about 170 μm, about 50 μm to about 180 μm, about 50 μm to about 190 μm, about 50 μm to about 200 μm, about 50 μm to about 250 μm, about 50 μm to about 300 μm, about 50 μm to about 500 μm, about 100 μm to about 120 μm, about 100 μm to about 150 μm, about 100 μm to about 160 μm, about 100 μm to about 170 μm, about 100 μm to about 180 μm, about 100 μm to about 190 μm, about 100 μm to about 200 μm, about 100 μm to about 250 μm, about 100 μm to about 300 μm, about 100 μm to about 500 μm, about 120 μm to about 150 μm, about 120 μm to about 160 μm, about 120 μm to about 170 μm, about 120 μm to about 180 μm, about 120 μm to about 190 μm, about 120 μm to about 200 μm, about 120 μm to about 250 μm, about 120 μm to about 300 μm, about 120 μm to about 500 μm, about 150 μm to about 160 μm, about 150 μm to about 170 μm, about 150 μm to about 180 μm, about 150 μm to about 190 μm, about 150 μm to about 200 μm, about 150 μm to about 250 μm, about 150 μm to about 300 μm, about 150 μm to about 500 μm, about 160 μm to about 170 μm, about 160 μm to about 180 μm, about 160 μm to about 190 μm, about 160 μm to about 200 μm, about 160 μm to about 250 μm, about 160 μm to about 300 μm, about 160 μm to about 500 μm, about 170 μm to about 180 μm, about 170 μm to about 190 μm, about 170 μm to about 200 μm, about 170 μm to about 250 μm, about 170 μm to about 300 μm, about 170 μm to about 500 μm, about 180 μm to about 190 μm, about 180 μm to about 200 μm, about 180 μm to about 250 μm, about 180 μm to about 300 μm, about 180 μm to about 500 μm, about 190 μm to about 200 μm, about 190 μm to about 250 μm, about 190 μm to about 300 μm, about 190 μm to about 500 μm, about 200 μm to about 250 μm, about 200 μm to about 300 μm, about 200 μm to about 500 μm, about 250 μm to about 300 μm, about 250 μm to about 500 μm, or about 300 μm to about 500 μm. In some cases, the distance d2 is about 50 μm, about 100 μm, about 120 μm, about 150 μm, about 160 μm, about 170 μm, about 180 μm, about 190 μm, about 200 μm, about 250 μm, about 300 μm, or about 500 μm. In some cases, the distance d2 is at least about 50 μm, about 100 μm, about 120 μm, about 150 μm, about 160 μm, about 170 μm, about 180 μm, about 190 μm, about 200 μm, about 250 μm, or about 300 μm. In some cases, the distance d2 is at most about 100 μm, about 120 μm, about 150 μm, about 160 μm, about 170 μm, about 180 μm, about 190 μm, about 200 μm, about 250 μm, about 300 μm, or about 500 μm.

[0104] In some cases, the one or more second target tweezer arrays comprise a rectangular array of tweezer spots formed with light at 459.5960(5) nm wavelength, a magic wavelength for theWSGR Docket No. 55436-726.6011S0 to3P0 clock transition, and a set of tweezer spots formed by light at 423.31 nm, which is near resonance with a transition between3P1to a higher-lying 6S8S3S1state. Both wavelengths provide large light-shifts to atoms in3P1, which prevents unwanted scattering from atoms loaded in said one or more second target tweezer arrays, and both wavelengths can be used interchangeably to this end. In some cases, systems and methods of the present disclosure provides a 35x35 site target array with 3.3 µm spacing, comprised of two separate rectangular arrays each using one of the two previously mentioned wavelengths. In some cases, systems and methods of the present disclosure have also operated with overlapped full-size arrays to achieve sufficient trap depth and obtained similar results. However, the overlapped condition may be more sensitive to alignments, and so it may be useful to use spatially separated arrays.

[0105] While translating the one or more first reservoir tweezer arrays adjacent to the one or more second target tweezer arrays, in some cases, both arrays may be illuminated with light resonant with the1S0 to3P1 F=3 / 2 mf =3 / 2 transition. This induces light-assisted collisions (LAC) between atoms in the one or more first reservoir tweezer arrays, resulting in either zero or one atom in each reservoir tweezer with about equal probability. In some cases, the 423.31 nm tweezer light illuminating the atoms in the one or more second target tweezer arrays prevents the atoms from scattering the light used to induce collisions, which might otherwise cause loss of existing atoms in said one or more second target tweezer arrays. Next, atoms from both the one or more first reservoir tweezer arrays and one or more second target tweezer arrays are transferred into the one or more vertically oriented standing-wave transport lattices and imaged, as described herein. In some cases, the imaging leaves the atoms at a temperature of about 20 µK, which may be too warm to efficiently transfer into the relatively shallow tweezers of the one or more first reservoir tweezer arrays and one or more second target tweezer arrays. Following each image, Doppler and Raman sideband cooling are applied to reach an average of ~ 0.1 motional quanta in each direction (see Raman Sideband Cooling (RSC) below). Once cooled, the atoms are transferred back into the one or more first reservoir tweezer arrays and one or more second target tweezer arrays, and a separate third one or more rearrangement tweezers derived from the same laser as the reservoir light is used to transfer atoms from filled sites of the one or more first reservoir tweezer arrays into empty sites of the one or more second target tweezer arrays. This comprises an example of a single loading cycle.

[0106] In order to shorten the time required to build up an atomic array, and to limit the losses associated with background gas collisions, systems and methods disclosed herein may, in some cases, load the MOT at the same time as performing operations in the science chamber. In some embodiments, this may be aided by the spatial separation between the one or more MOTs and science regions and by operating with static magnetic fields. In some cases, systems andWSGR Docket No. 55436-726.601methods of the present disclosure can simultaneously operate with the magnetic field gradients required by the one or more MOTs and the large 500 Gauss homogeneous bias field that may be for imaging and other operations required in the science regions. In some cases, the core-shell MOT configuration increases the MOT loading rate by about a factor of two, compared to a more standard sequential MOT configuration in a static magnetic field. The low scattering rate of 399 nm photons, combined with the large magnetic field and spatial separation between the one or more MOTs and science regions prevents light scattered in the one or more MOTs from being reabsorbed by atoms in the science region, which could cause heating or decoherence.

[0107] FIG.15B shows examples of the atom number increasing as a function of load cycles, with 105 sites in a single first reservoir tweezer array and 1225 sites in a single second target tweezer array. At first, the atom number increases about linearly as about 45 atoms are transferred into the second target tweezer array per cycle. The loading rate decreases slightly as the second target tweezer array fills, as some atoms from the first reservoir tweezer array are used to counteract loss. Once the number of vacancies in the second target tweezer array drops below the typical number of atoms in the first reservoir tweezer array, an equilibrium is reached with the vacancy fraction set by the per cycle probability of atom loss.

[0108] In FIG.15B, the atom number is inferred from the image taken before rearrangement, so the measured fill fraction is sensitive to all losses that occur between such images. In some cases, this fill fraction is 98%, with the largest single loss contribution coming from vacuum loss during the typical 300 ms time between images due to an about 30s vacuum lifetime. In some cases, systems and methods of the present disclosure can add additional diagnostic images directly after rearrangement to determine the atom number available for computation, and typical filling fractions of 99% are observed, again dominated by vacuum loss during the 150 ms typically allocated for rearrangement. An experimental control allocates a variable duration for rearrangement that depends on the number and length of moves to be performed, as well as the time required to compute the moves. The durations above may correspond to the example of a 1225-site array presented herein. Other examples of per-cycle loss are provided herein and include the handoff between the tweezer arrays and the lattice, and imaging loss and infidelity.

[0109] In some cases, systems and method disclosed herein provide an iterative method for assembling large arrays of individually controlled neutral atoms, suitable for quantum computation, simulation, and metrology. This approach decouples the final size of the array from the number of atoms that can be loaded at once and relieves the need to create many tweezer traps with sufficient depth for loading and imaging atoms. In particular, the traps of the one or more first reservoir tweezer arrays and one or more vertically oriented standing-wave transport lattices are much deeper than the traps of the one or more second target tweezer arrays,WSGR Docket No. 55436-726.601allowing for the use of large numbers of sites with moderate laser power. In some cases, such arrays have been observed with percent-level defects.

[0110] For quantum computing applications, iterative loading may, in some cases, be combined with mid-circuit measurement techniques in order to achieve continuous mid-circuit refilling of array defects. This capability may be useful for mitigating the effect of atom loss during execution of complex error-corrected circuits that may extend beyond the lifetime of an individual atom in the array. Examples of a Cavity-Enhanced Optical Lattice

[0111] FIG.16A shows an example of a cavity-enhanced optical lattice for rapid low-loss imaging. The diagram 1600A shows an example of the intersection of two cavity modes providing three-dimensional confinement for atoms within the field of view of a high-numerical- aperture imaging system. Confinement in the X and Y directions is provided by a self- intersecting standing-wave cavity 1610A, while confinement in the Z direction is provided by a separate self-intersecting running-wave cavity mode 1620A. Imaging light (not shown for clarity) is incident along the mode of the running-wave lattice.

[0112] FIG.16B shows an example of observed counts during a 7 ms long image of single atoms within an optical lattice, demonstrating well-resolved peaks for occupied and unoccupied sites. In some cases, per-image survival is 99.8(1)%.

[0113] FIG.16C€illustrates lattice homogeneity across the target array as characterized by light shifts on the1S0 mf = ½ to3P1 F= 3 / 2, mf = -½ transition. Atoms within 13.4 µm squares are averaged together for statistics and display clarity.

[0114] FIG.16D illustrate light shift contributions from each cavity averaged over rows. Markers and lines represent the measured and predicted profiles of the XY 1610D and Z 1620D cavities, with the prediction centered on the array.

[0115] FIG.15E illustrate light shift contributions from each cavity averaged over columns. Markers and lines represent the measured and predicted profiles of the XY 1610E and Z 1620E cavities, with the prediction centered on the array.

[0116] In some cases, rapid low-loss imaging of the locations of atoms may be useful to a repeated loading protocol. Because atoms may be heated by the photon scattering required for imaging, deep optical traps and imaging protocols with low equilibrium temperatures may be useful to prevent atom loss. Optical tweezers represent a method for achieving deep optical traps, as the tight foci can provide high intensities for moderate powers. The total power requirement for a large array of tweezers, however, may present scaling limitations due to limited available laser power. Cavity-enhanced optical lattices offer an alternative path toWSGR Docket No. 55436-726.601achieving many deep traps. While the laser intensity in a lattice is spread over a larger cross- sectional area than in optical tweezers both because it spans the regions between utilized sites, and because the Gaussian mode profile must be large compared to the size of the array to achieve uniform trapping, interference of many reflected paths in a cavity-supported lattice can enable more power-efficient generation of traps.

[0117] In some cases, systems and methods disclosed herein may employ two intersecting optical cavities to provide tight confinement in three dimensions, as shown in FIG.16A. Systems and methods disclosed herein may be integrated with the devices for cavity-based computing, such as those disclosed in International Publication WO2023 / 225227, which is incorporated by reference herein in its entirety.

[0118] The one or more first cavities 1610A, called the XY cavity, may, in some cases, be a retro-reflected four-mirror cavity whose mode intersects itself at the location of the atoms. The polarization may be oriented perpendicular to the plane of propagation, providing interference between the two crossed portions of the cavity mode. Compared to using two crossed non- interfering lattices, and assuming equal cavity finesse, this configuration can provide an eight- fold enhancement in trap depth (defined as the potential barrier between adjacent lattice sites) per unit of total laser power. The one or more first XY cavities may provide tight confinement in the directions perpendicular to the high numerical aperture optical axis used for the tweezers and imaging. The one or more second cavities 1620A, called the Z cavity, may be a four-mirror running wave cavity whose mode intersects itself at the location of the atoms at an angle, ɵ, from the XY plane. This may provide confinement along the tweezer axis. The polarization of the one or more second Z cavities may be oriented perpendicular to its intersection plane, enabling complete interference of the crossing modes. In some embodiments, each cavity has two highly reflecting mirrors, and two matched partially transmitting mirrors.

[0119] In some cases, the finesse of the one or more first XY cabinets may be about 1,500 (100) to about 5,000 (100). In some cases, the finesse of the one or more first XY cabinets may be about 1,500 (100) to about 2,000 (100), about 1,500 (100) to about 2,500 (100), about 1,500 (100) to about 2,750 (100), about 1,500 (100) to about 2,800 (100), about 1,500 (100) to about 2,850 (100), about 1,500 (100) to about 2,900 (100), about 1,500 (100) to about 2,950 (100), about 1,500 (100) to about 3,000 (100), about 1,500 (100) to about 3,500 (100), about 1,500 (100) to about 4,000 (100), about 1,500 (100) to about 5,000 (100), about 2,000 (100) to about 2,500 (100), about 2,000 (100) to about 2,750 (100), about 2,000 (100) to about 2,800 (100), about 2,000 (100) to about 2,850 (100), about 2,000 (100) to about 2,900 (100), about 2,000 (100) to about 2,950 (100), about 2,000 (100) to about 3,000 (100), about 2,000 (100) to about 3,500 (100), about 2,000 (100) to about 4,000 (100), about 2,000 (100) to about 5,000 (100),WSGR Docket No. 55436-726.601about 2,500 (100) to about 2,750 (100), about 2,500 (100) to about 2,800 (100), about 2,500 (100) to about 2,850 (100), about 2,500 (100) to about 2,900 (100), about 2,500 (100) to about 2,950 (100), about 2,500 (100) to about 3,000 (100), about 2,500 (100) to about 3,500 (100), about 2,500 (100) to about 4,000 (100), about 2,500 (100) to about 5,000 (100), about 2,750 (100) to about 2,800 (100), about 2,750 (100) to about 2,850 (100), about 2,750 (100) to about 2,900 (100), about 2,750 (100) to about 2,950 (100), about 2,750 (100) to about 3,000 (100), about 2,750 (100) to about 3,500 (100), about 2,750 (100) to about 4,000 (100), about 2,750 (100) to about 5,000 (100), about 2,800 (100) to about 2,850 (100), about 2,800 (100) to about 2,900 (100), about 2,800 (100) to about 2,950 (100), about 2,800 (100) to about 3,000 (100), about 2,800 (100) to about 3,500 (100), about 2,800 (100) to about 4,000 (100), about 2,800 (100) to about 5,000 (100), about 2,850 (100) to about 2,900 (100), about 2,850 (100) to about 2,950 (100), about 2,850 (100) to about 3,000 (100), about 2,850 (100) to about 3,500 (100), about 2,850 (100) to about 4,000 (100), about 2,850 (100) to about 5,000 (100), about 2,900 (100) to about 2,950 (100), about 2,900 (100) to about 3,000 (100), about 2,900 (100) to about 3,500 (100), about 2,900 (100) to about 4,000 (100), about 2,900 (100) to about 5,000 (100), about 2,950 (100) to about 3,000 (100), about 2,950 (100) to about 3,500 (100), about 2,950 (100) to about 4,000 (100), about 2,950 (100) to about 5,000 (100), about 3,000 (100) to about 3,500 (100), about 3,000 (100) to about 4,000 (100), about 3,000 (100) to about 5,000 (100), about 3,500 (100) to about 4,000 (100), about 3,500 (100) to about 5,000 (100), or about 4,000 (100) to about 5,000 (100). In some cases, the finesse of the one or more first XY cabinets may be about 1,500 (100), about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), about 4,000 (100), or about 5,000 (100). In some cases, the finesse of the one or more first XY cabinets may be at least about 1,500 (100), about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), or about 4,000 (100). In some cases, the finesse of the one or more first XY cabinets may be at most about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), about 4,000 (100), or about 5,000 (100).

[0120] In some cases, the finesse of the one or more second Z cabinets may be about 1,500 (100) to about 5,000 (100). In some cases, the finesse of the one or more second Z cabinets may be about 1,500 (100) to about 2,000 (100), about 1,500 (100) to about 2,500 (100), about 1,500 (100) to about 2,750 (100), about 1,500 (100) to about 2,800 (100), about 1,500 (100) to about 2,850 (100), about 1,500 (100) to about 2,900 (100), about 1,500 (100) to about 2,950 (100), about 1,500 (100) to about 3,000 (100), about 1,500 (100) to about 3,500 (100), about 1,500WSGR Docket No. 55436-726.601(100) to about 4,000 (100), about 1,500 (100) to about 5,000 (100), about 2,000 (100) to about 2,500 (100), about 2,000 (100) to about 2,750 (100), about 2,000 (100) to about 2,800 (100), about 2,000 (100) to about 2,850 (100), about 2,000 (100) to about 2,900 (100), about 2,000 (100) to about 2,950 (100), about 2,000 (100) to about 3,000 (100), about 2,000 (100) to about 3,500 (100), about 2,000 (100) to about 4,000 (100), about 2,000 (100) to about 5,000 (100), about 2,500 (100) to about 2,750 (100), about 2,500 (100) to about 2,800 (100), about 2,500 (100) to about 2,850 (100), about 2,500 (100) to about 2,900 (100), about 2,500 (100) to about 2,950 (100), about 2,500 (100) to about 3,000 (100), about 2,500 (100) to about 3,500 (100), about 2,500 (100) to about 4,000 (100), about 2,500 (100) to about 5,000 (100), about 2,750 (100) to about 2,800 (100), about 2,750 (100) to about 2,850 (100), about 2,750 (100) to about 2,900 (100), about 2,750 (100) to about 2,950 (100), about 2,750 (100) to about 3,000 (100), about 2,750 (100) to about 3,500 (100), about 2,750 (100) to about 4,000 (100), about 2,750 (100) to about 5,000 (100), about 2,800 (100) to about 2,850 (100), about 2,800 (100) to about 2,900 (100), about 2,800 (100) to about 2,950 (100), about 2,800 (100) to about 3,000 (100), about 2,800 (100) to about 3,500 (100), about 2,800 (100) to about 4,000 (100), about 2,800 (100) to about 5,000 (100), about 2,850 (100) to about 2,900 (100), about 2,850 (100) to about 2,950 (100), about 2,850 (100) to about 3,000 (100), about 2,850 (100) to about 3,500 (100), about 2,850 (100) to about 4,000 (100), about 2,850 (100) to about 5,000 (100), about 2,900 (100) to about 2,950 (100), about 2,900 (100) to about 3,000 (100), about 2,900 (100) to about 3,500 (100), about 2,900 (100) to about 4,000 (100), about 2,900 (100) to about 5,000 (100), about 2,950 (100) to about 3,000 (100), about 2,950 (100) to about 3,500 (100), about 2,950 (100) to about 4,000 (100), about 2,950 (100) to about 5,000 (100), about 3,000 (100) to about 3,500 (100), about 3,000 (100) to about 4,000 (100), about 3,000 (100) to about 5,000 (100), about 3,500 (100) to about 4,000 (100), about 3,500 (100) to about 5,000 (100), or about 4,000 (100) to about 5,000 (100). In some cases, the finesse of the one or more second Z cabinets may be about 1,500 (100), about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), about 4,000 (100), or about 5,000 (100). In some cases, the finesse of the one or more second Z cabinets may be at least about 1,500 (100), about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), or about 4,000 (100). In some cases, the finesse of the one or more second Z cabinets may be at most about 2,000 (100), about 2,500 (100), about 2,750 (100), about 2,800 (100), about 2,850 (100), about 2,900 (100), about 2,950 (100), about 3,000 (100), about 3,500 (100), about 4,000 (100), or about 5,000 (100).WSGR Docket No. 55436-726.601

[0121] In some cases, the design waists at the location of the atoms are about 100 µm to about 200 µm. In some cases, the design waists at the location of the atoms are about 100 µm to about 150 µm, about 100 µm to about 175 µm, about 100 µm to about 180 µm, about 100 µm to about 181 µm, about 100 µm to about 182 µm, about 100 µm to about 183 µm, about 100 µm to about 184 µm, about 100 µm to about 185 µm, about 100 µm to about 190 µm, about 100 µm to about 195 µm, about 100 µm to about 200 µm, about 150 µm to about 175 µm, about 150 µm to about 180 µm, about 150 µm to about 181 µm, about 150 µm to about 182 µm, about 150 µm to about 183 µm, about 150 µm to about 184 µm, about 150 µm to about 185 µm, about 150 µm to about 190 µm, about 150 µm to about 195 µm, about 150 µm to about 200 µm, about 175 µm to about 180 µm, about 175 µm to about 181 µm, about 175 µm to about 182 µm, about 175 µm to about 183 µm, about 175 µm to about 184 µm, about 175 µm to about 185 µm, about 175 µm to about 190 µm, about 175 µm to about 195 µm, about 175 µm to about 200 µm, about 180 µm to about 181 µm, about 180 µm to about 182 µm, about 180 µm to about 183 µm, about 180 µm to about 184 µm, about 180 µm to about 185 µm, about 180 µm to about 190 µm, about 180 µm to about 195 µm, about 180 µm to about 200 µm, about 181 µm to about 182 µm, about 181 µm to about 183 µm, about 181 µm to about 184 µm, about 181 µm to about 185 µm, about 181 µm to about 190 µm, about 181 µm to about 195 µm, about 181 µm to about 200 µm, about 182 µm to about 183 µm, about 182 µm to about 184 µm, about 182 µm to about 185 µm, about 182 µm to about 190 µm, about 182 µm to about 195 µm, about 182 µm to about 200 µm, about 183 µm to about 184 µm, about 183 µm to about 185 µm, about 183 µm to about 190 µm, about 183 µm to about 195 µm, about 183 µm to about 200 µm, about 184 µm to about 185 µm, about 184 µm to about 190 µm, about 184 µm to about 195 µm, about 184 µm to about 200 µm, about 185 µm to about 190 µm, about 185 µm to about 195 µm, about 185 µm to about 200 µm, about 190 µm to about 195 µm, about 190 µm to about 200 µm, or about 195 µm to about 200 µm. In some cases, the design waists at the location of the atoms are about 100 µm, about 150 µm, about 175 µm, about 180 µm, about 181 µm, about 182 µm, about 183 µm, about 184 µm, about 185 µm, about 190 µm, about 195 µm, or about 200 µm. In some cases, the design waists at the location of the atoms are at least about 100 µm, about 150 µm, about 175 µm, about 180 µm, about 181 µm, about 182 µm, about 183 µm, about 184 µm, about 185 µm, about 190 µm, or about 195 µm. In some cases, the design waists at the location of the atoms are at most about 150 µm, about 175 µm, about 180 µm, about 181 µm, about 182 µm, about 183 µm, about 184 µm, about 185 µm, about 190 µm, about 195 µm, or about 200 µm.

[0122] In some cases, the design waists at the location of the atoms are about 225 µm to about 280 µm. In some cases, the design waists at the location of the atoms are about 225 µm to about 250 µm, about 225 µm to about 260 µm, about 225 µm to about 265 µm, about 225 µm to aboutWSGR Docket No. 55436-726.601266 µm, about 225 µm to about 267 µm, about 225 µm to about 268 µm, about 225 µm to about 269 µm, about 225 µm to about 270 µm, about 225 µm to about 271 µm, about 225 µm to about 275 µm, about 225 µm to about 280 µm, about 250 µm to about 260 µm, about 250 µm to about 265 µm, about 250 µm to about 266 µm, about 250 µm to about 267 µm, about 250 µm to about 268 µm, about 250 µm to about 269 µm, about 250 µm to about 270 µm, about 250 µm to about 271 µm, about 250 µm to about 275 µm, about 250 µm to about 280 µm, about 260 µm to about 265 µm, about 260 µm to about 266 µm, about 260 µm to about 267 µm, about 260 µm to about 268 µm, about 260 µm to about 269 µm, about 260 µm to about 270 µm, about 260 µm to about 271 µm, about 260 µm to about 275 µm, about 260 µm to about 280 µm, about 265 µm to about 266 µm, about 265 µm to about 267 µm, about 265 µm to about 268 µm, about 265 µm to about 269 µm, about 265 µm to about 270 µm, about 265 µm to about 271 µm, about 265 µm to about 275 µm, about 265 µm to about 280 µm, about 266 µm to about 267 µm, about 266 µm to about 268 µm, about 266 µm to about 269 µm, about 266 µm to about 270 µm, about 266 µm to about 271 µm, about 266 µm to about 275 µm, about 266 µm to about 280 µm, about 267 µm to about 268 µm, about 267 µm to about 269 µm, about 267 µm to about 270 µm, about 267 µm to about 271 µm, about 267 µm to about 275 µm, about 267 µm to about 280 µm, about 268 µm to about 269 µm, about 268 µm to about 270 µm, about 268 µm to about 271 µm, about 268 µm to about 275 µm, about 268 µm to about 280 µm, about 269 µm to about 270 µm, about 269 µm to about 271 µm, about 269 µm to about 275 µm, about 269 µm to about 280 µm, about 270 µm to about 271 µm, about 270 µm to about 275 µm, about 270 µm to about 280 µm, about 271 µm to about 275 µm, about 271 µm to about 280 µm, or about 275 µm to about 280 µm. In some cases, the design waists at the location of the atoms are about 225 µm, about 250 µm, about 260 µm, about 265 µm, about 266 µm, about 267 µm, about 268 µm, about 269 µm, about 270 µm, about 271 µm, about 275 µm, or about 280 µm. In some cases, the design waists at the location of the atoms are at least about 225 µm, about 250 µm, about 260 µm, about 265 µm, about 266 µm, about 267 µm, about 268 µm, about 269 µm, about 270 µm, about 271 µm, or about 275 µm. In some cases, the design waists at the location of the atoms are at most about 250 µm, about 260 µm, about 265 µm, about 266 µm, about 267 µm, about 268 µm, about 269 µm, about 270 µm, about 271 µm, about 275 µm, or about 280 µm.

[0123] In some cases, the finesse of the XY and Z cavities are 2900(100) and 3000(100) respectively, and the design mode waists at the location of the atoms are 268 µm and 183 µm, respectively.

[0124] In some cases, the XY trap frequency is about 100 kHz to about 200 kHz. In some cases, the XY trap frequency is about 100 kHz to about 125 kHz, about 100 kHz to about 150 kHz, about 100 kHz to about 155 kHz, about 100 kHz to about 159 kHz, about 100 kHz to about 160WSGR Docket No. 55436-726.601kHz, about 100 kHz to about 161 kHz, about 100 kHz to about 165 kHz, about 100 kHz to about 170 kHz, about 100 kHz to about 180 kHz, about 100 kHz to about 200 kHz, about 125 kHz to about 150 kHz, about 125 kHz to about 155 kHz, about 125 kHz to about 159 kHz, about 125 kHz to about 160 kHz, about 125 kHz to about 161 kHz, about 125 kHz to about 165 kHz, about 125 kHz to about 170 kHz, about 125 kHz to about 180 kHz, about 125 kHz to about 200 kHz, about 150 kHz to about 155 kHz, about 150 kHz to about 159 kHz, about 150 kHz to about 160 kHz, about 150 kHz to about 161 kHz, about 150 kHz to about 165 kHz, about 150 kHz to about 170 kHz, about 150 kHz to about 180 kHz, about 150 kHz to about 200 kHz, about 155 kHz to about 159 kHz, about 155 kHz to about 160 kHz, about 155 kHz to about 161 kHz, about 155 kHz to about 165 kHz, about 155 kHz to about 170 kHz, about 155 kHz to about 180 kHz, about 155 kHz to about 200 kHz, about 159 kHz to about 160 kHz, about 159 kHz to about 161 kHz, about 159 kHz to about 165 kHz, about 159 kHz to about 170 kHz, about 159 kHz to about 180 kHz, about 159 kHz to about 200 kHz, about 160 kHz to about 161 kHz, about 160 kHz to about 165 kHz, about 160 kHz to about 170 kHz, about 160 kHz to about 180 kHz, about 160 kHz to about 200 kHz, about 161 kHz to about 165 kHz, about 161 kHz to about 170 kHz, about 161 kHz to about 180 kHz, about 161 kHz to about 200 kHz, about 165 kHz to about 170 kHz, about 165 kHz to about 180 kHz, about 165 kHz to about 200 kHz, about 170 kHz to about 180 kHz, about 170 kHz to about 200 kHz, or about 180 kHz to about 200 kHz. In some cases, the XY trap frequency is about 100 kHz, about 125 kHz, about 150 kHz, about 155 kHz, about 159 kHz, about 160 kHz, about 161 kHz, about 165 kHz, about 170 kHz, about 180 kHz, or about 200 kHz. In some cases, the XY trap frequency is at least about 100 kHz, about 125 kHz, about 150 kHz, about 155 kHz, about 159 kHz, about 160 kHz, about 161 kHz, about 165 kHz, about 170 kHz, or about 180 kHz. In some cases, the XY trap frequency is at most about 125 kHz, about 150 kHz, about 155 kHz, about 159 kHz, about 160 kHz, about 161 kHz, about 165 kHz, about 170 kHz, about 180 kHz, or about 200 kHz.

[0125] In some cases, the Z trap frequency is about 10 kHz to about 100 kHz. In some cases, the Z trap frequency is about 10 kHz to about 25 kHz, about 10 kHz to about 40 kHz, about 10 kHz to about 45 kHz, about 10 kHz to about 49 kHz, about 10 kHz to about 50 kHz, about 10 kHz to about 51 kHz, about 10 kHz to about 55 kHz, about 10 kHz to about 60 kHz, about 10 kHz to about 75 kHz, about 10 kHz to about 100 kHz, about 25 kHz to about 40 kHz, about 25 kHz to about 45 kHz, about 25 kHz to about 49 kHz, about 25 kHz to about 50 kHz, about 25 kHz to about 51 kHz, about 25 kHz to about 55 kHz, about 25 kHz to about 60 kHz, about 25 kHz to about 75 kHz, about 25 kHz to about 100 kHz, about 40 kHz to about 45 kHz, about 40 kHz to about 49 kHz, about 40 kHz to about 50 kHz, about 40 kHz to about 51 kHz, about 40 kHz to about 55 kHz, about 40 kHz to about 60 kHz, about 40 kHz to about 75 kHz, about 40 kHz toWSGR Docket No. 55436-726.601about 100 kHz, about 45 kHz to about 49 kHz, about 45 kHz to about 50 kHz, about 45 kHz to about 51 kHz, about 45 kHz to about 55 kHz, about 45 kHz to about 60 kHz, about 45 kHz to about 75 kHz, about 45 kHz to about 100 kHz, about 49 kHz to about 50 kHz, about 49 kHz to about 51 kHz, about 49 kHz to about 55 kHz, about 49 kHz to about 60 kHz, about 49 kHz to about 75 kHz, about 49 kHz to about 100 kHz, about 50 kHz to about 51 kHz, about 50 kHz to about 55 kHz, about 50 kHz to about 60 kHz, about 50 kHz to about 75 kHz, about 50 kHz to about 100 kHz, about 51 kHz to about 55 kHz, about 51 kHz to about 60 kHz, about 51 kHz to about 75 kHz, about 51 kHz to about 100 kHz, about 55 kHz to about 60 kHz, about 55 kHz to about 75 kHz, about 55 kHz to about 100 kHz, about 60 kHz to about 75 kHz, about 60 kHz to about 100 kHz, or about 75 kHz to about 100 kHz. In some cases, the Z trap frequency is about 10 kHz, about 25 kHz, about 40 kHz, about 45 kHz, about 49 kHz, about 50 kHz, about 51 kHz, about 55 kHz, about 60 kHz, about 75 kHz, or about 100 kHz. In some cases, the Z trap frequency is at least about 10 kHz, about 25 kHz, about 40 kHz, about 45 kHz, about 49 kHz, about 50 kHz, about 51 kHz, about 55 kHz, about 60 kHz, or about 75 kHz. In some cases, the Z trap frequency is at most about 25 kHz, about 40 kHz, about 45 kHz, about 49 kHz, about 50 kHz, about 51 kHz, about 55 kHz, about 60 kHz, about 75 kHz, or about 100 kHz.

[0126] In some cases, systems and methods disclosed herein may operate with XY trap frequencies near 160 kHz, and Z trap frequencies near 50 kHz, corresponding to 330 µK deep XY traps and 260 µK deep Z traps.

[0127] In some cases, the angle ɵ is about 0.1 ° to about 45 °. In some cases, the angle ɵ is about 0.1 ° to about 1 °, about 0.1 ° to about 5 °, about 0.1 ° to about 10 °, about 0.1 ° to about 15 °, about 0.1 ° to about 20 °, about 0.1 ° to about 25 °, about 0.1 ° to about 30 °, about 0.1 ° to about 35 °, about 0.1 ° to about 40 °, about 0.1 ° to about 44 °, about 0.1 ° to about 45 °, about 1 ° to about 5 °, about 1 ° to about 10 °, about 1 ° to about 15 °, about 1 ° to about 20 °, about 1 ° to about 25 °, about 1 ° to about 30 °, about 1 ° to about 35 °, about 1 ° to about 40 °, about 1 ° to about 44 °, about 1 ° to about 45 °, about 5 ° to about 10 °, about 5 ° to about 15 °, about 5 ° to about 20 °, about 5 ° to about 25 °, about 5 ° to about 30 °, about 5 ° to about 35 °, about 5 ° to about 40 °, about 5 ° to about 44 °, about 5 ° to about 45 °, about 10 ° to about 15 °, about 10 ° to about 20 °, about 10 ° to about 25 °, about 10 ° to about 30 °, about 10 ° to about 35 °, about 10 ° to about 40 °, about 10 ° to about 44 °, about 10 ° to about 45 °, about 15 ° to about 20 °, about 15 ° to about 25 °, about 15 ° to about 30 °, about 15 ° to about 35 °, about 15 ° to about 40 °, about 15 ° to about 44 °, about 15 ° to about 45 °, about 20 ° to about 25 °, about 20 ° to about 30 °, about 20 ° to about 35 °, about 20 ° to about 40 °, about 20 ° to about 44 °, about 20 ° to about 45 °, about 25 ° to about 30 °, about 25 ° to about 35 °, about 25 ° to about 40 °, about 25 ° to about 44 °, about 25 ° to about 45 °, about 30 ° to about 35 °, about 30 ° to about 40 °,WSGR Docket No. 55436-726.601about 30 ° to about 44 °, about 30 ° to about 45 °, about 35 ° to about 40 °, about 35 ° to about 44 °, about 35 ° to about 45 °, about 40 ° to about 44 °, about 40 ° to about 45 °, or about 44 ° to about 45 °. In some cases, the angle ɵ is about 0.1 °, about 1 °, about 5 °, about 10 °, about 15 °, about 20 °, about 25 °, about 30 °, about 35 °, about 40 °, about 44 °, or about 45 °. In some cases, the angle ɵ is at least about 0.1 °, about 1 °, about 5 °, about 10 °, about 15 °, about 20 °, about 25 °, about 30 °, about 35 °, about 40 °, or about 44 °. In some cases, the angle ɵ is at most about 1 °, about 5 °, about 10 °, about 15 °, about 20 °, about 25 °, about 30 °, about 35 °, about 40 °, about 44 °, or about 45 °.

[0128] In some cases, systems and methods disclosed herein may image atoms using the narrow-linewidth (e.g., about 180 kHz)1S0 to3P1 transition, which simultaneously provides cooling and scattering of photons for detection. Systems and methods disclosed herein may use a single beam, detuned several hundred kHz from the F= 3 / 2, mf = -½ state, with projection onto both the X and Z directions. For applications like array assembly where it may be advantageous to determine the presence of an atom but not the state of its nuclear spin (the qubit used for quantum information applications), systems and methods disclosed herein may apply optical pumping on the1S0 mf = -½ to3P1 F= 1 / 2, mf = 1 / 2 transition with light incident along the magnetic field (X direction). In an example of systems and methods of the present disclosure, some embodiments may collect 50 photons in 7 ms, and distinguish occupied from unoccupied sites with a fidelity of about 99.95%, as determined from the overlap error in a double-Gaussian fit to the bimodal distribution of counts. Atom loss during such an image was measured at 2(1) x 10-3. Spin-selective imaging may allow omission of the optical pumping, and use of thecombination of frequency and polarization selectivity within the 3P1 manifold to may create animbalance of scattering between the two nuclear spin states.

[0129] In some cases, the omission of the optical pumping does not modify the photon scattering rate appreciably but may introduce a finite spin flip probability of 4(1) x 10-3for some imaging parameters, measured as an additional apparent loss in repeated images of1S0 mf = ½. In some cases, these spin flips are due to the fact that the lattice polarization is perpendicular to the magnetic field, and so can induce spin-changing transitions within the3P1 manifold.

[0130] In order to improve uniformity of detuning of the imaging transition across the array and between different motional states, systems and methods disclosed herein may, in some cases, operate with a trapping wavelength that has about equal polarizability for the ground and excited states of the imaging transition a so-called magic wavelength, described herein. In some cases, such as the1S0 mf = ½ to3P1 mf =3 / 2 transition, a magic condition is observed near 783.8 nm.

[0131] The homogeneity of the optical lattice potential may be characterized by probing the1S0 mf = ½ to3P1 F= 3 / 2, mf = -1 / 2 transition, for which the excited state has about 40% higherWSGR Docket No. 55436-726.601polarizability than the ground state. A trap depth from the transition light shifts, measured using optical pumping from the1S0mf= ½ state, followed by spin-selective imaging such as in FIG. 16C . By varying the power in each cavity, their independent contributions may be extracted, displayed averaged over rows and columns in FIG.16D and FIG.16E. In some cases, over a 115 µm square array, peak deviations are observed within 20% for the Z lattice and 10% for the XY lattice. Examples of Raman Sideband Cooling

[0132] FIG.17A shows and example of Raman sideband cooling (RSC). Panel (a) illustrates a diagram of relevant directions and polarizations. RSC beams RSC1 and RSC2 are combined to cool the Z direction, while RSC1 and RSC3 are combined to cool the X and Y directions. Optical pumping (beam OP) is applied along the X direction, which is along the applied magnetic field.

[0133] FIG.17B shows and example of a level diagram for RSC cooling. Raman transitions detuned by 35 MHz from3P1 F= 3 / 2, mf = 1 / 2 are driven from1S0 mf = ½ to1S0 mf = -½, while reducing the number of motional quanta along the addressed direction(s) from n to n-1. Optical pumping through3P1 F= 1 / 2, mf = 1 / 2 returns atoms to1S0 mf = ½ to enable further cycles.

[0134] FIG.17Cshows an example of sideband spectra in the XY plane. The sideband imbalance indicates an average number of motional quanta along the differential momentum vector of the RSC1 and RSC3 beams of nxy = 0.08(1).

[0135] FIG.17D shows an example of sideband spectra in the Z direction. The sideband imbalance indicates an average number of motional quanta along the Z direction of nz = 0.12(1).

[0136] FIG.17E shows an example of heating in the lattice, as measured from the sideband imbalance in the XY plane, with a fit to an exponential growth profile, returning a time-constant of 190(40) ms.

[0137] In some cases, transferring atoms from the one or more vertically oriented standing-wave transport lattices into the shallow traps of the one or more second target tweezer arrays with low loss may use atoms that are much colder than the depth of the tweezer traps. By operating with colder atoms, it may be possible use shallower tweezers, and in turn scale to larger array sizes. To accomplish this, Raman sideband cooling (RSC) between the two ground nuclear spin states may be utilized. For the motional Raman transitions, two pairs of beams oriented along the X and Y directions may be used, and along the X and Z directions to provide cooling along all three directions, as illustrated in FIG.17A.The Raman transitions transfer atoms from1S0mf= ½ to1S0 mf = -½ state, and may be detuned from the motional carrier transition by the appropriate trap oscillation frequency in order to reduce the motional state by one quanta, asWSGR Docket No. 55436-726.601illustrated in FIG.17B. Optical pumping may be provided by a beam oriented along X and the magnetic field that addresses the1S0mf= -½ to1P1F= 1 / 2, mf= 1 / 2 transition.

[0138] In some cases, systems and methods of the present disclosure perform about 20 iterations of cooling, with each iteration consisting of a π pulse on the red motional sideband for each pair, followed by optical pumping. The pulse durations are 200 microseconds for the XY pair and 100 microseconds for the XZ pair (e.g., tuned to the Z direction sideband), and 50 microseconds for optical pumping. The total cooling sequence lasts 8 ms.

[0139] In some cases, an about 2 ms Doppler cooling pulse is applied prior to the RSC sequence using the same beam as for the imaging pulse, but with lower intensity and farther red-detuning. The Doppler cooling may result in about a 10 µK temperature in the cavity as measured by a release and recapture protocol. This about corresponds to 1 motional quanta occupancy in the cavity. Subsequent RSC reduces the average motional quanta to in each direction to ~0.1, as evident from the sideband imbalance following cooling as illustrated in FIG.17E.

[0140] In some cases, systems and methods of the present disclosure measuring heating in the XY plane while holding atoms in the one or more vertically oriented standing-wave transport lattices by performing Raman sideband spectroscopy to extract the average motional quantum number nxy as a function of time. In some cases, systems and methods of the present disclosure fit this quantity to a function representing exponential growth, as expected for parametric heating due to intensity fluctuations of the lattice and extract an exponential time-constant of 190(40) ms. In some embodiments, this may be due to conversion of laser frequency noise to amplitude by the cavity resonance and may be improved by optimizing the lock of lasers to the cavity. Examples of Per-Cycle Loss

[0141] FIG.18A shows an example characterization of per-cycle show an example of loss per pair of handoffs between a sample ~459 nm tweezer array and a sample transport lattice, measured by performing about 25 handoffs with cooling in between. The black line represents the estimated contribution from vacuum loss, with the red band representing day-to-day drifts in this value. Tweezer depths are normalized to their default values, which correspond to about 50 µK.

[0142] FIG.18B shows an example characterization of per-cycle show an example of loss per pair of handoffs between the a sample 423 nm tweezer array and a sample transport lattice, measured by performing 25 handoffs with cooling in between. The black line represents the estimated contribution from vacuum loss, with the red band representing day-to-day drifts in this value. Tweezer depths are normalized to their default values, which correspond to about 50 µK.WSGR Docket No. 55436-726.601

[0143] FIG.18C shows an example of a final vacancy fraction versus rearrangement tweezer power, normalized to the default tweezer depth of about 150 µK. For low powers, atoms may not be efficiently transferred from the one or more first reservoir tweezer arrays to the one or more second target tweezer arrays. For too high of powers, increased loss of loaded atoms is observed due to the one or more third rearrangement tweezer arrays passing nearby.

[0144] In some cases, vacuum loss due to the finite duration of loading cycles may be largest source of loss in the system, contributing percent-level per-cycle loss. Because the loading cycle can be of variable length, and because the vacuum level in the system can fluctuate from day to day, it may be difficult to estimate its exact contribution. However, in some examples typical values for the loading cycle duration and vacuum lifetime are observed to be 300 ms and 30 s respectively, so 1% represents a typical value for this loss. Disclosed herein are examples of other loss sources that contribute to a lesser degree, especially if parameters are not carefully optimized. In general, the loss mechanisms described herein may both limit the size of array that can be loaded (as the reservoir must be large enough to replace lost atoms) and limit the final fill fraction of the array.

[0145] As described herein, in some cases, imaging loss may be observed to be about 2(1) x 10-3where about 30% is the vacuum loss and the rest is the Raman scattering out of3P1 due to the trap light, and the discrimination infidelity is observed typically at or below the 10-3level. If an atom within one second target tweezer array is lost during the image after scattering enough photons to be identified as present, the defect may not be filled during the subsequent rearrangement step, leading to a defect in said second target tweezer array. If a site within one second target tweezer array is incorrectly identified as empty, an atom may be added to that site from one or more first reservoir tweezer arrays, and subsequently induce lossy collisions with the original occupant. Empty sites within one or more second target tweezer arrays or one or more first reservoir tweezer arrays that are mistakenly identified as full will lead to a previous defect not being repaired. However, because such defects may be rare, this mechanism may be less problematic. In some cases, systems and methods of the present disclosure operate with a discrimination threshold that balances the correct identification of empty and full sites. In a system where imaging losses and infidelity become dominant loss sources (a system with better vacuum), it may be advantageous to bias the threshold to minimize the more problematic forms of imaging error.

[0146] In some cases, each image requires handing off atoms from the one or more first reservoir tweezer arrays and one or more second target tweezer arrays into the one or more vertically oriented standing-wave transport lattices and back. In some cases, this process is sensitive to the alignment of the two or more arrays on the scale of a single lattice site, to theWSGR Docket No. 55436-726.601depth of the traps in the tweezer arrays, and to the temperature of the atoms. Atom temperature and alignment are described elsewhere herein. In some cases, systems and methods of the present disclosure isolate the effect of trap depth on handoff for cold atoms and well-aligned arrays by performing 25 subsequent handoff pairs between sample tweezers and a sample transport lattice. In some cases, systems and methods of the present disclosure perform Raman sideband cooling (RSC) each time the atoms are in the lattice between handoffs. For 459 nm sample tweezer depths below 75% of our typical operating conditions, significant loss may be observed. Above this power level, a constant loss rate of 0.0006(1) is observed, which is consistent with data taken with the handoff omitted, and with observed vacuum losses. The 423 nm sample tweezers show a similar behavior, though with a slightly higher loss rate of 0.0015(2) at high powers, which may be due at least in part to worse alignment or optical aberrations. If the sample tweezers are aligned to the trough of the XY cavity intensity, the loss can be as high as 0.02%. This fact explain at least in part the fluctuation of final filling fraction over long period of time as the sub-wavelength scale XY cavity to tweezer alignment fluctuates. It was observed the use of RSC does not necessarily improve the final filling fraction in an ideal operating condition but over longer cycles of loading it can help sustain the filling fraction, compared to the Doppler cooling. Apparently higher filling fraction can be observed for the RSC at lower tweezer depth.

[0147] In some cases, moving new atoms in the one or more second target tweezer arrays may cause loss of existing atoms in said one or more target tweezer arrays, as the one or more third rearrangement tweezers move near the occupied target array sites. Systems and methods of the present disclosure may address this possibility by measuring the final fill fraction of a sample target tweezer array versus the depth of a sample rearrangement tweezer. In one example, for a sample rearrangement tweezer that is too shallow, a lower loading rate is observed and so a lower final fill fraction of the sample target tweezer array. In another example, for too deep a sample rearrangement tweezer, a reduction in the final fill fraction of the sample target tweezer array is observed, as the rearrangement leads to loss of existing atoms. In yet another example, near typical operating conditions, a region with weak dependency of the fill fraction of the sample target tweezer array on the sample rearrangement tweezer power is observed. Alignment of Arrays

[0148] FIG.19 shows an example of the alignment of a sample first reservoir tweezer array and a second sample target tweezer array with a sample cavity lattice performed by monitoring atomic survival after repeated handoffs, while scanning the position offset of the first sample reservoir tweezer array and second sample target tweezer array. For the representative alignmentWSGR Docket No. 55436-726.601scan of the sample 459 nm tweezer array in FIG.19, 25 handoff pairs between the sample 459 nm tweezer array and sample cavity lattice were performed with no cooling in between. Data is averaged over the full utilized field of view. The presence of a visible fringe in the average image may indicate that the sites across the array can be aligned simultaneously.

[0149] In some cases, precise alignment of the different optical potentials may be useful for the performance of a repeated loading protocol. The alignment of the one or more target tweezer arrays to the one or more vertically oriented standing-wave transport lattices may be sensitive, as misalignment here may lead to increased handoff loss that may limit both the largest one or more target tweezer arrays that can be loaded, and the final fill fraction of the one or more target tweezer arrays.

[0150] In an example, alignment of the one or more first reservoir tweezer arrays and one or more second target tweezer arrays may comprise matching the spacing and tilt of the one or more vertically oriented standing-wave transport lattices, which may not drift appreciably over time. At a higher frequency, alignment of the one or more first reservoir tweezer arrays and one or more second target tweezer arrays with the one or more vertically oriented standing-wave transport lattices may additionally comprise aligning the X and Y offsets. Correct spacings and tilts may be obtained by populating the one or more vertically oriented standing-wave transport lattices with atoms transferred from an expanded but sparse version of a reservoir tweezer array described herein. In some cases, the atoms in the one or more vertically oriented standing-wave transport lattices may be imaged to determine the lattice grid, and then may be imaged in said expanded but sparse version of a reservoir tweezer array described herein to determine corrections to said expanded but sparse version of a reservoir tweezer array described herein. In another example, camera whose imaging system is corrected for chromatic shifts between different tweezer wavelengths may be used to register a sample target array tweezers to a sample reservoir array tweezers.

[0151] An example of translation alignment may be iteratively improved by performing repeated handoffs between a set of sample tweezers and a sample transport lattice and scanning the position offset of the sample tweezer arrays. In an example shown in FIG.19, a periodic modulation in the atomic survival is observed, which may be fit to determine improvements to the alignment. This process may be repeated to improve alignment of the sample tweezer arrays and sample transport lattice. Examples of Methods

[0152] FIG.20A shows an example method 2000A. In some cases, this method may comprise: generating an array of atoms comprising greater than 150 atoms and a fill factor of greater thanWSGR Docket No. 55436-726.60195% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers (block 2005A).

[0153] At block 2005A, in some cases, the fill factor is greater than 98% occupancy. In some cases, the array of atoms comprises greater than 500 atoms. In some cases, the array of atoms comprises greater than 1,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some cases, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some cases, the array of atoms comprises greater than 500 sites. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites.

[0154] FIG.20B shows an example method 2000B. In some cases, this method may comprise: generating an array of atoms comprising greater than 150 sites and a fill factor of greater than 95% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers (block 2005B).

[0155] At block 2005B, in some cases, the fill factor is greater than 98% occupancy. In some cases, the array of atoms comprises greater than 500 sites. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites. In some cases, each site comprises an atom. In some embodiments, each site comprises no more than two atoms. In some cases, the array of atoms comprises greater than 500 atoms. In some cases, the array of atoms comprises greater than 1,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some embodiments, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0156] FIG.20C shows an example method 2000C. In some cases, this method may comprise: generating an array of atoms comprising greater than 500 sites and a fill factor of greater than 50% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers (block 2005C).

[0157] At block 2005C, in some cases, the fill factor is greater than 60% occupancy. In some cases, the fill factor is greater than 90% occupancy. In some cases, the fill factor is greater than 95% occupancy. In some cases, the fill factor is greater than 98% occupancy. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites. In some cases, each site comprises an atom. In some cases, each site comprises no more than two atoms. In some cases, the array of atoms comprises greater than 500 atoms. In some cases, the array of atoms comprises greater than 1,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some cases, the array of atoms comprises a plurality of spatially distinct optical trapping sites. r comprise:WSGR Docket No. 55436-726.601

[0158] FIG.20D shows an example method 2000D. In some cases, this method may comprise: filling an array of atoms comprising greater than 1000 atoms a plurality loading cycles, wherein a fill fraction of said array of atoms is greater than 0.95 (block 2005D).

[0159] At block 2005D, in some cases, the fill fraction is greater than 0.98. In some cases, the array of atoms comprises greater than 1,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some cases, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites. In some cases, each site comprises an atom. In some cases, each site comprises no more than two atoms. In some cases, the plurality of loading cycles comprises greater than about 5 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 10 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 50 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 100 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 500 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 1,000 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 5,000 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 10,000 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 50,000 loading cycles. In some cases, the plurality of loading cycles comprises greater than about 100,000 loading cycles.

[0160] FIG.20E shows an example method 2000E. In some cases, this method may comprise: filling an array of atoms comprising greater than 1000 atoms, wherein a fill fraction of said array of atoms is greater than 0.95 (block 2005E); and maintaining said fill fraction over a plurality of loading cycles (block 2010E).

[0161] At block 2005E, in some cases, the fill fraction is greater than 0.98. In some cases, the array of atoms comprises greater than 5,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some cases, the array of atoms comprises a plurality of spatially distinct optical trapping sites. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites. In some cases, each site comprises an atom. In some cases, each site comprises no more than two atoms.

[0162] At block 2010E, in some cases, the plurality of loading cycles comprises greater than 5 loading cycles. In some cases, the plurality of loading cycles comprises greater than 500 loading cycles. In some cases, the plurality of loading cycles comprises greater than 5,000 loading cycles.WSGR Docket No. 55436-726.601

[0163] FIG.20F shows an example method 2000F. In some cases, this method may comprise: generating a first array of atoms comprising greater than 150 atoms and a fill factor of greater than 95%, wherein the first array of atoms is loaded from at least a second array, and wherein the second array is loaded from a reservoir while a plurality of atoms are in the first array (block 2005F).

[0164] At block 2005F, in some cases, the fill factor is greater than 98% occupancy. In some cases, the array of atoms comprises greater than 500 sites. In some cases, the array of atoms comprises greater than 1,000 sites. In some cases, the array of atoms comprises greater than 10,000 sites. In some cases, each site comprises an atom. In some cases, each site comprises no more than two atoms. In some cases, the array of atoms comprises greater than 500 atoms. In some cases, the array of atoms comprises greater than 1,000 atoms. In some cases, the array of atoms comprises greater than 10,000 atoms. In some cases, the array of atoms comprises a plurality of spatially distinct optical trapping sites.

[0165] In some cases, the method 2000F may further comprise: maintaining said fill factor over a plurality of loading cycles. In some cases, the plurality of loading cycles comprises greater than 5 loading cycles. In some cases, the plurality of loading cycles comprises greater than 500 loading cycles. In some cases, the plurality of loading cycles comprises greater than 5,000 loading cycles.

[0166] FIG.20G shows an example method 2000G. In some cases, this method may comprise: trapping atoms from an incident atomic beam into a magneto-optical trap (MOT) using a core- shell configuration, wherein the core-shell configuration comprises a shell at a first wavelength surround a core with a second wavelength (block 2005F); and transferring atoms from said MOT to an optical lattice, wherein said transferring is performed with optical tweezers (block 2010B).

[0167] At block 2005G, in some cases, a shell of the MOT comprises a first light near resonance with the 1S0 to 1P1 transition and a core of the MOT comprises a second light near resonance with the 1S0 to 3P1 transition. In some cases, the shell comprises 399 nm light near resonance with the 1S0 to 1P1 transition and the core comprises 556 nm light near resonance with the 1S0 to 3P1 transition. In some cases, a shell of the MOT is configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some cases, the atoms are Doppler cooled to about 10 µK. In some cases, a shell of the MOT comprises a first light near resonance with the 1S0 to 1P1 transition and a core of the MOT comprises a second light near resonance with the 1S0 to 3P1 transition. In some cases, the shell comprises 399 nm light near resonance with the 1S0 to 1P1 transition and the core comprises 556 nm light near resonance with the 1S0 to 3P1 transition. In some cases, a shell of the MOT isWSGR Docket No. 55436-726.601configured to capture atoms from the incident atomic beam and wherein a core of the MOT is configured to Doppler cool the atoms. In some cases, the atoms are Doppler cooled to about 10 µK. In some cases, the first site and the second site are in a reservoir array. In some cases, the first site and the second site are in a target array. In some cases, the first site is in a reservoir array and the second site is in a target array.

[0168] In some cases, the method 2000G may further comprise: generating a cavity-enhanced optical lattice, wherein the cavity enhanced optical lattice comprises a 3-D optical lattice. In some cases, the cavity enhanced optical lattice is formed from a standing wave optical lattice and a running wave optical lattice. In some cases, the method 2000G may further comprise: generating a core-shell magnetooptical trap (MOT). In some cases, the core-shell MOT comprises a ring at a first wavelength surround a central region with a second wavelength. In some cases, the method 2000G may further comprise: Doppler cooling one or more atoms in an array of atoms. In some cases, the method 2000G may further comprise: Raman sideband cooling one or more atoms in an array of atoms. In some cases, the method 2000G may further comprise: implementing a tweezer operation, wherein the tweezer operation comprises moving one or more atoms within an array of atom from a first site to a second site. In some cases, the method 2000G may further comprise: imaging one or more atoms in an array of atoms. In some cases, the method 2000G may further comprise: implementing a transport operation, wherein the transport operation is operable to move one or more atoms from a MOT to an array of atoms.

[0169] In some cases, one or more operations of any one of the methods 2000A-2000G of FIGs. 20A-20G may be performed in any order. Further, at least one of the one or more operations disclosed above with respect to any one of the methods 2000A-2000G of FIGs.20A-20G may be repeated, e.g., iteratively. Examples of Systems for Performing a Non-Classical Computation

[0170] 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.

[0171] 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,WSGR Docket No. 55436-726.601800, 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, 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 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.

[0172] 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.

[0173] 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.

[0174] 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 (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, 200 µs, 300 µs, 400 µs, 500 µs, 600 µs, 700 µs, 800 µs, 900 µs, 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, 600WSGR Docket No. 55436-726.601ms, 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 µs, 800 µs, 700 µs, 600 µs, 500 µs, 400 µs, 300 µs, 200 µs, 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, or less. Two or more atoms may be quantum mechanically 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.

[0175] 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,WSGR Docket No. 55436-726.601ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms.

[0176] 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. The 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%,WSGR Docket No. 55436-726.60199.95%, 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-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 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,WSGR Docket No. 55436-726.601gadolinium-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 that is within a range defined by any two of the preceding values.

[0177] 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.

[0178] 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.

[0179] 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 a3P1 or3P2 manifold. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3P1 or3P2 manifold of any atom described herein, such as a strontium-873P1 manifold or a strontium-873P2 manifold.

[0180] 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 and / 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 areWSGR Docket No. 55436-726.601hyperfine 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.

[0181] In some cases, the hyperfine states comprise nuclear spin states of a strontium-871S0 manifold and the qubit transition drives one or both of two nuclear spin states of strontium-871S0 to a state detuned from or within the3P2 or3P1 manifold. In some cases, the one-qubit transition is a two photon Raman transition between nuclear spin states of strontium-871S0 via a state detuned from or within the3P2 or3P1 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.

[0182] In some cases, the hyperfine states comprise nuclear spin states of ytterbium

[0183] 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.

[0184] 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.WSGR Docket No. 55436-726.601

[0185] 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 a 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).

[0186] 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 the3P1 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.

[0187] Qubits based on nuclear spin states in the electronic ground state may allow exploitation of long-lived metastable excited electronic states (such as a3P0state 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 theWSGR Docket No. 55436-726.601SLMs or AODs described herein. A shelving transition may comprise a transition between the1S0state in strontium-87 to the3P0or3P2state in strontium-87.

[0188] 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 clock 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.

[0189] 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 and / or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc.

[0190] 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.WSGR Docket No. 55436-726.601

[0191] 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 the1S0 state in strontium-87 to the1P1 state in strontium-87. The1P1 state in strontium-87 may fluoresce. The lower state of the qubit transition may comprise two nuclear spin states in the1S0 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.,3P0 state in strontium-87). In a second excitation, the imaging transition may be 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.

[0192] 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.

[0193] 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.

[0194] 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.WSGR Docket No. 55436-726.601

[0195] The vacuum units may be configured to maintain the system 200 at a pressure of at most about 10-6Pa, 9 x 10-7Pa, 8 x 10-7Pa, 7 x 10-7Pa, 6 x 10-7Pa, 5 x 10-7Pa, 4 x 10-7Pa, 3 x 10-7Pa, 2 x 10-7Pa, 10-7Pa, 9 x 10-8Pa, 8 x 10-8Pa, 7 x 10-8Pa, 6 x 10-8Pa, 5 x 10-8Pa, 4 x 10-8Pa, 3 x 10-8Pa, 2 x 10-8Pa, 10-8Pa, 9 x 10-9Pa, 8 x 10-9Pa, 7 x 10-9Pa, 6 x 10-9Pa, 5 x 10-9Pa, 4 x 10-9Pa, 3 x 10-9Pa, 2 x 10-9Pa, 10-9Pa, 9 x 10-10Pa, 8 x 10-10Pa, 7 x 10-10Pa, 6 x 10-10Pa, 5 x 10-10Pa, 4 x 10-10Pa, 3 x 10-10Pa, 2 x 10-10Pa, 10-10Pa, 9 x 10-11Pa, 8 x 10-11Pa, 7 x 10-11Pa, 6 x10-11Pa, 5 x 10-11Pa, 4 x 10-11Pa, 3 x 10-11Pa, 2 x 10-11Pa, 10-11Pa, 9 x 10-12Pa, 8 x 10-12Pa, 7 x 10-12Pa, 6 x 10-12Pa, 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 10-12Pa, 2 x 10-12Pa, 3 x 10-12Pa, 4 x 10-12Pa, 5 x 10-12Pa, 6 x 10-12Pa, 7 x 10-12Pa, 8 x 10-12Pa, 9 x 10-12Pa, 10-11Pa, 2 x 10-11Pa, 3 x 10-11Pa, 4 x 10-11Pa, 5 x 10-11Pa, 6 x 10-11Pa, 7 x 10-11Pa, 8 x 10-11Pa, 9 x 10-11Pa, 10-10Pa, 2 x 10-10Pa, 3 x 10-10Pa, 4 x 10-10Pa, 5 x 10-10Pa, 6 x 10-10Pa, 7 x 10-10Pa, 8 x 10-10Pa, 9 x 10-10Pa, 10-9Pa, 2 x 10-9Pa, 3 x 10-9Pa, 4 x 10-9Pa, 5 x 10-9Pa, 6 x 10-9Pa, 7 x 10-9Pa, 8 x 10-9Pa, 9 x 10-9Pa, 10-8Pa, 2 x 10-8Pa, 3 x 10-8Pa, 4 x 10-8Pa, 5 x 10-8Pa, 6 x 10-8Pa, 7 x 10-8Pa, 8 x 10-8Pa, 9 x 10-8Pa, 10-7Pa, 2 x 10-7Pa, 3 x 10-7Pa, 4 x 10-7Pa, 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.

[0196] 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.

[0197] 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.

[0198] 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. TheWSGR Docket No. 55436-726.601interior array may comprise a rectangular, square, rectangular prism, or cubic array of optical trapping sites.

[0199] 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).

[0200] 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 a 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.

[0201] 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 multi-qubit 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.

[0202] 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 RydbergWSGR Docket No. 55436-726.601atoms 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 (µm), 2 µm, 3 µm, 4 µm, 5 µm, 6 µm, 7 µm, 8 µm, 9 µm, 10 µm, or more from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance of at most about 10 µm, 9 µm, 8 µm, 7 µm, 6 µm, 5 µm, 4 µm, 3 µm, 2 µm, 1 µm, 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- 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.

[0203] 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.

[0204] 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-photonWSGR Docket No. 55436-726.601transition 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., 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.

[0205] 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.

[0206] 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 a1S0 ground state) to a Rydberg state (such as an n3S1 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 fieldsWSGR Docket No. 55436-726.601than 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.

[0207] 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).

[0208] 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.

[0209] 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.

[0210] 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 (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, or more. The pulse sequences may comprise a duration of at most about 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, 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,WSGR Docket No. 55436-726.60110 ns, or less. The pulse sequences may comprise a duration that is within a range defined by any two of the preceding values.

[0211] 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, 0.999993, 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.

[0212] 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 Mölmer-Sörenson 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.

[0213] 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 usingWSGR Docket No. 55436-726.601parametrization (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. Example of Cloud Computing

[0214] 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 herein (such as a network described herein with respect to FIG.1). The network may comprise a cloud computing network. Example of Optical Trapping Units

[0215] 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 211d, fifth optical trapping site 211e, sixth optical trapping site 211f, seventh optical trapping site 211g, eighth optical trapping site 211h, and ninth optical trapping site 211i, 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).

[0216] 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 (1D) array or a three-dimensional (3D) array.

[0217] 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.WSGR Docket No. 55436-726.601

[0218] 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 µm, 2 µm, 3 µm, 4 µm, 5 µm, 6 µm, 7 µm, 8 µm, 9 µm, 10 µm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 µm, 9 µm, 8 µm, 7 µm, 6 µm, 5 µm, 4 µm, 3 µm, 2 µm, 1 µm, 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.

[0219] 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 field 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 (1D) 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.

[0220] 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 1D 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.

[0221] 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 aWSGR Docket No. 55436-726.601resolution limit of the lasers. For example, the lasers can be configured to provide diffraction limited spot sizes for optical trapping.

[0222] 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 (Kr2) 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.

[0223] The lasers may comprise one or more metal-vapor lasers, such as one or more helium- cadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg) metal-vapor lasers, helium- selenium (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 (MnCl2) metal-vapor lasers.

[0224] 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 (ND:YVO4) 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 (Er:glass) lasers, erbium-ytterbium-codoped glass (Er / Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers.

[0225] 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 (AlGaInP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indiumWSGR Docket No. 55436-726.601gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.

[0226] 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, 2 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.

[0227] 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.WSGR Docket No. 55436-726.601

[0228] 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 (µJ), 2 µJ, 3 µJ, 4 µJ, 5 µJ, 6 µJ, 7 µJ, 8 µJ, 9 µJ, 10 µJ, 20 µJ, 30 µJ, 40 µJ, 50 µJ, 60 µJ, 70 µJ, 80 µJ, 90 µJ, 100 µJ, 200 µJ, 300 µJ, 400 µJ, 500 µJ, 600 µJ, 700 µJ, 800 µJ, 900 µJ, 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 µJ, 800 µJ, 700 µJ, 600 µJ, 500 µJ, 400 µJ, 300 µJ, 200 µJ, 100 µJ, 90 µJ, 80 µJ, 70 µJ, 60 µJ, 50 µJ, 40 µJ, 30 µJ, 20 µJ, 10 µJ, 9 µJ, 8 µJ, 7 µJ, 6 µJ, 5 µJ, 4 µJ, 3 µJ, 2 µJ, 1 µJ, 900 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.

[0229] The lasers may emit light having an average power of at least about 1 microwatt (µW), 2 µW, 3 µW, 4 µW, 5 µW, 6 µW, 7 µW, 8 µW, 9 µW, 10 µW, 20 µW, 30 µW, 40 µW, 50 µW, 60 µW, 70 µW, 80 µW, 90 µW, 100 µW, 200 µW, 300 µW, 400 µW, 500 µW, 600 µW, 700 µW, 800 µW, 900 µW, 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 µW, 800 µW, 700 µW, 600 µW, 500 µW, 400 µW, 300 µW, 200 µW, 100 µW, 90 µW, 80 µW, 70 µW, 60 µW, 50 µW, 40 µW, 30 µW, 20 µW, 10 µW, 9 µW, 8 µW, 7 µW, 6 µW, 5 µW, 4 µW, 3 µW, 2 µW, 1 µW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values.

[0230] 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 lightWSGR Docket No. 55436-726.601comprising 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, 1,370 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.

[0231] 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 x 10-13nm, 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 x 10-12nm, 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 x 10-11nm, 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 x 10-11nm, 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 xWSGR Docket No. 55436-726.60110-10nm, 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 about1 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 x 10-7nm, 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 10-10nm, 8 x 10-10nm, 7 x 10-10nm, 6 x 10-10nm, 5 x 10-10nm, 4 x 10-10nm, 3 x 10-10nm, 2 x 10-10nm, 1 x 10-10nm, 9 x 10-11nm, 8 x 10-11nm, 7 x 10-11nm, 6 x 10-11nm, 5 x 10-11nm, 4 x 10-11nm, 3 x 10-11nm, 2 x 10-11nm, 1 x 10-11nm, 9 x 10-12nm, 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 10-12nm, 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 10-14 nm, 7 x 10-14 nm, 6 x 10-14 nm, 5 x 10-14 nm, 4 x 10-14 nm, 3 x 10-14 nm, 2 x 10-14nm, 1 x 10-14nm, 9 x 10-15nm, 8 x 10-15nm, 7 x 10-15nm, 6 x 10-15nm, 5 x 10-15nm, 4 x 10-15nm, 3 x 10-15nm, 2 x 10-15nm, 1 x 10-15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.

[0232] 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 wavelength- dependent 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 theWSGR Docket No. 55436-726.601atoms. 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.

[0233] 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 θ may be tuned by selecting the polarization of the emitted light. For instance, when the emitted light is linearly polarized, the total polarizability α may be written as a sum of the scalar component αscalar and the tensor component αtensor:

[0234] By choosing θ 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.

[0235] 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).

[0236] 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.

[0237] 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.

[0238] 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 theWSGR Docket No. 55436-726.601microscope 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.

[0239] 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.

[0240] 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 may 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.

[0241] The optical trapping unit may comprise one or more spatial configuration artificial intelligence (AI) units configured to perform one or more AI 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 AI unit 216. Although depicted as comprising a single spatial configuration AI unit in FIG.3A, the optical trapping unit may comprise any number of spatial configuration AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0242] 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,WSGR Docket No. 55436-726.6018, 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.

[0243] The optical trapping unit may comprise one or more spatial arrangement artificial intelligence (AI) units configured to perform one or more AI 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 AI unit 218. Although depicted as comprising a single spatial arrangement AI unit in FIG.3A, the optical trapping unit may comprise any number of spatial arrangement AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial arrangement AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial arrangement AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0244] In some cases, the spatial configuration AI units and the spatial arrangement AI units may be integrated into an integrated AI unit. The optical trapping unit may comprise any number of integrated AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated AI units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated AI units.

[0245] 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.WSGR Docket No. 55436-726.601

[0246] 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.

[0247] 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.

[0248] 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 bring 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. Example of Electromagnetic Delivery Units

[0249] 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.

[0250] The electromagnetic delivery unit may comprise one or more microwave or radio- frequency (RF) energy sources, such as one or more magnetrons, klystrons, traveling-waveWSGR Docket No. 55436-726.601tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMPATT) 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.

[0251] The RF energy may comprise an average power of at least about 1 microwatt (µW), 2 µW, 3 µW, 4 µW, 5 µW, 6 µW, 7 µW, 8 µW, 9 µW, 10 µW, 20 µW, 30 µW, 40 µW, 50 µW, 60 µW, 70 µW, 80 µW, 90 µW, 100 µW, 200 µW, 300 µW, 400 µW, 500 µW, 600 µW, 700 µW, 800 µW, 900 µW, 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 µW, 800 µW, 700 µW, 600 µW, 500 µW, 400 µW, 300 µW, 200 µW, 100 µW, 90 µW, 80 µW, 70 µW, 60 µW, 50 µW, 40 µW, 30 µW, 20 µW, 10 µW, 9 µW, 8 µW, 7 µW, 6 µW, 5 µW, 4 µW, 3 µW, 2 µW, 1 µW, or less. The RF energy may comprise an average power that is within a range defined by any two of the preceding values.

[0252] 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 lightWSGR Docket No. 55436-726.601source 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.

[0253] 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.

[0254] The electromagnetic delivery unit may comprise one or more electromagnetic energy artificial intelligence (AI) units configured to perform one or more AI operations to selectively apply the electromagnetic energy to the atoms. For instance, the electromagnetic delivery unit may comprise AI unit 223. Although depicted as comprising a single AI unit in FIG.4, the electromagnetic delivery unit may comprise any number of AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0255] 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 single-qubit 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 (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, or more. Each single-qubit or two-qubit operation may comprise a duration of at most about 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, 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, 600WSGR Docket No. 55436-726.601kHz, 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.

[0256] 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 a3P0 or3P1 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, 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, or less. The Raman transitions may be detuned by a value that is within a range defined by any two of the preceding values.

[0257] 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 and / 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 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).WSGR Docket No. 55436-726.601

[0258] 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 configured 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.

[0259] 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 (µm), 1.5 µm, 2 µm, 2.5 µm 3 µm, 3.5 µm, 4 µm, 4.5 µm, 5 µm, 5.5 µm, 6 µm, 6.5 µm, 7 µm, 7.5 µm, 8 µm, 8.5 µm, 9 µm, 9.5 µm, 10 µm, or more. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at most about 10 µm, 9.5 µm, 9 µm, 8.5 µm, 8 µm, 7.5 µm, 7 µm, 6.5 µm, 6 µm, 5.5 µm, 5 µm, 4.5 µm, 4 µm, 3.5 µm, 3 µm, 2.5 µm, 2 µm, 1.5 µm, 1 µm, 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.WSGR Docket No. 55436-726.601Examples 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 D4^ width, the D86 width, and the like. For example, the beam may have a Gaussian beam waist of at least about 1.5 micrometers.

[0260] 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. The 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. Example of Integrated Optical Trapping Units and Electromagnetic Delivery Units

[0261] 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.

[0262] 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 theWSGR Docket No. 55436-726.601microscope 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 one-dimensional array of SLMs or AODs.

[0263] 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.

[0264] 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. Example of State Preparation Units

[0265] 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 toWSGR Docket No. 55436-726.601cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites.

[0266] 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.

[0267] 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 at 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 1D Zeeman slowers.

[0268] 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 1D, 2D, or 3D MOT.

[0269] 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, 710WSGR Docket No. 55436-726.601nm, 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.

[0270] 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 (µK), 90 µK, 80 µK, 70 µK, 60 µK, 50 µK, 40 µK, 30 µK, 20 µK, 10 µK, 9 µK, 8 µK, 7 µK, 6 µK, 5 µK, 4 µK, 3 µK, 2 µK, 1 µK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 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 µK, 2 µK, 3 µK, 4 µK, 5 µK, 6 µK, 7 µK, 8 µK, 9 µK, 10 µK, 20 µK, 30 µK, 40 µK, 50 µK, 60 µK, 70 µK, 80 µK, 90 µK, 100 µK, or more. The second temperature may be within a range defined by any two of the preceding values. The second MOT may comprise a 1D, 2D, or 3D MOT.

[0271] 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, 740WSGR Docket No. 55436-726.601nm, 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.

[0272] 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.

[0273] 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 most 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 µK, 9 µK, 8 µK, 7 µK, 6 µK, 5 µK, 4 µK, 3 µK, 2 µK, 1 µK, 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 µK, 2 µK, 3 µK, 4 µK, 5 µK, 6 µK, 7 µK, 8 µK, 9 µK, 10 µK, or more. The third temperature may be within a range defined by any two of the preceding values.

[0274] 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,WSGR Docket No. 55436-726.601780 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.

[0275] 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 optical 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,WSGR Docket No. 55436-726.601880 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.

[0276] 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 driving 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).

[0277] 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,WSGR Docket No. 55436-726.601860 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.

[0278] The coherent driving units may be configured to induce an RF transition between the non-equilibrium 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 or 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. Example of Controllers

[0279] The optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration AI units, spatial arrangement AI units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI 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 AI units, spatial arrangement AI units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI units, atom reservoirs,WSGR Docket No. 55436-726.601atom 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 AI units, spatial arrangement AI units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI units, atom reservoirs, atom movement units, or Rydberg excitation units. Example of Non-Classical Computers

[0280] In an aspect, 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.

[0281] In an aspect, 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. Example of Methods for Performing a Non-Classical Computation

[0282] In an aspect, 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.WSGR Docket No. 55436-726.601

[0283] FIG.6 shows a flowchart for an example of a first method 600 for performing a non- classical computation.

[0284] 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.

[0285] 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.

[0286] 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).

[0287] 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.

[0288] In an aspect, 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.WSGR Docket No. 55436-726.601

[0289] FIG.7 shows a flowchart for an example of a second method 700 for performing a non- classical computation.

[0290] 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.

[0291] 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.

[0292] 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).

[0293] 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.

[0294] In an aspect, 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.

[0295] FIG.8 shows a flowchart for an example of a third method 800 for performing a non- classical computation.

[0296] 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.WSGR Docket No. 55436-726.601The atoms may comprise any atoms described herein. The optical trapping sites may comprise any optical trapping sites described herein.

[0297] 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. Example of Parallel Addressing of Multi-Qubit Units

[0298] Direct excitation of strontium-87 from the ground state to Rydberg levels would require a laser with a wavelength of about 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 intermediate3P1 state. The about 7 kHz width of the3P1 state provides an effective balance between the two-photon effective Rabi rate and scattering via spontaneous decay from the3P1. FIG.9 shows an energy level structure for single-qubit and multi-qubit operations in strontium-87.

[0299] The optical system for single-qubit operations is also designed to work well for multi- qubit 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 the 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 about 100 µm x 100 µm was achieved, which is sufficient to address an array of about 1,000 atoms with a trapping site spacing of 3 µm. Methods for Error Corrected Quantum Computation

[0300] Systems and methods disclosed herein may generally relate to qubit loss during error correction. Within qubit loss during error correction, there may be at least two general pieces. Systems and methods described 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 be there is absent rather than incorrect. Systems and methods disclosed herein may be directed to identifying when an error is caused by a missing qubit.WSGR Docket No. 55436-726.601Systems and methods described 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.

[0301] Systems and methods disclosed herein may not generally modify the topology of the underlying surface code. Systems and methods described herein may improve upon methods of detecting atom loss by compressing the underlying protocol. Systems and methods disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate. Systems and methods disclosed herein may address un-induced erasure errors in addition to or alternatively to gate induced erasure errors.

[0302] Systems and methods of the present disclosure may improve upon other procedures at least because systems and methods of the present disclosure 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, methods and systems of the present disclosure may be used without changing the underlying surface code. Similarly, because changes to the underlying decoder and surface code are not needed, methods and systems of the present disclosure may be used with a wide variety of decoders and surface codes.

[0303] 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:2211.08468v1 [quant-ph] 15 Nov 2022, available at https: / / arxiv.org / pdf / 2211.08468.pdf, which is incorporated by reference herein in its entirety.

[0304] 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.

[0305] 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,WSGR Docket No. 55436-726.601arXiv:2302.03063v1 [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

[0306] 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.

[0307] 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.

[0308] 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 is known as syndrome extraction. Errors can occur at any point during the syndrome extraction process, including during readout of the ancilla qubits.

[0309] 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.WSGR Docket No. 55436-726.601Error Correction with Atom Loss

[0310] 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 generally employ repeated implementations of the circuit implementing the quantum computation. As the circuit is implemented and re-implemented statistics may be generated on what errors occurred. However, error correcting code implemented on systems with qubit loss may generally different than 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.

[0311] 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).

[0312] 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 operation 1020 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.

[0313] 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.

[0314] 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 non- interaction 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.WSGR Docket No. 55436-726.601

[0315] In an example, systems and methods described 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 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>.

[0316] 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, methods and system of the present disclosure also work when the 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.

[0317] 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.WSGR Docket No. 55436-726.601Examples of Identifying Qubit Loss

[0318] In an aspect, the present disclosure provides at least two methods of identifying qubit loss; however, various methods of identifying qubit loss may be integrated into methods and systems of the present disclosure. 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. Systems and methods for the implementation of a plurality of SWAP gates and modified knock-knock protocols are described in International Application PCT / US2024 / 018180, incorporated in its entirety herein for all purposes. Examples of Qubit Replacement

[0319] Methods and systems of the present disclosure 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 methods and systems herein may be combined with various types of qubits.

[0320] 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, atoms can be transferred between sites by tunneling / hopping between sites. In some cases, atoms can be transferred between sites by autonomous stabilization techniques.

[0321] 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).

[0322] 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 occupiedWSGR Docket No. 55436-726.601by 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

[0323] Systems and methods of the present disclosure 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 decode 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.

[0324] Systems and methods of the present disclosure 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.WSGR Docket No. 55436-726.601

[0325] Systems and methods of the present disclosure 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.

[0326] 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). Systems and methods of the present disclosure may not generally change the underlying topology of a topological code.

[0327] Systems and methods 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 and 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.

[0328] Systems and methods 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 and methods disclosed herein. A general 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.

[0329] Systems and methods 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 SteaneWSGR Docket No. 55436-726.601code, etc. Systems and methods 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. Systems and methods disclosed herein may be used with various qLDPC codes, for example, hypergraph product codes. A qLDPC code may be implemented as a stabilizer code.

[0330] Systems and methods 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. Methods and systems of the present disclosure 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. Systems and method of the present disclosure may update the matching graph passed to the decoder to account for a lost qubit.

[0331] 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.

[0332] 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.

[0333] To augment the decoding algorithm, systems and methods disclosed herein may update existing decoders to incorporate the change in error type. In some cases, systems and methods disclosed herein may update the matching graph passed to a decoder. In some cases, systems and methods 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.

[0334] 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 errorWSGR Docket No. 55436-726.601that occurs with a probability of 50%. The procedure may change slightly if data vs. ancilla qubits are lost.

[0335] Systems and methods described herein may be combined with systems and methods for reimplementing a qubit, as described in International Application PCT / US2024 / 018180, incorporated herein in its entirety for all purposes. Examples of Measurement Operation

[0336] 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 an 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.

[0337] In systems and methods of the present disclosure, 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.

[0338] 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 circuitWSGR Docket No. 55436-726.601may be adapted to retake or restart portions of the calculation implicating the lost qubit. As a consequence, systems and methods disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate.

[0339] 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

[0340] 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 graph 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.

[0341] 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.

[0342] In some cases, the non-classical computing system may comprise a plurality of qubits.

[0343] 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 atomWSGR Docket No. 55436-726.601movement 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.

[0344] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises trapp...

Claims

WSGR Docket No. 55436-726.601CLAIMS WHAT IS CLAIMED IS:

1. A method, comprising: (a) generating an array of atoms comprising greater than 150 atoms and a fill factor of greater than 95% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

2. The method of claim 1, wherein said fill factor is greater than 98% occupancy.

3. The method of claim 1, wherein said array of atoms comprises greater than 500 atoms.

4. The method of claim 3, wherein said array of atoms comprises greater than 1,000 atoms.

5. The method of claim 4, wherein said array of atoms comprises greater than 10,000 atoms.

6. The method of claim 1, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

7. The method of claim 6, wherein said array of atoms comprises greater than 500 sites.

8. The method of claim 7, wherein said array of atoms comprises greater than 1,000 sites.

9. The method of claim 8, wherein said array of atoms comprises greater than 10,000 sites.

10. A method, comprising: (a) generating an array of atoms comprising greater than 150 sites and a fill factor of greater than 95% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

11. The method of claim 10, wherein said fill factor is greater than 98% occupancy.

12. The method of claim 10, wherein said array of atoms comprises greater than 500 sites.

13. The method of claim 12, wherein said array of atoms comprises greater than 1,000 sites.

14. The method of claim 13, wherein said array of atoms comprises greater than 10,000 sites.

15. The method of claim 10, wherein each site comprises an atom.

16. The method of claim 15, wherein each site comprises no more than two atoms.

17. The method of claim 15 or 16, wherein said array of atoms comprises greater than 500 atoms.WSGR Docket No. 55436-726.60118. The method of claim 17, wherein said array of atoms comprises greater than 1,000 atoms.

19. The method of claim 18, wherein said array of atoms comprises greater than 10,000 atoms.

20. The method of claim 10, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

21. A method, comprising: (a) generating an array of atoms comprising greater than 500 sites and a fill factor of greater than 50% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

22. The method of claim 21, wherein said fill factor is greater than 60% occupancy.

23. The method of claim 22, wherein said fill factor is greater than 90% occupancy.

24. The method of claim 23, wherein said fill factor is greater than 95% occupancy.

25. The method of claim 24, wherein said fill factor is greater than 98% occupancy.

26. The method of claim 21, wherein said array of atoms comprises greater than 1,000 sites.

27. The method of claim 26, wherein said array of atoms comprises greater than 10,000 sites.

28. The method of claim 21, wherein each site comprises an atom.

29. The method of claim 28, wherein each site comprises no more than two atoms.

30. The method of claim 28 or 29, wherein said array of atoms comprises greater than 500 atoms.

31. The method of claim 30, wherein said array of atoms comprises greater than 1,000 atoms.

32. The method of claim 31, wherein said array of atoms comprises greater than 10,000 atoms.

33. The method of claim 21, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

34. A method, comprising: (a) filling an array of atoms comprising greater than 1000 atoms a plurality loading cycles, wherein a fill fraction of said array of atoms is greater than 0.

95.

35. The method of claim 34, wherein said fill fraction is greater than 0.

98.

36. The method of claim 34, wherein said array of atoms comprises greater than 1,000 atoms.WSGR Docket No. 55436-726.60137. The method of claim 36, wherein said array of atoms comprises greater than 10,000 atoms.

38. The method of claim 34, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

39. The method of claim 34, wherein said array of atoms comprises greater than 1,000 sites.

40. The method of claim 39, wherein said array of atoms comprises greater than 10,000 sites.

41. The method of claim 34, wherein each site comprises an atom.

42. The method of claim 34, wherein each site comprises no more than two atoms.

43. The method of claim 34, wherein said plurality of loading cycles comprises greater than 5 loading cycles.

44. The method of claim 43, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

45. The method of claim 44, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

46. A method, comprising: (a) filling an array of atoms comprising greater than 1000 atoms, wherein a fill fraction of said array of atoms is greater than 0.95; and (b) maintaining said fill fraction over a plurality of loading cycles.

47. The method of claim 46, wherein said fill fraction is greater than 0.

98.

48. The method of claim 46, wherein said array of atoms comprises greater than 5,000 atoms.

49. The method of claim 48, wherein said array of atoms comprises greater than 10,000 atoms.

50. The method of claim 46, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

51. The method of claim 46, wherein said array of atoms comprises greater than 1,000 sites.

52. The method of claim 51, wherein said array of atoms comprises greater than 10,000 sites.

53. The method of claim 46, wherein each site comprises an atom.

54. The method of claim 46, wherein each site comprises no more than two atoms.

55. The method of claim 46, wherein said plurality of loading cycles comprises greater than 5 loading cycles.WSGR Docket No. 55436-726.60156. The method of claim 55, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

57. The method of claim 56, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

58. A method, comprising: (a) generating a first array of atoms comprising greater than 150 atoms and a fill factor of greater than 95%, wherein said first array of atoms is loaded from at least a second array, and wherein said second array is loaded from a reservoir while a plurality of atoms are in said first array.

59. The method of claim 58, wherein said fill factor is greater than 98% occupancy.

60. The method of claim 58, wherein said array of atoms comprises greater than 500 sites.

61. The method of claim 60, wherein said array of atoms comprises greater than 1,000 sites.

62. The method of claim 61, wherein said array of atoms comprises greater than 10,000 sites.

63. The method of claim 58, wherein each site comprises an atom.

64. The method of claim 63, wherein each site comprises no more than two atoms.

65. The method of claim 63 or 64, wherein said array of atoms comprises greater than 500 atoms.

66. The method of claim 65, wherein said array of atoms comprises greater than 1,000 atoms.

67. The method of claim 66, wherein said array of atoms comprises greater than 10,000 atoms.

68. The method of claim 58, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

69. The method of claim 58, further comprising maintaining said fill factor over a plurality of loading cycles.

70. The method of claim 69, wherein said plurality of loading cycles comprises greater than 5 loading cycles.

71. The method of claim 70, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

72. The method of claim 71, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

73. A method for loading atoms into an optical trap, comprising:WSGR Docket No. 55436-726.601(a) trapping atoms from an incident atomic beam into a magneto-optical trap (MOT) using a core-shell configuration, wherein said core-shell configuration comprises a shell at a first wavelength surround a core with a second wavelength; and (b) transferring atoms from said MOT to an optical lattice, wherein said transferring is performed with optical tweezers.

74. The method of claim 73, wherein a shell of said MOT comprises a first light near resonance with said1S0 to1P1 transition and a core of said MOT comprises a second light near resonance with said1S0 to3P1 transition.

75. The method of claim 74, wherein said shell comprises 399 nm light near resonance with said1S0 to1P1 transition and said core comprises 556 nm light near resonance with said1S0 to3P1 transition.

76. The method of claim 73, wherein a shell of said MOT is configured to capture atoms from said incident atomic beam and wherein a core of said MOT is configured to Doppler cool said atoms.

77. The method of claim 76, wherein said atoms are Doppler cooled to about 10 µK.

78. The method of any preceding claim, wherein said method further comprises generating a cavity-enhanced optical lattice, wherein said cavity enhanced optical lattice comprises a 3-D optical lattice.

79. The method of claim 78, wherein said cavity enhanced optical lattice is formed from a standing wave optical lattice and a running wave optical lattice.

80. The method of any preceding claim, wherein said method further comprises generating a core-shell magnetooptical trap (MOT).

81. The method of claim 80, wherein said core-shell MOT comprises a ring at a first wavelength surround a central region with a second wavelength.

82. The method of claim 81, wherein a shell of said MOT comprises a first light near resonance with said1S0 to1P1 transition and a core of said MOT comprises a second light near resonance with said1S0 to3P1 transition.

83. The method of claim 82, wherein said shell comprises 399 nm light near resonance with said1S0 to1P1 transition and said core comprises 556 nm light near resonance with said1S0 to3P1 transition.

84. The method of claim 81, wherein a shell of said MOT is configured to capture atoms from said incident atomic beam and wherein a core of said MOT is configured to Doppler cool said atoms.

85. The method of claim 84, wherein said atoms are Doppler cooled to about 10 µK.WSGR Docket No. 55436-726.60186. The method of any preceding claim, wherein said method further comprises implementing a transport operation, wherein said transport operation is operable to move one or more atoms from a MOT to an array of atoms.

87. The method of any preceding claim, wherein said method further comprises implementing a tweezer operation, wherein said tweezer operation comprises moving one or more atoms within an array of atom from a first site to a second site.

88. The method of claim 87, wherein said first site and said second site are in a reservoir array.

89. The method of claim 87, wherein said first site and said second site are in a target array.

90. The method of claim 87, wherein said first site is in a reservoir array and said second site is in a target array.

91. The method of any preceding claim, wherein said method further comprises imaging one or more atoms in an array of atoms.

92. The method of any preceding claim, wherein said method further comprises Doppler cooling one or more atoms in an array of atoms.

93. The method of any preceding claim, wherein said method further comprises Raman sideband cooling one or more atoms in an array of atoms.

94. One or more non-transitory computer-readable media comprising machine- executable code comprising one or more instructions that, when executed, implement a method on a non-classical computer, wherein said non-classical computer is configured to execute said one or more instructions, said method comprising said method of any preceding claim.

95. A non-classical computing system, wherein said non-classical computing system is based on neutral atom qubits, wherein said non-classical computing system is operable to load atoms during a computing circuit, wherein said atoms loaded during said computing circuit are used in said computing circuit.

96. A non-classical computing system, said non-classical computing system comprising: an array of atoms comprising greater than 150 atoms and a fill factor of greater than 95% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

97. The non-classical computing system of claim 96, wherein said fill factor is greater than 98% occupancy.

98. The non-classical computing system of claim 96, wherein said array of atoms comprises greater than 500 atoms.WSGR Docket No. 55436-726.60199. The non-classical computing system of claim 98, wherein said array of atoms comprises greater than 1,000 atoms.

100. The non-classical computing system of claim 99, wherein said array of atoms comprises greater than 10,000 atoms.

101. The non-classical computing system of claim 96, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

102. The non-classical computing system of claim 101, wherein said array of atoms comprises greater than 500 sites.

103. The non-classical computing system of claim 102, wherein said array of atoms comprises greater than 1,000 sites.

104. The non-classical computing system of claim 103, wherein said array of atoms comprises greater than 10,000 sites.

105. A non-classical computing system, said non-classical computing system comprising: An array of atoms comprising greater than 150 sites and a fill factor of greater than 95% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

106. The non-classical computing system of claim 105, wherein said fill factor is greater than 98% occupancy.

107. The non-classical computing system of claim 105, wherein said array of atoms comprises greater than 500 sites.

108. The non-classical computing system of claim 107, wherein said array of atoms comprises greater than 1,000 sites.

109. The non-classical computing system of claim 108, wherein said array of atoms comprises greater than 10,000 sites.

110. The non-classical computing system of claim 105, wherein each site comprises an atom.

111. The non-classical computing system of claim 110, wherein each site comprises no more than two atoms.

112. The non-classical computing system of claim 110 or 111, wherein said array of atoms comprises greater than 500 atoms.

113. The non-classical computing system of claim 112, wherein said array of atoms comprises greater than 1,000 atoms.

114. The non-classical computing system of claim 113, wherein said array of atoms comprises greater than 10,000 atoms.WSGR Docket No. 55436-726.601115. The non-classical computing system of claim 105, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

116. A non-classical computing system, said non-classical computing system comprising: an array of atoms comprising greater than 500 sites and a fill factor of greater than 50% occupancy, wherein said plurality of atoms are trapped by one or both of a cavity-enhanced optical lattice or one or more optical tweezers.

117. The non-classical computing system of claim 116, wherein said fill factor is greater than 60% occupancy.

118. The non-classical computing system of claim 117, wherein said fill factor is greater than 90% occupancy.

119. The non-classical computing system of claim 118, wherein said fill factor is greater than 95% occupancy.

120. The non-classical computing system of claim 119, wherein said fill factor is greater than 98% occupancy.

121. The non-classical computing system of claim 116, wherein said array of atoms comprises greater than 1,000 sites.

122. The non-classical computing system of claim 121, wherein said array of atoms comprises greater than 10,000 sites.

123. The non-classical computing system of claim 116, wherein each site comprises an atom.

124. The non-classical computing system of claim 123, wherein each site comprises no more than two atoms.

125. The non-classical computing system of claim 123 or 124, wherein said array of atoms comprises greater than 500 atoms.

126. The non-classical computing system of claim 125, wherein said array of atoms comprises greater than 1,000 atoms.

127. The non-classical computing system of claim 126, wherein said array of atoms comprises greater than 10,000 atoms.

128. The non-classical computing system of claim 126, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

129. A non-classical computing system, said non-classical computing system comprising: an array of atoms comprising greater than 1000 atoms a plurality loading cycles, wherein a fill fraction of said array of atoms is greater than 0.95.WSGR Docket No. 55436-726.601130. The non-classical computing system of claim 129, wherein said fill fraction is greater than 0.

98.

131. The non-classical computing system of claim 129, wherein said array of atoms comprises greater than 1,000 atoms.

132. The non-classical computing system of claim 131, wherein said array of atoms comprises greater than 10,000 atoms.

133. The non-classical computing system of claim 129, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

134. The non-classical computing system of claim 129, wherein said array of atoms comprises greater than 1,000 sites.

135. The non-classical computing system of claim 134, wherein said array of atoms comprises greater than 10,000 sites.

136. The non-classical computing system of claim 129, wherein each site comprises an atom.

137. The non-classical computing system of claim 129, wherein each site comprises no more than two atoms.

138. The non-classical computing system of claim 129, wherein said plurality of loading cycles comprises greater than 5 loading cycles.

139. The non-classical computing system of claim 138, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

140. The non-classical computing system of claim 139, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

141. A non-classical computing system, said non-classical computing system comprising: an array of atoms comprising greater than 1000 atoms, wherein a fill fraction of said array of atoms is greater than 0.95, wherein said fill fraction is configured to remain substantially constant over a plurality of loading cycles.

142. The non-classical computing system of claim 141, wherein said fill fraction is greater than 0.

98.

143. The non-classical computing system of claim 141, wherein said array of atoms comprises greater than 1,000 atoms.

144. The non-classical computing system of claim 143, wherein said array of atoms comprises greater than 10,000 atoms.

145. The non-classical computing system of claim 141, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.WSGR Docket No. 55436-726.601146. The non-classical computing system of claim 141, wherein said array of atoms comprises greater than 1,000 sites.

147. The non-classical computing system of claim 146, wherein said array of atoms comprises greater than 10,000 sites.

148. The non-classical computing system of claim 141, wherein each site comprises an atom.

149. The non-classical computing system of claim 141, wherein each site comprises no more than two atoms.

150. The non-classical computing system of claim 141, wherein said plurality of loading cycles comprises greater than 5 loading cycles.

151. The non-classical computing system of claim 150, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

152. The non-classical computing system of claim 151, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

153. A non-classical computing system, said non-classical computing system comprising: a first array of atoms comprising greater than 150 atoms and a fill factor of greater than 95%, wherein said first array of atoms is loaded from at least a second array, and wherein said second array is loaded from a reservoir while a plurality of atoms are in said first array.

154. The non-classical computing system of claim 153, wherein said fill factor is greater than 98% occupancy.

155. The non-classical computing system of claim 153, wherein said array of atoms comprises greater than 500 sites.

156. The non-classical computing system of claim 155, wherein said array of atoms comprises greater than 1,000 sites.

157. The non-classical computing system of claim 156, wherein said array of atoms comprises greater than 10,000 sites.

158. The non-classical computing system of claim 153, wherein each site comprises an atom.

159. The non-classical computing system of claim 158, wherein each site comprises no more than two atoms.

160. The non-classical computing system of claim 158 or 159, wherein said array of atoms comprises greater than 500 atoms.

161. The non-classical computing system of claim 160, wherein said array of atoms comprises greater than 1,000 atoms.WSGR Docket No. 55436-726.601162. The non-classical computing system of claim 161, wherein said array of atoms comprises greater than 10,000 atoms.

163. The non-classical computing system of claim 153, wherein said array of atoms comprises a plurality of spatially distinct optical trapping sites.

164. The non-classical computing system of claim 153, further comprising an optical unit configured to maintain said fill factor over a plurality of loading cycles.

165. The non-classical computing system of claim 164, wherein said plurality of loading cycles comprises greater than 5 loading cycles.

166. The non-classical computing system of claim 165, wherein said plurality of loading cycles comprises greater than 500 loading cycles.

167. The non-classical computing system of claim 166, wherein said plurality of loading cycles comprises greater than 5,000 loading cycles.

168. A non-classical computing system, said non-classical computing system comprising: a magneto-optical trap (MOT), wherein said MOT is configured to trap atoms from an incident atomic beam into said MOT using a core-shell configuration, wherein said core-shell configuration comprises a ring at a first wavelength surrounding a central region with a second wavelength; an optical lattice; and optical tweezers, wherein said optical tweezers are configured to transfer atoms from said MOT to said optical lattice.

169. The non-classical computing system of claim 168, wherein a shell of said MOT comprises a first light near resonance with said1S0 to1P1 transition and a core of said MOT comprises a second light near resonance with said1S0 to3P1 transition.

170. The non-classical computing system of claim 169, wherein said shell comprises 399 nm light near resonance with said1S0 to1P1 transition and said core comprises 556 nm light near resonance with said1S0 to3P1 transition.

171. The non-classical computing system of claim 168, wherein a shell of said MOT is configured to capture atoms from said incident atomic beam and wherein a core of said MOT is configured to Doppler cool said atoms.

172. The non-classical computing system of claim 171, wherein said atoms are Doppler cooled to about 10 µK.

173. The non-classical computing system of any one of claims 168-172, wherein said optical unit is further configured to generate a cavity-enhanced optical lattice, wherein said cavity enhanced optical lattice comprises a 3-D optical lattice.WSGR Docket No. 55436-726.601174. The non-classical computing system of claim 173, wherein said cavity enhanced optical lattice is formed from a standing wave optical lattice and a running wave optical lattice.

175. The non-classical computing system of one of claims 168-174, wherein said optical unit is further configured to generate a core-shell magnetooptical trap (MOT).

176. The non-classical computing system of claim 175, wherein said core-shell MOT comprises a ring at a first wavelength surround a central region with a second wavelength.

177. The non-classical computing system of claim 176, wherein a shell of said MOT comprises a first light near resonance with said1S0 to1P1 transition and a core of said MOT comprises a second light near resonance with said1S0 to3P1 transition.

178. The non-classical computing system of claim 177, wherein said shell comprises 399 nm light near resonance with said1S0 to1P1 transition and said core comprises 556 nm light near resonance with said1S0 to3P1 transition.

179. The non-classical computing system of claim 176, wherein a shell of said MOT is configured to capture atoms from said incident atomic beam and wherein a core of said MOT is configured to Doppler cool said atoms.

180. The non-classical computing system of claim 179, wherein said atoms are Doppler cooled to about 10 µK.

181. The non-classical computing system of any one of claims 168-180, further comprising a transport unit configured to implement a transport operation, wherein said transport operation is operable to move one or more atoms from a MOT to an array of atoms.

182. The non-classical computing system of any one of claims 168-181, wherein said transport unit is further configured to implement a tweezer operation, wherein said tweezer operation comprises moving one or more atoms within an array of atom from a first site to a second site.

183. The non-classical computing system of claim 182, wherein said first site and said second site are in a reservoir array.

184. The non-classical computing system of claim 182, wherein said first site and said second site are in a target array.

185. The non-classical computing system of claim 182, wherein said first site is in a reservoir array and said second site is in a target array.

186. The non-classical computing system of any one of claims 168-185, further comprising an imaging unit configured to image one or more atoms in an array of atoms.

187. The non-classical computing system of any one of claims 168-186, further comprising a cooling unit configured to Doppler cool one or more atoms in an array of atoms.WSGR Docket No. 55436-726.601188. The non-classical computing system of any one of claims 168-187, wherein said cooling unit is further configured to Raman sideband cool one or more atoms in an array of atoms.

189. The method of any one of claims 73-93, further comprising moving said atoms from the MOT to a reservoir array via a moving optical lattice.

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