Systems and methods for networked qubit architectures

Abstract

A non-classical computer may comprise: an array of spatially distinct optical traps, wherein the array comprises a plurality of atoms; an optical cavity array, the optical cavity array comprising the array of spatially distinct optical modes; and an optical control hardware, wherein the optical control hardware reconfigurably combines optical paths from distinct atoms for entanglement generation.

Description

WSGR Docket No. 55436-755.601SYSTEMS AND METHODS FOR NETWORKED QUBIT ARCHITECTURES CROSS-REFERENCE

[0001] This application claims the benefit of U. S. Provisional Patent Application No.63 / 752,497, entitled “METHODS AND SYSTEMS FOR MICROCAVITY INTERCONNECTED ARCHITECTURES,” filed on January 31, 2025, and U.S. Provisional Patent Application No. 63 / 752,567, entitled “METHODS AND DEVICES FOR MEASUREMENT BASED QUANTUM COMPUTING WITH ATOMS AND CAVITIES,” filed on January 31, 2025, each of which is incorporated by reference herein in their entireties.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) = [ ] (the “zero state”) and |1) = (the “one state”). The two orthonormal basis states,{| 0), 11)}, together called the computational basis, span the two-dimensional linear vector (Hilbert) space of the qubit. The basis states can also be combined to form product basis states, e.g., 100), 101), 110), 111), 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 disclosure provides a non-classical computer, comprising: an array of spatially distinct optical traps, wherein the array comprises a plurality of atoms; an optical cavity array, the optical cavity array comprising an array of spatially distinct optical modes, wherein a cavity of the optical cavity array comprises an atom of the array of spatially distinct optical traps; and an optical control hardware, wherein the optical control hardware reconfigurably combines optical paths of the optical cavity array from at least two distinct atoms of the plurality of atoms for entanglement generation between the distinct atoms.WSGR Docket No. 55436-755.601

[0005] In some embodiments, the optical control hardware is configured to receive light from an atom at the first site within the optical cavity array and to direct the light to the second site within the optical cavity array. In some embodiments, atom-photon entanglement is generated at the first site, wherein a photon is redirected to the second site, and wherein at the second site an atom at the second site absorbs the photon to generate atom-atom entanglement between the first site and the second site. In some embodiments, atom-photon entanglement is generated at each of the first site and the second site, wherein a first photon is generated at the first site, wherein a second photon is generated at the first second site, and wherein atom-atom entanglement is generated by interfering the photons on a beam splitter and conditioning on two-photon detections. In some embodiments, the optical control hardware comprises: a telescope system and a plurality of light redirection optics. In some embodiments, the plurality of light redirection optics comprises a MEMS mirror array. In some embodiments, the plurality of light redirection optics comprises a plurality of individually tunable reflective optics. In some embodiments, the plurality of light redirection optics is configured to direct light from an atom at the first site within the optical cavity array and to a second site within the optical cavity array. In some embodiments, the telescope system comprises a local telescope and a global telescope. In some embodiments, the local telescope comprises a local telescope array, wherein each local telescope of the array is configured to redirect light from a block or subset of cavities of the array of optical cavities. In some embodiments, the global telescope is configured to control a size of an array of light beams each corresponding to a cavity of the array of optical cavities. In some embodiments, the optical cavity array comprises a first optical cavity array and a second optical cavity array, wherein the optical control hardware is configured to produce an optically generated Bell-pair distributed between a first site within the first optical cavity array and a second site within the second optical cavity array. In some embodiments, the first optical cavity and the second optical cavity are optically distinct. In some embodiments, the first optical cavity and the second optical cavity comprise distinct cavity optics. In some embodiments, the first optical cavity and the second optical cavity are spatially distinct. In some embodiments, the first optical cavity and the second optical cavity are connected by fiber bundle. In some embodiments, the first optical cavity and the second optical cavity are separated by a distance of greater than 1 meter. In some embodiments, the first optical cavity and the second optical cavity are within a single non-classical computing module. In some embodiments, the first optical cavity is within a first non-classical computing module and the second optical cavity is within a second non-classical computing module. In some embodiments, the optically generated Bellpair is configured to deterministically teleport qubits or gates between sites. In someWSGR Docket No. 55436-755.601embodiments, the optical control hardware is configured to enable all-to-all connectivity between spatially separate atoms.

[0006] In some embodiments, the optical control hardware is configured to receive light from an atom at the first site within the optical cavity array and to direct the light to the second site within the optical cavity array. In some embodiments, atom-photon entanglement is generated at the first site, wherein a photon is redirected to the second site, and wherein at the second site an atom at the second site absorbs the photon to generate atom-atom entanglement between the first site and the second site. In some embodiments, atom-photon entanglement is generated at each of the first site and the second site, wherein a first photon is generated at the first site, wherein a second photon is generated at the first second site, and wherein atom-atom entanglement is generated by interfering the photons on a beam splitter and conditioning on two-photon detections. In some embodiments, the optical control hardware comprises: a telescope system and a plurality of light redirection optics. In some embodiments, the plurality of light redirection optics comprises a MEMS mirror array. In some embodiments, the plurality of light redirection optics comprises a plurality of individually tunable reflective optics. In some embodiments, the plurality of light redirection optics is configured to direct light from an atom at the first site within the optical cavity array and to a second site within the optical cavity array. In some embodiments, the telescope system comprises a local telescope and a global telescope. In some embodiments, the local telescope comprises a local telescope array, wherein each local telescope of the array is configured to redirect light from a block or subset of cavities of the array of optical cavities. In some embodiments, the global telescope is configured to control a size of an array of light beams each corresponding to a cavity of the array of optical cavities. In some embodiments, the optical cavity array comprises a first optical cavity array and a second optical cavity array, wherein the optical control hardware is configured to produce an optically generated Bell-pair distributed between a first site within the first optical cavity array and a second site within the second optical cavity array. In some embodiments, the first optical cavity and the second optical cavity are optically distinct. In some embodiments, the first optical cavity and the second optical cavity comprise distinct cavity optics. In some embodiments, the first optical cavity and the second optical cavity are spatially distinct. In some embodiments, the first optical cavity and the second optical cavity are connected by fiber bundle. In some embodiments, the first optical cavity and the second optical cavity are separated by a distance of greater than 1 meter. In some embodiments, the first optical cavity and the second optical cavity are within a single non-classical computing module. In some embodiments, the first optical cavity is within a first non-classical computing module and the second optical cavity is within a second non-classical computing module. In some embodiments, the optically generated Bell-WSGR Docket No. 55436-755.601pair is configured to deterministically teleport qubits or gates between sites. In some embodiments, the optical control hardware is configured to enable all-to-all connectivity between spatially separate atoms.

[0007] In another aspect, the present disclosure provides a method of connecting spatially separate atoms within a trapped atom quantum computer, the method comprising: (a) trapping a plurality of atoms within an optical cavity array, the optical cavity array supporting a plurality of optical cavities comprising an array of spatially distinct optical traps; and (b) reconfigurably combining optical paths from distinct atoms for entanglement generation.

[0008] In some embodiments, (b) comprises one or more of receiving light from an atom at the first site within the optical cavity array and directing the light to the second site within the optical cavity array, or interfering light from an atom at the first site with light from an atom at the second site using a beamsplitter. In some embodiments, the method further comprises deterministically teleporting a qubit or a gate between the first site and the second site. In some embodiments, the method further comprises teleporting a gate between the first site and the second site, wherein the teleporting comprises, subsequent to (b), (i) applying local two-qubit gates at each of the first site and the second site, wherein each local two-qubit is between the qubit from the Bell-pair and a target qubit, and (ii) applying a single-qubit operation. In some embodiments, the method further comprises implementing a state purification operation. In some embodiments, the state purification operation comprises, subsequent to (b), (i) providing two remotely entangled Bell pairs; (ii) applying a local control-X gate at each of the two locations; and (iii) measuring one of the two qubits at each location. Ins some embodiments, the method further comprises implementing an error correction operation. In some embodiments, the error correction operation comprises syndrome extraction. In some embodiments, the syndrome extraction comprises transversal CNOT operations.

[0009] In another aspect, the present disclosure provides a method for networked quantum computing comprising: (a) providing a first plurality of qubits in a first set of optical cavities and a second plurality of qubits in a second set of optical cavities, wherein qubits of the first plurality of qubits and the second plurality of qubits both comprise atoms in an array of spatially distinct optical traps (b) distributing entanglement between the first plurality of qubits and the second plurality of qubits to form a distributed state of the plurality of qubits between the first set of optical cavities and the second set of optical cavities; and (c) performing a quantum computing operation based at least in part by interacting with the distributed state.

[0010] In some embodiments, the distributed state comprises a resource state. In some embodiments, the method further comprises continuously generating the resource state. In some embodiments, the method further comprises prior to (c), distilling the distributed state using nWSGR Docket No. 55436-755.601copies of the distributed state, wherein n is an integer greater than one. In some embodiments, n is three. In some embodiments, n is four or greater. In some embodiments, the quantum computing operation is a quantum gate operation. In some embodiments, the method further comprises at (c) teleporting a gate operation. In some embodiments, performing a quantum computation in (c) further comprises performing one or more quantum gates on a first qubit, a second qubit, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits. In some embodiments, the first qubit, the second qubit, or both are entangled with the distributed state. In some embodiments, the first qubit, the second qubit, or both are locally entangled with the distributed state. In some embodiments, the first qubit, the second qubit, or both are entangled with the distributed state using a distributed Bell pair. In some embodiments, the first qubit and the second qubit are in spatially distinct optical traps. In some embodiments, the first qubit and the second qubit are in non-neighboring optical traps. In some embodiments, the first qubit and the second qubit are in different optical cavity arrays. In some embodiments, the first qubit and the second qubit are in different quantum computing modules In some embodiments, performing a quantum computation in (c) further comprises performing one or more quantum gates on a third plurality of qubits, a fourth plurality of qubits, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are locally entangled with the distributed state. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state using a distributed Bell pair. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in spatially distinct optical traps. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in non-neighboring optical traps. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in different optical cavity arrays. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in different quantum computing modules In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are data qubits. In some embodiments, the quantum computing operation is an error correction operation. In some embodiments, the quantum computing operation is a logical operation. In some embodiments, the quantum computing operation is a transversal gate between logical qubits. In some embodiments, the transversal gate is a portion of a Steane style error correction code. In some embodiments, the transversal gate is a portion of a Knill style error correction code. In some embodiments, the first plurality of qubits, the second plurality of qubits, or both comprises a graph state in a first location, and wherein generating the distributedWSGR Docket No. 55436-755.601state in (b) further comprises generating a distributed graph state between the first location and a second location. In some embodiments, the first plurality of qubits, the second plurality of qubits, or both comprises a cluster state in a first location, and wherein generating the distributed state in (b) further comprises generating a distributed cluster state between the first location and a second location. In some embodiments, the distributed graph state comprises two or more entangled time slices. In some embodiments, the distributed cluster state comprises two or more entangled time slices. In some embodiments, performing a quantum computing operation in (c) comprises performing a measurement of a qubit state in a first time slice of the two or more entangled time slices, wherein the measurement influences a second time slice of the two or more entangled time slices. In some embodiments, the method further comprises performing a quantum circuit at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices. In some embodiments, the method further comprises performing a quantum computation at least in part by performing a sequence of measurements of qubit states across multiple time slices of the two or more entangled time slices.

[0011] In another aspect, the present disclosure provides a system for networked quantum computing comprising: a first plurality of qubits in a first set of optical cavities, a second plurality of qubits in a second set of optical cavities, wherein the first set of optical cavities and the second set of optical cavities are optically connected, and wherein qubits of the first plurality of qubits and the second plurality of qubits both comprise atoms in an array of spatially distinct optical traps; and a distributed state formed by distributed entanglement of the first plurality of qubits and the second plurality of qubits between the first set of optical cavities and the second set of optical cavities.

[0012] In some embodiments, the system further comprises interacting the distributed state with a resource state. In some embodiments, the resource state is continuously generated. In some embodiments, the distributed state is distilled using n copies of the distributed state, wherein n is an integer greater than one. In some embodiments, n is three. In some embodiments, n is four or greater. In some embodiments, the system is configured to perform a quantum gate operation. In some embodiments, the system is configured to perform one or more quantum gates on a first qubit, a second qubit, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits. In some embodiments, the first qubit, the second qubit, or both are entangled with the distributed state. In some embodiments, the first qubit, the second qubit, or both are locally entangled with the distributed state. In some embodiments, the first qubit, the second qubit, or both are entangled with the distributed state using a distributed Bell pair. In some embodiments, the first qubit and the second qubit are inWSGR Docket No. 55436-755.601spatially distinct optical traps. In some embodiments, the first qubit and the second qubit are in non-neighboring optical traps. In some embodiments, the first qubit and the second qubit are in different optical cavity arrays. In some embodiments, the first qubit and the second qubit are in different quantum computing modules In some embodiments, the system is configured to perform one or more quantum gates on a third plurality of qubits, a fourth plurality of qubits, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are locally entangled with the distributed state. In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state using a distributed Bell pair. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in spatially distinct optical traps. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in non-neighboring optical traps. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in different optical cavity arrays. In some embodiments, the third plurality of qubits and the fourth plurality of qubits are in different quantum computing modules In some embodiments, the third plurality of qubits, the fourth plurality of qubits, or both are data qubits. In some embodiments, the system is configured to perform an error correction operation. In some embodiments, the system is configured to perform a logical operation. In some embodiments, system is configured to perform a transversal gate between logical qubits. In some embodiments, the transversal gate is a portion of a Steane style method of error correction. In some embodiments, the transversal gate is a portion of a Knill style method of error correction. In some embodiments, the method further comprises distributing a graph state between a first location and a second location to produce a distributed graph state, at least in part by interacting the distributed state with the graph state, first plurality of qubits, the second plurality of qubits, or both comprises a graph state in a first location, and wherein the resource state comprises a distributed graph state between the first location and a second location. In some embodiments, the first plurality of qubits, the second plurality of qubits, or both comprises a cluster state in a first location, and generating the distributed state in (b) further comprises generating a distributed cluster state between the first location and a second location. In some embodiments, the distributed graph state comprises two or more entangled time slices. In some embodiments, the distributed cluster state comprises two or more entangled time slices. In some embodiments, the system is configured to perform a quantum computing operation based at least in part on a measurement of a qubit state in a first time slice of the two or more entangled time slices, wherein the measurement influences a second time slice of the two orWSGR Docket No. 55436-755.601more entangled time slices. In some embodiments, the system is configured to perform a quantum circuit at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices. In some embodiments, the system is configured to perform a quantum computation by performing a sequence of measurements of qubit states across multiple time slices of the two or more entangled time slices. In some embodiments, optically connecting the first set of optical cavities and the second set of optical cavities comprises transmitting light from the first set of optical cavities to the second set of optical cavities. In some embodiments, optically connecting the first set of optical cavities and the second set of optical cavities comprises interfering light from the first set of optical cavities and the second set of optical cavities on a beam splitter.

[0013] 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.

[0014] 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

[0015] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and / or take precedence over any such contradictory material.BRIEF DESCRIPTION OF THE DRAWINGS

[0016] 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:WSGR Docket No. 55436-755.601

[0017] FIG. 1 shows a computer control system that is programmed or otherwise configured to implement methods provided herein;

[0018] FIG. 2 shows an example of a system for performing a non-classical computation;

[0019] FIG. 3A shows an example of an optical trapping unit;

[0020] FIG. 3B shows an example of a plurality of optical trapping sites;

[0021] FIG. 3C shows an example of an optical trapping unit that is partially filled with atoms;

[0022] FIG. 3D shows an example of an optical trapping unit that is completely filled with atoms;

[0023] FIG. 4 shows an example of an electromagnetic delivery unit;

[0024] FIG. 5 shows an example of a state preparation unit;

[0025] FIG. 6 shows a flowchart for an example of a first method for performing a non-classical computation;

[0026] FIG. 7 shows a flowchart for an example of a second method for performing a non-classical computation;

[0027] FIG. 8 shows a flowchart for an example of a third method for performing a non-classical computation;

[0028] FIG. 9 shows an energy level structure for single-qubit and multi-qubit operations in strontium-87;

[0029] FIG. 10 shows an example method for error corrected non-classical computation;

[0030] 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;

[0031] FIG. 12 shows an example process for performing continuous, non-classical computation;

[0032] FIG. 13A shows example of a plurality of mirrors configured to provide a plurality of optical cavities;

[0033] 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;

[0034] FIG. 14 provides a non-limiting example of a cavity array, in accordance with some embodiments herein;

[0035] FIG. 15A provides a first embodiment for a polygonal mirror, in accordance with some embodiments herein;

[0036] FIG. 15B provides a second embodiment for a polygonal mirror, in accordance with some embodiments herein;

[0037] FIG. 15C provides an example of a retroreflector assembly, in accordance with some embodiments herein;WSGR Docket No. 55436-755.601

[0038] FIG. 16 provides an alternate embodiment of a cavity array, in accordance with some embodiments herein;

[0039] FIG. 17 provides a further embodiment of a cavity array, in accordance with some embodiments herein;

[0040] FIG. 18A provides an example of optical control hardware for distributed entanglement within an optical cavity array, in accordance with some embodiments herein;

[0041] FIG. 18B provides an example of optical control hardware for distributed entanglement between optical cavity arrays, in accordance with some embodiments herein;

[0042] FIG. 18C provides an example of a telescope system, in accordance with some embodiments herein;

[0043] FIG. 19 provides an example of optical control hardware for combining optical paths from different quantum computing modules, in accordance with some embodiments herein;

[0044] FIG. 20 provides an example of a method of connecting spatially separate atoms within a trapped atom quantum computer, in accordance with some embodiments herein;

[0045] FIG. 21 provides an example of a method for networked quantum computing, in accordance with some embodiments herein;

[0046] FIG. 22 provides an example of a method for distilling a distributed state, in accordance with some embodiments herein;

[0047] FIG. 23 provides an example of a quantum circuit for distilling a distributed state, in accordance with some embodiments herein;

[0048] FIG. 24 provides an alternate example of a quantum circuit for distilling a distributed state, in accordance with some embodiments herein;

[0049] FIG. 25A provides an example of a unit cell for distilling a distributed state, in accordance with some embodiments herein;

[0050] FIG. 25B provides an alternate example of a unit cell for distilling a distributed state, in accordance with some embodiments herein;

[0051] FIG. 26 provides an example of a method for teleporting gates, in accordance with some embodiments herein;

[0052] FIG. 27 provides an example of a method for measurement-based quantum computing, in accordance with some embodiments herein;

[0053] FIG. 28 provides an example of entangled time slices, in accordance with some embodiments herein; and

[0054] FIG. 29 provides an example of time-like fusions across a plurality of microcavity arrays.WSGR Docket No. 55436-755.601DETAILED DESCRIPTION

[0055] 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.Overview

[0056] Neutral atom quantum computing can provide several advantages over other quantum computing modalities, making neutral-atom-based technologies a promising candidate for nextgeneration computing applications. For example, neutral atom quantum computers may be readily scaled by trapping more atoms to replace or add to the qubits in a system. Neutral atom quantum computers also display favorable coherence times compared to other modalities, enabling longer and more reliable storage of information for more complex and error free computation. On the other hand, performing quantum computing operations on neutral atom quantum computers can be slow in some cases. This slowness may be due, at least in part, to the need to move atoms proximate each other to produce the atomic interactions making up these quantum computing operations. Recognized herein is the need for systems and methods for performing quantum computing operations in neutral atom quantum computing, while reducing the time cost associated with atom movement.

[0057] Systems and methods for networked qubit architectures, described herein, can reduce the time cost associated with atom movement in quantum logic gates, by distributing entanglement among non-neighboring qubits. This distributed entanglement may be used to create distributed states, which may in turn be used to perform quantum computing operations such as logic gates or error correction. The distributed state can provide connectivity between non-neighboring atoms, enabling atomic interactions without the need for moving the participating atoms proximate to each other.Systems for Optical Cavity Arrays

[0058] Provided herein are systems and methods for optical cavity arrays. In some cases, optical cavity arrays disclosed herein may be used to create a plurality of longitudinal modes within an optical cavity. These longitudinal modes may be transversely separated from each other, which can provide the advantage that each longitudinal mode may not be coupled to another. Using a system of optical devices, these transversely separated longitudinal modes may be arranged to form a plurality of transversely separated foci in a focal plane within the optical cavity. TheseWSGR Docket No. 55436-755.601transversely separated longitudinal modes forming the plurality of transversely separated foci are transversely non-degenerate.

[0059] In some cases, each focus of the plurality of transversely separated foci comprises an optical trapping site, such as the optical trapping sites 211 of the one or more optical trapping units 210 of the system 200, herein. The optical trapping sites may be formed by an optical tweezer array, such as the 2D array of FIG. 3A, described herein. The plurality of foci of the cavities herein may be overlapped with the plurality of optical trapping sites such that the traps are coincident with the optical traps.

[0060] In some cases, the optical cavities herein may be used to couple photons between optical foci of an optical cavity array. In some cases, these coupled photons between optical foci of an optical cavity array may be used to distribute entanglement between atoms or qubits located at each optical foci of an optical cavity array. In some cases, these coupled photons between optical foci of an optical cavity array may be used for readout operations for atoms or qubits located at each optical foci of an optical cavity array. In some cases, the optical cavities herein may be used to collect photons for networked quantum computing, herein. In some cases, the optical cavities herein are used in combination with optical control hardware, such as 1800A and 1800B of FIGs. 18A-18B herein, to reconfigurably distribute light from one optical focus to another optical focus. In some cases, reconfigurably distributing light may be used to create distributed entanglement between atoms or qubits located at different optical foci of the optical cavity array.

[0061] In some cases, the plurality of optical foci may be transversely separated to form an array. In some examples, the plurality of optical foci may be transversely separated by a distance of a few microns or more.

[0062] In some examples, the plurality of optical foci may be separated by about one micron or less. In some examples, the plurality of optical foci may be separated by about one micron to about 10 microns and increments therein. In some examples, the plurality of optical foci may be separated by about 10 microns. In some examples, the plurality of optical foci may be separated by about 10 microns to about 100 microns and increments therein. In some examples, the plurality of optical foci may be separated by about 100 microns. In some examples, the plurality of optical foci may be separated by about 100 microns or more.

[0063] In some cases, the systems and methods for optical cavity arrays, described herein, can generate an array of optical foci without the aberration sensitivity and resonator stability limitations which may be present when an array of optical foci is generated using a degenerate or “imaging” resonator. In some cases, systems and methods for optical cavity arrays, described herein, provide an array of optical foci having properties as if each optical foci was generated byWSGR Docket No. 55436-755.601its own optical resonator or optical cavity, each having its own unique path. Advantageously, systems and methods herein may provide an array of optical foci as if each optical foci was generated by its own optical resonator or optical cavity, while using bulk components.

[0064] In some cases, the systems and method for optical cavity arrays, described herein, can generated an array of optical foci extending transverse to the optical direction in one direction or two dimensions. In some cases, each transversely separated longitudinal mode forming the plurality of optical foci of the array is independently stable. Each transversely separated longitudinal mode may be operated far from any mode degeneracies, which may prevent susceptibility to mode mixing or aberration-induced instabilities, in some cases.

[0065] An optical cavity array may include a plurality of mirrors that form an optical cavity, a first lens system located within the optical cavity, and a second lens system located within the optical cavity. The first lens system has a first output facing a first mirror of the plurality of mirrors and a second output facing a second mirror of the plurality of mirrors. The second lens system has a second input facing the first input and a second output facing the second mirror. The first and second lens systems are configured such that the optical cavity supports longitudinal modes that are transversely non-degenerate, forming spatially separated waists that lie along a focal plane that is axially located between the first and second inputs. When the longitudinal modes are excited, the waists may be used as an array of optical dipole traps.Systems for Optical Cavity Arrays

[0066] In some cases, systems and methods for networked qubit architectures, described herein, provides systems for optical cavity arrays. Systems and methods for optical cavity arrays are described in, for instance, International Patent Publication No. WO2023 / 220415, which is incorporated by reference herein in its entirety for all purposes.

[0067] In some embodiments, an optical cavity array has an array of foci that are spatially separated by a few microns near the focal plane of an imaging system with a high numerical aperture (NA). In some cases, this configuration may advantageously avoid the aberration sensitivity and resonator stability limitations present when many foci are generating with a “degenerate” resonator (also known as an “imaging resonator”). Each focus may be thought of as being generated by its own optical resonator, or cavity, that has its own unique path.However, most (if not all) of the optical components are bulk, in some cases. These optical-resonator paths form an array that extends transversely to the optical axis in either one or two dimensions. Each optical-resonator path of the array is independently stable and may be operated far from any mode degeneracies that would render it susceptible to mode-mixing, aberration-induced instabilities, or both.WSGR Docket No. 55436-755.601

[0068] A non-limiting example of a cavity array 1400 is provided in FIG. 14. The optical cavity array 1400 may be used to generate an array of foci 1401, or waists, that are located on a focal plane 1402. In some cases, the foci 1401 extend along x. which is perpendicular to the optical axis (which is parallel z). For clarity in FIG. 14, only three foci 1401 are shown. However, there may be only two foci 1401, or more than three foci 1401, without departing from the scope hereof. While the foci 1401 are shown extending along x, the foci 1401 may alternatively or additionally extend along y. For clarity, each of the foci 1401 is enclosed by a small circle.

[0069] In some cases, the optical cavity array 1400 includes a high-NA retroreflector 1403 that is located on one side of the focal plane 1402. In one example, the high-NA retroreflector 1403 includes a planar retroreflecting mirror 1404, a first lens 1405 with a focal length f, and a first high-NA lens 1406 with a focal length f. The first lens 1405 may be located one focal length f in front of the mirror 1404 (i.e., parallel to the optical axis in the +z direction). The first lens 1405 may be a plano-convex lens, as shown in FIG. 14, or another type of lens (e.g., aspheric, biconvex lens, etc.). The first high-NA lens 1406 may be located one focal length f in front of (i.e., in the +z direction) the first lens 1405. The first high-NA lens 1406 may be an aspheric lens, microscope objective, compound lens systems (e.g., telescope), or another type of lens. Herein, the term “high NA” may mean values of the NA greater than or equal to about 0.5. However, the NA of any lens may be less than 0.5 without departing from the scope hereof.

[0070] In some cases, the optical cavity array 1400 also includes a finite-conjugate-ratio imaging system 1407 that is located on the other side of the focal plane (or atom plane, in some embodiments) 1402 from the high-NA retroreflector 1403. In some cases, the imaging system 1407 includes a second high-NA lens 1408 and a second lens 1409 that cooperate to image the focal plane 1402 onto an imaging plane (or conjugate plane, in some embodiments) 1410. In some cases, the optical cavity array 1400 also includes a retroreflector 1411 whose reflective face (or faces, in some embodiments) reflects the images back on themselves. The retroreflector 1411 may be positioned near the imaging plane 1410 but does not exactly coincide with it in some cases.

[0071] In FIG. 14, the formation of the foci 1401 may be illustrated by ray tracing. One of the foci 1401 is formed from rays illustrated as dashed lines. These rays indicate a first mode 1412(a) of the optical cavity array 1400 whose waist is said one of the foci 1403. A second of the foci 1401 may be formed from rays illustrated as solid lines (this focus lies on the optical axis). These rays indicate a second mode 1412(b) of the optical cavity array 1400 whose waist is said second of the foci 1401. A third of the foci 1401 may be formed from rays illustrated as dotted lines. These rays indicate a third mode 1412(c) of the optical cavity array 1400 whose waist is said third of the foci 1401. Thus, the terms "waist" and "focus" may be synonymousWSGR Docket No. 55436-755.601herein. It should be understood that the rays shown in FIG. 14 are for illustration only, and do not fully illustrate the shape (i.e., intensity distribution) of the modes 1412.

[0072] The modes 1412 of the optical cavity array 1400 can be excited by transmitting light through the mirror 1404, through the retroreflector 1411, or both. The light may be a single monochromatic laser beam, or several monochromatic laser beams that are transversely displaced from each other. The light may be any light source, described herein. In some cases, light in the optical cavity array 1400 can also leak out by transmitting through the mirror 1404, the retroreflector 1411, or both. Alternatively or additionally, light can be coupled out of the optical cavity array 1400 with a beam sampler. This beam sampler may be placed in a low-NA region to minimize aberrations, in some cases.

[0073] FIGs. 15A-15C provide non-limiting examples of polygonal mirrors, in accordance with some embodiments of the disclosure. FIG. 15A depicts a side view of a polygonal mirror 1500A that is one example of the retroreflector 1411. The polygonal mirror 1500A has a first face 1501 that forms a first oblique angle with the optical axis, a second face 1502 that is perpendicular to the optical axis, and a third face 1503 that forms a second oblique angle with the optical axis. The faces 1501, 1502, and 1503 may be positioned to retroreflect light thereon back onto itself. More specifically, the first oblique angle may be selected such that the third mode 1412(c) retroreflects off the first face 1501. Similarly, the second oblique angle may be selected such that the first model 1412(a) retroreflects off the third face 1503.

[0074] FIG. 15B depicts a side view of an alternative embodiment of a polygonal mirror 1500B that is similar to the polygonal mirror 1500A except that the second face 1502 is not recessed. Specifically, the second face 1502 is located farther in the +x direction in the polygonal mirror 1500 A, as compared to the polygonal mirror 1500B. In some cases, the polygonal mirror 1500B may be easier to fabricate than the polygonal mirror 1500A.

[0075] FIG. 15C depicts a side view of a cat' s-eye retroreflector array 1500C that is another example of the retroreflector 1411 of FIG. 14. In some cases, the cat’ s-eye array 1500C includes an array of microlenses 1504 that are all located at the same position along the optical axis (i.e., along z), but different transverse positions. In some cases, there is one microlens 1504 for each of the modes 1412. In some cases, the microlenses 1504 have the same focal length f". In some cases, the cat' s-eye array also includes a planar mirror 1505 located behind the microlenses 1504. In some cases, the planar mirror 1505 is positioned such that a reflective face 1506 of the planar mirror 1505 is approximately one focal length f ' away from the microlenses 1504

[0076] In another embodiment, the cat' s-eye retroreflector 1500C may be implemented with a spherical micromirror array (i.e., a one or two dimensional array of curved mirrors fabricated onWSGR Docket No. 55436-755.601a common substrate). One way to fabricate this spherical micromirror array would be to deposit a dielectric high-reflectivity coating on the spherical surface of a microlens array. In this case, each micromirror is convex.

[0077] In some cases, the polygonal mirror 1500A, polygonal mirror 1500B, or cat' s-eye array 1500C may be replaced with a single planar mirror if said mirror is placed precisely in the image plane 1410. While, in some cases, this may provide a simpler embodiment for fabricating the optical cavity array 1400, it may a fully degenerate cavity that is highly susceptible to mode mixing and all variety of aberrations that can mix the resonator modes, in some cases.

[0078] In another embodiment, the retroreflector 1411 may be a curved mirror. For example, the curved mirror may be a spherical concave mirror that faces the second lens 1409 and has radius of curvature denoted ROC. Assuming the second lens 1409 has a focal length f2, the distance between the curved mirror and the second lens 1409 may be approximately f2+ROC. This arrangement creates a stable array of foci 1401. Assuming ROC=1 cm, the diameter of the resulting field of view is approximately 200 μm, limited by the spherical aberrations of the curved mirror. Advantageously, the optical cavity array 1400 may be simpler and less expensive to fabricate with this curved mirror instead of the polygonal mirror 1500A, polygonal mirror 1500B, or cat's eye retroreflector 1500C, in some cases.

[0079] Depending upon the quality of the optics, it may be necessary to individually tune the resonators (into resonance with atoms for example). In this case a low-loss transmissive phase modulator could be installed near the image plane 1410 (i.e., between the image plane 1410 and second lens 1409). Alternatively, once the necessary phase shifts are determined, a custom antireflection-coated phase plate could be installed within the cavity, which would advantageously incur lower optical loss as compared to the phase modulator.

[0080] When the optical cavity array 1400 is excited with light (as described above), an optical dipole trap is formed at each of the foci 1401. The resulting array of optical dipole traps (also referred to as an array of optical tweezers) may be used to trap cold or ultracold atoms therein. The optical cavity array 1400 may be the same as the optical trapping unit 210 of the method 200, herein. The resulting array of optical dipole traps may be the same as the optical trapping sites 211 of the optical trapping unit 210, described herein. Advantageously, this array of optical dipole traps is located far any physical surface (e.g., the high-NA lenses 1406 and 1408), thereby ensuring that cold atoms will not be ejected upon colliding with such a surface. To further facilitate cold-atom trapping, the optical cavity array 1400 may be arranged such that the array of foci 1401 is located inside in an ultra-high vacuum environment. However, some conventional vacuum chambers are so large that the high-NA lenses 1406 and 1408 cannot be placed outside of the vacuum chamber, in some cases, as they will then be too far apart fromWSGR Docket No. 55436-755.601each other to produce foci 1401 (i.e., waists) that are tight enough for the application at hand. In such cases, the lenses 1406 and 1408 can be brought closer to each other by mounting both of them inside the vacuum chamber. In some cases, additional components of the optical cavity array 1400 may be mounted inside the vacuum chamber. Vacuum windows may be used to on the vacuum system to allow light to pass therethrough.

[0081] Once the atoms spatially trapped, they may then be driven, measured, coupled, or probed, as needed for the application at hand. The close proximity of the optical dipole traps to each other advantageously allows the atoms to directly couple to each other. Such coupling between individual trapped atoms could be used, for example, to construct highly parallelized entanglement, as described herein.

[0082] FIG. 16 provides a side view of an alternate embodiment of an optical cavity array 1600 that is similar to the optical cavity array 1400 except that the high-NA retroreflector 1403 has been replaced with a planar mirror 1601 whose optically reflective surface coincides with the focal plane 1402. In some cases, the foci 1401 may be located at the reflective surface of the mirror 1601, which coincides with the focal plane 1402. In some cases, affixed to the front face of the planar mirror 1601 is a sample of non-linear emitters 1602. Examples of such non-linear emitters 1602 include, but are not limited to, rare-earth ions, quantum dots, and solid-state color centers (e.g., silicon vacancy centers in diamond, nitrogen vacancy centers in diamond, etc.). Advantageously, no trapping techniques (e.g., magnetic or optical trapping of atoms) are needed to confine the emitters 1602 at this position, in some cases.

[0083] In embodiments, multiple quantum emitters in any of the present embodiments can be coupled to each other by employing beam steering optics (e.g., near the cat' s-eye array 1500C) to couple the modes 1412 together.

[0084] In some cases, systems and methods for optical cavity arrays herein may be used to provide an array of spatially distinct optical tweezers. A non-limiting example of providing spatially distinct optical tweezers using an optical cavity array is provided in FIG. 17, in accordance with some embodiments. In some cases, the spatially distinct optical tweezers may be used to confine atoms 1701 in order to perform quantum computing operations, described herein. For example, when the optical cavity array 1400 is excited with light an optical dipole trap is formed at each of the foci 1401. The resulting array of optical dipole traps (also referred to as an array of optical tweezers) may be used to trap atoms 1701 therein. Advantageously, this array of optical dipole traps may be located far any physical surface (e.g., the high-NA lenses 1406 and 1408), thereby ensuring that atoms 1701 will not be ejected upon colliding with such a surface. To further facilitate trapping of atoms 1701, the optical cavity array 1400 may be arranged such that the array of foci 1401 is located inside in an ultra-high vacuum environment.WSGR Docket No. 55436-755.601In some cases, however, some conventional vacuum chambers may be so large that the high-NA lenses 1406 and 1408 cannot be placed outside of the vacuum chamber, as they will then be too far apart from each other to produce foci 1401 (i.e., waists) that are tight enough for the application at hand (such as confining atoms in optical traps). In such cases, the high-NA lenses 1406 and 1408 can be brought closer to each other by mounting both of them inside the vacuum chamber. In some cases, additional components of the optical cavity array 1400 may be mounted inside the vacuum chamber. In some cases, vacuum windows may be used to on the vacuum system to allow light to pass therethrough.

[0085] Once the atoms 1701 are trapped in the optical tweezers, they may then be driven, measured, coupled, or probed, as needed to perform a quantum computing operation, described herein. For example, the atoms 1701 may be used in the method 600 of FIG. 6 for performing non-classical computation, described herein. For example, the atoms 1701 may be used in the method 700 of FIG. 7 for performing non-classical computation, described herein. For example, the atoms 1701 may be used in the method 800 of FIG. 8 for performing non-classical computation, described herein. In some cases, the atoms 1701 may be used in the process 1200 for performing continuous non-classical computation as provided in FIG. 12, herein.

[0086] In some cases, the close proximity of the optical dipole traps to each other can enable direct coupling or interaction between neighboring atoms. In some examples, this coupling or interaction between individual trapped atoms may be used to perform multi -qubit gates, to construct a multi-qubit quantum processor, or as quantum memory, as described herein.

[0087] In some cases, the optical cavity array 1400 may be excited using any light source described herein. For example, the optical cavity array 1400 may be excited using a light source which is the same or substantially the same as the light source described with reference to FIG.3A. In some cases, the optical cavity array 1400 is excited by one or more lasers described herein.

[0088] In some cases, the atoms 1701 are the same atoms described herein. In some cases, the atoms 1701 may comprise qubits, described herein. For example, the atoms 1701 may comprise neutral atoms. The atoms 1701 may comprise uncharged atoms.

[0089] In some examples, the atoms 1701 may comprise alkali atoms. In some examples, the atoms 1701 may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. In some examples, the atoms 1701 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. In some examples, the atoms 1701 may comprise alkaline earth atoms. In some examples, the atoms 1701 may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms,WSGR Docket No. 55436-755.601strontium (Sr) atoms, or barium (Ba) atoms. In some examples, the atoms 1701 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, orbarium-138 atoms. In some examples, the atoms 1701 may comprise rare earth atoms. In some examples, the atoms 1701 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. In some examples, the atoms 1701 may comprise scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium- 142 atoms, praseodymium- 141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium- 152 atoms, samarium- 154 atoms, europium-151 atoms, europium- 153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium-156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium- 156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium- 164 atoms, erbium -166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms.

[0090] In some examples, the atoms 1701 may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. In some examples, the atoms 1701 may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. In some examples, the atoms 1701 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. In some examples, the atoms 1701 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. In some examples, the atoms 1701 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. In some examples, the atoms 1701 may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc,WSGR Docket No. 55436-755.601Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, In some examples, the atoms 1701 may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium -45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium -136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium- 152 atoms, samarium- 154 atoms, europium-151 atoms, europium- 153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium-156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium- 156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium- 164 atoms, erbium -166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. In some examples, the atoms 1701 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,WSGR Docket No. 55436-755.601europium-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. In some examples, the atoms 1701 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 that is within a range defined by any two of the preceding values.WSGR Docket No. 55436-755.601Examples of Microcavity Interconnected Architectures

[0091] This disclosure hereby incorporates by reference for all purposes, International Patent Publication No. WO2023 / 220415; International Patent Publication No. WO2023 / 192167;Nickerson, N. H., et al., Freely Scalable Quantum Technologies Using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links, Phys. Rev. X 4, 041041 (2014); and Nigmatullin, R., et al., Minimally complex ion traps as modules for quantum communication and computing, arXiv: 1605.00111 [quant-ph], available at https: / / doi.org / 10.48550 / arXiv.1605.00111.

[0092] In one type of neutral atom quantum computing, atom-array -based quantum computing, atoms may be confined in an array of optical tweezers and used as qubits to perform quantum computing operations. In some cases, these arrays of optical tweezers may the same as those formed by optical cavity arrays, described herein. In some cases, these arrays of optical tweezers may confine atoms in arrays which are coincident with foci of the optical cavity arrays, described herein. In some cases, atoms may be trapped by any other means to be coincident with foci of the optical cavity arrays, described herein. One example strength of atom-array-based quantum computers is the ability to arbitrarily control the connectivity of the atoms in the array by moving the atoms around in space. This may be challenging for at least two reasons: (1) atoms have inertia, so moving them around in space may be relatively slow - requiring, in typical implementations, 100 μs or more; (2) moving the atoms around may necessitate high-bandwidth, high-dimensional control of light, for example, many optical tweezers may need to be moved independently.

[0093] Systems and method disclosed herein may provide a solution to the challenge of highdimensional optical control. Systems and methods of the present disclosure may use optical control hardware to generate distributed entanglement across an atom-array based quantum computer. For example, systems and method of the present disclosure may use optical control hardware to redirect optically generated Bell pairs between spatially separated atoms. In some cases, these distributed Bell pairs can be used to produce interaction between spatially separated atoms, rather than physically bringing the atoms together. In some cases, distributed Bell pairs can be used to deterministically teleport qubits or gates across the system, which combined with local Rydberg gates, can provide arbitrary connectivity between arrays of atoms or quantum computing modules. In combination, optical cavities may allow for efficient light collection, without atom moves (or without long-range atom moves), to achieve fully (or at least highly) connected quantum computing:

[0094] The present application provides systems, methods, non-classical computers, and methods of use thereof which may comprise an array of spatially distinct optical traps, wherein the array comprises a plurality of atoms; an optical cavity array, the optical cavity arrayWSGR Docket No. 55436-755.601comprising the array of spatially distinct optical modes; and an optical control hardware, wherein the optical control hardware reconfigurably combines optical paths from distinct atoms for entanglement generation. In some cases, the optical control hardware is configured to either (a) direct light from one atom to another to generate distributed entanglement through atomatom Bell pairs, or (b) to combine light from two atoms on shared detectors for heralded atomatom Bell pair generation.

[0095] An optical cavity array may comprise any example of an optical cavity array described herein, such as the optical cavity array 1400 provided in FIG. 4. An optical cavity array may comprise an example of the optical cavities or cavity arrays as described in International Patent Publication No. WO2023 / 220415; International Patent Publication No. WO2023 / 192167, which are incorporated by reference herein.

[0096] In some cases, systems and methods for networked qubit architectures, herein, can enable distributed entanglement between non-neighboring atoms or qubits in an optical cavity array. A non-limiting example of optical control hardware 1800A for distributed entanglement within an optical cavity array is provided in FIG. 18A, in accordance with some embodiments herein. In some cases, the optical control hardware 1800A is configured to receive light from an atom at a first site within the optical cavity array and to direct the light to the second site within the optical cavity array. In some cases, atom-photon entanglement is generated at the first site, wherein a photon is redirected to the second site, and wherein at the second site an atom at the second site absorbs the photon to generate distributed atom-atom entanglement between the first site in the optical cavity array and the second site in the optical cavity array. In some cases, atom-photon entanglement is generated at each of the first site and the second site, wherein a first photon is generated at the first site, wherein a second photon is generated at the second site, and wherein distributed atom-atom entanglement is generated by interfering the photons on a beam splitter and conditioning on two-photon detections. Advantageously, the optical control hardware 1800A may enable distributed entanglement between non-neighboring atoms in the optical cavity.

[0097] In some cases, systems and methods disclosed herein may leverage heralded Bell-pair generation for distributed entanglement. For example, distributed entanglement may be achieved by pitch and catch entanglement generation, where atom-photon entanglement is generated at atom A, the photon is captured in a cavity and directed to atom B, where its reabsorption (and the consequent atom-atom entanglement) is heralded by detection that the atom has left its initial state. In some cases, the photon may be directed from atom A to atom B by optical control hardware 1800A, described herein. Systems and methods disclosed herein may leverage heralded Bell-pair generation through, for example, two-photon post-selection, where atom-WSGR Docket No. 55436-755.601photon entanglement is generated at both atom A and atom B, and that entanglement is converted to atom-atom entanglement by interfering the photons on a beam splitter and conditioning on two-photon detections.

[0098] In some cases, systems and methods for networked qubit architectures, herein, can enable distributed entanglement between separate optical cavity arrays. A non-limiting example of optical control hardware 1800B for distributed entanglement between separate optical cavity arrays is provided in FIG. 18B, in accordance with some embodiments herein. In some cases, the optical cavity array comprises a first optical cavity array and a second optical cavity array, wherein the optical control hardware 1800B is configured to redirect an optically generated Bellpair from a first site within the first optical cavity array to a second site within the second optical cavity array. In some cases, the first optical cavity array and the second optical cavity array are optically distinct. In some cases, the first optical cavity array and the second optical cavity array comprise distinct cavity optics. In some cases, the first optical cavity array and the second optical cavity array are spatially distinct. In some cases, the first optical cavity array and the second optical cavity array are connected by fiber bundle. In some cases, the first optical cavity array and the second optical cavity array are located in separate quantum computing modules. In some cases, the first optical cavity array and the second optical cavity array are located in different vacuum chambers. In some cases, the first optical cavity array and the second optical cavity array are located in spatially distinct region. In some cases, the first optical cavity array and the second optical cavity array are separated by a distance of greater than 1 meter.

[0099] For example, the first optical cavity array and the second optical cavity array may be separated by a distance of less than about 1 millimeter (mm) to greater than about 100 meters (m) and increments therein. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 mm. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 mm to about 1 centimeter (cm) and increments therein. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 cm. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 cm to about 10 cm and increments therein. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 10 cm. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 10 cm to about 1 m and increments therein. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 m. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 1 m to about 10 m andWSGR Docket No. 55436-755.601increments therein. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 10 m. In some examples, the first optical cavity array and the second optical cavity array may be separated by a distance of about 10 m to about 100 m and increments therein. In some examples, the first optical cavity array and the second optical cavity array are separated by a distance of about 100 m or greater.

[0100] In some cases, the first optical cavity array and the second optical cavity array are connected over the distance by free space optics. In some cases, the first optical cavity array and the second optical cavity array are connected over the distance by optical fibers. In some cases, the first optical cavity array and the second optical cavity array are connected over the distance by a combination of free space optics and optical fibers. In practice, the first optical cavity array and the second optical cavity array may be connected over any distance where photons may be passed with sufficient fidelity to preserve the distributed entanglement, without departing from the present disclosure.

[0101] In some cases, the first optical cavity array and the second optical cavity array are within a single non-classical computing module. In some cases, the first optical cavity array is within a first non-classical computing module and the second optical cavity array is within a second non-classical computing module.

[0102] In some cases, the optical control hardware 1800A or 1800B comprises a telescope system. A non-limiting example of a telescope system is provided in FIG. 14C, in accordance with some embodiments herein. In some cases, the telescope system 1800C comprises a local telescope 1801 and a global telescope 1802. In some cases, the local telescope 1801 comprises a local telescope array, wherein each local telescope 1801 of the array is configured to redirect light from a block or subset of cavities of the array of optical cavities. In some cases, the global telescope 1802 is configured to control a size of an array of light beams each corresponding to a cavity of the array of optical cavities. In some cases, the telescope system 1800C comprises two or more local telescopes 1801. In some cases, the telescope system 1800C comprises two or more global telescopes 1802. In some cases, the telescope system 1800C comprises two or more local telescopes 1801 and two or more global telescopes 1802. While elements of the optical control hardware 1800A and 1800B, including the telescope system 1800C, local telescope 1801, and global telescope 1802 are depicted in FIGs. 18A-18B in a particular arrangement, it will be understood that any number of optical elements comprising the optical control hardware 1800A and 1800B may be present in any arrangement without departing from the present disclosure.

[0103] In some cases, the optical control hardware 1800A or 1800B comprises one or more light redirection optics. In some cases, the optical control hardware 1800A or 1800B comprises aWSGR Docket No. 55436-755.601telescope system and one or more light redirection optics. In some cases, the optical control hardware 1800A or 1800B comprises one or more telescope systems and one or more light redirection optics. In some cases, the optical control hardware 1800A or 1800B comprises one or more telescope systems and a plurality of light redirection optics. In some cases, the one or more light redirection optics 1803 is configured to redirect light within the optical control hardware 1800A or 1800B. In some cases, the one or more light redirection optics 1803 provides arbitrary control over one or more light paths passing through the optical control hardware 1800A or 1800B. In some cases, the one or more light redirection optics 1803 enables the reconfigurable combining of optical paths from different sites in an optical cavity array, described herein. In some cases, the one or more light redirection optics 1803 enables the reconfigurable combining of optical paths from different sites in spatially distinct optical cavity arrays, described herein.

[0104] In some cases, the one or more light redirection optics 1803 include one or more of mirrors, reflectors, prisms, lenses, deflectors, gratings, or fiber optics. In some cases, the one or more light redirection optics 1803 comprises one or more lightfield directing arrays (LDAs). In some cases, the one or more light redirection optics 1803 comprises one or more optical phased arrays (OP As). In some cases, the one or more light redirection optics 1803 comprises one or more micro-electro-mechanical systems (MEMS). In some cases, the one or more light redirection optics 1803 comprises one or more MEMS mirror arrays. In some cases, the one or more light redirection optics 1803 comprises one or more MEMS prism arrays. In some cases, the one or more light redirection optics 1803 comprises one or more MEMS lens arrays. In some cases, the one or more light redirection optics 1803 comprises one or more programmable optical semiconductors. In some cases, the one or more light redirection optics 1803 comprises one or more individually tunable reflective optics.

[0105] In some cases, the one or more light redirection optics 1803 enables the reconfigurable combining of optical paths from different quantum computing modules, described herein. A non-limiting example of optical control hardware 1900 comprising one or more light redirection optics 1803 for combining optical paths from different quantum computing modules is provided in FIG. 19, in accordance with some embodiments.

[0106] The present disclosure provides methods of use for the system described herein. For example, the present disclosure provides a method 2000 of connecting spatially separate atoms within a trapped atom quantum computer. A non-limiting example of a method 2000 for connecting spatially separate atoms within a trapped ion computer is provided in FIG. 20, in accordance with some embodiments. In some cases, the method 2000 may comprise trapping at 2001 a plurality of atoms within an optical cavity array, the optical cavity array supporting aWSGR Docket No. 55436-755.601plurality of optical cavities comprising an array of spatially distinct optical traps, and reconfigurably combining at 2002 optical paths from distinct atoms for entanglement generation.

[0107] In some cases, reconfigurably combining at 2002 comprises receiving light from an atom at the first site within the optical cavity array and directing the light to the second site within the optical cavity array.

[0108] In some cases, reconfigurably combining at 2002 comprises either (i) directing light from one atom to another to generate atom-atom bell pairs, or (ii) combining light from two atoms on shared detectors for heralded atom-atom bell pair generation.Examples of Distributed Entanglement

[0109] In an aspect, the present disclosure provides systems and methods for networked quantum computing. In some cases, the systems and methods for networked quantum computing, disclosed herein, employ optical control hardware described herein, such as 1800A and 1800B provided in FIG. 18.

[0110] The present disclosure provides methods of use for the system described herein. For instance, the present disclosure provides a method 2100 for networked quantum computing. A non-limiting example of a method 2100 for networked quantum computing is provided in FIG.21, in accordance with some embodiments. In some cases, the method 2100 may comprise providing at 2101 a first plurality of qubits in a first set of optical cavities, and a second plurality of qubits in a second set of optical cavities, wherein qubits of the first plurality of qubits and the second plurality of qubits both comprise atoms in an array of spatially distinct optical traps. In some cases, the method 2100 may comprise distributing entanglement at 2102 between the first plurality of qubits and the second plurality of qubits to form a distributed state of the plurality of qubits between the first set of optical cavities and the second set of optical cavities. In some cases, the method 2100 comprises performing at 2103 a quantum computing operation based at least in part by interacting with the distributed state.[OHl] In some cases, performing at 2103 a quantum computing operation may be the same as non-classical computations, described herein. For example, performing at 2103 a quantum computing operation may comprise the method 600 of FIG. 6 for performing non-classical computation, described herein. For example, performing at 2103 a quantum computing operation may comprise the method 700 of FIG. 7 for performing non-classical computation, described herein. For example, performing at 2103 a quantum computing operation may comprise the method 800 of FIG. 8 for performing non-classical computation, described herein. In some cases, performing at 2103 a quantum computing operation may comprise the process 1200 for performing continuous non-classical computation as provided in FIG. 12, herein.WSGR Docket No. 55436-755.601

[0112] In some cases, the first set of optical cavities and the second set of optical cavities are any optical cavity described herein. For example, the first set of optical cavities and the second set of optical cavities may comprise the optical cavity 1400 provided in FIG. 14 herein.

[0113] In some cases, the first plurality of qubits and the second plurality of qubits are in the same optical cavity array. In some cases, the first plurality of qubits and the second plurality of qubits are interspersed within the same region of a 2D array. In some cases, the first plurality of qubits and the second plurality of qubits are in different regions of a 2D layer of an atom array. In some cases, the first plurality of qubits and the second plurality of qubits are in separate optical cavity arrays.

[0114] In some cases, generating entanglement at 2102 comprises pitch and catch entanglement. In some cases, generating entanglement at 2102 comprises two-photon post-selection entanglement. In some cases, generating entanglement at 2102 comprises heralded Bell-pair generation for distributed entanglement. For example, distributed entanglement may be achieved by pitch and catch entanglement generation, where atom-photon entanglement is generated at atom A, the photon is captured in a cavity and directed to atom B, where its reabsorption (and the consequent atom-atom entanglement) is heralded by detection that the atom has left its initial state. In some cases, the photon may be directed from atom A to atom B by optical control hardware 1800A or 1800B, described herein. Systems and methods disclosed herein may leverage heralded Bell-pair generation thru, for example, two-photon post-selection, where atom-photon entanglement is generated at both atom A and atom B, and that entanglement is converted to atom-atom entanglement by interfering the photons on a beam splitter and conditioning on two-photon detections.Resource States

[0115] In some cases, the distributed state comprises a resource state. For example, a distributed Bell state may be consumed as a resource state to perform a quantum computing operation herein. In some cases, the distributed state may be interacted with another resource state to generate a distributed resource state. For example, a distributed Bell pair may be entangled with a graph state to produce a distributed graph state. A resource state, herein, may refer to a system of atoms or qubits which is consumed in the process of performing a quantum computing operation. For example, a distributed Bell pair may be a resource state, where the entangled state of the Bell pair is consumed to distribute entanglement across separate atoms or qubits in an optical cavity array, or across separate optical cavity arrays. Another example of a resource state is a graph state, which is formed by entangling multiple qubits or atoms. A further example of a resource state is a cluster state, which is formed by entangling multiple qubits or atoms in a regular grid. Resource states may be formed between nearest neighbors or may be formed acrossWSGR Docket No. 55436-755.601non-neighboring atoms of an optical cavity array, across multiple optical cavity arrays, or across multiple quantum computing modules. For example, a resource state may be formed by entangling nearest neighbors of an optical cavity array to form a graph state or cluster state or may be formed by entangling spatially distinct atoms or qubits using distributed entanglement. Resource states may be combined to form larger resource states. For example, two cluster states formed by entangling nearest neighbors of an array may be entangled using distributed entanglement, forming a larger cluster state.

[0116] In some cases, the particular arrangement of the resource state (which atoms or qubits are entangled with which other atoms or qubits) may determine the function performed by the resource state. As the resource state is consumed to perform a quantum computing operation, herein, the resource state may be continuously generated, such as at 2104 of the method 2100.

[0117] In some cases, resource states may enable space-time tradeoffs for performing quantum circuits. Space-time tradeoffs, herein, refer to the trade off between the number of physical qubits (space) and circuit depth (time) which may be achieved using resource states. For example, some quantum circuits which, on the one hand, may be performed as a sequence of sequential operations on a small number of qubits, may instead be performed in a smaller number of operations on a larger number of qubits. In this way, resource states can enable efficient performance of quantum circuits across atoms in optical cavity arrays, in particular, when distributed entanglement is used to cut down on or eliminate atom moves in the circuit. State Purification

[0118] In some cases, the distributed state may be generated in a probabilistic manner. Put simply, in some examples, not all distributed entanglement operations may succeed with absolute certainty. To produce a distributed state, distributed entanglement operations may be repeated more than once until successful. In some cases, systems and methods herein may employ heralded generation methods for producing a distributed state, such as heralded Bell pair generation described herein. Heralded methods can communicate that distributed state generation was successful, without collapsing entanglement of the distributed state.

[0119] Once a distributed state has been created, it may have a fidelity that is less than 100%, in some cases. Fidelity of a distributed state describes its proximity to an ideal, perfectly entangled state in state space. In the systems and methods herein, a high fidelity of a distributed state is important to increase successful completion of downstream operations, such as entangling states, and teleporting quantum gates. To produce a distributed state with a high likelihood of a high fidelity, multiple copies of a distributed state may be distilled to produce one high fidelity distributed state. A non-limiting example of a method 2200 for distilling a distributed state is provided in FIG. 22, in accordance with some embodiments. In some cases, the method 2100WSGR Docket No. 55436-755.601further comprises prior to performing the quantum computation at 2103, distilling the distributed state using n copies of the distributed state, wherein n is an integer greater than one.

[0120] A non-limiting example of a circuit 2300 for distilling distributed states is provided in FIG. 23, in accordance with some embodiments. In the example circuit 2300, n=3, and three copies of a distributed state are distilled to produce one high fidelity state. A non-limiting example of a circuit 2300 for distilling distributed states is provided in FIG. 23, in accordance with some embodiments. In the example circuit 2300, n=3, and three copies of a distributed state are distilled to produce one high fidelity state. In one example case, a circuit such as 2300 where n=3 may be performed in about 60 microseconds (μs), while a different example circuit where n=4 may be performed in about 40 μs. A non-limiting example of a circuit 2400 for distilling distributed states is provided in FIG. 24, in accordance with some embodiments. In the example circuit 2400, n=4, and four copies of a distributed state are distilled to produce one high fidelity.

[0121] In practice, two levels of purification may be useful in order to suppress the original infidelity quadratically, see for example, Nickerson, N. H., et al., Freely Scalable Quantum Technologies Using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links, Phys. Rev. X 4, 041041 (2014), which is incorporated by reference herein.

[0122] In some cases, the circuits for performing distillation of a distributed state may be performed using resource states. In some cases, the resources states used to perform distillation of a distributed state may be unit cells. A non-limiting example of unit cells for distillation of a distributed state where n=4 is provided in FIGs.25A-25B, in accordance with some embodiments. Based on the connectivity of this circuit, and nearest-neighbor interactions, the unit cell in this case would look like a square or row with an appendage. Placing the application qubit within the sub-square or sub-row could improve local connectivity between application qubits from different unit cells, enabling much faster multi-qubit operations for these nearby qubit pairs in some cases.

[0123] In some cases, the circuit diagrams in FIG. 23 and FIG.24 may use single site multiqubit addressing. Achieving this without movement may be implanted by things like: 2- or 3-photon multi-qubit; single-site clock; or single-site UV. Given that 1 purified, remote multiqubit gate may take ~50 μs in some cases, these addressing schemes should be parallelizable and fairly arbitrary across the array with a logical cycle time significantly shorter than 800 μs, in some cases.

[0124] While the above describes connectivity without atom movement, it may still be useful to move atoms for various applications. For example, it may be useful to rearrange atoms for reloading purposes. Thus, the methods described herein are not exclusive of some atomWSGR Docket No. 55436-755.601movement. Further there may be a continuum of cases where some atom movement is used based on the timescale of the operation needed.

[0125] In some cases, unit cells may be formed using entanglement between nearest neighbors of an optical cavity array. In some cases, unit cells may be formed using distributed entanglement across an optical cavity array. In some cases, unit cells may be formed using distributed entanglement across multiple optical cavity arrays. In some cases, unit cells may be formed using distributed entanglement across multiple quantum computing modules.Teleporting Gate Operations

[0126] In some cases, systems and methods herein provide for the teleportation of quantum gates using distributed entanglement. A non-limiting example of a method 2600 for teleporting gates is provided in FIG. 26, in accordance with some embodiments. In some cases, the method 2100 further comprises at 2103, teleporting a gate operation. In some cases, a single gate may be teleported using systems and methods herein. In some cases, multiple gates may be teleported simultaneously or contemporaneously. In some cases, all gates in a quantum circuit may be performed using teleported gates. In some cases, some gates in a quantum circuit may be performed locally through nearest neighbor atomic interactions and atom moves, while other gates in the circuit are performed by teleporting gates. Systems and method herein provide for a continuum of gate operations ranging from all gates being performed locally, to all gates being performed through teleported gates.

[0127] For example, returning to FIG. 19, each code block has atom movement for gates within block, but all inter-block gates may be done through optical interconnects. In some cases, the method further comprises implementing an error correction operation. In some cases, the error correction operation comprises syndrome extraction. In some cases, the syndrome extraction comprises transversal CNOT operations. In some cases, all, or a significant subset, of gates may be performed through optical interconnects. For example, there may be a case where Rydberg gates may not be needed in favor of optical interconnects. In some cases, each cavity comprises an atom which is coupled to a cavity and can do multi-qubit gates with a nearby atom.

[0128] For example, assume a high-fidelity bell pair is already shared between Alice and Bob. A gate can then be teleported if Alice and Bob each apply local two-qubit gates between their qubit from the Bell pair and their target qubit, followed by a single-qubit operation. This functionality may create high qubit connectivity and thereby serve as an alternative to qubit movement in an atom-based architecture.

[0129] In some cases, the method 2100 further comprises performing at 2602 one or more quantum gates on a first qubit, a second qubit, or both, based at least in part on a measured quantum state of the first plurality of qubits or the second plurality of qubits. In some cases, theWSGR Docket No. 55436-755.601first qubit, the second qubit, or both is entangled with the distributed state. In some cases, the first qubit, the second qubit, or both are locally entangled with the distributed state (i.e., via nearest neighbor interactions in an optical cavity array, via atom movement, or both). In some cases, the first qubit, the second qubit, or both are entangled with the distributed state using a distributed Bell pair. In some cases, the first qubit, the second qubit, or both are entangled with the distributed state using a distributed resource state.

[0130] In some cases, the method 2100 further comprises performing at 2603 one or more quantum gates on a third plurality of qubits, a fourth plurality of qubits, or both, based at least in part on a measured quantum state of the first plurality of qubits or the second plurality of qubits. In some cases, the third plurality of qubits, a fourth plurality of qubits, or both is entangled with the distributed state. In some cases, the third plurality of qubits, a fourth plurality of qubits, or both are locally entangled with the distributed state (i.e., via nearest neighbor interactions in an optical cavity array, via atom movement, or both). In some cases, the third plurality of qubits, a fourth plurality of qubits, or both are entangled with the distributed state using a distributed Bell pair. In some cases, the third plurality of qubits, a fourth plurality of qubits, or both are entangled with the distributed state using a distributed resource state. In some cases, the third plurality of qubits, a fourth plurality of qubits, or both are data qubits.

[0131] In some cases, the first qubit and the second qubit are located in spatially distinct optical traps. In some cases, the third plurality of qubits and the fourth plurality of qubits are located in spatially distinct optical traps. In some cases, the first qubit and the second qubit are located in non-neighboring optical traps. In some cases, the third plurality of qubits and the fourth plurality of qubits are located in non-neighboring optical traps. In some cases, the first qubit and the second qubit are located in spatially different optical cavity arrays. In some cases, the third plurality of qubits and the fourth plurality of qubits are located in different optical cavity arrays. In some cases, the first qubit and the second qubit are located in different quantum computing modules. In some cases, the third plurality of qubits and the fourth plurality of qubits are located in different quantum computing modules.Error Correction

[0132] In some cases, teleported gates described herein comprise one or more processes for error correction in a quantum computer. In some cases, teleported gates described herein comprise a portion of an error correcting code. In some cases, the error correcting code comprises a topological code. In some cases, the topological code is a stabilizer code. In some embodiments, the error correcting code is a surface code, a color code, a toric code, a shor style code, or a qLDPC code. In some cases, the color code is a Steane code. In some embodiments,WSGR Docket No. 55436-755.601the shor style code is a Bacon-shor code. In some embodiments, the qLDPC code is a hypergraph product code.

[0133] In some cases, the quantum computing operation is an error correction operation. In some cases, the quantum computing operation is a logical operation, performed by logical qubits, as a portion of an error correction operation. For example, in an error corrected quantum computation, the quantum computing operation may comprise an interaction between logical qubits while error correction operations within the logical qubit may comprise local two-qubit gates, such as Rydberg gates herein. Logical code blocks may be local while interactions between logical code blocks may be non-local and connected by cavities herein.

[0134] In some cases, the quantum computing operation is a transversal gate between logical qubits. In some cases, systems and method herein provide for efficient operation of transversal gates between logical qubits. Transversal gates may perform logical operations by performing the same operation over the physical qubits making up the logical qubits. Systems and methods for teleporting quantum gates, described herein, can enable transversal gates to be performed efficiently across multiple logical qubits which may be physically separated in an array.Transversal gates may be advantageous, as they don’t propagate errors within each code block comprising the logical qubits, allowing for fault tolerance within a quantum circuit.

[0135] In some cases, the transversal gate may be a portion of an error correction process, described herein. In some cases, the transversal gate may be a portion of a Steane style method of error correction. In some cases, the transversal gate may be a portion of a Knill style method of error correction. In some cases, teleported gates described herein may be used to perform the example method for error correction 1000 of FIG. 10, herein. In some cases, the system for error corrected quantum computing provided in FIG. 11 may be configured to employ teleported gates, described herein.Measurement-Based Quantum Computing

[0136] In some cases, resources states may be formed such that complex quantum circuits are carried out as a sequence of measurements of the resource state. In some cases, a dimension of a resource state may be time encoded, where earlier time slices of the resource state causally affect the state of later time slices as a means of passing information, performing quantum logic gates, and completing a quantum circuit. For example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may help to create a three-dimensional (3D) cluster state or other graph states, described herein. In some cases, entanglement in two dimensions may result from spatial proximity of qubits and local two-qubit gates, while entanglement in the third dimension may result from fusion operations between spatially distantWSGR Docket No. 55436-755.601qubits. In some cases, just a handful of time slices of the cluster state need to be sustained at any time.

[0137] Fusion operations are an example of a process that allows for the creation of entanglement between two separate resource states, described herein. Time-like fusion, on the other hand, can allow for the creation of entanglement between two resource states produced by the same physical device at two different time steps. In some cases, this involves creating entanglement between layers of resource states that are separated by two time steps. This process may allow for the creation of two copies of the same state, effectively doubling the number of logical qubits.

[0138] In some cases, the first plurality of qubits, the second plurality of qubits, or both comprises a graph state in a first location, and wherein generating the distributed state in 2102 further comprises generating a distributed graph state between the first location and a second location. In some cases, the first plurality of qubits, the second plurality of qubits, or both comprises a cluster state in a first location, and wherein generating the distributed state in 2102 further comprises generating a distributed cluster state between the first location and a second location. In some cases, the distributed graph state comprises two or more entangled time slices. In some cases, the distributed cluster state comprises two or more entangled time slices.

[0139] In some cases, the method 2100 further comprises performing measurement-based quantum computing, at least in part, by performing operations on multiple time slices of a distributed graph state or distributed cluster state in order to complete a quantum circuit. A nonlimiting example of a method 2700 for measurement-based quantum computing is provided in FIG. 27, in accordance with some embodiments. In some cases, performing a quantum operation at 2103 comprises performing a measurement of a qubit state in a first time slice of the two or more entangled time slices, wherein the measurement influences a second time slice of the two or more entangled time slices. In some cases, the method 2100 further comprises performing a quantum circuit at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices. In some cases, the method 2100 further comprises performing a quantum computation at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices

[0140] A non-limiting example 2800 of entangled time slices is provided in FIG. 28, in accordance with some embodiments. The example 2800 illustrates an example of a 2D matterqubit layout and interconnects. As illustrated, each vertex and X in a grid represents a matter qubit while edges represent pair-wise entanglement between those qubits. The vertices are data qubits and the X’s are qubits for fusion operations. The entanglement within each grid may beWSGR Docket No. 55436-755.601achieved with fast local multi-qubit operations, either with single-site addressing, or with global addressing and 4 small parallel movements of every other vertex qubit: up, down, left, and right. In some cases, for the global two-qubit approach, the X qubits may be of different atomic species.

[0141] As illustrated in 2800, the grids may represent separate atom arrays that are entangled sequentially (e.g., from left to right). Entanglement may be accomplished, in some cases, through distributed entanglement, described herein. In some cases, the X qubits in the tl array may first be used for fusion operations with the tO array and then reused for fusion operations with t2. In some cases, in between, these X qubits in tl may be used to reestablish entanglement with their diagonal counterparts.

[0142] As illustrated in the example 2800 shows the case where two-qubit gates within an atomarray or grid are implemented with nearest-neighbor connectivity. This is an example of a cluster state formed by nearest-neighbor entanglement in an optical cavity array, described herein. To obtain more general connectivity graphs between qubits in a single array, in some cases, optical control devices, described herein, may create Bell pairs within an array, or between separate optical cavity arrays. For example, a unit cell of several qubits may serve as a single effective qubit, with one serving as the data qubit, one serving as a remote Bell-pair generator, and two others providing Bell state purification, described herein. In some cases, the unit cell may be the same as or similar to Nigmatullin, R., Ballance, C. J., Beaudrap, N. de, & Benjamin, S. C. (2016). Minimally complex ion traps as modules for Quantum Communication and computing. New Journal of Physics, 18(10), 103028. https: / / doi.org / 10.1088 / 1367-2630 / 18 / 10 / 103028, which is incorporated by reference herein for all purposes.

[0143] In some cases, time-like fusion may be achieved across many atoms or qubits in an optical cavity array, or between separate optical cavity arrays. A non-limiting example of a system for time-like fusion is provided in FIG. 29, in accordance with some embodiments. Time-like fusion, herein, refers to a particular type of distributed entanglement, wherein different resource states generated by the same quantum emitter at different points in time are fused together using entanglement operations. In some cases, time-like fusion may be used to generate a 3D resource state used for measurement-based quantum computing. In the example, 2900 each optical cavity array 2901 may serve as a 2D plane of the cluster state. These planes may be entangled probabilistically (e.g., for measurement-based quantum computing), and sequentially following the arrow from one cavity array to the next.

[0144] In some cases, as an extension of this idea, microcavities may create fusion-based links between edges of the 2D arrays to increase the number of logical qubits in one timeslice. In suchWSGR Docket No. 55436-755.601case, each optical cavity array 2901 in the diagram may correspond to this group of edge-connected 2D arrays.

[0145] In some cases, the array loading time of many milliseconds may be a bottleneck for computations comprising large, distributed resource states. In some cases, the clock rate for the overall system may be the time to apply the parallel one-qubits and readout the atoms. This may be because the loading phase of some modules may overlap in time with the computation phase of others. For example, the limit then may become a question of how many modules can be built to work as an overall system.

[0146] Accordingly, to recover the filled array, it may be preferrable, in one example, to reload single atoms from a distant reservoir. In another example, it may be preferrable to drop the array after measurement and build a new array via enhanced loading. Because, in some cases, qubits may be discarded after the final measurement in this architecture, it may be beneficial to reload the array. For example, reloading may be with enhanced (e.g., about >96%) trap loading of ytterbium-171, or the like. Or, in some cases, it may be beneficial to move the imperfect array back to the dipole trap and load the empty sites using a shelving sequence.

[0147] In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may provide cavities to circumvent midcircuit atom reloading in gate-based quantum computing. For example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may perform many rounds of a gate-based QEC etc. on the 2D array. In some cases, when the atom loss probability hits some threshold, state teleportation to another array may be prepared via microcavities. For example, because this probably may implement bell state purification, this scheme may be more consistent with the approach where multi-qubit gates are teleported and local multi-qubit operations perform purification. In some cases, bouncing back and forth between the 2 arrays may be performed whenever atom loss gets too large.Examples of Systems for Performing a Non-Classical Computation

[0148] 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.

[0149] 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 comprise an array of optical tweezers, described herein. The optical trapping units may comprise an array of optical tweezers producedWSGR Docket No. 55436-755.601by an optical cavity, such as the optical cavity 1400 of FIG. 14, herein. The optical trapping units may be configured to generate a plurality of optical trapping sites. The optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites. For instance, the optical trapping units may be configured to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. The optical trapping units may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 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. In some cases, the optical trapping sites may be the same as the plurality of transversely separated optical foci 1401 of an optical cavity array 1400, herein. In some cases, the optical trapping sites may be coincident with the optical foci 1401.

[0150] 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.

[0151] 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. The atoms may be the same as atoms described elsewhere herein, such as the atoms 1701 of FIG. 17.

[0152] 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 mechanicallyWSGR Docket No. 55436-755.601entangled. Two or more atoms herein may be entangled using nearest-neighbor entanglement operations, herein. Two or more atoms herein may be entangled using distributed entanglement, described herein. 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, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100 ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms, 8 ms, 7 ms, 6 ms, 5 ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 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.

[0153] 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 -WSGR Docket No. 55436-755.601140 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.

[0154] 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,WSGR Docket No. 55436-755.601gadolinium- 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 of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium -23 atoms, potassium-39 atoms, potassium -40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-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,WSGR Docket No. 55436-755.601calcium-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 that is within a range defined by any two of the preceding values.

[0155] 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.

[0156] 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.

[0157] 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 a3P1or3P2manifold. The first and second atomic states may comprise first and second hyperfine states,WSGR Docket No. 55436-755.601respectively, on a3P1or3P2manifold of any atom described herein, such as a strontium-873P1manifold or a strontium-873P2manifold.

[0158] In some cases, the first and second atomic states are first and second hyperfine states of a first electronic state. Optical excitation may be applied between a first electronic state and a second electronic state. The optical excitation may excite the first hyperfine state or the second hyperfine state to the second electronic state. A single-qubit transition may comprise a two-photon transition between two hyperfine states within the first electronic state using a second electronic state as an intermediate state. To drive a single-qubit transition, a pair of frequencies, each detuned from a single-photon transition to the intermediate state, may be applied to drive a two-photon transition. In some cases, the first and second hyperfine states are hyperfine states of the ground electronic state. The ground electronic state may not decay by spontaneous or stimulated emission to a lower electronic state. The hyperfine states may comprise nuclear spin states.

[0159] In some cases, the hyperfine states comprise nuclear spin states of a strontium-871S0manifold and the qubit transition drives one or both of two nuclear spin states of strontium-871S0to a state detuned from or within the3P2or3P1manifold. In some cases, the one-qubit transition is a two photon Raman transition between nuclear spin states of strontium-871S0via a state detuned from or within the3P2or3P1manifold. 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.

[0160] In some cases, the hyperfine states comprise nuclear spin states of ytterbium.

[0161] 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.

[0162] 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 beWSGR Docket No. 55436-755.601separated 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.

[0163] 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).

[0164] 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 the3P1manifold may be many gigahertz from the single and two-WSGR Docket No. 55436-755.601qubit 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.

[0165] 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 crosstalk or to improve gate or detection fidelity. Such a storage or shelving process may be atom-selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the1S0state in strontium-87 to the3P0or3P2state in strontium-87.

[0166] 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.

[0167] 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 crosstalk. 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 crosstalk between neighboring qubits undergoing transitions. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc.

[0168] 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, avalancheWSGR Docket No. 55436-755.601diodes, 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.

[0169] 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 the1S0state in strontium-87 to the1P1state in strontium-87. The1P1state in strontium-87 may fluoresce. The lower state of the qubit transition may comprise two nuclear spin states in the1S0manifold. 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.,3P0state in strontium-87). In a second excitation, the imaging transition may be excited. The first transition may reduce crosstalk between neighboring atoms during computation. Fluorescence generated from the imaging transition may be collected on one or more readout optical units 230.

[0170] 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.

[0171] 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.

[0172] 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 vacuumWSGR Docket No. 55436-755.601pressure 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.

[0173] The vacuum units may be configured to maintain the system 200 at a pressure of at most about IO’6Pa, 9 x IO’7Pa, 8 x IO’7Pa, 7 x IO’7Pa, 6 x IO’7Pa, 5 x IO’7Pa, 4 x IO’7Pa, 3 x IO’7Pa, 2 x IO’7Pa, IO’7Pa, 9 x IO’8Pa, 8 x IO’8Pa, 7 x IO’8Pa, 6 x IO’8Pa, 5 x IO’8Pa, 4 x IO’8Pa, 3 x IO’8Pa, 2 x IO’8Pa, IO’8Pa, 9 x IO’9Pa, 8 x IO’9Pa, 7 x IO’9Pa, 6 x IO’9Pa, 5 x IO’9Pa, 4 x IO’9Pa, 3 x IO’9Pa, 2 x IO’9Pa, IO’9Pa, 9 x IO’10Pa, 8 x IO’10Pa, 7 x IO’10Pa, 6 x IO’10Pa, 5 x IO’10Pa, 4 x IO’10Pa, 3 x IO’10Pa, 2 x IO’10Pa, IO’10Pa, 9 x 10’11Pa, 8 x 10’11Pa, 7 x 10’11Pa, 6 xlO'11Pa, 5 x IO’11Pa, 4 x IO’11Pa, 3 x IO’11Pa, 2 x IO’11Pa, IO’11Pa, 9 x IO’12Pa, 8 x IO’12Pa, 7 x IO’12Pa, 6 x IO’12Pa, 5 x 10'12Pa, 4 x 10'12Pa, 3 x 10'12Pa, 2 x 10'12Pa, 10'12Pa, or lower. The vacuum units may be configured to maintain the system 200 at a pressure of at least about IO’12Pa, 2 x IO’12Pa, 3 x IO’12Pa, 4 x IO’12Pa, 5 x IO’12Pa, 6 x IO’12Pa, 7 x IO’12Pa, 8 x IO’12Pa, 9 x IO’12Pa, IO’11Pa, 2 x IO’11Pa, 3 x IO’11Pa, 4 x IO’11Pa, 5 x IO’11Pa, 6 x IO’11Pa, 7 x 10’11Pa, 8 x IO’11Pa, 9 x IO’11Pa, IO’10Pa, 2 x IO’10Pa, 3 x IO’10Pa, 4 x IO’10Pa, 5 x IO’10Pa, 6 x IO’10Pa, 7 x IO’10Pa, 8 x IO’10Pa, 9 x IO’10Pa, IO’9Pa, 2 x IO’9Pa, 3 x IO’9Pa, 4 x IO’9Pa, 5 x IO’9Pa, 6 x IO’9Pa, 7 x IO’9Pa, 8 x IO’9Pa, 9 x IO’9Pa, IO’8Pa, 2 x IO’8Pa, 3 x IO’8Pa, 4 x IO’8Pa, 5 x IO’8Pa, 6 x IO’8Pa, 7 x IO’8Pa, 8 x IO’8Pa, 9 x IO’8Pa, IO’7Pa, 2 x IO’7Pa, 3 x IO’7Pa, 4 x 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.

[0174] 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.

[0175] 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.WSGR Docket No. 55436-755.601

[0176] Alternatively or in addition, the atom reservoirs may comprise a portion of the optical trapping sites of the optical trapping units. A first subset of the optical trapping sites may be utilized for performing quantum computations and may be referred to as a set of computationally active optical trapping sites, while a second subset of the optical trapping sites may serve as an atom reservoir. For instance, the first subset of optical trapping sites may comprise an interior array of optical trapping sites, while the second subset of optical trapping sites comprises an exterior array of optical trapping sites surrounding the interior array. The interior array may comprise a rectangular, square, rectangular prism, or cubic array of optical trapping sites.

[0177] 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).

[0178] 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.

[0179] 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 entangledWSGR Docket No. 55436-755.601through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions.

[0180] The entanglement units may comprise one or more Rydberg units. The Rydberg units may be configured to electronically excite the at least first atom to a Rydberg state or to a superposition of a Rydberg state and a lower-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg units may be configured to induce one or more quantum mechanical entanglements between the Rydberg atoms or dressed Rydberg atoms and the at least second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (pm), 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance from the Rydberg atoms or dressed Rydberg atoms that is within a range defined by any two of the preceding values. The Rydberg units may be configured to allow the Rydberg atoms or dressed Rydberg atoms to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg units may be configured to induce the Rydberg atoms or dressed Rydberg atoms to relax to a lower-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.

[0181] 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 precedingWSGR Docket No. 55436-755.601values. For instance, the light may comprise one or more wavelengths that are within a range from 300 nm to 400 nm.

[0182] The Rydberg units may be configured to induce a two-photon transition to generate an entanglement. The Rydberg units may be configured to induce a two-photon transition to generate an entanglement between two atoms. The Rydberg units may be configured to selectively induce a two-photon transition to selectively generate an entanglement between two atoms. For instance, the Rydberg units may be configured to direct electromagnetic energy (such as optical energy) to particular optical trapping sites to selectively induce a two-photon transition to selectively generate the entanglement between the two atoms. The two atoms may be trapped in nearby optical trapping sites. For instance, the two atoms may be trapped in adjacent optical trapping sites. The two-photon transition may be induced using first and second light from first and second light sources, respectively. The first and second light sources may each comprise any light source described herein (such as any laser described herein). The first light source may be the same or similar to a light source used to perform a single-qubit operation described herein. Alternatively, different light sources may be used to perform a single-qubit operation and to induce a two-photon transition to generate an entanglement. The first light source may emit light comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from 650 nm to 700 nm). The second light source may emit light comprising one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., 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.

[0183] 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.

[0184] 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 a1So ground state) to aWSGR Docket No. 55436-755.601Rydberg state (such as an n3S1state, wherein n is a principal quantum number described herein). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transitions may be implemented using first and second laser sources, as described herein. The first laser source may emit pi-polarized light, which may not change the projection of atomic angular momentum along a magnetic field. The second laser may emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. The first and second qubit levels may be excited to Rydberg level using this polarization. However, the Rydberg levels may be more sensitive to magnetic fields than the ground state so that large splittings (for instance, on the order of 100s of MHz) may be readily obtained. This spectral selectivity may allow state selective excitation to Rydberg levels.

[0185] 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).

[0186] 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.

[0187] 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.WSGR Docket No. 55436-755.601

[0188] 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, 10 ns, or less. The pulse sequences may comprise a duration that is within a range defined by any two of the preceding values.

[0189] 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.

[0190] 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,” PhysicalWSGR Docket No. 55436-755.601Review 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.

[0191] The pulse sequences may further comprise one or more optimal control pulse sequences. The optimal control pulse sequences may be derived from one or more procedures, including gradient ascent pulse engineering (GRAPE) methods, Krotov’s method, chopped basis methods, chopped random basis (CRAB) methods, Nelder-Mead methods, gradient optimization using parametrization (GROUP) methods, genetic algorithm methods, and gradient optimization of analytic controls (GOAT) methods. For instance, the pulse sequences may be similar to those described in N. Khaneja et al., “Optimal Control of Coupled Spin Dynamics: Design of NMR Pulse Sequences by Gradient Ascent Algorithms,” Journal of Magnetic Resonance 172(2), 296-305 (2005); or J. T. Merrill et al., “Progress in Compensating Pulse Sequences for Quantum Computation,” Advances in Chemical Physics 154, 241-294 (2014), each of which is incorporated by reference in its entirety for all purposes.Examples of Cloud Computing

[0192] 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.Examples of Optical Trapping Units

[0193] FIG. 3A shows an example of an optical trapping unit 210. The optical trapping unit may be configured to generate a plurality 211 of spatially distinct optical trapping sites, as described herein. For instance, as shown in FIG. 3B, the optical trapping unit may be configured to generate a first optical trapping site 211a, second optical trapping site 211b, third optical trapping site 211c, fourth optical trapping site 21 Id, fifth optical trapping site 21 le, sixth optical trapping site 21 If, seventh optical trapping site 211g, eighth optical trapping site 21 Ih, and ninth optical trapping site 21 li, as depicted in FIG. 3A. The plurality of spatially distinct optical trapping sites may be configured to trap a plurality of atoms, such as first atom 212a, second atom 212b, third atom 212c, and fourth atom 212d, as depicted in FIG. 3A. As depicted in FIG.3B, each optical trapping site may be configured to trap a single atom. As depicted in FIG. 3B, some of the optical trapping sites may be empty (i.e., not trap an atom).WSGR Docket No. 55436-755.601

[0194] As shown in FIG. 3B, the plurality of optical trapping sites may comprise a two-dimensional (2D) array. The 2D array may be perpendicular to the optical axis of optical components of the optical trapping unit depicted in FIG. 3A. Alternatively, the plurality of optical trapping sites may comprise a one-dimensional (ID) array or a three-dimensional (3D) array.

[0195] 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.

[0196] Each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 pm, 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site maybe spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the preceding values.

[0197] 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. In some cases, the optical tweezers may be the same as those produced by an optical cavity 1400 of FIG. 14, herein. In some cases, the optical tweezers may be formed by an array of optical foci of an optical cavity 1400, described herein. 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 (ID) optical lattices, two-dimensional (2D) optical lattices, or three-dimensional (3D) optical lattices. For instance, the optical trapping sites may comprise one or more optical lattice sites of a 2D optical lattice, as depicted in FIG. 3B.

[0198] The optical lattices may be generated by interfering counter-propagating light (such as counter-propagating laser light) to generate a standing wave pattern having a periodic succession of intensity minima and maxima along a particular direction. A ID optical lattice may be generated by interfering a single pair of counter-propagating light beams. A 2D optical lattice may be generated by interfering two pairs of counter-propagating light beams. A 3D opticalWSGR Docket No. 55436-755.601lattice 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.

[0199] Returning to the description of FIG. 3A, the optical trapping unit may comprise one or more light sources configured to emit light to generate the plurality of optical trapping sites as described herein. For instance, the optical trapping unit may comprise a single light source 213, as depicted in FIG. 3A. Though depicted as comprising a single light source in FIG. 3A, the optical trapping unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources. The light sources may comprise one or more lasers. The lasers may be configured to operate at a resolution limit of the lasers. For example, the lasers can be configured to provide diffraction limited spot sizes for optical trapping.

[0200] 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.

[0201] The lasers may comprise one or more metal-vapor lasers, such as one or more heliumcadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg) metal-vapor lasers, heliumselenium (HeSe) metal-vapor lasers, helium-silver (HeAg) metal-vapor lasers, strontium (Sr) metal-vapor lasers, neon-copper (NeCu) metal-vapor lasers, copper (Cu) metal-vapor lasers, gold (Au) metal-vapor lasers, manganese (Mn) metal-vapor laser, or manganese chloride (MnCl₂) metal-vapor lasers.

[0202] 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-WSGR Docket No. 55436-755.601doped yttrium calcium oxoborate (Nd: YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm: YAG) lasers, ytterbium-doped ytrrium aluminum garnet (Yb: YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho: YAG) lasers, chromium-doped zinc selenide (Cr: ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce: LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce: LiCAF) lasers, erbium-doped glass (Erglass) lasers, erbium-ytterbium-codoped glass (Er / Yb:glass) lasers, uranium-doped calcium fluoride (U:CaF₂) lasers, or samarium-doped calcium fluoride (Sm:CaF₂) lasers.

[0203] The lasers may comprise one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGalnP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.

[0204] 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.

[0205] 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, 80WSGR Docket No. 55436-755.601MHz, 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.

[0206] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (pJ), 2 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.

[0207] 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, 800 W, 900 W, 1,000 W, or more. TheWSGR Docket No. 55436-755.601lasers 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.

[0208] The lasers may emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The lasers may emit light comprising one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm, 1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm, 1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm, 1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm, 1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm, 1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370 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 nm, 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 nm, 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 nm, 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 nm, 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,WSGR Docket No. 55436-755.601320 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.

[0209] 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 IO’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 IO'10nm, 2 x IO'10nm, 3 x IO'10nm, 4 x IO'10nm, 5 x IO'10nm, 6 x IO'10nm, 7 x IO'10nm, 8 x IO'10nm, 9 x IO'10nm, 1 x 10'9nm, 2 x 10'9nm, 3 x 10'9nm, 4 x 10'9nm, 5 x 10'9nm, 6 x 10'9nm, 7 x 10'9nm, 8 x 10'9nm, 9 x 10'9nm, 1 x 10'8nm, 2 x 10'8nm, 3 x 10'8nm, 4 x 10'8nm, 5 x 10'8nm, 6 x 10'8nm, 7 x 10'8nm, 8 x 10'8nm, 9 x 10'8nm, 1 x 10'7nm, 2 x 10'7nm, 3 x 10'7nm, 4 x 10'7nm, 5 x 10'7nm, 6 x 10'7nm, 7 x 10'7nm, 8 x 10'7nm, 9 x 10'7nm, 1 x 10'6nm, 2 x 10'6nm, 3 x 10'6nm, 4 x 10'6nm, 5 x 10'6nm, 6 x 10'6nm, 7 x 10'6nm, 8 x 10'6nm, 9 x 10'6nm, 1 x 10'5nm, 2 x 10'5nm, 3 x 10'5nm, 4 x 10'5nm, 5 x 10'5nm, 6 x 10'5nm, 7 x 10'5nm, 8 x 10'5nm, 9 x 10'5nm, 1 x 10'4nm, 2 x 10'4nm, 3 x 10'4nm, 4 x 10'4nm, 5 x 10'4nm, 6 x 10'4nm, 7 x 10'4nm, 8 x 10'4nm, 9 x 10'4nm, 1 x 10'3nm, or more. The lasers may emit light having a bandwidth of at most aboutl x 10'3nm, 9 x 10'4nm, 8 x 10'4nm, 7 x 10'4nm, 6 x 10'4nm, 5 x 10'4nm, 4 x 10'4nm, 3 x 10'4nm, 2 x 10'4nm, 1 x 10'4nm, 9 x 10'5nm, 8 x 10'5nm, 7 x 10'5nm, 6 x 10'5nm, 5 x 10'5nm, 4 x 10'5nm, 3 x 10'5nm, 2 x 10'5nm, 1 x 10'5nm, 9 x 10'6nm, 8 x 10'6nm, 7 x 10'6nm, 6 x 10'6nm, 5 x 10'6nm, 4 x 10'6nm, 3 x 10'6nm, 2 x 10'6nm, 1 x 10'6nm, 9 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 IO'10nm, 8 x IO'10nm, 7 x IO'10nm, 6 x IO'10nm, 5 x IO'10nm, 4 x IO'10nm, 3 x IO'10nm, 2 x IO'10nm, 1 x IO'10nm, 9 x 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 IO’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 10'14nm, 7 x 10'14nm, 6 x 10'14nm, 5 x 10'14nm, 4 x 10'14nm, 3 x 10'14nm, 2 x 10'14nm, 1 x 10'14nm, 9 x 10'15nm, 8 x 10'15nm, 7 x 10'15nm, 6 x 10'15nm, 5 x 10'15nm, 4 x 10'15WSGR Docket No. 55436-755.601nm, 3 x IO’15nm, 2 x IO'15nm, 1 x IO'15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.

[0210] The light sources may be configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms. A magic wavelength corresponding to an atom may comprise any wavelength of light that gives rise to equal or nearly equal polarizabilities of the first and second atomic states. The magic wavelengths for a transition between the first and second atomic states may be determined by calculating the wavelengthdependent polarizabilities of the first and second atomic states and finding crossing points. Light tuned to such a magic wavelength may give rise to equal or nearly equal differential light shifts in the first and second atomic states, regardless of the intensity of the light emitted by the light sources. This may effectively decouple the first and second atomic states from motion of the atoms. The magic wavelengths may utilize one or more scalar or tensor light shifts. The scalar or tensor light shifts may depend on magnetic sublevels within the first and second atomic states.

[0211] 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 a may be written as a sum of the scalar component (αscalarand the tensor component αtensor: (α = αscalar+ (3 cos θ² − 1)αtensor

[0212] By choosing 0 appropriately, the polarizability of the first and second atomic states may be chosen to be equal or nearly equal, corresponding to a zero or near zero differential light shift and the motion of the atoms may be decoupled.

[0213] 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. For 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 electrooptic deflectors (EODs) or electro-optic modulators (EOMs).WSGR Docket No. 55436-755.601

[0214] 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.

[0215] 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.

[0216] Alternatively or in addition, the OMs may comprise first and second AODs. The active regions of the first and second AODs may be imaged onto the back focal plane of the microscope objectives. The output of the first AOD may be optically coupled to the input of the second AOD. In this manner, the second AOD may make a copy of the optical output of the first AOD. This may allow for the generation of optical trapping sites in two or three dimensions.

[0217] 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.

[0218] 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.

[0219] The optical trapping unit may comprise one or more spatial configuration artificial intelligence (Al) units configured to perform one or more Al operations to determine the spatial configuration of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial configuration Al unit 216. Although depicted as comprising a single spatial configuration Al unit in FIG. 3A, the optical trapping unit may comprise any number of spatial configurationWSGR Docket No. 55436-755.601Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0220] The optical trapping unit may comprise one or more atom rearrangement units configured to impart an altered spatial arrangement of the plurality of atoms trapped with the optical trapping sites based on the one or more images obtained by the imaging unit. For instance, the optical trapping unit may comprise atom rearrangement unit 217. Although depicted as comprising a single atom rearrangement unit in FIG. 3A, the optical trapping unit may comprise any number of atom rearrangement units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atom rearrangement units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units.

[0221] The optical trapping unit may comprise one or more spatial arrangement artificial intelligence (Al) units configured to perform one or more Al operations to determine the altered spatial arrangement of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial arrangement Al unit 218. Although depicted as comprising a single spatial arrangement Al unit in FIG. 3A, the optical trapping unit may comprise any number of spatial arrangement Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial arrangement Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial arrangement Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0222] In some cases, the spatial configuration Al units and the spatial arrangement Al units may be integrated into an integrated Al unit. The optical trapping unit may comprise any number of integrated Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated Al units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated Al units.

[0223] 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 factorWSGR Docket No. 55436-755.601of 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.

[0224] 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.

[0225] 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.

[0226] 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.WSGR Docket No. 55436-755.601Examples of Electromagnetic Delivery Units

[0227] 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.

[0228] The electromagnetic delivery unit may comprise one or more microwave or radiofrequency (RF) energy sources, such as one or more magnetrons, klystrons, traveling-wave tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMP ATT) diodes, or masers. The electromagnetic energy may comprise microwave energy or RF energy. The RF energy may comprise one or more wavelengths of at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or more. The RF energy may comprise one or more wavelengths of at most about 10 km, 9 km, 8 km, 7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m, 500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m, 30 m, 20 m, 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm, 800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm, 80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or less. The RF energy may comprise one or more wavelengths that are within a range defined by any two of the preceding values.

[0229] 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, 800 W, 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,WSGR Docket No. 55436-755.60120 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.

[0230] The electromagnetic delivery unit may comprise one or more light sources, such as any light source described herein. For instance, the electromagnetic delivery unit may comprise light source 221. Although depicted as comprising a single light source in FIG. 4, the electromagnetic delivery unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources.

[0231] 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.

[0232] The electromagnetic delivery unit may comprise one or more electromagnetic energy artificial intelligence (Al) units configured to perform one or more Al operations to selectively apply the electromagnetic energy to the atoms. For instance, the electromagnetic delivery unit may comprise Al unit 223. Although depicted as comprising a single Al unit in FIG. 4, the electromagnetic delivery unit may comprise any number of Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0233] 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,WSGR Docket No. 55436-755.60110 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, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or more. The single-qubit or two-qubit operations may be applied with a repetition frequency of at most 1,000 kHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The single-qubit or two-qubit operations may be applied with a repetition frequency that is within a range defined by any two of the preceding values.

[0234] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations by inducing one or more Raman transitions between a first qubit state and a second qubit state described herein. The Raman transitions may be detuned from a3Po or3Pi line described herein. For instance, the Raman transitions may be detuned by at least about 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1 GHz, or more. The Raman transitions may be detuned by at most about 1 GHz, 900 MHz, 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.

[0235] Raman transitions may be induced on individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle or a frequency shift to a light beam based on an applied radio-frequency (RF) signal. The SLMWSGR Docket No. 55436-755.601or 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).

[0236] 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.

[0237] 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 10WSGR Docket No. 55436-755.601μm, 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.Examples of characteristic dimensions include, but are not limited to, a Gaussian beam waist, the full width at half maximum (FWHM) of the beam size, the beam diameter, the 1 / e2width, the D4σ width, the D86 width, and the like. For example, the beam may have a Gaussian beam waist of at least about 1.5 micrometers.

[0238] 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.Examples of Integrated Optical Trapping Units and Electromagnetic Delivery Units

[0239] 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 toWSGR Docket No. 55436-755.601deliver 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.

[0240] A single SLM or AOD may allow the implementation of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or in addition, two separate SLMs or AODs may be configured to each handle light with orthogonal polarizations. The light with orthogonal polarizations may be overlapped before the microscope objective. In such a scheme, each photon used in a two-photon transition described herein may be passed to the objective by a separate SLM or AOD, which may allow for increased polarization control. Qubit operations may be performed on a two-dimensional arrangement of atoms by bringing light from a first SLM or AOD into a second SLM or AOD that is oriented substantially orthogonally to the first SLM or AOD via an optical relay.Alternatively or in addition, qubit operations may be performed on a two-dimensional arrangement of atoms by using a one-dimensional array of SLMs or AODs.

[0241] 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.

[0242] 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 describedWSGR Docket No. 55436-755.601herein. Information about the intensity may be integrated into a feedback loop to stabilize the intensity.Examples of State Preparation Units

[0243] FIG. 5 shows an example of a state preparation unit 250. The state preparation unit may be configured to prepare a state of the plurality of atoms, as described herein. The state preparation unit may be coupled to the optical trapping unit and may direct atoms that have been prepared by the state preparation unit to the optical trapping unit. The state preparation unit may be configured to cool the plurality of atoms. The state preparation unit may be configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites.

[0244] 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.

[0245] 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.

[0246] 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. TheWSGR Docket No. 55436-755.601first 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.

[0247] The first MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0248] 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.

[0249] 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 ofWSGR Docket No. 55436-755.601at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0250] 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.

[0251] 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.

[0252] The sideband cooling units or Sisyphus cooling units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0253] 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 lightWSGR Docket No. 55436-755.601sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0254] 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 nonequilibrium 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).WSGR Docket No. 55436-755.601

[0255] The coherent driving units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0256] 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.WSGR Docket No. 55436-755.601Examples of Controllers

[0257] The optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units may include one or more circuits or controllers (such as one or more electronic circuits or controllers) that is connected (for instance, by one or more electronic connections) to the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units. The circuits or controllers may be configured to control the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration Al units, spatial arrangement Al units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy Al units, atom reservoirs, atom movement units, or Rydberg excitation units.Examples of Non-Classical Computers

[0258] In some cases, the present disclosure provides a non-classical computer comprising: a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprise at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, which non-classical operation includes a superposition between at least the first qubit state and the second qubit state; one or more entanglement units configured to quantum mechanically entangle at least a subset of the plurality of qubits in the superposition with at least another qubit of the plurality of qubits; and one or more readout optical units configured to perform one or more measurements of the one or more qubits, thereby obtaining a non-classical computation.

[0259] In some cases, the present disclosure provides a non-classical computer comprising a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites.WSGR Docket No. 55436-755.601Examples of Methods for Performing a Non-Classical Computation

[0260] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms comprising greater than 60 atoms; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state that is different from the first atomic state; (c) quantum mechanically entangling at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more optical measurements of the one or more superposition state to obtain the non-classical computation.

[0261] FIG. 6 shows a flowchart for an example of a first method 600 for performing a non-classical computation.

[0262] 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.

[0263] 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.

[0264] 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).

[0265] 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.

[0266] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) providing a plurality of qubits comprising greater than 60 atoms,WSGR Docket No. 55436-755.601each 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.

[0267] FIG. 7 shows a flowchart for an example of a second method 700 for performing a non-classical computation.

[0268] 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.

[0269] 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.

[0270] 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).

[0271] 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.WSGR Docket No. 55436-755.601

[0272] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) providing a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, and (b) using at least a subset of the plurality of qubits to perform the non-classical computation.

[0273] FIG. 8 shows a flowchart for an example of a third method 800 for performing a non-classical computation.

[0274] In a first operation 810, the method 800 may comprise providing a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites. The qubits may comprise any qubits described herein. The atoms may comprise any atoms described herein. The optical trapping sites may comprise any optical trapping sites described herein.

[0275] In a second operation 820, the method 800 may comprise using at least a subset of the plurality of qubits to perform a non-classical computation.Examples of Parallel Addressing of Multi-Qubit Units

[0276] Direct excitation of strontium-87 from the ground state to Rydberg levels would require a laser with a wavelength of approximately 218 nm. Alternatively, the Rydberg excitation operation can be performed using two-photon excitation combining 689 nm and 319 nm light, each detuned from the intermediate3Pi state. The approximately 7 kHz width of the3Pi state provides an effective balance between the two-photon effective Rabi rate and scattering via spontaneous decay from the3Pi. FIG. 9 shows an energy level structure for single-qubit and multi-qubit operations in strontium-87.

[0277] The optical system for single-qubit operations is also designed to work well for multiqubit gates. One of the single-qubit beams is used as one leg of the two-photon excitation scheme that drives transitions to the Rydberg electronic manifold. To satisfy the spatially-dependent frequency and phase matching condition, AODs are also used for the UV light.Importantly, the optical systems are matched so that the frequency shift of the UV light from one site to another is identical to that of the 689 nm light. The consequence of this constraint is that the performance of state-of-the-art UV AODs dictate the accessible field of view (FOV) for multi-qubit operations. Further, because one of the single-qubit beams is being used for multiqubit 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, andWSGR Docket No. 55436-755.601thus a larger FOV in the plane of the qubit array. An FOV of approximately 100 μm x 100 μm was achieved, which is sufficient to address an array of approximately 1,000 atoms with a trapping site spacing of 3 μm.Examples of Methods for Error Corrected Quantum Computation

[0278] The systems, the methods, the computer-readable media, and the techniques 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. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be directed to detecting qubit loss without destroying the data stored on the qubits. Instead of appearing as a gate measurement error, if a qubit is lost, the data that would be there is absent rather than incorrect. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be directed to identifying when an error is caused by a missing qubit. The systems, the methods, the computer-readable media, and the techniques disclosed herein may also be directed to modifying the decoder to handle loss events. For example, the error correcting code may be directed to updating the calculation to address for error. In some cases, knowing about the error may be needed in order to implement error correcting code. However, in other cases, the error correcting code may be modified to account for missing data without explicit knowledge that qubit is lost.

[0279] The systems, the methods, the computer-readable media, and the techniques disclosed herein may not generally modify the topology of the underlying surface code. The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon methods of detecting atom loss by compressing the underlying protocol. The systems, the methods, the computer-readable media, and the techniques disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate. The systems, the methods, the computer-readable media, and the techniques disclosed herein may address uninduced erasure errors in addition to or alternatively to gate induced erasure errors.

[0280] The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon other procedures at least because the systems, the methods, the computer-readable media, and the techniques disclosed herein may not comprise or require operations that modify the topology of the underlying surface code. In some cases, the underlying surface code may be unchanged. Instead, a matching graph passed to a decoder algorithm may be updated to account for a predicted probability distribution of a lost qubit. Because the matching graph passed to the decoder is updated, the underlying decoder may also be unchanged. Because the decoder is unchanged, the error correcting code may also not be changed. Accordingly, the systems, the methods, the computer-readable media, and theWSGR Docket No. 55436-755.601techniques disclosed herein may be used without changing the underlying surface code.Similarly, because changes to the underlying decoder and surface code are not needed, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with a wide variety of decoders and surface codes.

[0281] In some cases, tracking syndrome measurements may be used to detect loss events (e.g., defects). See, e.g., Siegel, A. et al., Adaptive Surface Code for Quantum Error Correction in the Presence of Temporary or Permanent Defects, arXiv:2211.08468vl [quant-ph] 15 Nov 2022, available at https: / / arxiv.org / pdf / 2211.08468.pdf, which is incorporated by reference herein in its entirety.

[0282] 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.

[0283] In some cases, noise structure in the hardware may be used for error correction. See e.g., Shay, K., et al., High threshold codes for neutral atom qubits with biased erasure errors, arXiv:2302.03063vl [quant-ph] 6 Feb 2023), which is incorporated by reference herein in its entirety. Similar to Wu, the atoms may still be present. Rather than addressing missing atoms or qubits, the above referenced application may turn gate errors into erasure errors.Error Correction

[0284] 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.

[0285] 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,WSGR Docket No. 55436-755.601there 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.

[0286] 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.

[0287] One method of error correction (Shor style) uses repeated rounds of syndrome extraction to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied. The decoding problem is commonly represented as a weighted graph or hypergraph. In this setting, each node in the graph corresponds to a collection of syndrome measurements. Such a collection of syndrome measurements is called a detector. Edges or hyperedges in the graph correspond to errors, and the weight of an edge corresponds to the likelihood of that error occurring. The occurrence of a given error may be expected cause the parity of all associated detectors to flip. The decoding problem can then be stated as follows: given a set of detectors (nodes) whose parities differ from those expected in the absence of error, determine the most likely set of physical errors (edges) that could cause the observed detections.Error Correction with Atom Loss

[0288] 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.

[0289] 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).WSGR Docket No. 55436-755.601

[0290] 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.

[0291] 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.

[0292] In systems not subject to atom loss, the errors may be discrete Pauli errors on physical qubits. But when qubits are stored on atoms, the atoms — and therefore the qubits they contain — can be lost. The effect on syndrome extraction in the presence of loss depends on hardware details. For neutral atoms using Rydberg gates, the effect of atom loss manifests as a noninteraction instead of a two-qubit gate. Practically, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom.

[0293] In an example, the systems, the methods, the computer-readable media, and the techniques disclosed herein may comprise a case in which a two-qubit interaction between a qubit and a lost qubit has an effect of a Pauli operation or identity operation on the qubit, as described in International Application PCT / US2024 / 018180, which is incorporated by reference in its entirety herein 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>.

[0294] 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 11>, 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 11>, 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 AWSGR Docket No. 55436-755.601or Atom B (which is not present). Accordingly, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom. While this example describes a case where the two-qubit interaction with a lost atom affects the Identity operation, the systems, the methods, the computer-readable media, and the techniques disclosed herein also work when 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.

[0295] The above works similarly if Atom A is also a lost qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit. For example, when the qubit is a lost qubit, a two-qubit gate between two lost qubits similarly affects the identity. Because the two-qubit operation between an atom a lost atom affects the identity. The protocol does not propagate errors (to first order) forward in time. For example, if the two-qubit operation is imperfect. The operation may propagate forward higher order errors in time.Examples of Identifying Qubit Loss

[0296] In some cases, the present disclosure provides at least two methods of identifying qubit loss; however, various methods of identifying qubit loss may be integrated into the systems, the methods, the computer-readable media, and the techniques disclosed herein. An example method of implementing an error correcting code which accounts for atom loss may comprise implementing a plurality of SWAP gates. Another example method of implementing an error correcting code which accounts for atom loss may comprise a modified knock-knock protocol. The systems, the methods, the computer-readable media, and the techniques disclosed herein for the implementation of a plurality of SWAP gates and modified knock-knock protocols may be the same as or similar to those described in International Application PCT / US2024 / 018180, which is incorporated by reference herein for all purposes.Examples of Qubit Replacement

[0297] The systems, the methods, the computer-readable media, and the techniques disclosed herein may replace qubits into a quantum circuit after a vacancy has been identified. In some cases, the qubit is an atomic qubit. In some cases, the qubit is atom trapped in a spatially distinct optical trapping site. Examples presented herein may be recite qubits comprising neutral atoms; however, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be combined with various types of qubits.WSGR Docket No. 55436-755.601

[0298] 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.

[0299] 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).

[0300] 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. 55436-755.601Examples of Modifying Decoding Algorithms

[0301] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used in connection with error correction methodologies for quantum computing systems. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correcting code. A decoder may 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.

[0302] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various error correcting codes. An error correcting code may be a Shor style code. For example, in a Shor style code, repeated rounds of syndrome extraction may be implemented to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied.

[0303] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various stabilizer codes. A stabilizer code may be an error correcting code which uses stabilizers. A stabilizer code may be a class of error correcting code. The class of stabilizer codes may include toric codes, surface codes, etc. By repeatedly measuring a quantum system using a complete set of commuting stabilizers, the system may be forced into a simultaneous and unique eigenstate of all the stabilizers. One can measure the stabilizers without perturbing the system; when the measurement outcomes change, this corresponds to one or more qubit errors, and the quantum state is projected by the measurements onto a different stabilizer eigenstate.

[0304] An error correcting code may comprise a topological code. The class of topological codes may overlap with the class of stabilizer codes. A topological code may comprise a surface code, a color code, a toric code, etc. A topological code may also be referred to as a homological code. A topological code may comprise an array or lattice of qubits arranged on a surface (or higher dimensional structure). The systems, the methods, the computer-readable media, and the techniques disclosed herein may not generally change the underlying topology of a topological code.WSGR Docket No. 55436-755.601

[0305] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various surface codes. A surface code may be implemented as a stabilizer code. For example, in the surface code literature, surface codes may comprise two types of qubits data qubits and measurement qubits (e.g., ancilla qubits). The data qubits may contain the information carried by the quantum circuit, whose error is to be corrected. The measurement qubits may be used to stabilize and manipulate the quantum state of the data qubit. In a surface code, the measurement qubits may comprise two types: measure-Z qubits 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.

[0306] The systems, the methods, the computer-readable media, and the techniques disclosed herein may employ surface codes. Surface codes disclosed herein may include, for example, variations upon the minimum-weight perfect matching algorithm to decode the surface code. However, many surface codes may be applicable to the systems, the methods, the computer-readable media, and the techniques disclosed herein. A 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.

[0307] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various color codes. A color code may be implemented as a stabilizer code. For example, a color code may comprise a Steane code, etc. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various Shor style codes, for example, a Bacon-shor code. A Shor style code may be implemented as a stabilizer code. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various qLDPC codes, for example, hypergraph product codes. A qLDPC code may be implemented as a stabilizer code.

[0308] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be used with various decoders. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correction code. A decoder may decode which errors occurred on which qubits. Once identified, these errors can be tracked and the information used to correct subsequent measurement outcomes using the classical control software. Decoder algorithms may include, for example, minimum-weight perfect matching, union find, tensor network decoder, belief propagation withWSGR Docket No. 55436-755.601ordered statistics decoder, maximum likelihood decoder, and look up table decoders. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be integrated with variations on the minimum-weight perfect matching such as sparse bloom and fusion blossom. A decoder may take in a matching graph. The systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to the decoder to account for a lost qubit.

[0309] 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.

[0310] 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.

[0311] To augment the decoding algorithm, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update existing decoders to incorporate the change in error type. In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to a decoder. In some cases, the systems, the methods, the computer-readable media, and the techniques disclosed herein may update the matching graph passed to a minimum-weight perfect matching decoder algorithm or any other decoder algorithm which takes in a matching graph.

[0312] In some examples, to update the matching graph, each node in the graph corresponds to a change-of-value of a particular stabilizer. Certain nodes are connected by edges corresponding to possible physical errors. These edges are weighted based on the likelihood of that particular error occurring. When an atom is lost and then replaced, the loss may be treated like a gate error that occurs with a probability of 50%. The procedure may change slightly if data vs. ancilla qubits are lost.

[0313] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be implement approaches for reimplementing a qubit, such as, as described in International Application PCT / US2024 / 018180, which is incorporated by reference herein for all purposes.WSGR Docket No. 55436-755.601Examples of Measurement Operation

[0314] 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.

[0315] In the systems, the methods, the computer-readable media, and the techniques disclosed herein, the fact of a missing qubit may not be immediately heralded. For example, it may become apparent that qubit is lost after completing a round of syndrome extraction, rather than immediately upon taking a measurement implicating a lost qubit. Once the round of syndrome extraction is complete, it may become apparent that there was a qubit loss, in order to proceed with the calculation in may be beneficial to flag a series of measurements taken during a window of time that includes a time when the qubit was missing. Each of these measurements may be flagged as untrustworthy. In some cases, the series of time which includes the flagged qubit may not be limited to the time in which the qubit is definitively lost. The series of time which includes the flagged qubit may include at least the time with the qubit was lost.

[0316] Advantageously, a qubit loss may be identified before measurement of the data qubits (and after a round of syndrome extraction). Since the data qubits have not yet been measured, it may be possible to continue on with a quantum circuit after replacing a lost qubit. The circuit may be adapted to retake or restart portions of the calculation implicating the lost qubit. As a consequence, the systems, the methods, the computer-readable media, and the techniques disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate.

[0317] 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 implementWSGR Docket No. 55436-755.601error 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

[0318] 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.

[0319] 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.

[0320] In some cases, the non-classical computing system may comprise a plurality of qubits.

[0321] In some cases, the non-classical computing system may comprise one or more electromagnetic delivery units. The electromagnetic delivery units may be configured to produce electromagnetic excitations to perform various operations, such as for example, atom movement, atom rearrangement, optical trapping, imaging, and various operations on atoms that may comprises portions of a nonclassical computation. Portions of a non-classical computation on trapped atoms may comprise optical pumping, entanglement operations, Rydberg operations, gate operations (e.g., one qubit operations, two qubit operations, etc.). In some cases, an atom movement unit may comprise an atom rearrangement unit. The components of a non-classical computing system are discussed herein above with respect to the operations they implement.

[0322] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises trapped atom qubits. In some cases, the trapped atom qubits comprise neutral atom qubits. In some cases, the neutral atom qubits comprise aWSGR Docket No. 55436-755.601Group II element or a Group Il-like element. In some cases, the Group II element or a Group II-like element comprises Ytterbium, Rubidium, Cesium, or Strontium. In some cases, the plurality of qubits comprises qubit states comprising nuclear spin states on the 'So manifold. In some cases, the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

[0323] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises a plurality of qubits, wherein the plurality of qubits comprises atomic qubits, and wherein an atom replacement operation is implemented using optical tweezers.

[0324] In some cases, the non-classical computing system may be configured to interact with a processor 1101. The processor may be classical processing system. The processor may be digital processing system.

[0325] In some cases, the system further comprises a processor configured to implement an error correcting code. In some cases, the processor is further configured to provide instructions to a non-classical computing system, wherein the non-classical computing system is configured to implement the instructions to: (i) identify that a qubit has been lost; (ii) replace the qubit; and (iii) reimplement the qubit into the circuit. In some cases, the processor is further configured to (iv) flag measurements taken while the qubit was missing as untrustworthy. In some cases, (i) comprises using a plurality of swap gates. In some cases, a swap gate within the plurality of swap gates is implemented as a plurality of CNOT gates. In some cases, the processor is further configured to provide instructions to the non-classical computing system to measure alternating atoms in a lattice; perform the plurality of swap gates to transfer data stored on data qubits to ancilla qubits; and measure the swapped data qubits to identify one or more lost atoms.

[0326] In some cases, (i) comprises using a modified knock-knock protocol, wherein the modified knock-knock protocol comprises: providing a first atom to be probed using a second atom, wherein the second atom is an ancilla qubit; preparing the second atom in a |+> state; applying a modified control-Z gate between the first atom and the second atom based on a Rydberg interaction; and rotating the second qubit back to a computational basis and performing a measurement. In some cases, (iii) comprises (A) use of a decoder algorithm, wherein the decoder algorithm takes in a graph and determines a set of edges. In some cases, prior to (A) the processor is further configured to update a matching graph passed to the decoder algorithm based on a predicted probability distribution of a lost qubit replaced in (ii). In some cases, (iii) comprises use of a minimum-weight perfect matching decoder algorithm. In some cases, the processor is further configured to update a matching graph passed to the minimum-weight perfect matching decoder algorithm based on a predicted probability distribution of a lost qubitWSGR Docket No. 55436-755.601replaced in (ii). In some cases, the processor is further configured to: if an ancilla qubit is lost, update the matching graph so that a node involving the ancilla qubit is connected by edges corresponding to the predicted probability distribution; and if a data qubit is lost, update the matching graph by assigning the predicted probability distribution to each node involving the data qubit. In some cases, each node involving the ancilla qubit is updated.

[0327] In some cases, the error correcting code is configured to be implemented during a quantum computation circuit. In some cases, the error correcting code is configured to be implemented without measurement of each or a plurality of data qubits. In some cases, the error correcting code is configured to be implemented substantially without loss of coherence of each or a plurality of data qubits. In some cases, processor is further configured to flag measurements taken during a window of time that includes a time when the lost qubit was missing as untrustworthy.

[0328] In some cases, the decoder comprises union find, tensor network decoder, belief propagation with ordered statistics decoder, maximum likelihood decoder, or a look up table decoder. In some cases, the decoder comprises minimum weight perfect matching. In some cases, the decoder comprises sparse blossom or fusion blossom. In some cases, the error correcting code comprises a topological code. In some cases, the topological code is a stabilizer code. In some cases, the error correcting code is a surface code, a color code, a toric code, a shor style code, or a qLDPC code. In some cases, the color code is a Steane code. In some cases, the shor style code is a Bacon-shor code. In some cases, the qLDPC code is a hypergraph product code. In some cases, each node in the matching graph corresponds to a change-of-value of a particular stabilizer and wherein pairs of nodes are connected by edges corresponding to possible physical errors. In some cases, the edges are weighted based on the likelihood of a particular error occurring. In some cases, atom loss is treated as a gate error that occurs with a probability of 50%.

[0329] In some cases, the processor is further configured to provide instructions to the non-classical computing system to perform a measurement operation, wherein the measurement operation is state selective. In some cases, the measurement operation comprises applying electromagnetic energy to a qubit to be measured, wherein the electromagnetic energy is configured to selectively drive the qubit to be measured from an initial state to an excited state in a presence of an applied magnetic field, wherein a selectivity of a transition to the excited state is based at least in part on a strength of the applied magnetic field. In some cases, the processor is further configured to determine that the qubit to be measured was in the initial state based at least in part on the qubit returning to the initial state by emission of a photon in response to the electromagnetic energy.WSGR Docket No. 55436-755.601Examples of Processes for Performing Continuous, Non-Classical Computations

[0330] FIG. 12 illustrates an example process 1200 for performing continuous, non-classical computation. In some examples, the process 1200 may be implemented with the science region and the reservoir region being sufficiently distinct to enable loading of the reservoir region with sufficiently low disturbance of atoms within the science region. One way to accomplish this is for the reservoir and science regions to not be sub-regions of the same array, but rather to be distinct arrays in the sense that they may be formed using different lasers or different optical elements. Further, in some cases, the reservoir may be loaded from a third “transport” array, rather than from a MOT. This last technique may eliminate a use for near-resonant light, and so may allow for continuous coherent operations to be performed during the reloading process.

[0331] At a high level, the process 1200 may comprise: loading a reservoir array, transferring atoms to a science array from the reservoir array, performing a computation / simulation using the science array, atomic loss occurring in the science array, refilling the science array from the reservoir array, and reloading the reservoir array. A science array may comprise atoms that are actively being used for the application (e.g., quantum computing, optical clocks, sensing, or any other application disclosed herein). A science array in a quantum computer may comprise data qubits and ancilla qubits. A reservoir array may comprise atoms that are not actively in use but which may be used at a later time to replace atoms lost from the science array, e.g., a lost ancilla qubit from a quantum computer.

[0332] In some cases, the process 1200 may begin with both the reservoir array and the science array being empty. Once atoms are loaded into the reservoir array, the atoms in the reservoir may then be imaged. In some examples, at least some of the atoms in the reservoir array may then transferred into the science array (for example using optical tweezers). The reservoir array can be reloaded to achieve a full or fuller reservoir array and enable further transfer of more atoms from the reservoir array into the science array. Once the science array is fully occupied (or occupied to a desired / predetermined amount), the computation / simulation may begin. During the computation / simulation, the science array may be periodically imaged to determine if and where atom loss has occurred. If an atom has been lost from a site in the science array, an atom from the reservoir array may be transferred to fill the site. This may continue provided there are sufficient atoms in the reservoir array. When there are not, new atoms may be loaded into the reservoir array, and the process continues. As illustrated in FIG. 12, the process 1200 may be iterative or repetitive in some examples, with potential repetition of one or more operations of the process 1200.

[0333] FIG. 12 shows an example method and system for performing continuous, non-classical computations. The method 1200 may comprise an operation 1210. Operation 1210 mayWSGR Docket No. 55436-755.601comprise loading a plurality of atoms into a reservoir array that includes a first plurality of spatially distinct optical trapping sites. The first plurality of optical trapping sites may be configured to trap the plurality of atoms. In some cases, the plurality of atoms are qubits in a non-classical computational system, such as a quantum computer, a quantum annealer, etc. In some cases, the plurality of atoms comprises atoms in an atomic clock.

[0334] The method 1200 may comprise an operation 1220. Operation 1220 may comprise transferring a first subset of the plurality of atoms from the reservoir array into a science array. The science array may include a second plurality of spatially distinct optical trapping sites. The second plurality of optical trapping sites may be configured to trap a plurality of atoms.

[0335] In some cases, at operation 1215, operations 1210 and 1220 may be repeated a number of times. The operations may be repeated until a science array comprises a sufficient fill factor for quantum computation, quantum simulation, clock operations, metrology operations, sensing operations, etc.

[0336] The method 1200 may comprise an operation 1230. Operation 1230 may comprise performing a first application using at least some of the first subset of the plurality of atoms in the science array. The application may be quantum computation, quantum simulation, clock operations, metrology operations, sensing operations, etc.

[0337] The method 1200 may comprise an operation 1240. Operation 1240 may comprise determining an atomic loss in one or more of the arrays. The operation may comprise determining an atomic loss number representing a difference between (i) a number of atoms in the first subset of the plurality of atoms and (ii) a number of atoms in a remaining subset of the first subset of the plurality of atoms that remain in the science array following the performing of the first non-classical computation. Atom loss may occur due to collisions with residual background gas, to leakage into un-trapped or otherwise undesirable internal states, due to heating associated with laser interactions, or other processes.

[0338] The method 1200 may comprise an operation 1250. Operation 1250 may comprise transferring a second subset of the plurality of atoms from the reservoir array into the science array. Operation 1250 may comprise a reloading operation. In some cases, the second subset of the plurality of atoms includes at least a number of atoms equal to the atomic loss number. In some cases, the second subset of the plurality of atoms includes a number of atoms less than the atomic loss number. The second subset may be transferred substantially without loss of a coherence of the plurality of atoms in the science array. The second subset may be transferred substantially without stopping an application in the science array.

[0339] In some cases, at operation 1255, operations 1230, 1240, and 1250 may be repeated a number of times. The operations may be repeated until a quantum computation, quantumWSGR Docket No. 55436-755.601simulation, clock operation, metrology operation, etc. is complete. The operations may be repeated while there are atoms in a reservoir to be filed into the science array.

[0340] The method 1200 may comprise an operation 1260. Operation 1260 may comprise reloading the reservoir array with additional atoms. The reservoir may be reloaded from an atom source. The atom source may be cooled atom source. In some examples, reservoir regions may be filled from a magneto-optical trap (MOT), from an atomic beam, from a thermal atomic gas, from another optical or other form of electromagnetic trap, or from any other source of atoms. In some examples, the initial loading of the science region may be direct (from any atomic source other than the reservoir array), from the reservoir array, or from a separate reservoir array than the one used for replenishing. In some examples, the reservoir region may be smaller, larger, or the same size / number of sites as the science region and similar techniques may be used to maintain an arbitrary number of atoms within each site of the science array.

[0341] In some cases, at operation 1265, operation 1260 may be repeated a number of times. Operation 1260 may be refilled a number of times to fill a reservoir array. The operation may be repeated such that an application in operation 1230 may be performed continuously.

[0342] In some cases, at operation 1275, operations 1255 and 1265 may be both be repeated in order to maintain a fill factor in the science array. In some cases, the method may comprise performing a second non-classical computation using at least some of one or both of (i) the remaining subset of the first subset of the plurality of atoms and (ii) the second subset of the plurality of atoms.

[0343] The present disclosure comprises various sub-operations of the method 1200. For example, one or more of the operations of the method 1200 may be removed. For example, one or more of the operations of the method 1200 may be repeated.Examples of Continuous Loading

[0344] A useful error-corrected quantum computer should remove entropy faster than it can enter. One source of entropy in a trapped atom quantum computer may be atom loss.Accordingly, it may be useful to conditionally refill sites in a trapped atom quantum computer continuously with the calculation. Continuous operation during a non-classical computation may comprise refilling a lost atom during operations of computation. For example, continuation operation in a gate-model quantum computer may comprise refilling lost atoms “mid-circuit” or between gate operations in a quantum computation. Continuous operation in a quantum simulator may comprise refilling an atom during the simulation. Continuous operation in a clock operation may comprise refilling an atom during operation of the clock. In general, continuousWSGR Docket No. 55436-755.601operation may comprise refilling an atom during the time in which the application is being implemented.

[0345] Continuous operation may comprise refilling an atom substantially without stopping the application. Substantially without stopping may comprise not performing recovery operations, such as repeating previous steps, to account for the atom loss. Such recovery operations may comprise repeating a calculation or a portion of a calculation to replace a “lost” portion.

[0346] Similarly, since each of quantum computation, quantum simulation, clock operations, metrology, and quantum sensing may utilize phenomena such as quantum coherence, it may be useful to maintain coherence while refilling atoms in an atom-based implementation of these applications (e.g., an atomic clock, a neutral atom quantum computer, etc.). For example, atoms may be refilled substantially without loss of coherence of atoms in the array. Substantially without loss of coherence may comprise contrast loss on the order of 10% or better on seconds time scale. E.g., less than 10% loss of contrast over 2 seconds, about 5% contrast loss over two seconds or better. Substantially without loss of coherence may comprise a contrast of better than 0.8 (maximum of 1) over 1 second.

[0347] The present disclosure provides systems, methods, computer-readable media, and techniques for continuous atom reloading. The systems, the methods, the computer-readable media, and the techniques disclosed herein may distinguish the science array and the reservoir array during atom transfer.

[0348] For example, the arrays may be physically distinguished. The systems, the methods, the computer-readable media, and the techniques disclosed herein may employ distinct sets or subsets of atoms within an array. For example, FIG. 12 and FIG. 14 show a science array and a reservoir array. In some cases, the science array is distinct from the reservoir array. In some cases, the science array is spatially distinct from the reservoir array. For example, the science array may be physically separated from the reservoir array. For example, the science array may be energetically separated from the reservoir array during atom movement. In some cases, both physical and energetic separation methods may be used to facilitate atom movement without disruption of the science array.

[0349] Physical separation of the science array and the reservoir array may be useful in at least some respects for example. If the science array and the reservoir array are physically distinct, the reservoir array can be more easily spatially separated from the science region. This can allow loading into the reservoir array without disturbing atoms in the science region while the reservoir region is being loaded. For example, a disturbance may occur from unwanted scattering, unwanted light shifts, etc. during transfer. In some cases, a separate optical system from the trap excitation may be used to move atoms from a first array to a second arrayWSGR Docket No. 55436-755.601disclosed herein. For example, the reservoir array may be loaded from a separate optical potential or array, which may disturb atoms in the science region if the reservoir and science arrays were too close. Using separate optical systems to generate the two arrays may be helpful for separating the science array and the reservoir array arrays. Using a separate (e.g., a third) optical system, for atom movement may further insulate the arrays.

[0350] In some cases, the reservoir and science regions may be separated either parallel or transverse to the axis along which imaging is performed. If the separation is parallel to the imaging axis, atoms may be transferred from reservoir to science region by means of translating the focus of focused trapping lasers, or by shifting the phase of a trapping optical lattice. In some cases, the transfer of the at least one atom that is from the reservoir array into the science array is a long-range transfer. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be combined with approaches for long range transport, such as those described in International Applications PCT / US2023 / 026595 and PCT / US2023 / 075948, which are incorporated herein in their entirety.

[0351] Electronic separation of the science array and the reservoir array may be useful in at least some respects for example. In some examples, the reservoir and science regions are distinguished by the internal or motional state occupied by the atoms (perhaps instead of being spatially separated). In some examples, the traps may be formed with spatially or temporally incoherent or coherent light, or by non-optical electromagnetic fields.

[0352] In some cases, coherence may be protected by applying a “hiding” excitations during or partially during atom reloading into the science array. A hiding excitation may comprise placing an atom being transferred or an atom already in an array into a dark state, a clock state, or another state forbidden by selection rules from an optical excitation used for the transferring. In some examples, hiding excitations may be applied to atoms in the science array during imaging or excitation of atoms to be moved into the science portion of the array. In some cases, the at least one atom that is transferred from the reservoir array into the science array is in a dark state. In some cases, an atom that is transferred from the reservoir array into the science array is in a dark state, a clock state, or another state forbidden by selection rules from an optical excitation used for the transferring. The systems, the methods, the computer-readable media, and the techniques disclosed herein may be combined with approaches for state selective movement, such as, as described in International Application PCT / US2023 / 075948 which is incorporated herein in its entirety.WSGR Docket No. 55436-755.601Examples of Methods and Systems for Alignment of Modules

[0353] Trapping arrays of atoms using laser beams may have various applications in the field of atomic physics. However, increasing the size of these arrays may be accompanied by more laser power or multiplying effective laser power with an optical cavity. Some optical cavities comprise two mirrors facing one another, e.g., a linear cavity, a folded cavity, a bowtie cavity, etc., while others may comprise a plurality of mirrors arranged to direct a beam in closed loop, e.g., a ring cavity. Optical cavities may generally allow a power to build up over multiple reflection within the cavity.

[0354] Optical cavities may be sensitive to alignment. For example, if a mirror is mispositioned, subsequent passes of the beam may not traverse the same optical path leading to leakage, divergence, and loss of amplification. In some cases, the beam directing elements may be fixed during manufacturing. While fixing beam directing elements decreases likelihood that an aligned optical cavity will fall out of alignment, it may be important to align the cavity optics precisely before fixing the optical elements in place.

[0355] In some cases, a spacer may hold several optical elements, e.g., mirrors, prisms, etc., in about the correct position and orientation. During manufacturing, they may be aligned before being affixed to the spacer. In the case of an optical cavity where the optical axis has a point of intersection, the optical alignment may create an overlap of multiple laser beams or optical passes of the same laser beam (either from multiple cavities or inside a single cavity). In the case of trapped array of atoms, micrometer precision or better may be advantageous. It may be advantageous to 1) measure the relative positions between beams with a high degree of precision, and to 2) adjust mirror positions to compensate for imperfect overlap.

[0356] The systems, the methods, the computer-readable media, and the techniques disclosed herein may be directed to measuring the relative positions between beams with a high degree of precision. This may allow for adjustment of mirror positions to compensate for imperfect overlap. The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon existing methods for cavity alignment in at least some cases. Methods of alignment that involve moving an obstruction may lack adequate precision or may not be scalable to multiple beams.

[0357] For example, moving slit methods can be used to align two cavity beams. In a moving slit alignment, a slit may be placed around the beams, and transmission of each cavity through the slit as the slit is moved up and down may be monitored. When the slit moves too far to either side it clips and finally completely blocks the cavity beam. When the transmission is maximized, the slit is exactly centered on the cavity beam, and if this maximum occurs at the same slit position for both cavity beams, then the beams are aligned in the direction of slit movement.WSGR Docket No. 55436-755.601

[0358] This measurement may be relatively simple for two beams, because as long as they lie in a single plane, two beams are guaranteed to intersect. However, it is difficult to extend this method to more than two beams (which do not necessarily have a single intersection point), multiple beams inside a single cavity (where the obstruction can unintentionally clip multiple beams at once), or beam arrangements that do not lie in a single plane. Also, it is slow to mechanically move a slit while monitoring transmission, and additionally clipping the modes can change their shape, altering the measurement. Some of these problems can be addressed by replacing the moving slit with a fine tip, see for example Cai (incorporated by reference above), but it remains difficult to infer the exact 3D geometry of the beams from the response of the cavity transmission to a moving obstruction.

[0359] The systems, the methods, the computer-readable media, and the techniques disclosed herein may use direct imaging of Rayleigh scattering from the beams onto one or more cameras. In some cases, the position of intracavity laser beams can be directly measured by imaging Rayleigh scattering from the beams onto one or more cameras. A beam propagating through air scatters a small amount of light outside the beam path, which can be collected by a standard imaging system. The amount of scattered light is relatively small, but it may be multiplied by the power buildup factor of an optical cavity, accordingly the method may be suitable for cavities.

[0360] In some cases, a cavity of the present disclosure may be used in combination with a frequency stabilized (alternatively, a wavelength stabilized) laser. A frequency stabilized laser may in some cases also be phase stabilized; however, some frequency stabilized lasers may not be phase stabilized. In some cases, a frequency stabilized laser may be an optical clock. In an optical clock, the frequency stability may be less than sub-Hertz. In some cases, a laser may be actively frequency stabilize. In some cases, a laser may be passively frequency stabilized.

[0361] In some cases, a frequency-stabilized laser may be resonant with a cavity of the present disclosure.

[0362] In some cases, a camera is used in combination with the cavity to monitor the Rayleigh scattered light. In some cases, a single camera may measure the projection of all beams onto a 2D plane. In some examples, imaging from at least one other angle allows extraction of the beam paths in 3D space. The combination of the two cameras may provide complete information about the cavity beams. This method may be nonintrusive. The method may not involve blocking any of the beams. In some cases, all beams can be measured simultaneously, in 3D space, within a single image exposure time.Cavity Spacers

[0363] FIG. 13A shows different views of a plurality of mirrors to provide a plurality of optical cavities, according to some embodiments. The plurality of mirrors may be contained or held inWSGR Docket No. 55436-755.601place within a cavity spacer 1301. The cavity spacer 1301 may be constructed such that the plurality of mirrors may be oriented in one operable configuration. The cavity spacer 1301 may be constructed such that the plurality of mirrors may be oriented in at least two or more operable configurations. In some cases, the plurality of mirrors may comprise fold end mirrors 1302.1 and 1302.2 and two end mirrors 1302.3 and 1302.4 of a standing wave cavity. A plurality of mirrors 1303.1 and 1303.2 may be in an operable configuration to enable the generation of a three optical trap interaction propagating within the standing wave cavity and light propagating within the running wave cavity at interaction that may lie at the center of the cavity or cavity spacer.

[0364] In some examples, FIG. 13A shows different views of a plurality of mirrors configured to provide a plurality of optical cavities, according to some embodiments. A cavity spacer 1301 may be configured to hold a plurality of mirrors 1302.1-1302.4 and 1303.1-1303.3, The cavity spacer can be a low thermal expansion glass, th...

Claims

WSGR Docket No. 55436-755.601CLAIMS WHAT IS CLAIMED IS:

1. A non-classical computer, comprising:an array of spatially distinct optical traps, wherein the array comprises a plurality of atoms;an optical cavity array, the optical cavity array comprising an array of spatially distinct optical modes, wherein a cavity of the optical cavity array comprises an atom of the array of spatially distinct optical traps; andan optical control hardware, wherein the optical control hardware reconfigurably combines optical paths of the optical cavity array from at least two distinct atoms of the plurality of atoms for entanglement generation between the distinct atoms.

2. The non-classical computer of claim 1, wherein the optical control hardware is configured to receive light from an atom at the first site within the optical cavity array and to direct the light to the second site within the optical cavity array.

3. The non-classical computer of claim 1, wherein atom-photon entanglement is generated at the first site, wherein a photon is redirected to the second site, and wherein at the second site an atom at the second site absorbs the photon to generate atom-atom entanglement between the first site and the second site.

4. The non-classical computer of claim 1, wherein atom-photon entanglement is generated at each of the first site and the second site, wherein a first photon is generated at the first site, wherein a second photon is generated at the first second site, and wherein atomatom entanglement is generated by interfering the photons on a beam splitter and conditioning on two-photon detections.

5. The non-classical computer of claim 1, wherein the optical control hardware comprises:a telescope system and a plurality of light redirection optics.

6. The non-classical computer of claim 5, wherein the plurality of light redirection optics comprises a MEMS mirror array.

7. The non-classical computer of claim 5, wherein the plurality of light redirection optics comprises a plurality of individually tunable reflective optics.

8. The non-classical computer of claim 5, wherein the plurality of light redirection optics is configured to direct light from an atom at the first site within the optical cavity array and to a second site within the optical cavity array.

9. The non-classical computer of claim 5, wherein the telescope system comprises a local telescope and a global telescope.WSGR Docket No. 55436-755.60110. The non-classical computer of claim 9, wherein the local telescope comprises a local telescope array, wherein each local telescope of the array is configured to redirect light from a block or subset of cavities of the array of optical cavities.

11. The non-classical computer of claim 9, wherein the global telescope is configured to control a size of an array of light beams each corresponding to a cavity of the array of optical cavities.

12. The non-classical computer of claim 1, wherein the optical cavity array comprises a first optical cavity array and a second optical cavity array, wherein the optical control hardware is configured to produce an optically generated Bell-pair distributed between a first site within the first optical cavity array and a second site within the second optical cavity array.

13. The non-classical computer of claim 12, wherein the first optical cavity and the second optical cavity are optically distinct.

14. The non-classical computer of claim 13, wherein the first optical cavity and the second optical cavity comprise distinct cavity optics.

15. The non-classical computer of claim 12, wherein the first optical cavity and the second optical cavity are spatially distinct.

16. The non-classical computer of claim 15, wherein the first optical cavity and the second optical cavity are connected by fiber bundle.

17. The non-classical computer of claim 15, wherein the first optical cavity and the second optical cavity are separated by a distance of greater than 1 meter.

18. The non-classical computer of claim 1, wherein the first optical cavity and the second optical cavity are within a single non-classical computing module.

19. The non-classical computer of claim 1, wherein the first optical cavity is within a first non-classical computing module and the second optical cavity is within a second non- classical computing module.

20. The non-classical computer of claim 1, wherein the optically generated Bell-pair is configured to deterministically teleport qubits or gates between sites.

21. The non-classical computer of claim 1, wherein the optical control hardware is configured to enable all-to-all connectivity between spatially separate atoms.

22. A method of connecting spatially separate atoms within a trapped atom quantum computer, the method comprising:(a) trapping a plurality of atoms within an optical cavity array, the optical cavity array supporting a plurality of optical cavities comprising an array of spatially distinct optical traps; andWSGR Docket No. 55436-755.601(b) reconfigurably combining optical paths from distinct atoms for entanglement generation.

23. The method of claim 22, wherein (b) comprises one or more of receiving light from an atom at the first site within the optical cavity array and directing the light to the second site within the optical cavity array, or interfering light from an atom at the first site with light from an atom at the second site using a beamsplitter.

24. The method of claim 22, wherein the method further comprises deterministically teleporting a qubit or a gate between the first site and the second site.

25. The method of claim 24, wherein the method further comprises teleporting a gate between the first site and the second site, wherein the teleporting comprises, subsequent to (b), (i) applying local two-qubit gates at each of the first site and the second site, wherein each local two-qubit is between the qubit from the Bell-pair and a target qubit, and (ii) applying a single-qubit operation.

26. The method of claim 22, wherein the method further comprises implementing a state purification operation.

27. The method of claim 26, wherein the state purification operation comprises, subsequent to (b), (i) providing two remotely entangled Bell pairs; (ii) applying a local control-X gate at each of the two locations; and (iii) measuring one of the two qubits at each location.

28. The method of claim 22, wherein the method further comprises implementing an error correction operation.

29. The method of claim 28, wherein the error correction operation comprises syndrome extraction.

30. The method of claim 29, wherein the syndrome extraction comprises transversal CNOT operations.

31. A method for networked quantum computing comprising:(a) providing a first plurality of qubits in a first set of optical cavities and a second plurality of qubits in a second set of optical cavities, wherein qubits of the first plurality of qubits and the second plurality of qubits both comprise atoms in an array of spatially distinct optical traps(b) distributing entanglement between the first plurality of qubits and the second plurality of qubits to form a distributed state of the plurality of qubits between the first set of optical cavities and the second set of optical cavities; and(c) performing a quantum computing operation based at least in part by interacting with the distributed state.WSGR Docket No. 55436-755.60132. The method of claim 31, further comprising interacting the distributed state with a resource state.

33. The method of claim 32, further comprising continuously generating the resource state.

34. The method of claim 31, further comprising prior to (c), distilling the distributed state using n copies of the distributed state, wherein n is an integer greater than one.

35. The method of claim 34, wherein n is three.

36. The method of claim 34, wherein n is four or greater.

37. The method of claim 31, wherein the quantum computing operation is a quantum gate operation.

38. The method of claim 31, further comprising at (c) teleporting a gate operation.

39. The method of claim 31, wherein performing a quantum computation in (c) further comprises performing one or more quantum gates on a first qubit, a second qubit, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits.

40. The method of claim 39, wherein the first qubit, the second qubit, or both are entangled with the distributed state.

41. The method of claim 40, wherein the first qubit, the second qubit, or both are locally entangled with the distributed state.

42. The method of claim 40, wherein the first qubit, the second qubit, or both are entangled with the distributed state using a distributed Bell pair.

43. The method of claim 39, wherein the first qubit and the second qubit are in spatially distinct optical traps.

44. The method of claim 39, wherein the first qubit and the second qubit are in nonneighboring optical traps.

45. The method of claim 39, wherein the first qubit and the second qubit are in different optical cavity arrays.

46. The method of claim 39, wherein the first qubit and the second qubit are in different quantum computing modules47. The method of claim 31, wherein performing a quantum computation in (c) further comprises performing one or more quantum gates on a third plurality of qubits, a fourth plurality of qubits, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits.

48. The method of claim 47, wherein the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state.WSGR Docket No. 55436-755.60149. The method of claim 48, wherein the third plurality of qubits, the fourth plurality of qubits, or both are locally entangled with the distributed state.

50. The method of claim 48, wherein the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state using a distributed Bell pair.

51. The method of claim 47, wherein the third plurality of qubits and the fourth plurality of qubits are in spatially distinct optical traps.

52. The method of claim 47, wherein the third plurality of qubits and the fourth plurality of qubits are in non-neighboring optical traps.

53. The method of claim 47, wherein the third plurality of qubits and the fourth plurality of qubits are in different optical cavity arrays.

54. The method of claim 47, wherein the third plurality of qubits and the fourth plurality of qubits are in different quantum computing modules55. The method of claim 47, wherein the third plurality of qubits, the fourth plurality of qubits, or both are data qubits.

56. The method of claim 31, wherein the quantum computing operation is an error correction operation.

57. The method of claim 31, wherein the quantum computing operation is a logical operation.

58. The method of claim 31, wherein the quantum computing operation is a transversal gate between logical qubits.

59. The method of claim 58, wherein the transversal gate is a portion of a Steane style error correction code.

60. The method of claim 58, wherein the transversal gate is a portion of a Knill style error correction code.

61. The method of claim 31, wherein the first plurality of qubits, the second plurality of qubits, or both comprises a graph state in a first location, and wherein generating the distributed state in (b) further comprises generating a distributed graph state between the first location and a second location.

62. The method of claim 31, wherein the first plurality of qubits, the second plurality of qubits, or both comprises a cluster state in a first location, and wherein generating the distributed state in (b) further comprises generating a distributed cluster state between the first location and a second location.

63. The method of claim 61, wherein the distributed graph state comprises two or more entangled time slices.WSGR Docket No. 55436-755.60164. The method of claim 62, wherein the distributed cluster state comprises two or more entangled time slices.

65. The method of claim 63 or 64, wherein performing a quantum computing operation in (c) comprises performing a measurement of a qubit state in a first time slice of the two or more entangled time slices, wherein the measurement influences a second time slice of the two or more entangled time slices.

66. The method of claim 65, further comprising performing a quantum circuit at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices.

67. The method of claim 65, further comprising performing a quantum computation at least in part by performing a sequence of measurements of qubit states across multiple time slices of the two or more entangled time slices.

68. A system for networked quantum computing comprising:a first plurality of qubits in a first set of optical cavities,a second plurality of qubits in a second set of optical cavities, wherein the first set of optical cavities and the second set of optical cavities are optically connected, and wherein qubits of the first plurality of qubits and the second plurality of qubits both comprise atoms in an array of spatially distinct optical traps; anda distributed state formed by distributed entanglement of the first plurality of qubits and the second plurality of qubits between the first set of optical cavities and the second set of optical cavities.

69. The system of claim 68, wherein further comprising a resource state, wherein the resource state is interacted with the distributed state.

70. The system of claim 69, wherein the resource state is continuously generated.

71. The system of claim 68, wherein the distributed state is distilled using n copies of the distributed state, wherein n is an integer greater than one.

72. The system of claim 71, wherein n is three.

73. The system of claim 71, wherein n is four or greater.

74. The system of claim 68, wherein the system is configured to perform a quantum gate operation.

75. The system of claim 68, wherein the system is configured to perform one or more quantum gates on a first qubit, a second qubit, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits.

76. The system of claim 75, wherein the first qubit, the second qubit, or both are entangled with the distributed state.WSGR Docket No. 55436-755.60177. The system of claim 76, wherein the first qubit, the second qubit, or both are locally entangled with the distributed state.

78. The system of claim 76, wherein the first qubit, the second qubit, or both are entangled with the distributed state using a distributed Bell pair.

79. The system of claim 75, wherein the first qubit and the second qubit are in spatially distinct optical traps.

80. The system of claim 75, wherein the first qubit and the second qubit are in nonneighboring optical traps.

81. The system of claim 75, wherein the first qubit and the second qubit are in different optical cavity arrays.

82. The system of claim 75, wherein the first qubit and the second qubit are in different quantum computing modules83. The system of claim 68, wherein the system is configured to perform one or more quantum gates on a third plurality of qubits, a fourth plurality of qubits, or both, based at least in part on a measured quantum state of the first plurality of qubits and the second plurality of qubits.

84. The system of claim 83, wherein the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state.

85. The system of claim 84, wherein the third plurality of qubits, the fourth plurality of qubits, or both are locally entangled with the distributed state.

86. The system of claim 84, wherein the third plurality of qubits, the fourth plurality of qubits, or both are entangled with the distributed state using a distributed Bell pair.

87. The system of claim 83, wherein the third plurality of qubits and the fourth plurality of qubits are in spatially distinct optical traps.

88. The system of claim 83, wherein the third plurality of qubits and the fourth plurality of qubits are in non-neighboring optical traps.

89. The system of claim 83, wherein the third plurality of qubits and the fourth plurality of qubits are in different optical cavity arrays.

90. The system of claim 83, wherein the third plurality of qubits and the fourth plurality of qubits are in different quantum computing modules91. The system of claim 83, wherein the third plurality of qubits, the fourth plurality of qubits, or both are data qubits.

92. The system of claim 68, wherein the system is configured to perform an error correction operation.

93. The system of claim 68, wherein the system is configured to perform a logical operation.WSGR Docket No. 55436-755.60194. The system of claim 68, wherein system is configured to perform a transversal gate between logical qubits.

95. The system of claim 94, wherein the transversal gate is a portion of a Steane style method of error correction.

96. The system of claim 94, wherein the transversal gate is a portion of a Knill style method of error correction.

97. The system of claim 69, wherein the method further comprises distributing a graph state between a first location and a second location to produce a distributed graph state, at least in part by interacting the distributed state with the graph state, first plurality of qubits, the second plurality of qubits, or both comprises a graph state in a first location, and wherein the resource state comprises a distributed graph state between the first location and a second location.

98. The system of claim 69, wherein the first plurality of qubits, the second plurality of qubits, or both comprises a cluster state in a first location, and wherein generating the distributed state in (b) further comprises generating a distributed cluster state between the first location and a second location.

99. The system of claim 97, wherein the distributed graph state comprises two or more entangled time slices.

100. The system of claim 98, wherein the distributed cluster state comprises two or more entangled time slices.

101. The system of claim 99 or 100, wherein the system is configured to perform a quantum computing operation based at least in part on a measurement of a qubit state in a first time slice of the two or more entangled time slices, wherein the measurement influences a second time slice of the two or more entangled time slices.

102. The system of claim 101, wherein the system is configured to perform a quantum circuit at least in part by performing a sequence of measurements of the qubit state across multiple time slices of the two or more entangled time slices.

103. The system of claim 101, wherein the system is configured to perform a quantum computation by performing a sequence of measurements of qubit states across multiple time slices of the two or more entangled time slices.

104. The system of claim 68, wherein optically connecting the first set of optical cavities and the second set of optical cavities comprises transmitting light from the first set of optical cavities to the second set of optical cavities.WSGR Docket No. 55436-755.601105. The system of claim 68, wherein optically connecting the first set of optical cavities and the second set of optical cavities comprises interfering light from the first set of optical cavities and the second set of optical cavities on a beam splitter.

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