Systems and methods for high-fidelity universal gates in the ytterbium-171 ground state nuclear spin qubit

Abstract

Arrays of optically trapped neutral atoms are a promising architecture for the realization of quantum computers. In order to run increasingly complex algorithms, it is advantageous to demonstrate high-fidelity and flexible gates between long-lived and highly coherent qubit states. In this work, we demonstrate a universal high-fidelity gate-set with individually controlled and parallel application of single-qubit gates and two-qubit gates operating on the ground-state nuclear spin qubit in arrays of tweezer-trapped Ytterbium-171 atoms. We utilize the long lifetime, flexible control, and high physical fidelity of our system to characterize native gates using single and two-qubit Clifford and symmetric subspace randomized benchmarking circuits with more than 200 CZ gates applied to one or two pairs of atoms. We measure our two-qubit entangling gate fidelity to be 99.72(3)% (99.40(3)%) with (without) post-selection. In addition, we introduce a simple and optimized method for calibration of multi-parameter quantum gates. These results represent important milestones towards executing complex and general quantum computation with neutral atoms.

Description

Attorney Docket No. 55436-754.601SYSTEMS AND METHODS FOR HIGH-FIDELITY UNIVERSAL GATES IN THE YTTERBIUM-171 GROUND STATE NUCLEAR SPIN QUBITCROSS-REFERENCE

[0001] This application claims the benefit of U. S. Provisional Patent Application No.63 / 722,017, entitled “SYSTEMS AND METHODS FOR HIGH-FIDELITY UNIVERSAL GATES IN THE YTTERBIUM-171 GROUND STATE NUCLEAR SPIN QUBIT,” filed on November 18, 2024, which is incorporated by reference herein in its entirety.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., |00), 101), 110), |11), each called a quantum register. Generally, n qubits are represented by a superposition state vector in 2ndimensional Hilbert space.SUMMARY

[0004] In an aspect, the present disclosure provides a method for inducing a transition between a first state and a second state comprising: (a) providing a shaped composite pulse; and (b) driving a transition between said first state and said second state with said shaped composite pulse, wherein said first state and said second state are separated in energy by an optical frequency.

[0005] In some embodiments, said transition is an optical transition. In some embodiments, said first state or said second state is a metastable state. In some embodiments, said metastable state is a clock state. In some embodiments, said optical transition is part of a quantum computation. In some embodiments, said optical transition is part of a two-qubit gate. In some embodiments, said two-qubit gate comprises a two-step process. In some embodiments, a first step of said two-step process is said optical transition to a metastable state, and wherein a second step of said two-stepAttorney Docket No. 55436-754.601process is a transition to a Rydberg state. In some embodiments, the method further comprises performing said two-step process. In some embodiments, said two-qubit gate is a controlled-Z gate. In some embodiments, said controlled-Z gate comprises a fidelity of greater than 99.5%. In some embodiments, said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some embodiments, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) postselection using a benchmark sequence that is insensitive to single-qubit phases. In some embodiments, said quantum computation comprises individually addressable single-qubit gates. In some embodiments, said quantum computation comprises nuclear spin qubits in neutral atoms. In some embodiments, said neutral atoms are alkaline earth or alkaline earth like atoms. In some embodiments, said neutral atoms comprise two valence electrons in an outer s-shell. In some embodiments, said neutral atoms are 171 -Yb atoms. In some embodiments, said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse is configured to induce a population inversion between said first state and said second state. In some embodiments, said shaped composite pulse comprise a length of about 130 microseconds. In some embodiments, the method further comprises setting a phase profile and an amplitude profile of said shaped composite pulse. In some embodiments, said shaped composite pulse is a pulse or a pulse sequence. In some embodiments, the method further comprises inducing a transition between a third state and a fourth state, wherein said third state and said fourth state are separated in energy by a second optical frequency, and wherein a difference between said optical frequency and said second optical frequency is a radio frequency. In some embodiments, inducing said transition between said third state and said fourth state is performed, at least in part, using a second shaped composite pulse.

[0006] In another aspect, the present disclosure provides a method for inducing a population inversion between optically accessible states comprising: (a) setting a phase profile and an amplitude profile of a pulse or a pulse sequence; and (b) directing said pulse or said pulse sequence toward a quantum system to drive said population inversion between said optically accessible states.

[0007] In some embodiments, said optically accessible states are separated in energy by an optical frequency. In some embodiments, a state of said optically accessible states is a metastable state. In some embodiments, said metastable state is a clock state. In some embodiments, said population inversion between said optically accessible states is part of a quantum computation. In some embodiments, said population inversion is part of a two-qubit gate. In some embodiments,Attorney Docket No. 55436-754.601said two-qubit gate comprises a two-step process. In some embodiments, a first step of said two-step process is an optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state. In some embodiments, the method further comprises performing said two-step process. In some embodiments, said two-qubit gate is a controlled-Z gate. In some embodiments, said controlled-Z gate comprises a fidelity of greater than 99.5%. In some embodiments, said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some embodiments, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases. In some embodiments, said quantum computation comprises individually addressable single-qubit gates. In some embodiments, said quantum computation comprises nuclear spin qubits in neutral atoms. In some embodiments, said neutral atoms are alkaline earth or alkaline earth like atoms. In some embodiments, said neutral atoms comprise two valence electrons in an outer s-shell. In some embodiments, said neutral atoms are 171 -Yb atoms. In some embodiments, said pulse or said pulse sequence comprises a shaped composite pulse. In some embodiments, said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse comprise a length of about 130 microseconds.

[0008] In another aspect, the present disclosure provides a system for inducing a transition between a first state and a second state comprising: a pulse shaper configured to receive an optical pulse and to provide a shaped composite pulse; and a two-state system, wherein said shaped composite pulse is directed to said two state system to drive a transition between states of said two-state system, wherein said states are separated in energy by an optical frequency.

[0009] In some embodiments, a state of said two-state system is a metastable state. In some embodiments, said metastable state is a clock state. In some embodiments, both states of said two state system are promoted to metastable states. In some embodiments, said two-state system is qubit. In some embodiments, said optical transition is part of a two-qubit gate. In some embodiments the system further comprises instructions which when delivered to said pulse shaper are configured to implement a quantum computation comprising said two-qubit gate. In some embodiments, said two-qubit gate comprises a two-step process. In some embodiments, a first step of said two-step process is said optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state. In some embodiments, said quantum computation comprises individually addressable single-qubit gates. In some embodiments, said two-qubit gate is a controlled-Z gate. In some embodiments, said controlled-ZAttorney Docket No. 55436-754.601gate comprises a fidelity of greater than 99.5%. In some embodiments, said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some embodiments, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases. In some embodiments, said qubit is a nuclear spin qubit in a neutral atom. In some embodiments, said neutral atom is an alkaline earth or alkaline earth like atom. In some embodiments, said neutral atom comprises two valence electrons in an outer s-shell. In some embodiments, said neutral atom is a 171-Yb atom. In some embodiments, said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 -XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence. In some embodiments, said shaped composite pulse comprise a length of about 130 microseconds. In some embodiments, said pulse shape is configured to receive instructions which when executed set a phase profile and an amplitude profile of said shaped composite pulse. In some embodiments, said shaped composite pulse is a pulse or a pulse sequence. In some embodiments, the system further comprises a second two state system comprising states separated in energy by a second optical frequency, and wherein a difference between said optical frequency of said two state system and said second optical frequency of said second two state system is a radiofrequency. In some embodiments, a second composite pulse is directed to said second two state system to drive a transition between states of said two state system.

[0010] In another aspect, the present disclosure provides a quantum computing system comprising: a plurality of qubits, wherein said plurality of qubits comprise alkaline earth or alkaline earth like atoms; and a controlled-Z fidelity of greater than 99.5% fidelity.

[0011] In some embodiments, said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some embodiments, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) postselection using a benchmark sequence that is insensitive to single-qubit phases.

[0012] In another aspect, the present disclosure provides a quantum computing system comprising: a plurality of qubits, wherein said plurality of qubits comprise alkaline earth or alkaline earth like atoms; and a single-qubit Clifford Randomized Benchmarking (CRB) fidelity of greater than 99.9%.

[0013] In some embodiments, said single-qubit CRB fidelity is a fidelity of 99.963(2)%. In some embodiments, said quantum computing system is a trapped atom quantum computing system. In some embodiments, said trapped atom quantum computing system is a neutral atom quantum computing system. In some embodiments, said neutral atom quantum computing system is basedAttorney Docket No. 55436-754.601on nuclear spin qubits. In some embodiments, said neutral atom quantum computing system is based on two valence electron atoms. In some embodiments, said neutral atom quantum computing system comprises Ytterbium-171 atoms. In some embodiments, atoms are continuously reloaded from a reservoir, wherein the reservoir does not comprise spatially distinct optical trapping sites. In some embodiments, a qubit is measured mid-circuit substantially without loss of coherence.

[0014] 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.

[0015] 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

[0016] 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

[0017] 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 will 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:

[0018] FIG. 1 shows a computer control system that is programmed or otherwise configured to implement methods provided herein;

[0019] FIG. 2 shows an example of a system for performing a non-classical computation.

[0020] FIG. 3A shows an example of an optical trapping unit;Attorney Docket No. 55436-754.601

[0021] FIG. 3B shows an example of a plurality of optical trapping sites;

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

[0023] FIG. 3D shows an example of an optical trapping unit that is completely filled with atoms;

[0024] FIG. 4 shows an example of an electromagnetic delivery unit;

[0025] FIG. 5 shows an example of a state preparation unit;

[0026] FIG. 6 shows a flowchart for an example of a first method for performing a non-classical computation;

[0027] FIG. 7 shows a flowchart for an example of a second method for performing a non-classical computation;

[0028] FIG. 8 shows a flowchart for an example of a third method for performing a non-classical computation;

[0029] FIG. 9 shows an energy level structure for single-qubit and multi-qubit operations in strontium-87;

[0030] FIG. 10 shows an example method for error corrected non-classical computation;

[0031] 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;

[0032] FIG. 12 shows an example process for performing continuous, non-classical computation;

[0033] FIG. 13A shows example of a plurality of mirrors configured to provide a plurality of optical cavities;

[0034] 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;

[0035] FIG. 14 provides a non-limiting example of a level diagram for site selective readout in Yb-171, in accordance with some embodiments;

[0036] FIG. 15A provides a non-limiting example of clock transition for atoms trapped in a science array, in accordance with some embodiments;

[0037] FIG. 15B provides a non-limiting example of clock state lifetime for atoms trapped in a science array, in accordance with some embodiments;

[0038] FIG. 15C provides a non-limiting example of clock laser frequency noise, in accordance with some embodiments;

[0039] FIG. 15D provides a non-limiting example of coherence decay in a spin locking experiment, in accordance with some embodiments;

[0040] FIG. 16A provides a non-limiting example of Rydberg state lifetime after a pi pulse into the Rydberg state, in accordance with some embodiments;Attorney Docket No. 55436-754.601

[0041] FIG. 16B provides a non-limiting example of Clock-Rydberg measurement by Ramsey decay, in accordance with some embodiments;

[0042] FIG. 16C provides T2 echo decay with traps on, in accordance with some embodiments;

[0043] FIG. 16D provides T2 echo decay without traps on, in accordance with some embodiments;

[0044] FIG. 17 provides a non-limiting example of a method for inducing a transition between a first state and a second state, in accordance with some embodiments;

[0045] FIG. 18 provides a non-limiting example of a level diagram of the nuclear spin states of Yb-171, in accordance with some embodiments;

[0046] FIG. 19 provides a non-limiting example of a system for performing quantum operations in Yb-171, in accordance with some embodiments;

[0047] FIG. 20 provides a non-limiting example of an averaged randomized benchmarking (RB) curve for seven interactive zone (IZ) sites, in accordance with some embodiments;

[0048] FIG. 21 A provides a non-limiting example of shelving fidelity for an Xcllpulse, in accordance with some embodiments;

[0049] FIG. 21B provides a non-limiting example of frequency noise, in accordance with some embodiments;

[0050] FIG. 21C provides a non-limiting example of measuring clock pulse performance, in accordance with some embodiments;

[0051] FIG. 22A provides a non-limiting example of a Rydberg blockade mechanism, in accordance with some embodiments;

[0052] FIG. 22B provides a non-limiting example of fidelity over a ID scan of eigenvectors, in accordance with some embodiments;

[0053] FIG. 22C provides a non-limiting example of improved performance compared to scans of raw gate parameters, in accordance with some embodiments;

[0054] FIG. 23A provides a non-limiting example of two-qubit Clifford gates executed 20 times per random circuit, in accordance with some embodiments;

[0055] FIG. 23B provides a non-limiting example of symmetric subspace benchmarking of the CZ gate, in accordance with some embodiments;

[0056] FIG. 24 provides a non-limiting example of a method for inducing a population inversion between optically accessible states, in accordance with some embodiments;

[0057] FIG. 25 provides a non-limiting example of a system for inducing a transition between a first state and a second state, in accordance with some embodiments;

[0058] FIG. 26 provides a non-limiting example of an error budget, in accordance with some embodiments;Attorney Docket No. 55436-754.601

[0059] FIG. 27 provides a non-limiting example of circuits for a 2Q Clifford RB experiment, in accordance with some embodiments; and

[0060] FIG. 28 provides a non-limiting example of 2Q Clifford RB success probabilities for a range of criteria, in accordance with some embodiments.DETAILED DESCRIPTION OF THE INVENTION

[0061] While various embodiments of the systems and methods are 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 systems and methods described herein may be employed.Overview

[0062] In many cases, error-corrected quantum computation requires the ability to perform high-fidelity gate operations and readout on large numbers of physical-qubits. Toward this end, platforms utilizing individually controlled neutral atoms may employ techniques to assemble large arrays of atoms. In some cases, as many as a thousand atomic qubits may be assembled in an array, such as the atomic arrays described in M. Norcia, H. Kim, W. Caimcross, M. Stone, A. Ry ou, M. Jaffe, M. Brown, et al., Iterative assembly of 171 Yb atom arrays with cavity -enhanced optical lattices, PRX Quantum 5, 030316 (2024)., G. Pichard, D. Lim, E. Bloch, J. Vaneecloo, L. Bourachot, G.-J. Both, G. Meriaux, S. Dutartre, R. Hostein, J. Paris, B. Ximenez, A. Signoles, A. Browaeys, T. Lahaye, and D. Dreon, Rearrangement of individual atoms in a 2000-site optical-tweezer array at cryogenic temperatures, Phys. Rev. Appl. 22, 024073 (2024)., and H. J.Manetsch, G. Nomura, E. Bataille, K. H. Leung, X. Lv, and M. Endres, A tweezer array with 6100 highly coherent atomic qubits (2024), arXiv: 2403.12021., each of which is incorporated by reference herein in its entirety. In some cases, assembling such arrays of atomic qubits can enable error-corrected quantum computation by providing a large number of physical qubits.

[0063] In some cases, platforms using individually controlled neutral atoms may additionally employ techniques for mid-circuit measurement, such as those mid-circuit measurement techniques described in K. Singh, C. Bradley, S. Anand, V. Ramesh, R. White, and H. Bemien, Mid-circuit correction of correlated phase errors using an array of spectator qubits, Science 380, 1265 (2023)., E. Deist, Y.-H. Lu, J. Ho, M. K. Pasha, J. Zeiher, Z. Yan, and D. M. Stamper-Kum, Mid-circuit cavity measurement in a neutral atom array, Phys. Rev. Lett. 129, 203602 (2022)., T. M. Graham, L. Phuttitarn, R. Chinnarasu, Y. Song, C. Poole, K. Jooya, J. Scott, A. Scott, P. Eichler, and M. Saffman, Midcircuit measurements on a single-species neutral alkali atom quantum processor, Phys. Rev. X 13, 041051 (2023)., M. A. Norcia, W. B. Cairncross, H. Kim, et al., Midcircuit qubit measurement and rearrangement in a Ytterbium-171 atomic array, Phys. Rev. X 13, 041034Attorney Docket No. 55436-754.601(2023)., J. W. Lis, A. Senoo, W. F. McGrew, F. Ronchen, A. Jenkins, and A. M. Kaufman, Midcircuit operations using the omg architecture in neutral atom arrays, Phys. Rev. X 13, 041035 (2023)., and D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, etal., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024)., each of which is incorporated by reference herein in its entirety. In one example, high-fidelity single qubit gates with arbitrary local control have been demonstrated in atoms featuring hyperfine qubits, such as described in D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, et al., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024)., which is incorporated by reference herein in its entirety. In another example, high-fidelity single qubit gates with arbitrary local control have been demonstrated in atoms featuring nuclear spin qubits, such as described in K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N.Crisosto, J. P. King, S. S. Kondov, K. Kotru, S. C. Larsen, etal., Assembly and coherent control of a register of nuclear spin qubits, Nature Communications 13, 2779 (2022)., which is incorporated by reference herein in its entirety. In some cases, two-qubit gates with fidelity above 99%, such the Surface Code threshold as described in A. M. Stephens, Fault-tolerant thresholds for quantum error correction with the surface code, Phys. Rev. A 89, 022321 (2014)., which is incorporated by reference herein in its entirety, have been demonstrated in a number of systems. In one example, two-qubit gates with fidelity above 99% have been demonstrated for hyperfine qubits, such as hyperfine systems described in H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bemien, M. Greiner, V. Vuletic, H. Pichler, and M. D. Lukin, Parallel implementation of high-fidelity multi qubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019)., and S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletic, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023)., each of which is incorporated by reference herein in its entirety. In another example, two-qubit gates with fidelity above 99% have been demonstrated for optical qubits, such as optical systems described in I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, High-fidelity entanglement and detection of alkaline-earth Rydberg atoms, Nature Physics 16, 857 (2020), which is incorporated by reference herein in its entirety. In a further example, two-qubit gates with fidelity above 99% have been demonstrated for metastable nuclear spin qubits, such as metastable nuclear spin systems described in M. Peper, Y. Li, D. Y. Knapp, M. Bileska, S. Ma, G. Liu, P. Peng, B. Zhang, S. P. Horvath, A. P. Burgers, and J. D. Thompson, Spectroscopy and modeling of Ytterbium-171 Rydberg states for high-fidelity two-qubit gates (2024), arXiv:2406.01482., which is incorporated by reference herein in its entirety.Attorney Docket No. 55436-754.601

[0064] Among the different optically trapped neutral atom platforms, ground-state nuclear spin qubits can demonstrate long coherence times due to a high degree of insensitivity to environmental perturbations such, in some cases. For example, ground-state nuclear spin qubits may demonstrate a high degree of insensitivity to trap light-shifts and magnetic fields, such as described in K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N. Crisosto, J. P. King, S. S. Kondov, K. Kotru, S. C. Larsen, et al., Assembly and coherent control of a register of nuclear spin qubits, Nature Communications 13, 2779 (2022)., which is incorporated by reference herein in its entirety. In another example, ground-state nuclear spin qubits may demonstrate a near-infinite lifetime with respect to decay, such as described in A. Jenkins, J. W. Lis, A. Senoo, W. F. McGrew, and M. Kaufman, Ytterbium nuclear-spin qubits in an optical tweezer array, Phys. Rev. X 12, 021027 (2022)., which is incorporated by reference herein in its entirety.

[0065] In many cases, nuclear spin ground state qubits may provide certain advantages compared to other qubit systems, such as metastable nuclear spin qubits, optical qubits, and hyperfine qubits. In one example, metastable nuclear spin qubits or optical qubits may suffer from relatively short lifetimes in comparison to nuclear spin ground state qubits due, at least in part, to trap Raman scattering. In another example, hyperfine qubits in alkali atoms may demonstrate a relatively high sensitivity to trap light-shifts when compared to nuclear spin ground state qubits.

[0066] Recognized herein is the need for systems and methods which combine the capabilities of high-fidelity gates with the ability for atomic qubit rearrangement.

[0067] In some cases, error correction schemes may be implemented which utilize physical qubits efficiently, such as error correction schemes described in D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, etal., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024)., Q. Xu, J. P. Bonilla Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays, Nature Physics, 1 (2024)., and Y. Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, Long-range-enhanced surface codes, Phys. Rev. A 110, 022607 (2024)., each of which is incorporated by reference herein in their entirety. In some cases, universal, gate-based quantum computation comprises the use of a maximally entangling two-qubit gate. Some examples of maximally entangling two qubit gates include the CNOT gate, such as described in J. I. Cirac and P. Zoller, Quantum computations with cold trapped ions, Phys. Rev. Lett. 74, 4091 (1995)., the XX gate, such as described in A. Sorensen and K. Molmer, Quantum computation with ions in thermal motion, Phys. Rev. Lett.82, 1971 (1999)., or the CZ gate, such as described in E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, Observation of Rydberg blockadeAttorney Docket No. 55436-754.601between two atoms, Nature Physics 5, 110 (2009)., each of which is incorporated by reference herein in their entirety. In some cases, universal, gate-based computation further comprises the use of single qubit gates. In some cases, the use of single qubit gates may present certain challenges in neutral atom systems, at least in part due to closely spaced trapped qubits. In some cases, in order to mitigate these challenges, global single-qubit operations have been used in such neutral atom systems. These global single-qubit gates may be capable of characterizing certain gate errors, such as described in S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletic, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023)., C. H. Baldwin, B. J. Bjork, J. P. Gaebler, D. Hayes, and D. Stack, Subspace benchmarking high-fidelity entangling operations with trapped ions, Phys. Rev. Res. 2, 013317 (2020), and R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and linear response modeling of high-fidelity Rydberg gates (2024), arXiv:2407.20184., each of which is incorporated by reference herein in their entireties. In some cases, however, certain aspects of gate fidelity may not be measurable using global single-qubit gates. In some cases, a more accurate measurement of overall gate fidelity may be provided by two-qubit Clifford benchmarking, such as described in J. Emerson, R. Alicki, and K. Zyczkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005)., and C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80, 012304 (2009)., each of which is incorporated by reference herein in their entireties. In some cases, two-qubit Clifford benchmarking provides advantages when compared to benchmarking using global single-qubit gates.

[0068] Recognized herein is the need for systems and methods which can implement two-qubit Clifford benchmarking in neutral atom systems to provide highly accurate benchmarking that is not susceptible to the additional errors using benchmarking based on global single-qubit gates.Examples of Methods for Inducing Transitions Between Two Quantum States

[0069] In one aspect, systems and methods of the present disclosure provide a method for inducing a transition between quantum states. A non-limiting example of a method for inducing a transition between a first state and a second state 1700 is provided in FIG. 17, in accordance with some embodiments. In some cases, the method 1700 comprises providing at 1705 a shaped composite pulse and driving at 1710 an optical transition between the first state and the second state with the shaped composite pulse.

[0070] In some cases, a standard optical pulse, such as a square optical pulse, may comprise a Gaussian or semi -Gaussian profile. In some cases, a shaped optical pulse, herein, comprises an opticalAttorney Docket No. 55436-754.601pulse that has been modified to have a specific temporal profile, spectral profile, or both, which can be different from the standard Gaussian or semi-Gaussian profile. In some cases, shaping an optical pulse can involve adjusting parameters such as the intensity, duration, or spectral phase of an optical source to create optical pulses with a specific temporal profile, spectral profile, or both. In some cases, shaped optical pulses may be configured to specifically excite specific optical transitions. In some cases, shaped optical pulses may be further configured to suppress unwanted effects of optical transitions, which may be present with excitation using a standard optical pulse. Some examples of shaped optical pulses include sech2optical pulses, parabolic optical pulses, square optical pulses, chirped optical pulses, raised-cosine optical pulses, Blackman optical pulses, and the like.

[0071] In some cases, a composite optical pulse, herein, comprises a sequence of individual optical pulses which are configured to implement a specific optical transition. In some cases, the individual optical pulses of a composite optical pulse each comprise well-defined relative phases. In some cases, controlling the sequence and relative phase of each individual optical pulse of a composite optical pulse can allow for more precise control of an optical transition, at least in part by mitigating for errors in pulse area, pulse duration, or pulse frequency. In some cases, composite pulses may be configured to implement a specific phase shift to an atom or qubit. Some examples of composite pulses include BB1 pulses, CORPSE pulses, SCROFULOUS pulses, Ulla pulses, Ullb pulses, Y90-X180-Y90 pulses, and the like.

[0072] In some cases, the shaped composite pulse comprises a Blackman shaped pulse. In some cases, the shaped composite pulse comprises a Y90 pulse. In some cases, the shaped composite pulse comprises an XI 80 pulse. In some cases, the shaped composite pulse comprises one or more Y90 and X180 pulse. For example, the shaped composite pulse may comprise a Y90 - X180 - Y90 pulse sequence. For example, the shaped composite pulse may comprise any pulse described in M. H. Levitt, R. Freeman, NMR Population Inversion Using a Composite Pulse, Journal of Magnetic Resonance, 33, 2, 473-476, (1979)., which is incorporated by reference herein in its entirety. In some cases, the shaped composite pulse comprises a pi pulse. In some cases, the shaped composite pulse is configured to induce a population inversion between the first state and the second state.

[0073] In some cases, an optical pulse herein may be defined as a rotation with respect to a specific axis, for example Y90 or X180. In one naming convention for this case, the first letter refers to the axis of rotation, for example X, Y, or Z. In such a naming convention, the following numbers refer to the number of degrees of a rotation, for example 90 degrees or 180 degrees. In another naming convention, an optical pulse herein may be referred to by the rotation in radians. For example, a 90 degree pulse may also be referred to as a π / 2 pulse, and a 180 degree pulse may also be referred to as a 7t pulse. In some cases, an optical pulse herein may comprise any pulse described in, for instance, M.Attorney Docket No. 55436-754.601H. Levitt, R. Freeman, NMR Population Inversion Using a Composite Pulse, Journal of Magnetic Resonance, 33, 2, 473-476, (1979)., which is incorporated by reference herein in its entirety.

[0074] In some cases, the shaped composite pulse comprises a length of about 130 microseconds. In some examples, the shaped composite pulse comprises a length of about 10 microseconds. In some examples, the shaped composite pulse comprises a length of about 10 microseconds to about 50 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 50 microseconds. In some examples, the shaped composite pulse comprises a length of about 50 microseconds to about 100 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 100 microseconds. In some examples, the shaped composite pulse comprises a length of about 100 microseconds to about 150 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 150 microseconds. In some examples, the shaped composite pulse comprises a length of about 150 microseconds to about 200 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 200 microseconds. In some examples, the shaped composite pulse comprises a length of about 200 microseconds to about 300 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 300 microseconds. In some examples, the shaped composite pulse comprises a length of about 300 microseconds to about 500 microseconds and increments therein. In some examples, the shaped composite pulse comprises a length of about 500 microseconds or more.

[0075] In some cases, providing at 1705 a shaped composite pulse provides particular advantage and utility by supplying an optical pulse configured to selectively excite specific optical transitions, while mitigating errors in pulse area, pulse duration, or pulse frequency, and suppressing unwanted effects which may be present with excitation using a standard optical pulse.

[0076] In some cases, providing at 1705 a shaped composite pulse comprises any pulse sequence, described herein. For example, the shaped composite pulse may comprise a pulse sequence provided by an electromagnetic delivery unit, described herein.

[0077] In some cases, the shaped composite pulse is a pulse or a pulse sequence, described herein. In some cases, the pulse sequences of a shaped composite pulse, herein, 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.Attorney Docket No. 55436-754.601

[0078] The pulse sequences may be configured to decrease the duration of time required to implement multi -qubit operations, as described herein. 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.

[0079] 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.

[0080] The pulse sequences may enable the implementation of multi-qubit operations on non-adiabatic timescales while maintaining effectively adiabatic dynamics. For instance, the pulse sequences may comprise one or more of shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, counterdiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences. For instance, the pulse sequences may be similar to those described in M. V. Berry, “Transitionless Quantum Driving,” Journal of Physics A: Mathematical and Theoretical 42(36), 365303 (2009), www.doi.org / 10.1088 / 1751-8113 / 42 / 36 / 365303; Y.-Y. Jau et al., “Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction,” Nature Physics 12(1), 71-74 (2016); T. Keating et al., “Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing,” Physical Review A 91, 012337 (2015); A. Mitra et al., “Robust Mblmer-Sbrenson Gate for Neutral Atoms UsingAttorney Docket No. 55436-754.601Rapid 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.

[0081] 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.

[0082] In some cases, providing at 1705 a shaped composite pulse may be done, at least in part, using any optical source, described herein. In some cases, providing at 1705 a shaped composite pulse may be done, at least in part, using any light source, described herein. In some cases, providing at 1705 a shaped composite pulse may be done, at least in part, using any laser, described herein.

[0083] 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 (An) excimer lasers, krypton dimer (Kn) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe ) 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.

[0084] 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 (MnCE) metal-vapor lasers.Attorney Docket No. 55436-754.601

[0085] The lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal -doped crystal lasers, or metal -doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd: YAG) lasers, neodymium / chromium doped yttrium aluminum garnet (Nd / Cr: YAG) lasers, erbium-doped yttrium aluminum garnet (Er: YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd: YLF) lasers, neodymium-doped yttrium orthovanadate (Nd:YVO₄ ) lasers, neodymium-doped yttrium calcium oxoborate (Nd: YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm: YAG) lasers, ytterbium -doped ytrrium aluminum garnet (Yb: YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho: YAG) lasers, chromium-doped zinc selenide (Cr: ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce: LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce: LiCAF) lasers, erbium-doped glass (Er:glass) lasers, erbium-ytterbium-codoped glass (Er / Yt:glass) lasers, uranium-doped calcium fluoride (U: CaF2) lasers, or samarium-doped calcium fluoride (Sm: CaF2) lasers.

[0086] 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.

[0087] 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.Attorney Docket No. 55436-754.601

[0088] The lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values.

[0089] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (pJ), 2 pj, 3 pj, 4 pj, 5 pj, 6 pj, 7 pj, 8 pj, 9 pj, 10 pj, 20 pj, 30 pj, 40 pj, 50 pj, 60 pj, 70 pj, 80 pj, 90 pj, 100 pj, 200 pj, 300 pj, 400 pj, 500 pj, 600 pj, 700 pj, 800 pj, 900 pj, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 pj, 800 pj, 700 pj, 600 pj, 500 pj, 400 pj, 300 pj, 200 pj, 100 pj, 90 pj, 80 pj, 70 pj, 60 pj, 50 pj, 40 pj, 30 pj, 20 pj, 10 pj, 9 pj, 8 pj, 7 pj, 6 pj, 5 pj, 4 pj, 3 pj, 2 pj, 1 pj, 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.Attorney Docket No. 55436-754.601

[0090] The lasers may emit light having an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values.

[0091] 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,Attorney Docket No. 55436-754.6011,160 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, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values.

[0092] 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 about 1 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'10Attorney Docket No. 55436-754.601nm, 5 x IO'10nm, 4 x IO'10nm, 3 x IO'10nm, 2 x IO'10nm, 1 x IO'10nm, 9 x IO'11nm, 8 x IO'11nm, 7 x IO'11nm, 6 x IO'11nm, 5 x IO'11nm, 4 x IO'11nm, 3 x IO'11nm, 2 x IO'11nm, 1 x IO'11nm, 9 x IO’12nm, 8 x 10'12nm, 7 x 10'12nm, 6 x 10'12nm, 5 x 10'12nm, 4 x 10'12nm, 3 x 10'12nm, 2 x IO’12nm, 1 x 10'12nm, 9 x 10'13nm, 8 x 10'13nm, 7 x 10'13nm, 6 x 10'13nm, 5 x 10'13nm, 4 x 10'13nm, 3 x 10'13nm, 2 x 10'13nm, 1 x 10'13nm, 9 x 10'14nm, 8 x 10'14nm, 7 x 10'14nm, 6 x 10'14nm, 5 x 10'14nm, 4 x 10'14nm, 3 x 10'14nm, 2 x 10'14nm, 1 x 10'14nm, 9 x IO'15nm, 8 x IO’15nm, 7 x IO'15nm, 6 x IO'15nm, 5 x IO'15nm, 4 x IO'15nm, 3 x IO'15nm, 2 x IO'15nm, 1 x IO’15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.

[0093] 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.

[0094] In some cases, the method 1700 comprises driving at 1710 an optical transition between a first state and a second state with the shaped composite pulse. In some cases, the first state and the second state are different quantum states of an atom. In some cases, the first state is a ground state of an atom, and the second state is an excited state of an atom. In some cases, the first state and the second state are separated in energy by an optical frequency. In some cases, the first state is an excited state, and the second state is a higher-lying excited state of an atom. In some cases, the first state or the second state comprises a metastable state of an atom. In some cases, driving at 1710 comprises one or more intermediate states between the first state and the second state.

[0095] In some cases, the first state or the second state comprises one or more nuclear spin states of an atom. In some cases, the first state or the second state comprises one or more electron states of an atom. In some cases, the first state or the second state comprises one or more hyperfine states of an atom. In some cases, the first state or the second state comprises one or more Zeeman states of an atom. In some cases, the first state or the second state comprises one or more Rydberg states of an atom.Attorney Docket No. 55436-754.601

[0096] In some cases, two states separated by an optical frequency, described herein, refers to two distinct energy levels or quantum states, where the distinct energy levels or quantum states comprise an energy difference between them corresponding to the energy of a photon at a specific optical frequency. In some cases, the photon at a specific optical frequency comprises any optical frequency of infrared electromagnetic radiation. In some cases, the photon at a specific optical frequency comprises any optical frequency of visible light. In some cases, the photon at a specific optical frequency comprises any optical frequency of ultraviolet light.

[0097] In some examples, an optical frequency of two states separated by an optical frequency, herein, is at least about 1012Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is between about 1012Hz to about 1013Hz. In some examples, an optical frequency of two states separated by an optical frequency, herein, is about 1013Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is between about 1013Hz to about 1014Hz. In some examples, an optical frequency of two states separated by an optical frequency, herein, is about 1014Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is between about 1014Hz to about 1015Hz. In some examples, an optical frequency of two states separated by an optical frequency, herein, is about 1015Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is between about 1015Hz to about 1016Hz. In some examples, an optical frequency of two states separated by an optical frequency, herein, is about 1016Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is between about 1016Hz to about 1017Hz. In some examples, an optical frequency of two states separated by an optical frequency herein, is greater than about 1017Hz.

[0098] In some cases, the first state or the second state is a metastable state. In some cases, the metastable state is a clock state, described herein. In some cases, the optical transition at 1710 is part of a quantum computation, described herein. In some cases, the optical transition at 1710 is part of a two-qubit gate, described herein. In some cases, the two-qubit gate comprises a two-step process. In some cases, the two-step process comprises the optical transition at 1710 to a metastable state, and the second step comprises a transition to a Rydberg state. In some cases, the method 1700 further comprises performing at 1715 a two-step process, described herein, wherein the optical transition at 1710 is part of a two-qubit gate, wherein the two-qubit gate comprises the two-step process, and wherein a first step of the two-step process comprises an optical transition to a metastable state, and a second step of the two step process comprises a transition to a Rydberg state.

[0099] In some cases, driving at 1710 an optical transition between a first state and a second state with the shaped composite pulse comprises a transition between different quantum states of oneAttorney Docket No. 55436-754.601or more atoms, described herein. In some cases, the atoms are neutral atoms. In some cases, the atoms are alkaline earth atoms. In some cases, the atoms are alkaline earth-like atoms. In some cases, the atoms comprise two valence electrons in an outer shell. In some cases, the atoms are 171-Yb atoms.

[0100] One or more atoms may comprise alkali atoms. One or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium -45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium -145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium- 175 atoms, or lutetium- 176 atoms.

[0101] 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 ofAttorney Docket No. 55436-754.601elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium -6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium- 156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium- 164 atoms, erbium -166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium- 176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 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-Attorney Docket No. 55436-754.60142 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium- 130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, 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,Attorney Docket No. 55436-754.601holmium-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.

[0102] 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.

[0103] The first atomic state and second atomic state may comprise two distinct electronic states. For instance, the first and second atomic states may comprise first and second electronic states on a multiplet manifold, such as a triplet manifold. The first and second atomic states may comprise first and second electronic states, respectively, on a1So,3Po,3Pi, or3P2 manifold. The first and second atomic states may comprise first and second electronic states, respectively, on a1So,3Po,3Pi, or3P2 manifold of any atom described herein. For example, the first and second atomic states may comprise first and second electronic states on a ytterbium-1711So manifold, a ytterbium-1713Po manifold, a strontium-873Pi manifold, or a strontium-873P2 manifold. In some cases, the distinct electronic states may be in two distinct hyperfine sublevels.Examples of Qubit Gate Schemes in Strontium

[0104] 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.

[0105] In some cases, the electronic states comprise nuclear spin states of a strontium-87xSo manifold and the qubit transition drives one or both of two nuclear spin states of strontium-87xSo to a state detuned from or within the3P2 or3Pi manifold. In some cases, the one-qubit transition is a two photon Raman transition between nuclear spin states of strontium-87 'So via a state detuned from or within the3P2 or3Pi manifold. In some cases, the nuclear spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift mayAttorney Docket No. 55436-754.601be 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.

[0106] Qubits based on nuclear spin states in the electronic ground state may allow exploitation of long-lived metastable excited electronic states (such as a3Po state in strontium-87) for qubit storage. Atoms may be selectively transferred into such a state to reduce cross-talk or to improve gate or detection fidelity. Such a storage or shelving process may be atom -selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the1So state in strontium-87 to the3Po or3P2 state in strontium-87.

[0107] 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. In one non-limiting example, the first state and the second state of the method 1700 are energy levels of strontium-87, as depicted in FIG. 9.

[0108] In some cases, the electronic states comprise nuclear spin states of ytterbium.

[0109] 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 or ytterbium -171.

[0110] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1 / 2 (such as a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 7 / 2 spin state, may also drive mN = 7 / 2 to mN = 5 / 2, mN = 5 / 2 to mN = 3 / 2, mN = 3 / 2 to mN = 1 / 2, mN = 1 / 2 to mN = -1 / 2, mN = -1 / 2 to mN = -3 / 2, mN = -3 / 2 to mN = -5 / 2, mN = -5 / 2 to mN = -7 / 2, and mN = -7 / 2 to mN = -9 / 2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transferAttorney Docket No. 55436-754.601atoms 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.[OHl] In some cases, 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).

[0112] Stark shifting of the nuclear spin manifold may shift neighboring nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. Stark shifting may decrease leakage from the first and second nuclear spin state to other states in the nuclear spin manifold. Starks shifts may be achievable up to 100s of kHz for less than 10 mW beam powers. Upper state frequency selectivity may decrease scattering from imperfect polarization control. Separation of different angular momentum states in the3Pi manifold may be many gigahertz from the single and two-qubit gate light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for two-qubit transitions (e.g., how quickly the transition can be driven) may be faster than the decoherence rate. Scattering from the intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve fidelity of two-qubit transitions.Attorney Docket No. 55436-754.601Examples of Qubit Gate Schemes in Ytterbium

[0113] A non-limiting example of a level diagram of the nuclear spin states of Ytterbium-171 is provided in FIG. 18, in accordance with some embodiments. As depicted in FIG. 18, excitation to and from a metastable clock state |c) = |3Po, mf = -1 / 2) may form an intermediate state of a sequential excitation scheme. In some cases, the excitation to and from a metastable clock state is per-formed via a Xfkshelving pulse, or any shaped composite pulse described herein, designed and calibrated to transfer as much population as possible be-tween |1) and |c). For example, the excitation may be performed by any shaped composite pulse which provides the composite effect of a Xfkpulse. A combination of frequency and polarization-selectivity can provide stateselectivity for this excitation process, and thus for the two-qubit gate. In some cases, from the clock state, a pulse of ultraviolet (UV), 302 nm, o- -polarized light is applied to drive the |c) <-> |r) = |653Si, F = 3 / 2, mf = -3 / 2) transition. The UV pulse phase and amplitude profile may be chosen to ensure that every atomic pair returns to its initial state after the pulse, while pairs of neighboring clock atoms acquire an additional n phase shift due to the Rydberg blockade mechanism, such as described in D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M.D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000)., which is incorporated by reference herein in its entirety. Finally, atoms may be returned to |1) with a second Xfkpulse, having acquired a conditional phase that implements a CZ gate. In one nonlimiting example, the first state and the second state of the method 1700 are energy levels of ytterbium, as depicted in FIG. 18.

[0114] In one non-limiting example, systems and methods herein may be employed on a system comprising Ytterbium-171. In some cases, the method 1700 further comprises setting at 1720 a phase profile and an amplitude profile of the shaped composite pulse. In some cases, setting at 1720 comprises adjusting the spectral or temporal properties of the shaped composite pulse to perform a specific function. In some cases, setting at 1720 is done using any controller described herein. In some cases, setting at 1720 is done using any computer system described herein.

[0115] In one non-limiting example, systems and methods herein may be employed on a system comprising Ytterbium-171.Examples of Systems for Performing Single and Two-Qubit Gates on Ytterbium-171

[0116] In one example of systems and methods herein, operations may be performed on the quantum states of the ground-state nuclear spin of Ytterbium-171 atoms through individually controlled single-qubit gate operations and two-qubit operations based on sequential state selective coherent excitation to a long-lived clock state and Rydberg state. In some cases, systems and methods herein may enable a controlled-Z (CZ) fidelity of 99.72(3)% (99.40(3)%) with (without) postAttorney Docket No. 55436-754.601selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments, and a singlequbit CRB fidelity of 99.963(2)%. Using a benchmark sequence that is insensitive to single-qubit phases (relevant to situations where such phases can be canceled between pairs of gates by using echo techniques as in [9, 13]) and antisymmetric errors may further enable a CZ gate fidelity of 99.84(6)% (99.56(5)%) with (without) post-selection. In some cases, the combination of these spatially selective high-fidelity gates with the previously demonstrated continuous loading and mid-circuit measurement may enable new demonstrations of quantum error correction and complex circuits among many physical-qubits, as described herein, and additionally in Microsoft Azure Quantum and Atom Computing, Submitted to arxiv / quant-ph on November 19th(2024)., which is incorporated by reference herein for all purposes.

[0117] One example of a system for performing quantum operations in Yterbium-171 consists of an array of optically trapped single Ytterbium-171 atoms 1900, as depicted in FIG. 19. In some cases, the array of Yterbium-171 atoms 1900 may be substantially the same as those described in M. Norcia, H. Kim, W. Cairncross, M. Stone, A. Ryou, M. Jaffe, M. Brown, et al., Iterative assembly of 171 Yb atom arrays with cavity-enhanced optical lattices, PRX Quantum 5, 030316 (2024)., and M. A. Norcia, W. B. Cairncross, H. Kim, et al., Midcircuit qubit measurement and rearrangement in a171Yb atomic array, Phys. Rev. X 13, 041034 (2023)., each of which is incorporated by reference herein in its entirety. In some cases, the nuclear spin ground states1So, mf = -1 / 2 (mf = 1 / 2) are used to encode the |0) (|1)) states of our qubit, as shown in FIG. 18.

[0118] In some cases, systems and methods herein may be configured to isolate the performance of gate operations. Single and two-qubit gates may be performed in the science optical tweezers formed with 460 nm light (which can provide state-insensitive trapping for the1So ^3Po optical clock transition used in our two-qubit gates), while state preparation and state-sensitive, nondestructive readout are performed in the reservoir 483 nm tweezers (which can provide stateinsensitive trapping for the1So ^3Pi inter-combination line). In order to enhance the data-rate while limiting the potential for inhomogeneity between sites, two-qubit gates may be performed within one or two pairs of traps in a single row of the array, that is refilled from a larger reservoir of atoms after readout. In some cases, this operation may be performed in the interaction zone (IZ) 1905 of the array 1900.Examples of Single-Qubit Gates

[0119] In some cases, systems and methods herein may employ a single-qubit addressing scheme. In one example of a single-qubit addressing scheme, local and parallel control of the pulse area and phase of rotations applied to the nuclear spin qubits of multiple atoms in the IZ 1905 may be enabled. In such an example the Raman beams may be red detuned by 5 GHz of theAttorney Docket No. 55436-754.6011So ^3Pi, F = 1 / 2 transition depicted in FIG. 18, resulting in a two-photon Raman Rabi rate of ~ 2TI *7 kHz. The phase of the applied gate may be set by the differential phase of the two Raman beams, and can be controlled arbitrarily, in some cases.

[0120] One example of a single-qubit (IQ) gate-set used herein consists of Zn / 2and ^7r / 2oPerations, where Zn / 2operations are performed virtually via frame-tracking and used to update the phase of the next applied Appulse. In some cases, these single-qubit gates may be substantially the same or similar to those described in D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017)., which is incorporated by reference herein in its entirety. In some cases, the performance of these gates may be characterized using Clifford Randomized Benchmarking (CRB) experiments, such as those described in J. P. Gaebler, A. M. Meier, T. R. Tan, R. Bowler, Y. Lin, D. Hanneke, J. D. Jost, J. P. Home, E. Knill, D. Leibfried, and D. J. Wineland, Randomized benchmarking of multiqubit gates, Phys. Rev. Lett. 108, 260503 (2012)., E.Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011)., and E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012)., each of which is incorporated by reference herein in their entireties. In one example, CRB of single-qubit gates herein may yield an average fidelity of 99.963(2)% per Clifford gate, averaged over seven IZ sites 1905. A non-limiting example of an averaged randomized benchmarking (RB) curve for seven IZ sites 1905 is depicted in FIG. 20, with the CRB circuit executed in parallel among the IZ atoms.

[0121] In some cases, calibrating the Xn / 2gate requires setting the pulse area by tuning the product of Rabi rates associated with the individual Raman beams, and zeroing differential lightshifts by tuning the intensity-ratio of the two Raman beams. In some cases, pulse sequences may be used which are the same or similar to the Robust Phase Estimation (RPE) methods described in S. Kimmel, G. H. Low, and T. J. Yoder, Robust calibration of a universal single-qubit gate set via robust phase estimation, Phys. Rev. A 92, 062315 (2015), which is incorporated by reference herein in its entirety. In some cases, local beam alignment calibrations via single beam AC-Stark shift measurements may be used to align each set of Raman beams to the atoms. Typical gate performance may be limited by quasi-static drifts in alignment and beam in-tensities, not fundamental processes such as intermediate state scattering. In some cases, all calibrations and corrections are applied to individual qubits.Attorney Docket No. 55436-754.601Examples of Two-Qubit Gates

[0122] In some cases, two-qubit (2Q) gates may be performed by state-selectively exciting pairs of atoms to high-lying Rydberg states via a two-step process to apply a symmetric CZ gate. In some cases, two-qubit gates herein may differ from previously demonstrated two-photon excitation schemes, such as those described in H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletic, H. Pichler, and M. D. Lukin, Parallel implementation of high-fidelity multi qubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019)., T. Wilk, A. Ga aetan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, Entanglement of two individual neutral atoms using Rydberg blockade, Phys. Rev. Lett. 104, 010502 (2010)., T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, M. Ebert, and M. Saffman, Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array, Phys. Rev. Lett. 123, 230501 (2019)., and S. Ma, A. P. Burgers, G. Liu, J. Wilson, B. Zhang, and J. D. Thompson, Universal gate operations on nuclear spin qubits in an optical tweezer array of171Yb atoms, Phys. Rev. X 12, 021028 (2022)., each of which is incorporated by reference herein in their entireties, where the drive lasers are applied simultaneously and detuned from a short-lived intermediate state. In some cases, two-qubit gates herein may operate with sequential, resonant excitation to and from a long-lived intermediate state. This may provide several key advantages. It may allow maximization of the Rabi frequency to the short-lived Rydberg state given power constraints, which in turn may reduce the effects of Rydberg decay. In addition, by using a long-lived intermediate state, scattering from this state may be reduced as well. Because the excitation to the intermediate state is relatively slow, in some cases, moderate differential light-shifts on the narrow transition may be used herein to prevent atoms from participating in a gate, providing opportunities for site-selective addressing while using global gate lasers.

[0123] Excitation to and from the metastable clock state |c) = |3Po, rm = -1 / 2) that forms the intermediate state of the sequential excitation scheme may be performed via a Xclkπshelving pulse, designed and calibrated to transfer as much population as possible be-tween |1) and |c). A combination of frequency and polarization-selectivity can provide state- selectivity for this excitation process, and thus for the two-qubit gate. From the clock state, a pulse of ultraviolet (UV), 302 nm, ^--polarized light is applied to drive the |c) |r) = |653Si, F = 3 / 2, r = -3 / 2) transition, such as depicted in FIG. 18. In some cases, the UV pulse phase and amplitude profile may be chosen to ensure that every atomic pair returns to its initial state after the pulse, while pairs of neighboring clock atoms acquire an additional n phase shift due to the Rydberg blockade mechanism. Finally, atoms may be returned to |1) with a second Xclkπpulse, having acquired a conditional phase that implements a CZ gate.Attorney Docket No. 55436-754.601

[0124] During the application two-qubit gates in the example system 1900, described herein, the atoms may be trapped in 460 nm optical tweezers within the IZ 1905 with at a trap frequency ωx / (2π) = 50kHz. Non-participating reservoir atoms may be maintained in |0), and do not couple to |c), in some cases.

[0125] In some cases, ensuring optimal performance of the two-qubit gate further comprises minimizing population and phase errors arising from the clock shelving and unshelving pulses. The clock pulses may be applied to a Doppler-sensitive (single-photon) transition and can be relatively slow (Ωclk / (2π) « 7kHz) compared to the trap frequency. In some cases, this relative speed difference can make them sensitive to a specific set of errors, wherein finite atomic temperature leads to a spread in Rabi frequencies between motional states, and atoms can be coupled to other motional states of the trap. In some cases, laser phase and amplitude noise near the Rabi frequency, as well as trap-induced decay from the clock state can also degrade performance. In some cases, quasi-static errors in clock laser detuning can lead to qubit phase shifts during the gate.

[0126] In order to mitigate the effects of atomic motion, atoms may be cooled near their motional ground state along the direction of the clock laser. In one example, 3D gray molasses may be used to cool atoms to n = 0.25(10) along the x-direction. Examples of cooling with 3D gray molasses are described in J. W. Lis, A. Senoo, W. F. McGrew, F. Röbchen, A. Jenkins, and A. M. Kaufman, Midcircuit operations using the omg architecture in neutral atom arrays, Phys. Rev.X 13, 041035 (2023)., which is incorporated by reference herein in its entirety. In one example, for a Lamb-Dicke parameter q = 0.26, this can limit the clock-shelving fidelity to < 99.8% for a single Ti pulse. With two clock shelving pulses per qubit per CZ gate, clock shelving errors can limit CZ fidelity significantly, in some cases. To further reduce sensitivity to any effect that causes spread in Rabi rates, and atomic temperature, shaped composite pulses (SCPs), described herein, may be employed. In some cases, the smooth pulse shape reduces unwanted frequency components that can induce motional state changing transitions, and the composite pulse is designed to be robust to pulse-area errors.

[0127] In some cases, using SCPs may present some challenges. In one example, SCPs may be longer than a square pulse, making them more susceptible to laser frequency noise, Raman scattering and quasi-static detunings. In some cases, a Blackman shaped Yclkπ / 2~— Yclkπ / 2pulse offers enough robustness to mitigate these challenges, while balancing the im-pact from frequency noise. This SCP pulse may be referred to as the YXYclkpulse. In some cases, the Xclkπoperation comprises a YXYclkpulse.

[0128] A non-limiting example of shelving fidelity for Xclkπpulses is provided in FIG. 21A for atoms starting on state |1). In some cases, typical shelving fidelities per pulse can exceed 99.85%Attorney Docket No. 55436-754.601in these conditions. Most of the error may be due to population left in |c), and a smaller loss or decay to the other ground state due to Raman scattering of the 460 nm light. In one example, each Blackman-shaped 7t-clock pulse lasts 130 ps, and is calibrated using RPE techniques, described herein. In a further example the laser frequency noise may be calibrated on an optical self-heterodyne fiber interferometer setup, such as described in F. Kefelian, H. Jiang, P.Lemonde, and G. Santarelli, Ultralow-frequency-noise stabilization of a laser by locking to an optical fiber-delay line, Opt. Lett. 34, 914 (2009)., which is incorporated by reference herein in its entirety. In some examples, spin locking atomic measurements may be employed that map laser frequency noise into changes of atomic coherence, such as described in R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and linear response modeling of high-fidelity Rydberg gates (2024), arXiv:2407.20184., and R. Finkelstein, R. B.-S. Tsai, X. Sun, P. Scholl, S. Direkci, T. Gefen, J. Choi, A. L. Shaw, and M. Endres, Universal quantum operations and ancilla-based read-out for tweezer clocks, Nature 634, 321 (2024)., each of which is incorporated by reference herein in their entireties. In some cases, the frequency range of interest spans from a few kHz to 20 kHz, where typical laser locks have limited gain. In some cases, the interferometer reports a larger frequency noise than the one inferred from the atomic measurement, as provided in FIG. 21B. In some cases, this disagreement may be due to a combination of excess acoustic noise and limitations on the noise floor of the interferometer. In such cases, the contribution of laser frequency noise may be bounded to the one extracted in the spin locking experiments.

[0129] In some cases, clock pulse performance on arbitrary nuclear spin qubit states may be calibrated using systems and methods described in S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletic, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023)., which is incorporated by reference herein in its entirety. In some cases, the method comprises initializing atom pairs in |00), and applying N blocks that consist of (i) a common random Haar-distributed IQ rotation Rrandon each qubit, (ii) a two qubit unitary U on the atomic pair, (iii) an echo pulse (X„) on the qubit space, and (iv) an additional application of U. A non-limiting example of calibration of clock pulse performance is provided in FIG. 21C. In the example, each block uses a different Rrand, and satisfies that U — Xn— U does not create entanglement. After the N blocks are applied, a deterministic IQ rotation Rf, pre-calculated under the assumption that U is ideal, returns atoms to the |00) state. Readout may be performed in the two-qubit computational basis. By progressively constructing our CZ gate from different U gates, for instance identity, XclkπXclkπor finally a full CZ gate, different error sources may be identified. In some cases, this protocol may be referred toAttorney Docket No. 55436-754.601as U-global echo randomized benchmarking sequence (U-GERB). In some cases, as compared to the 2Q CRB described herein, this U-GERB protocol applies the same gates to atoms within a pair and so constitutes a symmetric subspace benchmark.

[0130] A non-limiting example of a characteristic clock-GERB curve for U = XclkπXclkπin is provided in FIG. 21C. In some cases, this measurement is subjected to errors from the eight IQ gates used. In some cases, this may be characterized in the case where U is the identity operation and measure a contribution of 0.32(2)% per atomic pair per GERB block. For this set, characteristic of a well-tuned system, a pre-selected fidelity for the shelve-unshelve clock sequence (circles) of 99.80(1)% per pair may be measured, after removing the IQ error. In the depicted example, the post-selected fidelity of the clock shelve-unshelve sequence on atoms that remain on the qubit subspace at the end of the circuit is 99.94(2)% (squares), which may be mostly affected by decoherence be-tween the optical and ground state qubits. The difference between the pre- and post-selected fidelities points to leakage (atoms that remained in |c)), and loss. By re-cycling atoms from the3Po state and imaging them after-wards, a pair survival probability (triangles) of 99.979(4)% may be measured after each shelve-unshelve pulse.

[0131] In some cases, once clock shelving is complete, entangling gates are performed via the Rydberg blockade mechanism, coupling atoms from the clock state to the |r) state with a single global beam at 301.9 nm. A non-limiting example of the Rydberg blockade mechanism is provided in FIG. 22A. In some cases, the Rydberg blockade mechanism employed herein may be substantially the same or similar to that described in M. Saffman, Quantum computing with atomic qubits and Rydberg interactions: progress and challenges, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 202001 (2016)., which is incorporated by reference herein in its entirety. In the example system 1900, an ultraviolet laser system enables Rydberg Rabi rates Ωryd> 2π × 15 MHz. In some cases, a smaller Rabi rate may be chosen to remain far below the Rydberg interaction energy U / h = 2π × 160 MHz. In the example system 1900, for most Rydberg operations the 460 nm tweezers are kept on, as they provide a trapping potential for the |r) state due to the ion core polarizability and have minimal impact on gate performance. In some cases, the Rydberg state lifetime may be 65(3) ps at a typical trap depth.

[0132] In some cases, CZ gates are implemented with an approximation of the time-optimal gate, using the sinusoidal phase parametrization φ(t) = A cos(ωt — φ₀) + At, with a square pulse of length T. In some cases, these CZ gates may be similar to those described in S. Jandura and G. Pupillo, Time-optimal two-and three qubit gates for Rydberg atoms, Quantum 6, 712 (2022) employing the parameterization of S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T.Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletic, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atomAttorney Docket No. 55436-754.601quantum computer, Nature 622, 268 (2023)., each of which is incorporated by reference herein in their entireties. In some cases, optimization over the phase parameters guarantees that the |01 ) (| 10)) states undergo nearly closed rotations through the |0r)(|r0)) states, ideally leaving no population in the Rydberg state and picking up a single-qubit phase φ01(φ10), while the |1 ideally leaves no population in | W ) = (|cr) + |rc)) / 2 and picks up a different phase φ11. The latter may be decomposed into the sum of single-qubit and entangling phases as φ1±= φent+ φ01+ φ10- Choosing gate parameters so that φent= π, the final unitary is Uo= diag(1, eeiφ, eiφ, —ei(φ+φ)), which can be con-verted to the ideal CZ gate Uo= diag(1,1,1,—1) with virtual single-qubit Z rotations.

[0133] While optimal phase profiles may be readily obtained in simulation, experimental imperfections shift the optimum, requiring calibration of the control parameters. In some cases, the example system 1900 may implement an optimized calibration by diagonalizing the simulated Hessian matrix of the gate error. In some cases, the resulting eigenvectors are nearly decoupled, so that near-optimal fidelity can be reached with a single ID scan along each eigenvector. A non-limiting example of fidelity over a ID scan of each eigenvector for the example system 1900 is provided in FIG.22B. In some cases, this strategy maintains the robustness and clarity of ID scans, while greatly improving convergence compared to scans of the raw gate parameters, as depicted in FIG. 22C.

[0134] In some cases, the gate is calibrated using an echoed metric such as the population of the desired state after applying CZN— Xn— CZN, with N gates on either side of the Xnpulse, as shown in FIG. 22B. In some cases, the echo removes any dependence on the single-qubit phases, which are obtained separately by robust phase estimation. It is straightforward to incorporate effects like finite blockade and pulse rise time into the model, improving the initial estimate of optimal parameters and the decoupling of the eigen-vectors.

[0135] In some cases, the systems and methods herein can provide particular advantage in performing two qubit gates. In some cases, the optical transition at 1710 is a part of a two-qubit gate. In some cases, the two qubit gate is a controlled-Z gate. In some cases, the systems and methods herein can provide improved fidelity in performing a controlled-Z two-qubit gate. In one example, systems and methods herein can enable a controlled-Z gate with a fidelity of greater than 99.5%. In another example, systems and methods herein can enable a controlled-Z gate with a fidelity of 99.72(3)% with post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments, and a controlled-Z gate with a fidelity of 99.40(3)% without post selection from two-qubit Clifford Randomized Benchmarking experiments. In a further example, systems and methods herein can enable a controlled-Z gate with a fidelity of 99.84(6)% with postselection using a benchmark sequence that is insensitive to single-qubit phases, and a controlled-Attorney Docket No. 55436-754.601Z gate with a fidelity of 99.56(5)% without post-selection using a benchmark sequence that is insensitive to single-qubit phases.

[0136] Clifford Randomized Benchmarking (CRB) refers to a protocol to measure the average error rate of quantum gate operations, at least in part, by running a random sequence of Clifford gates configured to return a quantum system to an initial state. The set of all Clifford gates form the Clifford group, a group of operators generated by three fundamental quantum gates, the Hadamard gate (H), the phase gate (5), and the controlled-NOT (CNOT) gate. A member of the Clifford group (C) obeys the following transformation property CPCP=P where P is a Pauli operator. That is, a Clifford gate, when conjugated with a Pauli operator, may result in another Pauli operator. In some cases, Clifford gates may play a crucial role in quantum error detection and quantum error correction. In CRB, a series of Clifford gates may be applied to a quantum system such that the sequence of Clifford gates is configured to return the final state of quantum system to a target initial state. The fidelity of the final state of the quantum system may be measured with respect to the target initial. In some cases, the fidelity is a function of the length of the sequence of Clifford gates and may be fit to a function of exponential decay. In some cases, this serves to amplify gate errors over the sequence of Clifford gates, making it easier to detect small errors with higher accuracy. CRB may be more sensitive to gate error than single-qubit gate-based error detection and is favorable over single-qubit based fidelity measurement in many cases.

[0137] Clifford gates are the basis for many error-correcting protocols envisioned for universal fault-tolerant quantum computation, as described in D. Gottesman, An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation (2009), arXiv:0904.2557, which is incorporated by reference herein in its entirety. The Clifford group may be sufficiently complex that most errors (even coherent errors) will be perfectly depolarized under the assumption of fixed Clifford-error channels associated with each Clifford gate. In some cases, this can enable unbiased comparisons across different experimental platforms, even if the implementation of the Clifford operators may vary, such as described in J. P. Gaebler, A. M. Meier, T. R. Tan, R.Bowler, Y. Lin, D. Hanneke, J. D. Jost, J. P. Home, E. Knill, D. Leibfried, and D. J. Wineland, Randomized Benchmarking of Multiqubit Gates, Physical Review Letters 108, 260503 (2012)., which is incorporated by reference herein in its entirety. In some cases, the ability to perform local, independent single-qubit operations is both important for use in quantum algorithms, and enables fidelity benchmarks that average over the full Hilbert space of the qubits, as opposed to only the symmetric subspace as has been the case in recent gate benchmarking approaches with tweezer-trapped neutral atoms described in S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini,Attorney Docket No. 55436-754.601M. Greiner, V. Vuletic, and M. D. Lukin, High-fidelity parallel entangling gates on a neutralatom quantum computer, Nature 622, 268 (2023)., M. Peper, Y. Li, D. Y. Knapp, M. Bileska, S.Ma, G. Liu, P. Peng, B. Zhang, S. P. Horvath, A. P. Burgers, and J. D. Thompson, Spectroscopy and modeling of171Yb Rydberg states for high-fidelity two-qubit gates (2024),arXiv:2406.01482, R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and linear response modeling of high-fidelity Rydberg gates (2024), arXiv:2407.20184., and S. Ma, G. Liu, P. Peng, B. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D. Thompson, High-fidelity gates and mid-circuit erasure conversion in an atomic qubit, Nature 622, 279 (2023)., each of which is incorporated by reference herein in their entireties. In some cases, physical error sources, such as de-cay from the Rydberg state, entangling phase, or single-qubit phase errors, will contribute to circuit errors with slightly different weights.

[0138] In the example system 1900, two-qubit Clifford gates are implemented by performing the IQ and 2Q gates, described herein. In one example, for each realization of the experiment, twenty random quantum circuits with a given depth are generated with the Qiskit Experiments package described in A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, et al., Quantum computing with qiskit, arXiv preprint arXiv:2405.08810 (2024)., which is incorporated by reference herein in its entirety. A non-limiting example of two-qubit Clifford gates executed 20 times per random circuit is provided in FIG. 23 A, for the example system 1900. By fitting the data to an exponential decay curve, an average two-qubit Clifford gate fidelity with (without) post-selection of 99.40(4)% (98.93(4)%) is measured. Accounting for the average number of native 2Q and IQ gates per Clifford gate (~ 1.51 and ~ 4.36 respectively), and assuming a depolarizing error model, a CZ fidelity of 99.72(3)% (99.40(3)%) with (without) post-selection may be extracted. Atom loss is measured to be 0.11(1)% per CZ gate and is dominated by Rydberg state decoherence.

[0139] This 2Q CRB measurement may be compared to a CZ-GERB measurement, that represents a symmetric subspace benchmark of our CZ gate. A non-limiting example of subspace benchmarking of the CZ gate for the example system 1900 is provided in FIG. 23B. In this example, the presence of echoes within each GERB block makes this circuit insensitive to singlequbit phase offsets, and more robust against quasi-static drifts. Using the CZ-GERB circuit fidelity (under the assumption of no anti-symmetric errors) of 99.84(6)% (99.56(5)%) with (without) post-selection on atoms remaining in the qubit subspace is measured. In some cases, a large fraction of the difference between the fidelity metrics may be attributed to quasi-static errors of the clock laser detuning, which maps into single-qubit phase errors.Attorney Docket No. 55436-754.601

[0140] The excess loss (0.11(3)%) measured in CZ-GERB in FIG. 23B relative to clock-GERB in FIG. 21C may contribute half of the additional pre-selected infidelity measured between the two metrics, 0.20(2)% and 0.44(5)% respectively in the example system 1900. Errors in entangling or single-qubit phases between consecutive CZ gates can lead to excess infidelity, in some cases. Table I summarizes the observed infidelities.1Q-GERB Clock-GERB CZ-GERB 2Q-CRBPre-selected 0.16(1)% 0.20(2)% 0.44(5)% 0.60(3)% infidelityPot-selected 0.16(1)% 0.06(2)% 0.16(6)% 0.28(3)% infidelityPair Loss 0.00(1)% 0.021(4)% 0.13(3)% 0.11(1)%TABLE I. Infidelity contribution per CZ gate as measured by GERB and CRB. Clock-GERB, CZ-GERB, and 2Q CRB, infidelities are corrected by the error of a single IQ gate and the average number of IQ gates in each circuit.

[0141] In some cases, systems and methods herein enable high-fidelity single- and two-qubit gates on ground state nuclear spin qubits, as demonstrated by the example system 1900. While the demonstration of two-qubit gates was restricted to one or two pairs of interaction sites in the example 1900, the methods demonstrated here can also be applied to larger arrays with arbitrary connectivity by utilizing larger numbers of tweezer traps and coherent movement of atoms. These extensions may benefit from the long coherence time and insensitivity to light-shifts of the ground-state nuclear spin qubit. In some cases, care must be taken to minimize atomic heating during movement. In a complementary approach, the narrow linewidth of the clock shelving transition may enable the use of local light-shifts to modify connectivity within static arrays of atoms (though connectivity in this approach may be limited to nearby atoms). By combining these techniques with mid-circuit measurements, described herein, and continuous loading techniques, described herein, as well as erasure conversion provided by state-selective, nondestructive measurement

[0053] , the high-fidelity and flexible gates demonstrated herein may enable the execution of complex error-corrected quantum circuits. In some cases, non-destructive measurement techniques herein may be similar to those described in K. Sahay, J. Jin, J. Claes, J. D. Thompson, and S. Puri, High-threshold codes for neutral-atom qubits with biased erasure errors, Phys. Rev. X 13, 041013 (2023)., which is incorporated by reference herein in its entirety.

[0142] In some cases, improvements may be made to the example system 1900, such as improvements in clock laser quasi-static and fast frequency noise described in R. Finkelstein, R. B.-S. Tsai, X. Sun, P. Scholl, S. Direkci, T. Gefen, J. Choi, A. L. Shaw, and M. Endres,Attorney Docket No. 55436-754.601Universal quantum operations and ancilla-based read-out for tweezer clocks, Nature 634, 321 (2024)., and L. Li, W. Huie, N. Chen, B. DeMarco, and J. P. Covey, Active cancellation of servoinduced noise on stabilized lasers via feedforward, Phys. Rev. Appl. 18, 064005 (2022).

[0143] H. Bombin, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015)., each of which is incorporated by reference herein in their entireties. Additional improvements may be made through investigation of the complex Rydberg state manifold, as described in M. Peper, Y. Li, D. Y. Knapp, M. Bileska, S. Ma, G. Liu, P. Peng, B. Zhang, S. P. Horvath, A. P. Burgers, and J. D. Thompson, Spectroscopy and modeling of171Yb Rydberg states for high-fidelity two-qubit gates (2024), arXiv:2406.01482., which is incorporated by reference herein in its entirety. In some cases, the high-fidelity measured for the example system 1900, and the prospect of physical-qubit-number efficient encodings and single-shot-fault tolerant schemes described in H. Bombin, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015), which is incorporated by reference herein in its entirety, may enable particular advantage and utility enabled by all-to-all connectivity through movement.Example of a Method for Inducing Population Inversion Between Optically Accessible States

[0144] In some cases, the present disclosure provides a method for inducing a population inversion between optically accessible states 2400, provided in FIG. 24. In some cases, the method comprises setting at 2405 a phase profile and an amplitude profile of a pulse or a pulse sequence, and directing at 2410 the pulse or pulse sequence towards a quantum system to drive the population inversion between the optically accessible states.

[0145] In some cases, the optically accessible states are separated in energy by an optical frequency. In some cases, the optically accessible states comprise the first state and the second state of method 1700, described herein. In some cases, a state of the optically accessible states is a metastable state. In some cases, the optically accessible states comprise a two-state system. In some cases, both states of the two-state system are promoted to metastable states. In some cases, the metastable state is a clock state. In some cases, the population inversion between the optically accessible states is part of a quantum computation, described herein. In some cases, the population inversion is part of a two-qubit gate, described herein. In some cases, the two-qubit gate comprises a two-step process, described herein. In some cases, a first step of said two-step process is an optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state. In some cases, the method 2400 further comprises performing at 2415 a two-step process, described herein. In some cases, the two-qubit gate is a controlled-Z gate, described herein. In some cases, the controlled-Z gate comprises a fidelity of greater than 99.5%. In some cases, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%)Attorney Docket No. 55436-754.601with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some cases, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

[0146] In some cases, the population inversion is part of a quantum computation, described herein. In some cases, the quantum computation comprises individually addressable single-qubit gates, described herein. In some cases, the quantum computation comprises nuclear spin qubits in neutral atoms. In some cases, the neutral atoms are alkaline or alkaline earth-like atoms. In some cases, the neutral atoms comprise two valence electrons in an outer s-shell. In some cases, the neutral atoms are Ytterbium-171 atoms. In some cases, the atoms or qubits of method 2400 comprise any atom or qubit, described herein.

[0147] In some cases, a pulse or pulse sequence of the method 2400 comprises any pulse sequence described herein. In some cases, the pulse or pulse sequence of the method 2400 comprises any shaped composite pulse (SCP) described herein. In some cases, the shaped composite pulse is a pi pulse. In some cases, the shaped composite pulse comprises a length of about 130 microseconds. In some cases, the shaped composite pulse comprises any length described herein.Examples of a Systems for Inducing a Transition Between a First State and a Second State

[0148] In some cases, the present disclosure provides a system 2500 for inducing a transition between a first state 2505 and a second state 2510, provided in FIG. 25. In some cases, the system 2500 comprises a pulse shaper 2515 configured to receive an optical pulse 2520 and to provide a shaped composite pulse 2525. In some cases, the optical pulse 2520 is provided by any light source, described herein. In some cases, the optical pulse 2520 is provided by any laser, described herein. In some cases, the shaped composite pulse 2525 is any shaped composite pulse described herein. For example, the shaped composite pulse 2525 may be the same shaped composite pulse provided in the method 1700. For example, the shaped composite pulse 2525 may be the same shaped composite pulse provided in the method 2400.

[0149] In some cases, the shaped composite pulse 2525 is directed to the two-state system 2530.In some cases, the two-state system comprises the first state 2505 and the second state 2510. In some cases, the first state 2505 is any quantum state described herein. For example, the first state 2505 may be the first state of the method 1700. For example, the first state 2505 may be the first state of the method 2400. For example, the first state 2505 may be any state of the example system 1900. In some cases, the second state 2510 is any quantum state described herein. For example, the second state 2510 may be the second state of the method 1700. For example, the second state 2510 may be the second state of the method 2400. For example, the second state 2510 may be any state of the example system 1900. In some cases, a state of the two-state systemAttorney Docket No. 55436-754.6012530 is a metastable state. In some cases, the metastable state is a clock state, described herein. In some cases, both states of the two-state system 2530 are promoted to metastable states by the composite pulse 2525. In some cases, the two-state system 2530 comprises a qubit, described herein.

[0150] In some cases, the shaped composite pulse 2525 is configured to drive an optical transition between states of the two-state system 2530. In some cases, the optical transition is part of a two-qubit gate, described herein. In some cases, the optical transition is the same optical transition as the method 1700. In some cases, the optical transition is the same optical transition as the method 2400. In some cases, the two-qubit gate comprises a two-step process, described herein. In some cases, a first step of said two-step process is an optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state. In some cases, the system 2500 may be used to perform a two-step process, described herein. In some cases, the two-qubit gate is a controlled-Z gate, described herein. In some cases, the controlled-Z gate comprises a fidelity of greater than 99.5%. In some cases, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some cases, said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

[0151] In some cases, the system 2500 may be used to perform a quantum computation, described herein. In some cases, the quantum computation comprises individually addressable single-qubit gates, described herein. In some cases, the quantum computation comprises nuclear spin qubits in neutral atoms. In some cases, the neutral atoms are alkaline or alkaline earth -like atoms. In some cases, the neutral atoms comprise two valence electrons in an outer s-shell. In some cases, the neutral atoms are Ytterbium-171 atoms. In some cases, the atoms or qubits of the system 2500 comprise any atom or qubit, described herein.

[0152] In some cases, a pulse or pulse sequence 2525 comprises any pulse sequence described herein. In some cases, the pulse or pulse sequence 2525 comprises any shaped composite pulse (SCP) described herein. In some cases, the shaped composite pulse is a pi pulse. In some cases, the shaped composite pulse comprises a length of about 130 microseconds. In some cases, the shaped composite pulse comprises any length described herein.

[0153] In some cases, the present disclosure provides a quantum computing system comprising a plurality of qubits, wherein the plurality of qubits comprise alkaline earth or alkaline earth-like atoms, and a controlled-Z gate of fidelity greater than 99.5% fidelity. In some cases, the controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments. In some cases, the controlled-Z gateAttorney Docket No. 55436-754.601fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

[0154] In some cases, the present disclosure provides a quantum computing system comprising a plurality of qubits, wherein the qubits comprise alkaline earth or alkaline earth-like atoms, and a single-qubit Clifford Randomized Benchmarking fidelity of greater than 99.9%. In some cases, the single-qubit CRB fidelity of 99.963(2)%.

[0155] In some cases, any system herein comprises a trapped atom quantum computing system. In some cases, any system herein comprises a neutral atom quantum computing system. In some cases, any system herein is based on nuclear spin qubits, described herein. In some cases, any system herein is based on two valence electron atoms. In some cases, any system herein comprises Ytturbium-171 atoms. In some cases, any system herein comprises any atom, described herein.

[0156] In some cases, any system herein is continuously reloaded from a reservoir, described herein. In some cases, the reservoir does not comprise spatially distinct optical trapping sites. In some cases, a qubit is measured mid-circuit substantially without loss of coherence.

[0157] In some cases, any system herein comprises a reservoir array 260 of the system 200. 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.

[0158] 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.

[0159] In some cases, any system herein 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, acoustooptic deflectors (AODs), or spatial light modulators (SLMs).

[0160] In some cases, the system 2500 further comprises instructions 2535 which when delivered to the pulse shaper 2515, are configured to implement a quantum computation, described herein,Attorney Docket No. 55436-754.601using a two-qubit gate, described herein. In some cases, the instructions 2535 may be provided by any computer system, described herein.Examples of Experimental Methods and Sources of Error in an Example Ytterbium-171 System

[0161] In some cases, systems and methods herein may be applied to a sample system, such as the sample Ytterbium-171 system 1900.

[0162] In some cases, two-qubit gates circuit depths scans repeat a circuit about 20 times and sample 10 different circuits for CZ-GERB and 20 different circuits for 2Q-CRB. The value assigned as the measured probability is the weighted average over the different circuit realizations. Error bars at each circuit depth are assigned as the standard error of the mean among the different realizations. For circuit depth scans depicted herein, all the decay functions may be fitted to exponential functions apx+ b, where x represents the depth of the circuit, a and p are fitting parameters, and b is fixed to a predetermined value. Typically b is related to the single atom subspace dimensionality d and the total number of qubits in the measurement basis

[0022] , All infidelities, 1 - F, are calculated from the decay fit as(d2— l)p + 11 - F = 1 - « (1 - p)(l - d)

[0163] Confidence intervals on the error rates and fidelities may be reported based on the standard deviation errors of the fitting parameters taking into account the actual spread of the experimental data.

[0164] For the IQ CRB measurement, b = / i is used, as this is measured in the single atom basis. For the clock-GERB experiments presented in FIG. 21C different values of b are chosen for the different cases. For the pre-selected fidelity, analyzed in the two-qubit measurement basis, b = 1 / 9 is fixed because each atom in a pair can be in states |0), |1), or |c) and leakage is larger than loss. For the post-selected fidelity on the qubit-subspace each pair atom can be in states |0) or |1), so b is fixed at b = 1 / 4. Finally, the loss measurement represents any other possibility, so b = 0.

[0165] For the CZ-GERB and 2Q Clifford depth curves in FIGs. 23A-23B b is fixed at b = 0 for the pair survival and pre-selected fidelities, but b is fixed at b = 1 / 4 for the qubit-subspace postselected fidelity. One may set b = 0 in this case is because atom loss in the readout is a possible state in both curves, and once the circuit fails producing the desired output, the state will never go back to its initial state.

[0166] In some cases, the contribution of known error sources using a super-operator simulator with measured experimental values as inputs may be estimated. In one example herein, errors are averaged over time, and represent an understanding of the typical operational state of a system or method herein.Attorney Docket No. 55436-754.601

[0167] In some cases, errors may be separated between the clock operation and the UV operation. In the clock, six primary error sources may be present in some cases: (1) clock frequency drift, (2) fast laser frequency noise, (3) trap scattering out of the clock state to the qubit subspace, (4) finite clock temperature, (5) differential light shift due to the shaped clock pulse between the |1) and |c) state, and (6) loss from the science tweezers. In the UV, error contribution may be present in some cases from: (1) finite Rydberg lifetime, (2) Rydberg decoherence, and (3) pulse repeatability.

[0168] During a calibration cycle, the clock frequency may be susceptible to drift. In some cases, this clock drift may be estimated by a Gaussian distributed static detuning with width 33 Hz for the clock operation based on experimental measurements of this frequency drift. Using a measurement based on detuning several minutes after calibration, this error source may be attributed an upper bound for the true error due to this effect. This can contribute 0.126% infidelity, with 0.007% of this due to leakage, in some cases.

[0169] In some cases, the clock laser frequency noise may be estimated based upon a smoothed fit to experimental data of a spin-lock measurement, and the implied two-sided phase power spectral density from exponential decays in the spin locking experiment shown in FIG. 21B. This fit goes down to 10 Hz, and thus may be distinct from the slow time scale drift. For a given realization of the phase noise power spectral densitywe sample the laser phase.4>(t) = 2 Js(|1(fjAfkcos (2TT fkt + c|)fc)kwith (pkchosen randomly and Af some small step in frequency space. This phase adds simply to the desired phase of the pulses, in some examples. The clock phase noise contributes 0.116% to the infidelity and 0.088% to the leakage rate, in some cases.

[0170] In some cases, to model the effects of atom temperature on the clock shelving fraction, only the modification to the resonant, carrier Rabi rate due to the Debye-Waller effect may be considered. Here, the Rabi rate for the nth motional eigenstate along the direction of the driving laser is reduced by e-??2 / 2Lri(j]2), with q ~ 0.26 the Lamb-Dicke parameter and Lnthe nth Laguerre polynomial. In order to approximate the calibration procedure which takes place at this same temperature, the thermally average Rabi rate may be set to give a perfect pulse area for the given pulse duration. In such a case, colder than average atoms will be over-rotated while warmer than average atoms will be under-rotated. In one example where n = 0.25, the motional eigenstates are sampled independently for both atoms. In such an example, temperature effects in the clock cause 0.013% infidelity entirely in the form of leakage.

[0171] In some cases, the trap scatters atoms out of the clock state due to far off resonant Raman transitions. This scattering may observed to have a lifetime of 1.06 s, in some cases. In order toAttorney Docket No. 55436-754.601simplify the analysis, the leakage scattering may be modeled herein as having a branching ratio of 50% to each atomic ground state, since the lifetime of the main intermediate state3Pi in the decay pathway is short compared to the clock Rabi rate. This may contribute 0.019% to infidelity, with 0.008% leaking back to the clock state, in this example. The observed 5 s 1 / e loss contributes to 0.005% leakage, in this example.

[0172] In some cases, due to the AC Stark shifts from far off-resonant states, the clock operation shifts the differential frequency between |1) and |c) by an average of 125 Hz over the course of the clock pulse. In one example, the detuning may be calibrated so that the time-averaged detuning is 0, but the light shift varies over the course of the pulse, and the shaped clock pulse will only be resonant at its average Rabi rate. In this example, this contributes 0.009% error, entirely as leakage since the average detuning is 0.

[0173] In some cases, UV errors may be modeled using a Rydberg lifetime as 65 ps, where an atom lost from the Rydberg state is assumed to be permanently lost. This leads to a loss probability of 0.075% during the CZ gate. Assuming the measured T2results from quasi-static detuning errors, the root mean square (rms) detuning can be calculated as 2TT X Arms= —.T2 Simulation of the time-optimal gate with a static detuning error yields an infidelity 2.9(2TTA / (1)2, for a total infidelity 2.9 / (T2fl)2= 0.007% in this example. Similarly, the predicted loss is 0.001% in this example. In this example, there is a measured atom loss of 0.11(3)% and 0.09(1)% for GERB and CRB after accounting for clock loss (see Table I, above). The small discrepancy between the predictions of the model and RB measurements can potentially be explained by gate parameter drift or excess decay of the entangled Rydberg state.

[0174] In one example, a 0.4 % variation in the time optimal pulse area is modeled to be constant over the duration of the UV gate as measured in the experiment. This contributes error at 0.007%, mostly due to the 0.005% loss in this example.

[0175] In general, when calculating the fidelity for an arbitrary CZ gate, one may be free to choose a single-qubit phase as seen fit. In one example, the single qubit phase may be chosen which minimizes each error independently, in order to avoid overestimating coherent errors. In such an example, the parameters of the time-optimal gate may be re-calibrated for a finite Rydberg blockade of 160 MHz.

[0176] A non-limiting example of contributions of each error source with 1 — F > 104is provided in FIG. 26 for an example system. Here, the height of each bar represents the decrease in average gate fidelity due to this error. The darker portion of the bar gives the probability for either atom to be outside of the qubit subspace following the CZ gate, i.e., at least one atom is either leaked or lost. The lighter portion alone is the portion of the error not due to leakage or loss. The bars are further shaded according to whether they occur on the clock operation lighterAttorney Docket No. 55436-754.601or the UV operation darker Along with these error sources, the experimentally measured CZ CRB and GERB infidelities are depicted, as well as the numerically modeled totals for both leakage and loss, and the decrease in fidelity AF. This error budget predicts an infidelity of 0.375% with at least one atom leaked or lost 0.211% of the time.

[0177] When realizing a CZ-GERB experiment, effects such as single-qubit phase errors may not be relevant due to the echo on both qubits present in each GERB block. Therefore, most of the CZ-GERB error may be assumed to be caused by leakage, loss, and any gate parameter calibration errors in this case. For other metrics such as the Clifford RB ones, additional errors that are not echoed, for example that affect the single-qubit phase, contribute to the measurement.

[0178] In some cases, other potential effects that can contribute to this error budget, some are relatively small, i.e. clock laser intensity noise, Doppler effects, finite blockade, but others are less known, for example the effect of atom motion in an imperfect trapping potential on clock operations, and a complete understanding of how Rydberg pair states affects the CZ gate under practical operational conditions that involves multiple lasers and background fields.

[0179] In some cases, a standard method of analyzing Clifford Randomized Benchmarking experiments is to compute the average success probability over all circuits at a fixed Clifford depth m, and then to fit these values to a single exponential in m. In another example, an additional analysis may be performed that takes more of the details of each circuit into account. This may be desirable in some cases, as there is significant spread in the number of native gates used to implement any given single Clifford gate, such as shown in FIG. 27. In this example, a simple model is used in which we treat loss, leakage, and errors in the computational subspace as constituting three independent failure modes, and sup-pose that for each there is a fixed probability of occurrence pi (pi) each time an Xn / 2(CZ) gate is applied. Moreover, we assume that these failures are irreversible, in the sense that once a circuit has failed, it will not later succeed.

[0180] This model indicates that we should fit the success probabilities of the circuits to the form P = A p^ / 2p^Zwhere A, pi, and pi are model parameters, #Xn / 2is the number of instances of the Xn / 2gate in the circuit, and similarly for CZ gates. The success probabilities are computed in four ways, corresponding to different choices of success and post-selection criteria:1. Number of instances with no atom loss, divided by total number of shots. Fitting this data gives estimates for pi and pi corresponding to the atom loss error mechanism. In the main text this is referred to as pair loss.2. Number of instances atoms are in either nuclear spin qubit state, i.e., not in the clock state or lost, divided by number of instances without loss. This estimates the leakage failureAttorney Docket No. 55436-754.601rates. This choice of success and post-selection criteria is not used in the main text but allows separating out estimates of the three independent error mechanisms.3. Number of successful |00) outcomes, divided by number of occasions either pair, is measured in any computational basis state. This estimates the qubit subspace failure rates, as in the main text. In the body text this is referred to as the post-selected case.4. Number of successful |00) outcomes, divided by total number of shots. This estimates the overall failure rates. In the main text this is referred to as pre-selected data.

[0181] Performing these fits using the curve fit function from scipy. optimize, we obtain the values in Table II. Within the model described above, the second and third choices for success and post-selection criteria correspond to conditional probabilities, and the fourth probability ought to be the product of the first three.Loss Leakage Qubit errors TotalPl 1.0000 1.0000.9987 1.0000 p2.9988.9983 1.0000.9931TABLE II. Fit parameters for each of the three error mechanisms resulting from the fine-grained analysis of CRB in terms of number of Xn / 2and CZ gates in a circuit, as described in the text.

[0182] For both the leakage and loss estimations, the success probability pi is estimated to be 1, consistent with these errors occurring only when entangling gates are implemented. On the other side, / ?2 is estimated as 1 for errors restricted to the computational basis, which is consistent with our interpretation that the CZ gate does not transfer population between computational basis states and is the sole responsible of loss and leakage. We notice that the best fit our model to the pre-selected data estimates that all the errors are accounted by for by errors on the CZ gates. However, we have measured nonzero error on our single-qubit gates. This just points to the fact that some of the assumptions made are not totally correct.

[0183] From these estimates of success probabilities for the two nontrivial native gates, we can obtain an estimate for the average success probability of a single Clifford gate via the definition fgPclif = Pl * p2where « 4.36, n2« 1.51 are the average number of IQ and 2Q gates per Clifford in our sampled circuits (the superscript stands for fine-grained). The inferred success probabilities p^ yfor the four different analysis cases described before can be found in Table III, with good agreement with the analysis presented in the main text.

[0184] Independently fiting the data to the coarse-grained model of the formP = Ap^Attorney Docket No. 55436-754.601is shown in FIG. 28, with the fitted success probabilities for each case as shown in Table III. We obtain good agreement between coarse- and fine-grained estimates as described in Table III. The product of success rates for the three error mechanisms is close to the total success rate, validating the notion that atom loss, leakage to the clock state, and qubit subspace errors may be treated as roughly independent error processes at the level of Clifford gates.Loss Leakage Qubit errors Total.9982.9974.9942.9896 PclifPclif.9982.9974.9940.9897TABLE III. Comparison between estimates of the success rates for each of three error mechanisms between the usual analysis of CRB in terms of Clifford depth and the fine-grained analysis described in the text.

[0185] In some cases, the failure-rate model is physically well-motivated in the case of loss errors, as an atom, once lost, will not return. For leakage errors, it should be accurate in the case of short circuits, as the amplitude for returning to the qubit subspace from the clock state is small given high fidelity clock rotations, in some cases. Moreover, the circuits may randomize the phase between the qubit and clock state, so that this return probability is further suppressed. In the case of shorter circuits, this model may be fairly accurate.Example of Systems for Performing a Non-Classical Computation

[0186] 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.

[0187] The system 200 may comprise one or more trapping units 210. The trapping units may comprise one or more optical trapping units. The optical trapping units may comprise any optical trapping unit described herein, such as an optical trapping unit described herein with respect to FIG. 3A. The optical trapping units may be configured to generate a plurality of optical trapping sites. The optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites. For instance, the optical trapping units may be configured to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. The optical trapping units may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 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,Attorney Docket No. 55436-754.601500, 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.

[0188] 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.

[0189] 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.

[0190] One or more atoms of the plurality of atoms may comprise qubits, as described herein (for instance, with respect to FIG. 4). Two or more atoms may be quantum mechanically entangled. Two or more atoms may be quantum mechanically entangled with a coherence lifetime of at least about 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, 6 ms, 7 ms, 8 ms, 9 ms, 10 ms, 20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms, 300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2 s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or more. Two or more atoms may be quantum mechanically entangled with a coherence lifetime of at most about 10 s, 9 s, 8 s, 7 s, 6 s, 5 s, 4 s, 3 s, 2 s, 1 s, 900 ms, 800 ms, 700 ms, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100 ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms, 8 ms, 7 ms, 6 ms, 5 ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, or less. Two or more atoms may be quantum mechanically entangled with a coherence lifetime that is within a range defined by any two of the precedingAttorney Docket No. 55436-754.601values. One or more atoms may comprise neutral atoms. One or more atoms may comprise uncharged atoms.

[0191] One or more atoms may comprise alkali atoms. One or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium -45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium -145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium- 175 atoms, or lutetium- 176 atoms.

[0192] 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 selectedAttorney Docket No. 55436-754.601from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium -6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium- 156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium- 164 atoms, erbium -166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium- 176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium- 41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium- 42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms,Attorney Docket No. 55436-754.601barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium-157 atoms, gadolinium- 158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium- 158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, Ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176Attorney Docket No. 55436-754.601atoms, 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.

[0193] 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.

[0194] 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.

[0195] The first atomic state may comprise a first hyperfine electronic state and the second atomic state may comprise a second hyperfine electronic state that is different from the first hyperfine electronic state. For instance, the first and second atomic states may comprise first and second hyperfine states on a multiplet manifold, such as a triplet manifold. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi or3P2 manifold. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi or3P2 manifold of any atom described herein, such as a strontium-873Pi manifold or a strontium-873P2 manifold.

[0196] 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.

[0197] In some cases, the hyperfine states comprise nuclear spin states of a strontium-87 'So manifold and the qubit transition drives one or both of two nuclear spin states of strontium-87xSo to a state detuned from or within the3P2 or3Pi manifold. In some cases, the one-qubit transitionAttorney Docket No. 55436-754.601is a two photon Raman transition between nuclear spin states of strontium-87 'So via a state detuned from or within the3P2 or3Pi manifold. In some cases, the nuclear spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift may be driven off resonance with any, all, or a combination of a single-qubit transition, a two-qubit transition, a shelving transition, an imaging transition, etc.

[0198] In some cases, the hyperfine states comprise nuclear spin states of ytterbium.

[0199] 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.

[0200] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1 / 2 (such as a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 7 / 2 spin state, may also drive mN = 7 / 2 to mN = 5 / 2, mN = 5 / 2 to mN = 3 / 2, mN = 3 / 2 to mN = 1 / 2, mN = 1 / 2 to mN = -1 / 2, mN = -1 / 2 to mN = -3 / 2, mN = -3 / 2 to mN = -5 / 2, mN = -5 / 2 to mN = -7 / 2, and mN = -7 / 2 to mN = -9 / 2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 5 / 2 spin state, may also drive mN = 7 / 2 to mN = 3 / 2, mN = 5 / 2 to mN = 1 / 2, mN = 3 / 2 to mN = -1 / 2, mN = 1 / 2 to mN = -3 / 2, mN = -1 / 2 to mN = -5 / 2, mN = -3 / 2 to mN = -7 / 2, and mN = -5 / 2 to mN = -9 / 2. Such a transition may thus not be selective for inducing transitions between particular spin states on the nuclear spin manifold.

[0201] 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, theAttorney Docket No. 55436-754.601light 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).

[0202] Stark shifting of the nuclear spin manifold may shift neighboring nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. Stark shifting may decrease leakage from the first and second nuclear spin state to other states in the nuclear spin manifold. Starks shifts may be achievable up to 100s of kHz for less than 10 mW beam powers. Upper state frequency selectivity may decrease scattering from imperfect polarization control. Separation of different angular momentum states in the3Pi manifold may be many gigahertz from the single and two-qubit gate light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for two-qubit transitions (e.g., how quickly the transition can be driven) may be faster than the decoherence rate. Scattering from the intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve fidelity of two-qubit transitions.

[0203] Qubits based on nuclear spin states in the electronic ground state may allow exploitation of long-lived metastable excited electronic states (such as a3Po state in strontium-87) for qubit storage. Atoms may be selectively transferred into such a state to reduce cross-talk or to improve gate or detection fidelity. Such a storage or shelving process may be atom -selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the1So state in strontium-87 to the3Po or3P2 state in strontium-87.

[0204] 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 groundAttorney Docket No. 55436-754.601state from the clock state and imaging again. In some cases, a magic wavelength transition is used to drive the clock transition.

[0205] The clock light for shelving can be atom -selective or not atom-selective. In some cases, the clock transition is globally applied (e.g., not atom selective). A globally applied clock transition may include directing the light without passing through a microscope objective or structuring the light. In some cases, the clock transition is atom-selective. Clock transition which are atom-selective may potentially allow us to improve gate fidelities by minimizing cross-talk. For example, to reduce cross talk in an atom, the atom may be shelved in the clock state where it may not be affected by the light. This may reduce cross-talk between neighboring qubits undergoing transitions. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc.

[0206] The system 200 may comprise one or more readout units 230. The readout units may comprise one or more readout optical units. The readout optical units may be configured to perform one or more measurements of the one or more superposition states to obtain the non-classical computation. The readout optical units may comprise one or more optical detectors. The detectors may comprise one or more photomultiplier tubes (PMTs), photodiodes, avalanche diodes, single-photon avalanche diodes, single-photon avalanche diode arrays, phototransistors, reverse-biased light emitting diodes (LEDs), charge coupled devices (CCDs), or complementary metal oxide semiconductor (CMOS) cameras. The optical detectors may comprise one or more fluorescence detectors. The readout optical unit may comprise one or more objectives, such as one or more objective having a numerical aperture (NA) of at least about 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, or more. The objective may have an NA of at most about 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, or less. The objective may have an NA that is within a range defined by any two of the preceding values.

[0207] The one or more readout optical units 230 may make measurements, such as projective measurements, by applying light resonant with an imaging transition. The imaging transition may cause fluorescence. An imaging transition may comprise a transition between the 'So state in strontium-87 to theJPi state in strontium-87. ThexPi state in strontium-87 may fluoresce. The lower state of the qubit transition may comprise two nuclear spin states in the 'So manifold. The one or more states may be resonant with the imaging transition. A measurement may comprise two excitations. In a first excitation, one of the two lower states may be excited to the shelving state (e.g.,3Po state in strontium-87). In a second excitation, the imaging transition may be excited. The first transition may reduce cross-talk between neighboring atoms duringAttorney Docket No. 55436-754.601computation. Fluorescence generated from the imaging transition may be collected on one or more readout optical units 230.

[0208] 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.

[0209] 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.

[0210] The vacuum units may comprise any combination of vacuum pumps described herein. For instance, the vacuum units may comprise one or more roughing pumps (such as a scroll pump) configured to provide a first stage of rough vacuum pumping. The roughing vacuum pumps may be configured to pump gases out of the system 200 to achieve a low vacuum pressure condition. For instance, the roughing pumps may be configured to pump gases out of the system 200 to achieve a low vacuum pressure of at most about 103Pascals (Pa). The vacuum units may further comprise one or more high-vacuum pumps (such as one or more ion pumps, getter pumps, or both) configured to provide a second stage of high vacuum pumping or ultra -high vacuum pumping. The high-vacuum pumps may be configured to pump gases out of the system 200 to achieve a high vacuum pressure of at most about 10'3Pa or an ultra-high vacuum pressure of at most about 10'6Pa once the system 200 has reached the low vacuum pressure condition provided by the one or more roughing pumps.

[0211] The vacuum units may be configured to maintain the system 200 at a pressure of at most about IO’6Pa, 9 x IO’7Pa, 8 x IO’7Pa, 7 x IO’7Pa, 6 x IO’7Pa, 5 x IO’7Pa, 4 x IO’7Pa, 3 x IO’7Pa, 2 x IO’7Pa, IO’7Pa, 9 x IO’8Pa, 8 x IO’8Pa, 7 x IO’8Pa, 6 x IO’8Pa, 5 x IO’8Pa, 4 x IO’8Pa, 3 x IO’8Pa, 2 x IO’8Pa, IO’8Pa, 9 x IO’9Pa, 8 x IO’9Pa, 7 x IO’9Pa, 6 x IO’9Pa, 5 x IO’9Pa, 4 x 10’9Pa, 3 x IO’9Pa, 2 x IO’9Pa, IO’9Pa, 9 x IO’10Pa, 8 x IO’10Pa, 7 x IO’10Pa, 6 x IO’10Pa, 5 x IO’10Pa, 4 x IO’10Pa, 3 x IO’10Pa, 2 x IO’10Pa, IO’10Pa, 9 x 10’11Pa, 8 x 10’11Pa, 7 x 10’11Pa, 6 xlO’11Pa, 5 x 10’11Pa, 4 x 10’11Pa, 3 x 10’11Pa, 2 x 10’11Pa, 10’11Pa, 9 x IO’12Pa, 8 x IO’12Pa, 7 x IO’12Pa, 6 x IO’12Pa, 5 x 10'12Pa, 4 x 10'12Pa, 3 x 10'12Pa, 2 x 10'12Pa, 10'12Pa, or lower. The vacuum units may be configured to maintain the system 200 at a pressure of at least about 10'12Pa, 2 x IO’12Pa, 3 x IO’12Pa, 4 x W12Pa, 5 x IO’12Pa, 6 x IO’12Pa, 7 x IO’12Pa, 8 x IO’12Pa, 9 xAttorney Docket No. 55436-754.601IO’12Pa, 10’11Pa, 2 x 10’11Pa, 3 x IO’11Pa, 4 x IO’11Pa, 5 x IO’11Pa, 6 x IO’11Pa, 7 x IO’11Pa, 8 x IO’11Pa, 9 x IO’11Pa, IO’10Pa, 2 x IO’10Pa, 3 x IO’10Pa, 4 x IO’10Pa, 5 x IO’10Pa, 6 x IO’10Pa, 7 x IO’10Pa, 8 x IO’10Pa, 9 x IO’10Pa, IO’9Pa, 2 x IO’9Pa, 3 x IO’9Pa, 4 x IO’9Pa, 5 x IO’9Pa, 6 x IO’9Pa, 7 x IO’9Pa, 8 x IO’9Pa, 9 x IO’9Pa, IO’8Pa, 2 x IO’8Pa, 3 x IO’8Pa, 4 x IO’8Pa, 5 x 10’8Pa, 6 x IO’8Pa, 7 x IO’8Pa, 8 x IO’8Pa, 9 x IO’8Pa, IO’7Pa, 2 x IO’7Pa, 3 x IO’7Pa, 4 x IO’7Pa, 5 x 10'7Pa, 6 x 10'7Pa, 7 x 10'7Pa, 8 x 10'7Pa, 9 x 10'7Pa, 10'6Pa, or higher. The vacuum units may be configured to maintain the system 200 at a pressure that is within a range defined by any two of the preceding values.

[0212] 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.

[0213] 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.

[0214] 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.

[0215] 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).

[0216] 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,Attorney Docket No. 55436-754.601the 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.

[0217] The entanglement units may also be configured to quantum mechanically entangle at least a subset of the atoms with at least another atom to form one or more multi-qubit units. The multiqubit units may comprise two-qubit units, three-qubit units, four-qubit units, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For instance, a two-qubit unit may comprise a first atom quantum mechanically entangled with a second atom, a three-qubit unit may comprise a first atom quantum mechanically entangled with a second and third atom, a four-qubit unit may comprise a first atom quantum mechanically entangled with a second, third, and fourth atom, and so forth. The first, second, third, or fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first, second, third, or fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first, second, third, and fourth atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions.

[0218] 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 dressedAttorney Docket No. 55436-754.601Rydberg 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.

[0219] The Rydberg units may comprise one or more light sources (such as any light source described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths may be selected to correspond to a wavelength that forms the Rydberg atoms or dressed Rydberg atoms. For instance, the light may comprise one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. The light may comprise one or more wavelengths of at most about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 300 nm to 400 nm.

[0220] 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 lightAttorney Docket No. 55436-754.601sources may emit light having substantially equal and opposite spatially-dependent frequency shifts.

[0221] 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.

[0222] State selective excitation of atoms to Rydberg levels may enable the implementation of multi -qubit operations. The multi -qubit operations may comprise two-qubit operations, three-qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or more. Two-photon transitions may be used to excite atoms from a ground state (such as a 'So ground state) to a Rydberg state (such as an n3Si state, wherein n is a principal quantum number described herein). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transitions may be implemented using first and second laser sources, as described herein. The first laser source may emit pi -polarized light, which may not change the projection of atomic angular momentum along a magnetic field. The second laser may emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. The first and second qubit levels may be excited to Rydberg level using this polarization. However, the Rydberg levels may be more sensitive to magnetic fields than the ground state so that large splittings (for instance, on the order of 100s of MHz) may be readily obtained. This spectral selectivity may allow state selective excitation to Rydberg levels.

[0223] 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).

[0224] 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 electromagneticAttorney Docket No. 55436-754.601delivery 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.

[0225] 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.

[0226] 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.

[0227] 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,Attorney Docket No. 55436-754.6010.999, 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.

[0228] The pulse sequences may enable the implementation of multi-qubit operations on non-adiabatic timescales while maintaining effectively adiabatic dynamics. For instance, the pulse sequences may comprise one or more of shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, counterdiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences. For instance, the pulse sequences may be similar to those described in M. V. Berry, “Transitionless Quantum Driving,” Journal of Physics A: Mathematical and Theoretical 42(36), 365303 (2009), www.doi.org / 10.1088 / 1751-8113 / 42 / 36 / 365303; Y.-Y. Jau et al., “Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction,” Nature Physics 12(1), 71-74 (2016); T. Keating et al., “Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing,” Physical Review A 91, 012337 (2015); A. Mitra et al., “Robust Mblmer- Sorenson 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.

[0229] 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

[0230] 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.Attorney Docket No. 55436-754.601Examples of Optical Trapping Units

[0231] 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).

[0232] 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.

[0233] 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.

[0234] 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.

[0235] The optical trapping sites may comprise one or more optical tweezers. Optical tweezers may comprise one or more focused laser beams to provide an attractive or repulsive force to hold or move the one or more atoms. The beam waist of the focused laser beams may comprise a strong electric field gradient. The atoms may be attracted or repelled along the electric field gradient to the center of the laser beam, which may contain the strongest electric field. The optical trapping sites may comprise one or more optical lattice sites of one or more opticalAttorney Docket No. 55436-754.601lattices. 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.

[0236] The optical lattices may be generated by interfering counter -propagating light (such as counter-propagating laser light) to generate a standing wave pattern having a periodic succession of intensity minima and maxima along a particular direction. A ID optical lattice may be generated by interfering a single pair of counter-propagating light beams. A 2D optical lattice may be generated by interfering two pairs of counter-propagating light beams. A 3D optical lattice may be generated by interfering three pairs of counter-propagating lights beams. The light beams may be generated by different light sources or by the same light source. Therefore, an optical lattice may be generated by at least about 1, 2, 3, 4, 5, 6, or more light sources or at most about 6, 5, 4, 3, 2, or 1 light sources.

[0237] 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.

[0238] 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 (An) excimer lasers, krypton dimer (Kn) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe ) 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.

[0239] 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)Attorney Docket No. 55436-754.601metal-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.

[0240] The lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal -doped crystal lasers, or metal -doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd: YAG) lasers, neodymium / chromium doped yttrium aluminum garnet (Nd / Cr: YAG) lasers, erbium-doped yttrium aluminum garnet (Er: YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd: YLF) lasers, neodymium-doped yttrium orthovanadate (ND:YVO4) lasers, neodymium-doped yttrium calcium oxoborate (Nd: YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm: YAG) lasers, ytterbium -doped ytrrium aluminum garnet (Yb: YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho: YAG) lasers, chromium-doped zinc selenide (Cr: ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce: LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce: LiCAF) lasers, erbium-doped glass (Er:glass) lasers, erbium-ytterbium-codoped glass (Er / Yt:glass) lasers, uranium-doped calcium fluoride (U: CaF2) lasers, or samarium-doped calcium fluoride (Sm: CaF2) lasers.

[0241] 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.

[0242] 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, 40Attorney Docket No. 55436-754.601fs, 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.

[0243] The lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values.

[0244] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (pJ), 2 pj, 3 pj, 4 pj, 5 pj, 6 pj, 7 pj, 8 pj, 9 pj, 10 pj, 20 pj, 30 pj, 40 pj, 50 pj, 60 pj, 70 pj, 80 pj, 90 pj, 100 pj, 200 pj, 300 pj, 400 pj, 500 pj, 600 pj, 700 pj, 800 pj, 900 pj, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 pj, 800 pj, 700 pj, 600 pj, 500 pj, 400 pj, 300 pj, 200 pj, 100 pj, 90 pj, 80 pj, 70 pj, 60 pj, 50 pj, 40 pj, 30 pj, 20 pj, 10 pj, 9 pj, 8 pj, 7 pj, 6 pj, 5 pj, 4 pj, 3 pj, 2 pj, 1 pj, 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 lasersAttorney Docket No. 55436-754.601may emit light having a pulse energy that is within a range defined by any two of the preceding values.

[0245] The lasers may emit light having an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values.

[0246] 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,Attorney Docket No. 55436-754.6011,340 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, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values.

[0247] 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 about 1 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 10'8Attorney Docket No. 55436-754.601nm, 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 IO’9nm, 3 x 10'9nm, 2 x l0’9n m, 1 x 10'9nm, 9 x IO'10nm, 8 x 1 O'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 10’10nm, 9 x IO'11nm, 8 x IO'11nm, 7 x IO'11nm, 6 x IO'11nm, 5 x IO'11nm, 4 x IO'11nm, 3 x 10’11nm, 2 x IO'11nm, 1 x IO'11nm, 9 x IO’12nm, 8 x 10'12nm, 7 x IO’12nm, 6 x 10'12nm, 5 x 10’12nm, 4 x IO’12nm, 3 x 10'12nm, 2 x IO’12nm, 1 x 10'12nm, 9 x 10'13nm, 8 x 10'13nm, 7 x 10’13nm, 6 x 10'13nm, 5 x 10'13nm, 4 x 10'13nm, 3 x 10'13nm, 2 x 10'13nm, 1 x 10'13nm, 9 x 10’14nm, 8 x 10'14nm, 7 x 10'14nm, 6 x 10'14nm, 5 x 10'14nm, 4 x IO’14nm, 3 x IO’14nm, 2 x 10’14nm, 1 x 10'14nm, 9 x IO'15nm, 8 x IO’15nm, 7 x IO'15nm, 6 x IO’15nm, 5 x IO'15nm, 4 x 10"15nm, 3 x IO’15nm, 2 x IO'15nm, 1 x IO’15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.

[0248] 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.

[0249] For instance, group III atoms and metastable states of alkaline earth or alkaline earth-like atoms may possess relatively large tensor shifts whose angle relative to an applied magnetic field may be tuned to cause a situation in which scalar and tensor shifts balance and give a zero or near zero differential light shift between the first and second atomic states. The angle 0 may be tuned by selecting the polarization of the emitted light. For instance, when the emitted light is linearly polarized, the total polarizability a may be written as a sum of the scalar component ascaiar and the tensor component ( tensor •OL ^scalar + O COS 0 ty&tensor

[0250] 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.

[0251] 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 opticalAttorney Docket No. 55436-754.601trapping unit may comprise an OM 214 configured to generate the plurality of optical trapping sites. Although depicted as comprising one OM in FIG. 3A, the optical trapping unit may comprise any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMs may comprise one or more digital micromirror devices (DMDs). The OMs may comprise one or more liquid crystal devices, such as one or more liquid crystal on silicon (LCoS) devices. The OMs may comprise one or more spatial light modulators (SLMs). The OMs may comprise one or more acousto-optic deflectors (AODs) or acousto-optic modulators (AOMs). The OMs may comprise one or more electro-optic deflectors (EODs) or electro-optic modulators (EOMs).

[0252] 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.

[0253] 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.

[0254] 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.

[0255] 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.

[0256] 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 CMOSAttorney Docket No. 55436-754.601cameras. 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.

[0257] The optical trapping unit may comprise one or more spatial configuration artificial intelligence (Al) units configured to perform one or more Al operations to determine the spatial configuration of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial configuration Al unit 216. Although depicted as comprising a single spatial configuration Al unit in FIG. 3A, the optical trapping unit may comprise any number of spatial configuration Al units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration Al units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration Al units. The Al operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0258] 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.

[0259] 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.

[0260] 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.

[0261] 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 factorAttorney Docket No. 55436-754.601may 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.

[0262] 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.

[0263] 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.

[0264] 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), andAttorney Docket No. 55436-754.601(iii) may be performed iteratively until a large filling factor is achieved. Operation (iii) may comprise translating the moves identified in operation (ii) to waveforms that may be sent to an arbitrary waveform generator (AWG) and using the AWG to drive AODs to move the atoms. The set of moves may be determined using the Hungarian algorithm described in W. Lee et al, “Defect-Free Atomic Array Formation Using Hungarian Rearrangement Algorithm,” Physical Review A 95, 053424 (2017), which is incorporated herein by reference in its entirety for all purposes.Examples of Electromagnetic Delivery Units

[0265] 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.

[0266] 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, I 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.

[0267] The RF energy may comprise an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW,Attorney Docket No. 55436-754.601800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 Watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The RF energy may comprise an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or less. The RF energy may comprise an average power that is within a range defined by any two of the preceding values.

[0268] 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.

[0269] 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.

[0270] 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.Attorney Docket No. 55436-754.601

[0271] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) on the qubits described herein. The electromagnetic delivery unit may be configured to apply one or more two-qubit operations (such as one or more two-qubit gate operations) on the two-qubit units described herein. Each singlequbit or two-qubit operation may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, or more. Each single-qubit or two-qubit operation may comprise a duration of at most about 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. Each single-qubit or two-qubit operation may comprise a duration that is within a range defined by any two of the preceding values. The single-qubit or two-qubit operations may be applied with a repetition frequency of at least 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or more. The single-qubit or two-qubit operations may be applied with a repetition frequency of at most 1,000 kHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The single-qubit or two-qubit operations may be applied with a repetition frequency that is within a range defined by any two of the preceding values.

[0272] 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, 500Attorney Docket No. 55436-754.601kHz, 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.

[0273] Raman transitions may be induced on individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle or a frequency shift to a light beam based on an applied radio-frequency (RF) signal. The SLM or AOD may be combined with an optical conditioning system that images the SLM or AOD active region onto the back focal plane of a microscope objective. The microscope objective may perform a spatial Fourier transform on the optical field at the position of the SLM or AOD. As such, angle (which may be proportional to RF frequency) may be converted into position. For example, applying a comb of radio frequencies to an AOD may generate a linear array of spots of light at a focal plane of the objective, with each spot having a finite extent determined by the characteristics of the optical conditioning system (such as the point spread function of the optical conditioning system).

[0274] 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.-n-Attorney Docket No. 55436-754.601

[0275] The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at least about 10 nm, 50 nm, 75 nm, 100 nm, 125 nm, 150 nm, 175 nm, 200 nm, 225 nm, 250 nm, 275 nm, 300 nm, 325 nm, 350 nm, 375 nm, 400 nm, 425 nm, 450 nm, 475 nm, 500 nm, 525 nm, 550 nm, 575 nm, 600 nm, 625 nm, 650 nm, 675 nm, 700 nm, 725 nm, 750 nm, 775 nm, 800 nm, 825 nm, 850 nm, 875 nm, 900 nm, 925 nm, 950 nm, 975 nm, 1 micrometer (pm), 1.5 pm, 2 pm, 2.5 pm 3 pm, 3.5 pm, 4 pm, 4.5 pm, 5 pm, 5.5 pm, 6 pm, 6.5 pm, 7 pm, 7.5 pm, 8 pm, 8.5 pm, 9 pm, 9.5 pm, 10 pm, or more. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at most about 10 pm, 9.5 pm, 9 pm, 8.5 pm, 8 pm, 7.5 pm, 7 pm, 6.5 pm, 6 pm, 5.5 pm, 5 pm, 4.5 pm, 4 pm, 3.5 pm, 3 pm, 2.5 pm, 2 pm, 1.5 pm, 1 pm, 975 nm, 950 nm, 925 nm, 900 nm, 875 nm, 850 nm, 825 nm, 800 nm, 775 nm, 750 nm, 725 nm, 700 nm, 675 nm, 650 nm, 625 nm, 600 nm, 575 nm, 550 nm, 525 nm, 500 nm, 475 nm, 450 nm, 425 nm, 400 nm, 375 nm, 350 nm, 325 nm, 300 nm, 275 nm, 250 nm, 225 nm, 200 nm, 175 nm, 150 nm, 125 nm, 100 nm, 75 nm, 25 nm, 10 nm, or less. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension as defined by any two of the proceeding values. For example, the beam can have a characteristic dimension of about 1.5 micrometers to about 2.5 micrometers. Examples of characteristic dimensions include, but are not limited to, a Gaussian beam waist, the full width at half maximum (FWHM) of the beam size, the beam diameter, the 1 / e2width, the D4s width, the D86 width, and the like. For example, the beam may have a Gaussian beam waist of at least about 1.5 micrometers.

[0276] 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 maximumAttorney Docket No. 55436-754.601resolution 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

[0277] The optical trapping units and electromagnetic delivery units described herein may be integrated into a single optical system. A microscope objective may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit described herein and to deliver light for trapping atoms generated by an optical trapping unit described herein.Alternatively or in addition, different objectives may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit and to deliver light from trapping atoms generated by an optical trapping unit.

[0278] A single SLM or AOD may allow the implementation of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or in addition, two separate SLMs or AODs may be configured to each handle light with orthogonal polarizations. The light with orthogonal polarizations may be overlapped before the microscope objective. In such a scheme, each photon used in a two-photon transition described herein may be passed to the objective by a separate SLM or AOD, which may allow for increased polarization control. Qubit operations may be performed on a two-dimensional arrangement of atoms by bringing light from a first SLM or AOD into a second SLM or AOD that is oriented substantially orthogonally to the first SLM or AOD via an optical relay. Alternatively or in addition, qubit operations may be performed on a two-dimensional arrangement of atoms by using a onedimensional array of SLMs or AODs.

[0279] 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.

[0280] 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 theAttorney Docket No. 55436-754.601optical trapping units or electromagnetic delivery units described herein). Such intensity control may be maintained by an optical subsystem that measures the intensity of light emitted by the various light sources, allowing closed-loop control of the intensity. Each light source may be coupled to an intensity actuator, such as an intensity servo control. The actuator may comprise an acousto-optic modulator (AOM) or electro-optic modulator (EOM). The intensity may be measured using an optical detector, such as a photodiode or any other optical detector described herein. Information about the intensity may be integrated into a feedback loop to stabilize the intensity.Examples of State Preparation Units

[0281] 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.

[0282] 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.

[0283] 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. TheAttorney Docket No. 55436-754.601second velocity may be within a range defined by any two of the preceding values. The Zeeman slowers may comprise ID Zeeman slowers.

[0284] The state preparation unit may comprise a first magneto-optical trap (MOT) 252. The first MOT may be configured to cool the atoms to a first temperature. The first temperature may be at most about 10 millikelvin (mK), 9 mK, 8 mK, 7 mK, 6 mK, 5 mK, 4 mK, 3 mK, 2 mK, 1 mK, 0.9 mK, 0.8 mK, 0.7 mK, 0.6 mK, 0.5 mK, 0.4 mK, 0.3 mK, 0.2 mK, 0.1 mK, or less. The first temperature may be at least about 0.1 mK, 0.2 mK, 0.3 mK, 0.4 mK, 0.5 mK, 0.6 mK, 0.7 mK, 0.8 mK, 0.9 mK, 1 mK, 2 mK, 3 mK, 4 mK, 5 mK, 6 mK, 7 mK, 8 mK, 9 mK, 10 mK, or more. The first temperature may be within a range defined by any two of the preceding values. The first MOT may comprise a ID, 2D, or 3D MOT.

[0285] 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.

[0286] The state preparation unit may comprise a second MOT 253. The second MOT may be configured to cool the atoms from the first temperature to a second temperature that is lower than the first temperature. The second temperature may be at most about 100 microkelvin (pK), 90 pK, 80 pK, 70 pK, 60 pK, 50 pK, 40 pK, 30 pK, 20 pK, 10 pK, 9 pK, 8 pK, 7 pK, 6 pK, 5 pK, 4 pK, 3 pK, 2 pK, 1 pK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, or less. The second temperature may be at least about 100 nK, 200 nK, 300Attorney Docket No. 55436-754.601nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 pK, 2 pK, 3 pK, 4 pK, 5 pK, 6 pK, 7 pK, 8 pK, 9 pK, 10 pK, 20 pK, 30 pK, 40 pK, 50 pK, 60 pK, 70 pK, 80 pK, 90 pK, 100 pK, or more. The second temperature may be within a range defined by any two of the preceding values. The second MOT may comprise a ID, 2D, or 3D MOT.

[0287] The second MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0288] 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.

[0289] 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 sidebandAttorney Docket No. 55436-754.601cooling units or Sisyphus cooling units may be configured to use sideband cooling to cool the atoms from the second temperature to a third temperature that is lower than the second temperature. The third temperature may be at most about 10 pK, 9 pK, 8 pK, 7 pK, 6 pK, 5 pK, 4 pK, 3 pK, 2 pK, 1 pK, 900 nK, 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, 90 nK, 80 nK, 70 nK, 60 nK, 50 nK, 40 nK, 30 nK, 20 nK, 10 nK, or less. The third temperature may be at most about 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80 nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 pK, 2 pK, 3 pK, 4 pK, 5 pK, 6 pK, 7 pK, 8 pK, 9 pK, 10 pK, or more. The third temperature may be within a range defined by any two of the preceding values.

[0290] 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.

[0291] 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 equilibriumAttorney Docket No. 55436-754.601distribution of atomic states to a non-equilibrium atomic state. For instance, the optical pumping units may be configured to emit light to optically pump the atoms from an equilibrium distribution of atomic states to a single pure atomic state. The optical pumping units may be configured to emit light to optically pump the atoms to a ground atomic state or to any other atomic state. The optical pumping units may be configured to optically pump the atoms between any two atomic states. The optical pumping units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0292] The state preparation unit may comprise one or more coherent driving units. For instance, the state preparation unit may comprise coherent driving unit 256. Although depicted as comprising a coherent driving unit in FIG. 5, the state preparation may comprise any number of coherent driving units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more coherent driving units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 coherent driving units. The coherent driving units may be configured to coherently drive the atoms from the non-equilibrium state to the first or second atomic states described herein. Thus, the atoms may be optically pumped to an atomic state that is convenient to access (for instance, based on availability of light sources that emit particular wavelengths or based on other factors) and then coherently driven to atomic states described herein that are useful for performing quantum computations. The coherent driving units may be configured to induce a single photon transition between the non-equilibrium stateAttorney Docket No. 55436-754.601and 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).

[0293] 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.

[0294] The coherent driving units may be configured to induce an RF transition between the nonequilibrium state and the first or second atomic state. The coherent driving units may comprise one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. For instance, the coherent driving units may comprise one or more RF sources (such as any RF source described herein) configured to emit RF radiation. The RF radiation may comprise one or more wavelengths of at least about 10 centimeters (cm), 20 cm, 30 cm, 40 cm, 50 cm, 60 cm, 70 cm, 80 cm, 90 cm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, or more. The RF radiation may comprise one or more wavelengths of at most about 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 90 cm, 80 cm, 70 cm, 60 cm, 50 cm, 40 cm, 30 cm, 20 cm, 10 cm, or less. The RF radiation may comprise one or more wavelengths that are within a range defined by any two of the preceding values. Alternatively or in addition, the coherent driving units may comprise one or more light sources (such as any lightAttorney Docket No. 55436-754.601sources described herein) configured to induce a two-photon transition corresponding to the RF transition.Examples of Controllers

[0295] 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

[0296] 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.Attorney Docket No. 55436-754.601

[0297] In some cases, the present disclosure provides a non-classical computer comprising a plurality of qubits comprising greater than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites.Examples of Methods for Performing a Non-Classical Computation

[0298] 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.

[0299] FIG. 6 shows a flowchart for an example of a first method 600 for performing a non-classical computation.

[0300] 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.

[0301] 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.

[0302] 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).

[0303] 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.Attorney Docket No. 55436-754.601

[0304] In some cases, the present disclosure provides a method for performing a non-classical computation, comprising: (a) providing a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprise at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; (b) applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, which non-classical operation includes a superposition between at least the first qubit state and the second qubit state; (c) quantum mechanically entangling at least a subset of the plurality of qubits in the superposition with at least another qubit of the plurality of qubits; and (d) performing one or more optical measurements of the one or more qubits, thereby obtaining said the-classical computation.

[0305] FIG. 7 shows a flowchart for an example of a second method 700 for performing a non-classical computation.

[0306] 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 comprises 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.

[0307] 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.

[0308] 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).Attorney Docket No. 55436-754.601

[0309] 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.

[0310] 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.

[0311] FIG. 8 shows a flowchart for an example of a third method 800 for performing a non-classical computation.

[0312] 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.

[0313] 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

[0314] 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 scatering via spontaneous decay from the3Pi. FIG. 9 shows an energy level structure for single-qubit and multi -qubit operations in strontium-87.

[0315] The optical system for single-qubit operations is also designed to work well for multiqubit gates. One of the single-qubit beams is used as one leg of the two-photon excitation scheme that drives transitions to the Rydberg electronic manifold. To satisfy the spatially-dependent frequency and phase matching condition, AODs are also used for the UV light. Importantly, the optical systems are matched so that the frequency shift of the UV light from one site to another is identical to that of the 689 nm light. The consequence of this constraint is that the performance of state-of-the-art UV AODs dictate the accessible field of view (FOV) for multi -qubit operations. Further, because one of the single-qubit beams is being used for multi-qubit operations (and the two single-qubit beams are matched), the FOV for single-qubit operations will be the same. A figure of merit for UV AODs is the product of the active aperture and the RF bandwidth of theAttorney Docket No. 55436-754.601device. For a fixed beam size in the back focal plane of the objective, increasing either of these quantities results in a larger scan angle of the beams, and thus a larger FOV in the plane of the qubit array. An FOV of approximately 100 pm x 100 pm was achieved, which is sufficient to address an array of approximately 1,000 atoms with a trapping site spacing of 3 pm.

[0316] In some cases, the systems and methods herein may be configured to perform quantum operations on the nuclear spin states of ytterbium.

[0317] A non-limiting example of performing quantum operations on the nuclear spin states of ytterbium is provided in FIG. 14, in accordance with some embodiments. As depicted in FIG. 14, excitation to and from a metastable clock state |c) = |3Po, mf = —1 / 2) may form an intermediate state of a sequential excitation scheme. In some cases, the excitation to and from a metastable clock state is per-formed via a Xlkshelving pulse, designed and calibrated to transfer as much population as possible be-tween |1) and |c). A combination of frequency and polarization-selectivity can provide state-selectivity for this excitation process, and thus for the two-qubit gate. In some cases, from the clock state, a pulse of ultraviolet (UV), 302 nm, o--polarized light is applied to drive the |c) |r) = |653Si, F = 3 / 2, mf = -3 / 2) transition. The UV pulse phase and amplitude profile may be chosen to ensure that every atomic pair returns to its initial state after the pulse, while pairs of neighboring clock atoms acquire an additional n phase shift due to the Rydberg blockade mechanism, such as described in D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000)., which is incorporated by reference herein in its entirety. Finally, atoms may be returned to |1) with a second Xlkpulse, having acquired a conditional phase that implements a CZ gate.Examples of Methods for Error Corrected Quantum Computation

[0318] 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 beAttorney Docket No. 55436-754.601needed 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.

[0319] 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.

[0320] The systems, the methods, the computer-readable media, and the techniques disclosed herein may improve upon other procedures at least because the systems, the methods, the computer-readable media, and the techniques disclosed herein may not comprise or require operations that modify the topology of the underlying surface code. In some cases, the underlying surface code may be unchanged. Instead, a matching graph passed to a decoder algorithm may be updated to account for a predicted probability distribution of a lost qubit. Because the matching graph passed to the decoder is updated, the underlying decoder may also be unchanged. Because the decoder is unchanged, the error correcting code may also not be changed. Accordingly, the systems, the methods, the computer-readable media, and the techniques disclosed herein may 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.

[0321] 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.

[0322] 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.

[0323] 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,Attorney Docket No. 55436-754.601arXiv: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

[0324] 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.

[0325] The process of detecting and correcting errors on the logical qubits amounts to measuring parities of a predetermined set of operators that act on the physical qubits and using the measured parities to diagnose and correct errors. In a simple example, a parity measurement checks the equality of two qubits to return a true or false answer, which can be used to determine whether a correction needs to occur. Additional measurements can be made for a system greater than two qubits. Since the physical qubits cannot be measured directly without collapsing the state of the logical system, these parities are measured using ancillary qubits (i.e., ancilla). Thus, there may be two types of physical qubits: data qubits on which the logical information is stored and ancilla qubits which are used to extract the desired parity checks.

[0326] 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.

[0327] 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 LossAttorney Docket No. 55436-754.601

[0328] 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.

[0329] 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).

[0330] 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.

[0331] 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.

[0332] 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.

[0333] 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 aAttorney Docket No. 55436-754.601qubit and a lost qubit has an effect of a Pauli operation or identity operation on the qubit, as described in International Application PCT / US2024 / 018180, which is incorporated by reference in its entirety herein in its entirety for all purposes. In such a case, a first atom A and second atom B may be neighboring qubits. The qubits may be trapped ion qubits. The qubits may be trapped atom qubits. In some cases, one qubit acquires a phase conditioned on state-selective excitation of the other. In some cases, the state selective excitation is from a state |1> to a state |r>.

[0334] In some cases, a state |r> is a Rydberg state. In some cases, a state |r> is a Rydberg state of a neutral atom qubit. If at A is excited to a Rydberg state, then Atom B (if present) experiences a shift due to the Rydberg interaction. In some cases, an optical excitation may be tuned to a frequency difference between the |1> state and the Rydberg state. If Atom A is in state |1>, then Atom A is at least transiently driven to the Rydberg state and Atom B (if present) experiences a shift due to the Rydberg interaction. If Atom B is not present and Atom A is in state |1>, there is no Shift to Atom B. If Atom A is in state |0> and Atom B is present, then the energy gap is too large, and nothing happens to Atom A or Atom B. If Atom A is in state |0> and Atom B is not present, then the energy gap is still too large, and nothing happens to Atom A or Atom B (which isn’t present). Accordingly, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom. While this example describes a case where the two-qubit interaction with a lost atom affects the Identity operation, the systems, the methods, the computer-readable media, and the techniques disclosed herein also work when 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.

[0335] 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

[0336] 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 compriseAttorney Docket No. 55436-754.601implementing a plurality of SWAP gates. Another example method of implementing an error correcting code which accounts for atom loss may comprise a modified knock-knock protocol. The systems, the methods, the computer-readable media, and the techniques disclosed herein for the implementation of a plurality of SWAP gates and modified knock-knock protocols may be the same as or similar to those described in International Application PCT / US2024 / 018180, which is incorporated by reference herein in its entirety for all purposes.Examples of Qubit Replacement

[0337] 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.

[0338] 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.

[0339] 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).

[0340] 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 aAttorney Docket No. 55436-754.601filling factor of less than 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, the atom rearrangement unit may attain a filling factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atom rearrangement unit may attain a filling factor of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atom rearrangement unit may attain a filling factor that is within a range defined by any two of the preceding values.Examples of Modifying Decoding Algorithms

[0341] 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.

[0342] 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.

[0343] 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 aAttorney Docket No. 55436-754.601quantum 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.

[0344] 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.

[0345] 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.

[0346] 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.

[0347] 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, theAttorney Docket No. 55436-754.601computer-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.

[0348] The systems, the methods, ...

Claims

1. Attorney Docket No. 55436-754.6012.CLAIMS WHAT IS CLAIMED IS:

1. A method for inducing a transition between a first state and a second state comprising:4.(a) providing a shaped composite pulse; and5.(b) driving a transition between said first state and said second state with said shaped composite pulse, wherein said first state and said second state are separated in energy by an optical frequency.

2. The method of claim 1, wherein said transition is an optical transition.

3. The method of claim 1, wherein said first state or said second state is a metastable state.

4. The method of claim 3, wherein said metastable state is a clock state.

5. The method of claim 1, wherein said optical transition is part of a quantum computation.

6. The method of claim 5, wherein said optical transition is part of a two-qubit gate.

7. The method of claim 6, wherein said two-qubit gate comprises a two-step process.

8. The method of claim 7, wherein a first step of said two-step process is said optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state.

9. The method of claim 8, further comprising performing said two-step process.

10. The method of claim 6, wherein said two-qubit gate is a controlled-Z gate.

11. The method of claim 10, wherein said controlled-Z gate comprises a fidelity of greater than 99.5%.

12. The method of claim 11, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments.

13. The method of claim 12, wherein said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

14. The method of claim 5, wherein said quantum computation comprises individually addressable single-qubit gates.

15. The method of claim 5, wherein said quantum computation comprises nuclear spin qubits in neutral atoms.

16. The method of claim 15, wherein said neutral atoms are alkaline earth or alkaline earth like atoms.Attorney Docket No. 55436-754.60117. The method of claim 15, wherein said neutral atoms comprise two valence electrons in an outer s-shell.

18. The method of claim 15, wherein said neutral atoms are 171-Yb atoms.

19. The method of claim 1, wherein said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.

20. The method of claim 1, wherein said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.

21. The method of claim 1, wherein said shaped composite pulse is configured to induce a population inversion between said first state and said second state.

22. The method of claim 1, wherein said shaped composite pulse comprise a length of about 130 microseconds.

23. The method of claim 1, further comprising setting a phase profile and an amplitude profile of said shaped composite pulse.

24. The method of claim 1, wherein said shaped composite pulse is a pulse or a pulse sequence.

25. A method for inducing a population inversion between optically accessible states comprising:29.(a) setting a phase profile and an amplitude profile of a pulse or a pulse sequence; and30.(b) directing said pulse or said pulse sequence toward a quantum system to drive said population inversion between said optically accessible states.

26. The method of claim 25, wherein said optically accessible states are separated in energy by an optical frequency.

27. The method of claim 25, wherein a state of said optically accessible states is a metastable state.

28. The method of claim 27, wherein said metastable state is a clock state.

29. The method of claim 25, wherein said population inversion between said optically accessible states is part of a quantum computation.

30. The method of claim 29, wherein said population inversion is part of a two-qubit gate.

31. The method of claim 30, wherein said two-qubit gate comprises a two-step process.

32. The method of claim 31, wherein a first step of said two-step process is an optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state.Attorney Docket No. 55436-754.60133. The method of claim 32, further comprising performing said two-step process.

34. The method of claim 30, wherein said two-qubit gate is a controlled-Z gate.

35. The method of claim 34, wherein said controlled-Z gate comprises a fidelity of greater than 99.5%.

36. The method of claim 35, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments.

37. The method of claim 35, wherein said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

38. The method of claim 29, wherein said quantum computation comprises individually addressable single-qubit gates.

39. The method of claim 29, wherein said quantum computation comprises nuclear spin qubits in neutral atoms.

40. The method of claim 39, wherein said neutral atoms are alkaline earth or alkaline earth like atoms.

41. The method of claim 39, wherein said neutral atoms comprise two valence electrons in an outer s-shell.

42. The method of claim 39, wherein said neutral atoms are 171-Yb atoms.

43. The method of claim 25, wherein said pulse or said pulse sequence comprises a shaped composite pulse.

44. The method of claim 43, wherein said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.

45. The method of claim 43, wherein said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.

46. The method of claim 43, wherein said shaped composite pulse comprise a length of about 130 microseconds.

47. A system for inducing a transition between a first state and a second state comprising:52.a pulse shaper configured to receive an optical pulse and to provide a shaped composite pulse; and53.a two-state system, wherein said shaped composite pulse is directed to said two state system to drive a transition between states of said two-state system, wherein said states are separated in energy by an optical frequency. Attorney Docket No. 55436-754.60148. The system of claim 47, wherein a state of said two-state system is a metastable state.

49. The system of claim 48, wherein said metastable state is a clock state.

50. The system of claim 48, wherein both states of said two state system are promoted to metastable states.

51. The system of claim 47, wherein said two-state system is qubit.

52. The system of claim 51, wherein said optical transition is part of a two-qubit gate.

53. The system of claim 52, further comprising instructions which when delivered to said pulse shaper are configured to implement a quantum computation comprising said two-qubit gate.

54. The system of claim 53, wherein said two-qubit gate comprises a two-step process.

55. The system of claim 54, wherein a first step of said two-step process is said optical transition to a metastable state, and wherein a second step of said two-step process is a transition to a Rydberg state.

56. The system of claim 53, wherein said quantum computation comprises individually addressable single-qubit gates.

57. The system of claim 53, wherein said two-qubit gate is a controlled-Z gate.

58. The system of claim 57, wherein said controlled-Z gate comprises a fidelity of greater than 99.5%.

59. The system of claim 58, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments.

60. The system of claim 58, wherein said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

61. The system of claim 51, wherein said qubit is a nuclear spin qubit in a neutral atom.

62. The system of claim 61, wherein said neutral atom is an alkaline earth or alkaline earth like atom.

63. The system of claim 61, wherein said neutral atom comprises two valence electrons in an outer s-shell.

64. The system of claim 61, wherein said neutral atom is a 171 -Yb atom.

65. The system of claim 47, wherein said shaped composite pulse comprises a Blackman shaped pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.Attorney Docket No. 55436-754.60166. The system of claim 47, wherein said shaped composite pulse is a pi pulse, optionally comprising a Y90 - XI 80 - Y90 pulse sequence.

67. The system of claim 47, wherein said shaped composite pulse comprise a length of about 130 microseconds.

68. The system of claim 47, wherein said pulse shape is configured to receive instructions which when executed set a phase profile and an amplitude profile of said shaped composite pulse.

69. The system of claim 47, wherein said shaped composite pulse is a pulse or a pulse sequence.

70. A quantum computing system comprising:77.a plurality of qubits, wherein said plurality of qubits comprise alkaline earth or alkaline earth like atoms; and78.a controlled-Z fidelity of greater than 99.5% fidelity.

71. The system of claim 70, wherein said controlled-Z fidelity is 99.72(3)% (99.40(3)%) with (without) post selection from two-qubit Clifford Randomized Benchmarking (CRB) experiments.

72. The system of claim 70, wherein said controlled-Z gate fidelity is 99.84(6)% (99.56(5)%) with (without) post-selection using a benchmark sequence that is insensitive to single-qubit phases.

73. A quantum computing system comprising:82.a plurality of qubits, wherein said plurality of qubits comprise alkaline earth or alkaline earth like atoms; and83.a single-qubit Clifford Randomized Benchmarking (CRB) fidelity of greater than 99.9%.

74. The system of claim 73, wherein said single-qubit CRB fidelity is a fidelity of 99.963(2)%.

75. The system of any one of claims 70-74, wherein said quantum computing system is a trapped atom quantum computing system.

76. The system of claim 75, wherein said trapped atom quantum computing system is a neutral atom quantum computing system.

77. The system of claim 76, wherein said neutral atom quantum computing system is based on nuclear spin qubits.

78. The system of claim 76, wherein said neutral atom quantum computing system is based on two valence electron atoms.Attorney Docket No. 55436-754.60179. The system of claim 77 or 78, wherein said neutral atom quantum computing system comprises Ytterbium-171 atoms.

80. The system of claim any one of claims 70-79, wherein atoms are continuously reloaded from a reservoir, wherein the reservoir does not comprise spatially distinct optical trapping sites.

81. The system of claim any one of claims 70-80, wherein a qubit is measured midcircuit substantially without loss of coherence82. The method of claim 1, further comprising inducing a transition between a third state and a fourth state, wherein said third state and said fourth state are separated in energy by a second optical frequency, and wherein a difference between said optical frequency and said second optical frequency is a radio frequency.

83. The method of claim 82, wherein said inducing said transition between said third state and said fourth state is performed, at least in part, by using a second shaped composite pulse.

84. The system of claim 47, further comprising a second two state system comprising states separated in energy by a second optical frequency, and wherein a difference between said optical frequency of said two state system and said second optical frequency of said second two state system is a radiofrequency.

85. The system of claim 84, wherein a second composite pulse is directed to said second two state system to drive a transition between states of said two state system.

Images