Scalable neutral atom based quantum computing

Abstract

To provide a method and system for performing non-classical computations.SOLUTION: An optical system 200 use, as a whole, a plurality of spatially distinct optical trapping sites to trap a plurality of atoms, one or more electromagnetic delivery units 220 to apply electromagnetic energy to one or more atoms of the plurality to induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state, one or more entanglement units to quantum mechanically entangle 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, and one or more readout optical units 230 to perform measurements of the superposition states to obtain the non-classical computation.SELECTED DRAWING: Figure 2

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 62 / 984,174, filed March 2, 2020, which is incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY FUNDED RESEARCH

[0002] This invention was made with U.S. government support under Small Business Research and Development Incentive Program Grants Nos. 1843926 and 1951188 awarded by the National Science Foundation. The U.S. government has certain rights in this invention. [Background technology]

[0003] Quantum computers typically utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on data. Quantum computers can differ from digital electronic computers, which are based on transistors. For example, while digital computers must encode data into binary digits (bits), where each bit is always in one of two distinct states (0 or 1), quantum computing uses quantum bits (qubits), which can be in a superposition of states. Summary of the Invention [Problem to be solved by the invention]

[0004] There is recognized herein a need for methods and systems for performing non-classical computations.

[0005] The present disclosure provides systems and methods for implementing non-classical or uncharged atomic techniques utilizing atoms (e.g., neutral or uncharged atoms). The atoms may be optically trapped in large arrays. The quantum mechanical states of the atoms (e.g., atomic hyperfine states or nuclear spin states) may be configured to serve as quantum bit (qubit) basis states. The qubit states may be manipulated by interaction with light, radio frequency, or other electromagnetic radiation, thereby performing non-classical or quantum computation. [Means for solving the problem]

[0006] In one aspect, the present disclosure provides a method for performing non-classical computation, the method comprising: (a) activating a non-classical computation unit including (i) one or more capture units that capture a plurality of atoms, the plurality of atoms being qubits; (ii) one or more first electromagnetic delivery units; (iii) one or more Rydberg units; (iv) one or more second electromagnetic delivery units; and (v) one or more readout units; (b) applying first electromagnetic energy to one or more atoms of the plurality of atoms using the one or more first electromagnetic delivery units, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state; and (c) using the one or more Rydberg units to read out the one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state. (d) using one or more second electromagnetic delivery units, applying second electromagnetic energy comprising one or more pulse sequences to the one or more multi-qubit units, whereby implementing one or more multi-qubit gate operations on the one or more multi-qubit units, where the one or more pulse sequences comprise one or more non-adiabatic pulses that maintain adiabatic dynamics; and (e) using one or more readout units, performing one or more measurements of the one or more superposition states, thereby obtaining a non-classical computation.The one or more pulse sequences may include one or more members selected from the group consisting of a shortcut to adiabaticity (STA) pulse sequence, a transition-free quantum drive (TQD) pulse sequence, a superadiabatic pulse sequence, an inverse adiabatic drive pulse sequence, a derivative removal by adiabatic gate (DRAG) pulse sequence, and a weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequence. The first electromagnetic delivery unit and the second electromagnetic delivery unit may be the same. The first electromagnetic delivery unit and the second electromagnetic delivery unit may be different. The one or more pulse sequences may further include one or more optimally controlled pulse sequences. The one or more optimal control pulse sequences may be derived from one or more procedures selected from the group consisting of gradient ascent pulse engineering (GRAPE), Krotov, chopped-basis, chopped random basis (CRAB), Nelder-Mead, gradient optimization using parameterization (GROUP), genetic algorithm, and gradient optimization of analytic controls (GOAT). The one or more pulse sequences may have a duration of at least 10 nanoseconds (ns). The one or more pulse sequences may have a duration of up to 100 microseconds (μs). The one or more multi-qubit gate operations may have a fidelity of at least 0.9. The one or more multi-qubit gate operations may have a fidelity of up to 0.999999.The method may further include using one or more Rydberg units to electronically excite at least one atom of the one or more atoms in the one or more superposition states to a Rydberg state, thereby forming one or more Rydberg atoms, or using one or more Rydberg units to sequentially drive the one or more atoms in the one or more superposition states through transitions to the Rydberg state, thereby forming one or more dressed Rydberg atoms. The method may further include using one or more Rydberg units to induce one or more quantum mechanical entanglements between the one or more Rydberg atoms or dressed Rydberg atoms and at least one other atom, wherein the at least one other atom is located at a distance of 10 micrometers (μm) or less from the one or more Rydberg atoms or dressed Rydberg atoms. The method may further include driving one or more Rydberg atoms or dressed Rydberg atoms into a lower-energy atomic state using one or more Rydberg units, thereby forming one or more multi-qubit units. The one or more trapping units may include one or more optical trapping units. The method may further include generating a plurality of spatially distinct optical trapping sites using one or more optical trapping units. The method may further include trapping up to one atom of the plurality of atoms using each optical trapping site of the plurality of spatially distinct optical trapping sites. One or more of the multi-qubit units may include a two-qubit unit. One or more of the multi-qubit gate operations may include a two-qubit gate operation. In some embodiments, (c) may further include implementing one or more multi-qubit gate operations on the one or more multi-qubit units.In some embodiments, in (c), quantum mechanically entanglement of at least a subset of the one or more atoms using one or more Rydberg units can include using one or more pulse sequences. In some embodiments, the one or more pulse sequences can include one or more non-adiabatic pulses that maintain adiabatic dynamics. In some embodiments, generation of entanglement by the Rydberg unit can be a gated operation. In some embodiments, the pulse shape or pulse sequence can be applied by the Rydberg unit.

[0007] In another aspect, the present disclosure provides a system for performing non-classical computation, the system including one or more capture units, one or more first electromagnetic delivery units, one or more Rydberg units, one or more second electromagnetic delivery units, one or more readout units, and one or more controllers coupled to the one or more capture units, the one or more first electromagnetic delivery units, the one or more Rydberg units, the one or more second electromagnetic delivery units, and the one or more readout units, wherein the one or more controllers (i) instruct the one or more units to capture a plurality of atoms, the plurality of atoms being a plurality of qubits; (ii) instruct the one or more first electromagnetic delivery units to apply a first electromagnetic energy to one or more atoms of the plurality of atoms, thereby causing the one or more atoms to transition between a first atomic state and at least a second atomic state different from the first atomic state. (iii) instructing the one or more Rydberg units to quantum-mechanically entangle 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 to form one or more multi-qubit units; (iv) implementing one or more multi-qubit gate operations on the one or more multi-qubit units by directing the one or more second electromagnetic delivery units to apply second electromagnetic energy comprising one or more pulse sequences to the one or more multi-qubit units, the one or more pulse sequences comprising one or more non-adiabatic pulses that maintain adiabatic dynamics; and (v) instructing the one or more readout units to perform one or more measurements of the one or more superposition states, thereby obtaining a non-classical computation.

[0008] Although described herein with respect to a multi-qubit Rydberg gate system, the methods and systems of the present disclosure may be used in a variety of gate systems. For example, the methods and systems described herein may be applied to single-qubit gates. In another example, 87 Two nuclear spin states of Sr (e.g., m F = -9 / 2, and m F =-7 / 2 1 In systems encoded in the S0 state (e.g., Figure 15A), residual nuclear spin states may exist outside the qubit subspace, and the population of these states may constitute leakage errors. The two-photon transition driving the single-qubit gate may also drive leakage transitions to nuclear spin states outside the qubit subspace. By selectively Stark-shifting the nuclear spin states, the leakage transitions can be shifted away from the frequency of the qubit transition. As described elsewhere herein, the use of DRAG pulses, performed on the two-photon transitions, suppresses leakage transitions, thereby enabling fast, high-fidelity single-qubit gates.

[0009] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in the 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 present disclosure. Accordingly, the drawings and description should be regarded as illustrative in nature and not restrictive.

[0010] Citation by reference 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 that the publications and patents or patent applications incorporated by reference conflict with the disclosure contained in the specification, the present specification is intended to supersede and / or supersede such conflicting material.

[0011] 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 referred to herein as "Figure" and "FIG."). [Brief explanation of the drawings]

[0012] [Figure 1] FIG. 1 illustrates a computer control system programmed or otherwise configured to carry out the methods provided herein. [Figure 2] FIG. 1 illustrates an example of a system for performing non-classical computation. [Figure 3A] FIG. 1 illustrates an example of a light capture unit. [Figure 3B] FIG. 10 shows an example of multiple light trapping sites. [Figure 3C] FIG. 1 shows an example of a light-trapping unit partially filled with atoms. [Figure 3D] FIG. 1 shows an example of a light-trapping unit completely filled with atoms. [Figure 4] FIG. 1 illustrates an example of an electromagnetic delivery unit. [Figure 5] FIG. 10 illustrates an example of a state preparation unit. [Figure 6] FIG. 2 shows a flowchart of an example of a first method for performing non-classical computation. [Figure 7]FIG. 10 shows a flowchart of an example of a second method for performing non-classical computation. [Figure 8] FIG. 10 shows a flowchart of an example of a third method for performing non-classical computation. [Figure 9] FIG. 1 illustrates an example qubit containing the 3P2 state of strontium-87. [Figure 10A] Stark shift simulation of the 1S0 hyperfine state of strontium-87. [Figure 10B] Stark shift simulation of the 1S0 hyperfine state of strontium-87. [Figure 11A] Simulation of single-qubit control by the Stark shift. [Figure 11B] Simulation of single-qubit control by the Stark shift. [Figure 12A] FIG. 1 shows an example of an array of captured light generated by an SLM. [Figure 12B] FIG. 1 shows an example of an array of captured light generated by an SLM. [Figure 13] FIG. 1 illustrates an optical system for delivering four different wavelengths. [Figure 14] FIG. 1 illustrates the trapping and cooling of strontium-87 and strontium-88 atoms using red-light magnetic trapping (MOT). [Figure 15A] FIG. 1 illustrates the energy level structure for single-qubit and multi-qubit operations in strontium-87. [Figure 15B] FIG. 1 illustrates an optical system for delivering light to perform single-qubit and multi-qubit operations on multiple trapped atoms in parallel. [Figure 15C] FIG. 1 illustrates an optical system configured to dynamically generate and control beams using a single electro-optic modulator (EOM) and two acousto-optic deflectors (AODs), for each beam, each driven by an RF signal from an arbitrary waveform generator. [Figure 16A] FIG. 1 shows a simulation of two atoms in an initial diatomic state. [Figure 16B] FIG. 10 shows a simulation of two atoms in an initial two-atom state with an inverse adiabatic driving field applied to perform a transition-less quantum driving gate. [Figure 16C] Figure 1 shows a numerical simulation of a two-atom system with Rydberg blockade passing through a Rydberg dressing gate. [Figure 16D] FIG. 1 shows an example of a differential adiabatic gated (DRAG) pulse. [Figure 16E] Figure 1 shows a simulated example of optimizing the DRAG pulse parameters to minimize error. [Figure 17A] FIG. 1 shows a calibration image of a fully filled 7×7 array of light capture sites. [Figure 17B] FIG. 1 shows the labeling of filled and unfilled light capture sites in a 7×7 array. [Figure 17C] FIG. 1 shows 25×25 pixel binning around each light capture site in a 7×7 array. [Figure 17D] FIG. 1 shows the identification of each capture site in a 7×7 array as filled or unfilled. [Figure 17E] FIG. 10 illustrates the transfer from a filled optical trapping site to an unfilled optical trapping site, avoiding collisions between atoms. DETAILED DESCRIPTION OF THE INVENTION

[0013] While various embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed.

[0014] Unless otherwise specified, all technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. As used in this specification and the appended claims, the singular forms "a," "an," and "the" include plural references unless the context clearly dictates otherwise. References to "or" herein are intended to encompass "and / or" unless otherwise specified.

[0015] Whenever the terms "at least," "greater than," or "greater than or equal to" appear before or after the first number in a series of two or more numbers, the terms "at least," "greater than," or "greater than or equal to" apply to each number in the series. For example, 1, 2, or 3 or more is equivalent to 1 or more, 2 or more, or 3 or more.

[0016] Whenever the terms "no more than," "less than," "less than or equal to," or "at most" appear before or after the first number in a series of two or more numbers, the terms "no more than," "less than," "less than," or "at most" apply to each number in the series. For example, 3, 2, or 1 or less is equivalent to 3 or less, 2 or less, or 1 or less.

[0017] Where values ​​are described as ranges, such disclosure will be understood to include disclosure of all possible subranges within such ranges, as well as specific numerical values ​​falling within such ranges, whether or not a specific numerical value or specific subrange is explicitly stated.

[0018] As used herein, like letters refer to like elements.

[0019] As used herein, the terms “artificial intelligence,” “artificial intelligence procedure,” “artificial intelligence operation,” and “artificial intelligence algorithm” generally refer to any system or computational procedure that takes one or more actions to improve or maximize the likelihood of successfully achieving a goal. The term “artificial intelligence” may include “generative modeling,” “machine learning” (ML), and / or “reinforcement learning” (RL).

[0020] As used herein, the terms “machine learning,” “machine learning procedure,” “machine learning operation,” and “machine learning algorithm” generally refer to any system or analytical and / or statistical procedure that progressively improves computer performance of a task. Machine learning may include a machine learning algorithm. A machine learning algorithm may be a trained algorithm. Machine learning (ML) may include one or more supervised, semi-supervised, or unsupervised machine learning techniques. For example, an ML algorithm may be a trained algorithm trained by supervised learning (e.g., various parameters are determined as weights or scaling coefficients). ML may include one or more of regression analysis, regularization, classification, dimensionality reduction, ensemble learning, meta-learning, association rule learning, cluster analysis, anomaly detection, deep learning, or ultra-deep learning.ML includes k-means, k-means clustering, k-nearest neighbors, learning vector quantization, linear regression, nonlinear regression, least squares regression, partial least squares regression, logistic regression, stepwise regression, multivariate adaptive regression splines, ridge regression, principal component regression, least absolute shrinkage and selection operation, least angle regression, canonical correlation analysis, factor analysis, independent component analysis, linear discriminant analysis, multidimensional scaling, non-negative matrix factorization, principal component analysis, principal coordinate analysis, projection pursuit, Sammon mapping, stochastic neighborhood embedding of t distribution, AdaBoosting, boosting, gradient boosting, bootstrap aggregation, ensemble averaging, decision trees, conditional decision trees, boosted decision trees, gradient boosted decision trees, random forests, stacked generalization, These may include, but are not limited to, a Bayesian network, a Bayesian belief network, a naive Bayes, a Gaussian naive Bayes, a multinomial naive Bayes, a hidden Markov model, a hierarchical hidden Markov model, a support vector machine, an encoder, a decoder, an autoencoder, a stacked autoencoder, a perceptron, a multilayer perceptron, an artificial neural network, a feedforward neural network, a convolutional neural network, a recurrent neural network, a long short-term memory, a deep belief network, a deep Boltzmann machine, a deep convolutional neural network, a deep recurrent neural network, or a generative adversarial network.

[0021] As used herein, the terms “reinforcement learning,” “reinforcement learning procedure,” “reinforcement learning operation,” and “reinforcement learning algorithm” generally refer to any system or computational procedure that performs one or more actions to reinforce or maximize some concept of cumulative reward for interacting with an environment. An agent executing a reinforcement learning (RL) procedure may receive positive or negative reinforcement, called an “instantaneous reward,” for performing one or more actions in an environment, thereby moving itself and the environment into various new states.

[0022] An agent's goal may be to enhance or maximize some notion of cumulative reward. For example, an agent's goal may be to enhance or maximize a "discounted reward function" or an "average reward function." A "Q function" may represent the maximum cumulative reward that can be obtained from a state and an action taken in that state. A "value function" and a "generalized benefit estimator" may represent the maximum cumulative reward that can be obtained from a state given the optimal or best choice of action. RL may utilize any one or more of such notions of cumulative reward. As used herein, such a function may be referred to as a "cumulative reward function." Thus, calculating the best or optimal cumulative reward function may be equivalent to finding the agent's best or optimal policy.

[0023] The interaction of an agent with its environment may be formulated as one or more Markov Decision Processes (MDPs). RL procedures may not assume knowledge of an exact mathematical model of the MDP. The MDP may be completely unknown, partially known, or completely known to the agent. RL procedures may lie within a spectrum between two ranges: "model-based" or "model-free" with respect to prior knowledge of the MDP. As such, RL procedures may target large MDPs where accurate methods may be infeasible or unavailable due to the unknown or stochastic nature of the MDP.

[0024] RL procedures may be implemented using one or more computer processors described herein. The digital processing unit may utilize agents that train, store, and later deploy "policies" to enhance or maximize cumulative reward. Policies may be explored (e.g., searched) as far as possible or over a desired period of time. Such optimization problems may be solved by saving approximations to optimal policies, saving approximations to cumulative reward functions, or both. In some cases, RL procedures may store one or more tables of approximations to such functions. In other cases, RL procedures may utilize one or more "function approximations."

[0025] Examples of function approximators may include neural networks (such as deep neural networks) and probabilistic graphical models (such as Boltzmann machines, Helmholtz machines, and Hopfield networks). A function approximator may create a parameterization that approximates the cumulative reward function. Optimizing the function approximation with respect to the parameterization may consist of perturbing the parameters in a direction that enhances or maximizes the cumulative reward and thus enhances or optimizes the policy (such as with policy gradient methods), or by perturbing the function approximator to approach satisfying Bellman's optimality criterion (such as with time lag methods).

[0026] During training, the agent may perform actions within the environment to acquire more information about the environment and about the appropriate or best choice of policy for survival or better utility. The agent's actions may be randomly generated (e.g., especially in the early stages of training) or may be prescribed by another machine learning paradigm (such as supervised learning, imitation learning, or any other machine learning procedure described herein). The agent's actions may be refined by selecting actions that are closer to the agent's perception of what the reinforced or optimal policy is. Various training strategies may lie on a spectrum between two ranges: off-policy and on-policy, with respect to choosing between exploration and exploitation.

[0027] As used herein, the terms "nonclassical computation," "nonclassical procedure," "nonclassical operation," or any "nonclassical computer" generally refer to any method or system for performing a computational procedure outside the paradigm of classical computing. A nonclassical computation, nonclassical procedure, nonclassical operation, or nonclassical computer may include quantum computation, quantum procedure, quantum operation, or quantum computer.

[0028] As used herein, the terms “quantum computation,” “quantum procedure,” “quantum operation,” and “quantum computer” generally refer to any method or system for performing computations using quantum mechanical operations in a Hilbert space represented by a quantum device (such as unitary transformations on a quantum channel or completely positive trace-preserving (CPTP) maps). Thus, quantum computation and classical (or digital) computation may be similar in the following aspect: both computations may involve a sequence of instructions performed on input information and providing an output. Various paradigms of quantum computing may decompose a quantum operation into a sequence of elementary quantum operations that simultaneously affect a subset of qubits in a quantum device. Quantum operations may be selected, for example, based on their locality or ease of physical implementation. A quantum procedure or computation may consist of a sequence of such instructions, which may represent different quantum evolutions on a quantum device for various applications. For example, procedures for calculating or simulating quantum chemistry can represent quantum states and electron spin-orbit annihilation and creation operators by using qubits (e.g., two-level quantum systems) and universal quantum gate sets (e.g., Hadamard, controlled-uncontrolled (CNOT), and π / 8 rotation) through the so-called Jordan-Wigner or Bravi-Kitaev transformations.

[0029] Additional examples of quantum procedures or computations may include optimization procedures such as quantum approximate optimization algorithms (QAOA) and finding quantum minima. QAOA may involve rotating single qubits and performing entanglement gates on multiple qubits. In quantum adiabatic computations, instructions may carry out probabilistic or non-probabilistic evolutionary paths from an initial quantum system to a final quantum system.

[0030] Quantum-inspired procedures may include simulated annealing, parallel tempering, master equation solvers, Monte Carlo procedures, etc. Quantum-classical or hybrid algorithms or procedures may include procedures such as variational quantum eigensolvers (VQEs) and variational and adiabatically navigated quantum eigensolvers (VanQver).

[0031] The quantum computer may include one or more adiabatic quantum computers, quantum gate arrays, one-way quantum computers, topological quantum computers, quantum Turing machines, quantum annealers, Ising solvers, or gate models of quantum computing.

[0032] As used herein, the term "adiabatic" refers to any process taking place in a quantum mechanical system in which the parameters of the Hamiltonian change slowly compared to the natural timescale of the system's evolution.

[0033] As used herein, the term "non-adiabatic" refers to any process-performing quantum mechanical system in which the parameters of the Hamiltonian change rapidly compared to, or on a timescale similar to, the natural timescale of the system's evolution.

[0034] A system for performing non-classical computations In one aspect, the present disclosure provides a system for performing non-classical computations. The system may include one or more optical trapping units configured to generate 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 including more than 60 atoms; one or more electromagnetic delivery units configured to apply 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 different from the first atomic state; one or more entanglement units configured to quantum mechanically entangle 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 one or more readout optical units configured to perform one or more measurements of the one or more superposition states to obtain a non-classical computation.

[0035] 2 illustrates an example system 200 for performing non-classical computation. The non-classical computation may include quantum computation. The quantum computation may include gate-model quantum computation.

[0036] The system 200 may include one or more capture units 210. The capture units may include one or more light capture units. The light capture units may include any light capture units described herein, such as the light capture units described herein with respect to FIG. 3A. The light capture units may be configured to generate multiple light capture sites. The light capture units may be configured to generate multiple spatially distinct light capture sites. For example, the light capture units may generate multiple spatially distinct light capture sites. For example, the light capture units may generate multiple spatially distinct light capture sites, such as 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 ... It may be configured to generate 0, 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 light capture sites. Light capture units are available in sizes up to approximately 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, 1 The light capture unit may be configured to generate 0,000, 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 light capture sites. The light capture unit may be configured to capture a number of light capture sites that fall within a range defined by any two of the foregoing values.

[0037] The light trapping unit can be configured to trap a plurality of atoms. For example, the light trapping unit can 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 number of light capture units is 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 unit may be configured to trap a number of atoms within a range defined by any two of the aforementioned values.

[0038] Each optical capture site of the optical capture unit may be configured to capture at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. Each optical capture site may be configured to capture up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each optical capture site may be configured to capture a number of atoms within a range defined by any two of the aforementioned values. Each optical capture site may be configured to capture a single atom.

[0039] One or more atoms of the plurality of atoms may include a qubit as described herein (e.g., with respect to FIG. 4). Two or more atoms may be quantum mechanically entangled. The two or more atoms may be quantum entangled for at least about 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100 μs, 200 μs, 300 μs, 400 μs, 500 μs, 600 μs, 700 μs, 800 μs, 900 μs, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, They may be quantum mechanically entangled with a coherence lifetime of 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 longer. Two or more atoms can have a maximum of approximately 10s, 9s, 8s, 7s, 6s, 5s, 4s, 3s, 2s, 1s, 900ms, 800ms, 700ms, 600ms, 500ms, 400ms, 300ms, 200ms, 100ms, 90ms, 80ms, 70ms, 60ms, 50ms, 40ms, 30ms, 20ms, 10ms, 9ms, 8ms, 7ms, 6ms, 5ms, 4ms, 3ms, 2ms , 1 ms, 900 μs, 800 μs, 700 μs, 600 μs, 500 μs, 400 μs, 300 μs, 200 μs, 100 μs, 90 μs, 80 μs, 70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4 μs, 3 μs, 2 μs, 1 μs, or shorter. Two or more atoms may be quantum mechanically entangled with a coherence lifetime within a range defined by any two of the aforementioned values. One or more atoms may include neutral atoms. One or more atoms may include uncharged atoms.

[0040] The one or more atoms may include an alkali atom. The one or more atoms may include a lithium (Li) atom, a sodium (Na) atom, a potassium (K) atom, a rubidium (Rb) atom, or a cesium (Cs) atom. The one or more atoms may include a lithium-6 atom, a lithium-7 atom, a sodium-23 atom, a potassium-39 atom, a potassium-40 atom, a potassium-41 atom, a rubidium-85 atom, a rubidium-87 atom, or a cesium-133 atom. The one or more atoms may include an alkaline earth atom. The one or more atoms may include a beryllium (Be) atom, a magnesium (Mg) atom, a calcium (Ca) atom, a strontium (Sr) atom, or a barium (Ba) atom. The one or more atoms may include 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-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, or barium-138 atoms. The one or more atoms may include a rare earth atom. The one or more atoms may include a scandium (Sc) atom, a yttrium (Y) atom, a lanthanum (La) atom, a cerium (Ce) atom, a praseodymium (Pr) atom, a neodymium (Nd) atom, a samarium (Sm) atom, a europium (Eu) atom, a gadolinium (Gd) atom, a terbium (Tb) atom, a dysprosium (Dy) atom, a holmium (Ho) atom, an erbium (Er) atom, a thulium (Tm) atom, an ytterbium (Yb) atom, or a lutetium (Lu) atom.The one or more atoms may be selected from the group consisting of 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-1 59 atom, dysprosium-156 atom, dysprosium-158 atom, dysprosium-160 atom, dysprosium-161 atom, dysprosium-162 atom, dysprosium-163 atom, dysprosium-164 atom, erbium-162 atom, erbium-164 atom, erbium-166 atom, erbium-167 atom, erbium-168 atom, erbium-170 atom, holmium-165 atom, thulium-169 atom, ytterbium-168 atom, ytterbium-170 atom, ytterbium-171 atom, ytterbium-172 atom, ytterbium-173 atom, ytterbium-174 atom, ytterbium-176 atom, lutetium-175 atom, or lutetium-176 atom.

[0041] The plurality of atoms may include 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 include 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 include 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 include 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 include 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. The atoms may comprise rare earth atoms. For example, the plurality of atoms may be 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. Atoms, Rubidium 85 atoms, Rubidium 87 atoms, Cesium 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 134 atoms, Barium 135 atoms, Barium 136 atoms, Barium 137 atoms, Barium 138 atoms, Scandium 45 atoms,Yttrium-89 atom, Lanthanum-139 atom, Cerium-136 atom, Cerium-138 atom, Cerium-140 atom, Cerium-142 atom, Praseodymium-141 atom, Neodymium-142 atom, Neodymium-143 atom, Neodymium-145 atom, Neodymium-146 atom, Neodymium-148 atom, Samarium-144 atom, Samarium-149 atom, Samarium-150 atom, Samarium-152 atom, Samarium-154 atom, Europium-151 atom, Europium-153 atom, Gadolinium-154 atom, Gadolinium-155 atom, Gadolinium-156 atom, Gadolinium-157 atom, Gadolinium-158 atom, Gadolinium-160 atom, Terbium-159 atom, Dysprosium 156 atom, dysprosium-158 atom, dysprosium-160 atom, dysprosium-161 atom, dysprosium-162 atom, dysprosium-163 atom, dysprosium-164 atom, erbium-162 atom, erbium-164 atom, erbium-166 atom, erbium-167 atom, erbium-168 atom, erbium-170 atom, holmium-165 atom, thulium-169 atom, ytterbium-168 atom, ytterbium-170 atom, ytterbium-171 atom, ytterbium-172 atom, ytterbium-173 atom, ytterbium-174 atom, ytterbium-176 atom, lutetium-175 atom, or lutetium-176 atom. Atoms may be enriched to isotopic abundances of up to 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. 6 atoms of lithium, 7 atoms of lithium, 23 atoms of sodium, 39 atoms of potassium, 40 atoms of potassium, 41 atoms of potassium, 85 atoms of rubidium, 87 atoms of rubidium, 133 atoms of cesium, 9 atoms of beryllium, 24 atoms of magnesium, 25 atoms of magnesium, 26 atoms of magnesium, 40 atoms of calcium, 42 atoms of calcium, 43 atoms of calcium, 44 atoms of calcium, 46 atoms of calcium, 48 atoms of calcium, 84 atoms of strontium, 86 atoms of strontium,Strontium-87 atom, Strontium-88 atom, Barium-130 atom, Barium-132 atom, Barium-134 atom, Barium-135 atom, Barium-136 atom, Barium-137 atom, Barium-138 atom, Scandium-45 atom, Yttrium-89 atom, Lanthanum-139 atom, Cerium-136 atom, Cerium-138 atom, Cerium-140 atom, Cerium-142 atom, Praseodymium-141 atom, Neodymium-142 atom, Neodymium-143 atom, Neodymium-145 atom, Neodymium-146 atom, Neodymium-148 atom, Samarium-144 atom, Samarium-149 atom, Samarium-150 atom, Samarium-152 atom, Samarium-154 atom, Europium-151 atom, Europium-153 atom, Gadolinium-154 atom, Gadolinium-155 atom, Gadolinium-156 atom, Gadolinium-157 atom, Gadolinium-158 atom, Gadolinium-159 atom, Gadolinium-160 atom, Gadolinium-161 atom, Gadolinium-162 atom, Gadolinium-163 atom, Gadolinium-164 atom, Gadolinium-165 atom, Gadolinium-166 atom, Gadolinium-167 atom, Gadolinium-168 atom, Gadolinium-169 ...9 atom, Gadolinium-169 atom, Gadolinium-160 atom, Ga dolinium-157 atom, gadolinium-158 atom, gadolinium-160 atom, terbium-159 atom, dysprosium-156 atom, dysprosium-158 atom, dysprosium-160 atom, dysprosium-161 atom, dysprosium-162 atom, dysprosium-163 atom, dysprosium-164 atom, erbium-162 atom, erbium-164 atom, erbium-166 atom, erbium-167 atom, erbium-168 atom, erbium-169 atom, erbium-170 atom, erbium-171 atom, erbium-172 atom, erbium-173 atom, erbium-174 atom, erbium-175 atom, erbium-176 atom, erbium-177 atom, erbium-178 atom, erbium-179 ... The ion may include thulium-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. The plurality of atoms may be selected from the group consisting of lithium 6 atoms, lithium 7 atoms, sodium 23 atoms, potassium 39 atoms, potassium 40 atoms, potassium 41 atoms, rubidium 85 atoms, rubidium 87 atoms, cesium 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 134 atoms, and the like, enriched to an isotopic abundance within a range defined by any two of the foregoing values.Barium-135 atom, barium-136 atom, barium-137 atom, barium-138 atom, scandium-45 atom, yttrium-89 atom, lanthanum-139 atom, cerium-136 atom, cerium-138 atom, cerium-140 atom, cerium-142 atom, praseodymium-141 atom, neodymium-142 atom, neodymium-143 atom, neodymium-145 atom, neodymium-146 atom, neodymium-148 atom, samarium-144 atom, samarium-149 atom, samarium-150 atom, samarium-152 atom, samarium-154 atom, europium-151 atom, europium-153 atom, gadolinium-154 atom, gadolinium-155 atom, gadolinium-156 atom, gadolinium-157 atom, gadolinium-158 atom, gadolinium-159 atom, gadolinium-160 atom, gadolinium-161 atom, gadolinium-162 atom, gadolinium-163 atom, gadolinium-164 atom, gadolinium-165 atom, gadolinium-166 atom, gadolinium-167 atom, gadolinium-168 atom, gadolinium-169 atom, gadolinium-170 atom, gadolinium-171 atom, gadolinium-172 atom, gadolinium-173 atom, gadolinium-174 atom, gadolinium-175 atom, gadolinium-176 atom, gadolinium-177 atom, gadolinium-178 atom, gadolinium-179 atom, gadolinium-180 atom, gadolinium-181 atom, gadolinium- The thulium-160 atom may include a terbium-159 atom, a dysprosium-156 atom, a dysprosium-158 atom, a dysprosium-160 atom, a dysprosium-161 atom, a dysprosium-162 atom, a dysprosium-163 atom, a dysprosium-164 atom, an erbium-162 atom, an erbium-164 atom, an erbium-166 atom, an erbium-167 atom, an erbium-168 atom, an erbium-170 atom, a holmium-165 atom, a thulium-169 atom, an ytterbium-168 atom, an ytterbium-170 atom, an ytterbium-171 atom, an ytterbium-172 atom, an ytterbium-173 atom, an ytterbium-174 atom, an ytterbium-176 atom, a lutetium-175 atom, or a lutetium-176 atom.

[0042] System 200 may include one or more first electromagnetic delivery units 220. The first electromagnetic delivery unit may include any electromagnetic delivery unit described herein, such as the electromagnetic delivery unit described herein with respect to FIG. 4. The first electromagnetic delivery unit 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 superpositions of a first atomic state and a second atomic state different from the first atomic state.

[0043] 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 the second atomic state may be elevated in energy relative to the ground atomic state of the atom. The first atomic state or the second atomic state may be equal in energy to the ground atomic state of the atom.

[0044] The first atomic state may include a first hyperfine electronic state, and the second atomic state may include a second hyperfine electronic state that is different from the first hyperfine electronic state. For example, the first and second atomic states may include first and second hyperfine states on a multiplet manifold, such as a triplet manifold. The first and second atomic states may be 3 P1 or 3 The first and second hyperfine states can be included on the P2 manifold, respectively. The first and second atomic states are strontium-87 3 P1 variant or strontium-87 3 of any atom described herein, such as the P2 manifold. 3 P1 or 3 It may contain the first and second hyperfine states on the P2 manifold, respectively.

[0045] Figure 9 shows the 3 The left panel of Figure 9 shows an example of a qubit containing a P2 state. 3 The right panel of Figure 9 shows the rich energy level structure of the P2 state. The right panel of Figure 9 shows the P2 state of strontium-87, which is insensitive (to first order) to changes in magnetic field around 70 Gauss. 3 Potential qubit transitions within the P2 state are shown.

[0046] In some cases, the first and second atomic states are first and second hyperfine states of the first electronic state. Optical excitation may be applied between the first electronic state and the second electronic state. The optical excitation may excite the first hyperfine state and / or the second hyperfine state to the second electronic state. The single-qubit transition may include a two-photon transition between two hyperfine states within the first electronic state, using the second electronic state as an intermediate state. To drive the single-qubit transition, a pair of frequencies detuned from the single-photon transition to the intermediate state may be applied to drive the 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 to a lower electronic state by spontaneous or stimulated emission. The hyperfine state may include a nuclear spin state. In some cases, the hyperfine state may be a strontium-87 1 The qubit transition involves the nuclear spin state of the S0 manifold, and the strontium-87 1 One or both of the two nuclear spin states of S0 can be 3 P2 or 3 Drives into detuned states from or within the P1 manifold. In some cases, one-qubit transitions occur at strontium-87 1 is a two-photon Raman transition between the nuclear spin states of S0, 3 P2 or 3 via states detuned from or within the P1 manifold. In some cases, the nuclear spin state may be Stark-shifted. The Stark shift may be optically driven. The optical Stark shift may be off-resonance with any, all, or a combination of single-qubit transitions, two-qubit transitions, shelving transitions, imaging transitions, etc.

[0047] The first atomic state can include a first nuclear spin state, and the second atomic state can include a second nuclear spin state different from the first nuclear spin state. The first and second atomic states can include first and second nuclear spin states, respectively, of quadrupolar nuclei. The first and second atomic states can include first and second nuclear spin states, respectively, of spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nuclei. The first and second atomic states can include first and second nuclear spin states, respectively, of any atom described herein, such as the first and second spin states of strontium-87.

[0048] For first and second nuclear spin states associated with nuclei containing spins greater than 1 / 2 (such as spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nuclei), transitions between the first and second nuclear spin states may involve transitions between other spin states on the nuclear spin manifold. For example, for a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all nuclear spin levels may be separated by equal energy. Thus, for example, m N =9 / 2 spin state to m N Transitions designed to move atoms to the =7 / 2 spin state (such as Raman transitions) are N =7 / 2 to m N =5 / 2, m N =5 / 2 to m N =3 / 2, m N =3 / 2 to m N =1 / 2, m N =1 / 2 to m N =-1 / 2, m N =-1 / 2 to m N =-3 / 2, m N =-3 / 2 to m N =-5 / 2, m N =-5 / 2 to m N = -7 / 2, and m N =-7 / 2 to m N It can also be driven to -9 / 2, m N is the nuclear spin state. Similarly, for example, m N =9 / 2 spin state to mN Transitions designed to move atoms to the =5 / 2 spin state (such as Raman transitions) are also N =7 / 2 to m N =3 / 2, m N =5 / 2 to m N =1 / 2, m N =3 / 2 to m N =-1 / 2, m N =1 / 2 to m N =-3 / 2, m N =-1 / 2 to m N =-5 / 2, m N =-3 / 2 to m N = -7 / 2, and m N =-5 / 2 to m N = -9 / 2. Therefore, such transitions may not be selective for inducing transitions between specific spin states on the nuclear spin manifold.

[0049] Alternatively, it may be desirable to implement selective transitions between specific first and second spin states on the nuclear spin manifold. This can be achieved by providing light from a light source that provides an AC Stark shift and pushes adjacent nuclear spin states out of resonance with the transition between the desired transition between the first and second nuclear spin states. For example, N =-9 / 2 and m N If transitions from the first and second nuclear spin states with π = -7 / 2 are desired, the light will N = -5 / 2 spin state, thereby providing an AC Stark shift N =-7 / 2 and m N =-5 / 2 state transitions are significantly reduced. Similarly, m N =-9 / 2 and m N If transitions from the first and second nuclear spin states with ≈-5 / 2 are desired, the light will N = -1 / 2 spin state, thereby providing an AC Stark shift N =-5 / 2 and m N=-1 / 2 state. This effectively creates a two-level subsystem within the nuclear spin manifold that is isolated from the rest of the nuclear spin manifold, greatly simplifying the dynamics of the qubit system. The nuclear spin manifold (e.g., for a spin 9 / 2 nucleus, m N =-9 / 2 and m N =-7 / 2, m N =7 / 2 and m N =9 / 2, m N =-9 / 2 and m N =-5 / 2, or m N =5 / 2 and m N It may be advantageous to use nuclear spin states near the edge of the nuclear spin manifold (e.g., m N =-5 / 2 and m N =-3 / 2 or m N =-5 / 2 and m N =-1 / 2) can be used to implement two AC Stark shifts (e.g., m N =-7 / 2 and m N =-1 / 2 or m N =-9 / 2 and m N =3 / 2).

[0050] Stark shifts of the nuclear spin manifold can shift adjacent nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and the second electronic state or states detuned therefrom. Stark shifts can reduce leakage from the first and second nuclear spin states of the nuclear spin manifold into other states. Stark shifts can potentially be achieved up to hundreds of kHz with beam powers less than 10 mW. Frequency selectivity of the upper states can reduce scattering due to imperfect polarization control. 3The separation of different angular momentum states within the P1 manifold can be several gigahertz, numerically larger than single- and two-qubit gated states. Leakage into other states of the nuclear spin manifold can lead to decoherence. The Rabi frequency of the two-qubit transition (i.e., how fast the transition can be driven) can be faster than the decoherence rate. Scattering from intermediate states in the two-qubit transition can cause decoherence. Detuning from the intermediate state can improve the fidelity of the two-qubit transition.

[0051] Qubits based on the nuclear spin state of the electronic ground state require long-lived metastable excited electronic states (e.g., strontium-87) for qubit storage. 3 This may allow for the utilization of states such as the P0 state. Atoms may be selectively transferred to such states to reduce crosstalk or improve gating or detection fidelity. Such storage or shelving processes may be atom-selective using the SLM or AOD described herein. Shelving transitions are typically used for strontium-87. 1 Strontium-87 from the S0 state 3 P0 or 3 This may include a transition to the P2 state.

[0052] Clock transitions (also referred to herein as "shelving transitions" or "storage transitions") may be qubit-state selective. The upper state of a clock transition may have a very long natural lifetime, e.g., greater than 1 second. The linewidth of a clock transition may be much narrower than the energy spacing of the qubits. This may enable direct spectral resolution. An ensemble may be transferred from one of the qubit states to the clock state. This allows individual qubit states to be read out individually by first transferring the ensemble from one qubit state to the clock state, performing imaging on the qubit, and then transferring the ensemble from the clock state back to the ground state and imaging again. In some cases, magic wavelength transitions are used to drive the clock transitions.

[0053] The shelving clock light may or may not be atom-selective. In some cases, the clock transitions are applied globally (e.g., not atom-selective). Globally applied clock transitions may involve directing or structuring the light without passing through a microscope objective. In some cases, the clock transitions are atom-selective. Atom-selective clock transitions may minimize crosstalk and improve gate fidelity. For example, to reduce atomic crosstalk, atoms may be shelved to a clock state that may not be affected by the light. This may reduce crosstalk between neighboring qubits during the transition. To implement atom-selective clock transitions, the light passes through one or more microscope objectives and / or is configured with one or more of a spatial light modulator, a digital micromirror device, a crossed acousto-optic deflector, etc.

[0054] The system 200 may include one or more readout units 230. The readout unit may include one or more readout light units. The readout light units may be configured to perform one or more measurements of one or more superposition states to obtain non-classical calculations. The readout light units may include one or more photodetectors. The detectors may include 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 photodetectors may include one or more fluorescence detectors. The readout light unit may include one or more objective lenses, such as one or more objective lenses 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 lenses may have NAs of up to 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 lenses may have NAs within a range defined by any two of the aforementioned values.

[0055] One or more readout light units 230 can perform measurements, such as projection measurements, by applying light that is resonant with the imaging transition. The imaging transition can produce fluorescence. The imaging transition is the fluorescence of strontium-87. 1 Strontium-87 from the S0 state 1 This may include a transition to the P1 state.1 The P1 state can emit fluorescence. The substates of the qubit transition are 1 The measurement may include two nuclear spin states of the S0 manifold. One or more states may be resonant with the imaging transition. The measurement may include two excitations. In the first excitation, one of the two substates may be excited to a shelving state (e.g., strontium-87). 3 P0 state). In the second excitation, an imaging transition can be excited. The first transition can reduce crosstalk between neighboring atoms during calculation. Fluorescence generated from the imaging transition can be collected by one or more readout optical units 230.

[0056] The imaging unit can be used to determine if one or more atoms have been lost from the trap. The imaging unit can be used to observe the array of atoms in the trap.

[0057] The system 200 may include one or more vacuum units 240. The one or more vacuum units may include one or more vacuum pumps. The vacuum unit may include 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 include one or more wet (e.g., oil-sealed) or dry roughing vacuum pumps. The vacuum unit may include one or more high vacuum pumps, such as one or more cryosorption pumps, diffusion pumps, turbomolecular pumps, molecular drag pumps, turbodrag hybrid pumps, cryogenic pumps, ion pumps, or getter pumps.

[0058] The vacuum unit may include any combination of vacuum pumps described herein. For example, the vacuum unit may include one or more roughing pumps (such as scroll pumps) configured to provide a first stage of rough vacuum pumping. The roughing vacuum pumps may be configured to evacuate gases from the system 200 to achieve a rough vacuum pressure condition. For example, the roughing pumps may pump gases from the system 200 to achieve a rough vacuum pressure condition up to about 10 3 The vacuum unit may be configured to achieve a low vacuum pressure of up to about 10 Pascals (Pa). The vacuum unit may further include 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 or ultra-high vacuum pumping. The high vacuum pumps evacuate gases from the system 200 once the system 200 reaches the low vacuum pressure provided by the one or more roughing pumps, to achieve a low vacuum pressure of up to about 10 Pascals (Pa). -3 High vacuum pressure of up to 10 Pa or -6 The vacuum pump can be configured to achieve an ultra-high vacuum pressure of 100 Pa.

[0059] The vacuum unit can be used to reduce the system load by approximately 10 -6 Pa, 9 x 10 -7 Pa, 8 × 10 -7 Pa, 7 × 10 -7 Pa, 6 × 10 -7 Pa, 5 × 10 -7 Pa, 4 × 10 -7 Pa, 3 × 10 -7 Pa, 2 × 10 -7 Pa, 10 -7 Pa, 9 x 10 -8 Pa, 8 × 10 -8 Pa, 7 × 10 -8 Pa, 6 × 10 -8 Pa, 5 × 10 -8 Pa, 4 × 10 -8 Pa, 3 × 10 -8 Pa, 2 × 10 -8 Pa, 10 -8 Pa, 9 × 10 -9 Pa, 8 × 10 -9 Pa, 7 × 10 -9 Pa, 6 × 10-9 Pa, 5 × 10 -9 Pa, 4 × 10 -9 Pa, 3 × 10 -9 Pa, 2 × 10 -9 Pa, 10 -9 Pa, 9 × 10 -10 Pa, 8 × 10 -10 Pa, 7 × 10 -10 Pa, 6 × 10 -10 Pa, 5 × 10 -10 Pa, 4 × 10 -10 Pa, 3 × 10 -10 Pa, 2 × 10 -10 Pa, 10 -10 Pa, 9 × 10 -11 Pa, 8 × 10 -11 Pa, 7 × 10 -11 Pa, 6 × 10 -11 Pa, 5 × 10 -11 Pa, 4 × 10 -11 Pa, 3 × 10 -11 Pa, 2 × 10 -11 Pa, 10 -11 Pa, 9 × 10 -12 Pa, 8 × 10 -12 Pa, 7 × 10 -12 Pa, 6 × 10 -12 Pa, 5 × 10 -12 Pa, 4 × 10 -12 Pa, 3 × 10 -12 Pa, 2 × 10 -12 Pa, 10 -12 The vacuum unit may be configured to maintain the system 200 at a pressure of at least about 10 Pa, or lower. -12 Pa, 2 × 10 -12 Pa, 3 × 10 -12 Pa, 4 × 10 -12 Pa, 5 × 10 -12 Pa, 6 × 10 -12 Pa, 7 × 10 -12 Pa, 8 × 10 -12 Pa, 9 × 10 -12 Pa, 10 -11 Pa, 2 × 10 -11 Pa, 3 × 10 -11 Pa, 4 × 10 -11 Pa, 5 × 10 -11 Pa, 6 × 10 -11 Pa, 7 × 10-11 Pa, 8 × 10 -11 Pa, 9 × 10 -11 Pa, 10 -10 Pa, 2 × 10 -10 Pa, 3 × 10 -10 Pa, 4 × 10 -10 Pa, 5 × 10 -10 Pa, 6 × 10 -10 Pa, 7 × 10 -10 Pa, 8 × 10 -10 Pa, 9 × 10 -10 Pa, 10 -9 Pa, 2 × 10 -9 Pa, 3 × 10 -9 Pa, 4 × 10 -9 Pa, 5 × 10 -9 Pa, 6 × 10 -9 Pa, 7 × 10 -9 Pa, 8 × 10 -9 Pa, 9 × 10 -9 Pa, 10 -8 Pa, 2 × 10 -8 Pa, 3 × 10 -8 Pa, 4 × 10 -8 Pa, 5 × 10 -8 Pa, 6 × 10 -8 Pa, 7 × 10 -8 Pa, 8 × 10 -8 Pa, 9 × 10 -8 Pa, 10 -7 Pa, 2 × 10 -7 Pa, 3 × 10 -7 Pa, 4 × 10 -7 Pa, 5 × 10 -7 Pa, 6 × 10 -7 Pa, 7 × 10 -7 Pa, 8 × 10 -7 Pa, 9 × 10 -7 Pa, 10 -6 The vacuum unit may be configured to maintain the system 200 at a pressure within a range defined by any two of the aforementioned values.

[0060] System 200 may include one or more state preparation units 250. The state preparation units may include any of the state preparation units described herein, such as the state preparation units described herein with respect to Figure 5. The state preparation units may be configured to prepare states for multiple atoms.

[0061] The system 200 may include one or more atom reservoirs 260. The atom reservoirs may be configured to provide one or more replacement atoms to replace one or more atoms at one or more optical trapping sites when atoms are lost from the optical trapping sites. The atom reservoirs may be spatially separated from the optical trapping unit. For example, the atom reservoirs may be located at a distance from the optical trapping unit.

[0062] Alternatively or additionally, the atomic reservoir may comprise a portion of the optical trapping sites of the optical trapping unit. A first subset of the optical trapping sites may be utilized to perform quantum computations and may be referred to as the computationally active set of optical trapping sites, while a second subset of the optical trapping sites may function as the atomic reservoir. For example, the first subset of the optical trapping sites may comprise an internal array of optical trapping sites, and the second subset of the optical trapping sites may comprise an external array of optical trapping sites surrounding the internal array. The internal array may comprise a rectangular, square, rectangular prism, or cubic array of optical trapping sites.

[0063] System 200 may include one or more atom transfer units 270. The atom transfer units may be configured to transfer one or more replacement atoms from one or more atom reservoirs to one or more optical trapping sites. For example, the one or more atom transfer units may include one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs).

[0064] The system 200 may include one or more entanglement units 280. The entanglement unit 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 when quantum mechanically entangled. Alternatively or additionally, the first or second atom may not be in a superposition state when quantum mechanically entangled. The first atom and the second atom may be quantum mechanically entangled by one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The entanglement unit may be configured to quantum mechanically entangle any number of atoms described herein.

[0065] The entanglement unit may be configured to quantum mechanically entangle at least a subset of atoms with at least another atom to form one or more multiqubit units. A multiqubit unit may include a two-qubit unit, a three-qubit unit, a four-qubit unit, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For example, a two-qubit unit may include a first atom quantum mechanically entangled with a second atom, a three-qubit unit may include a first atom quantum mechanically entangled with a second and third atom, a four-qubit unit may include a first atom quantum mechanically entangled with a second, third, and fourth atom, and so on. The first, second, third, or fourth atom may be in a superposition state when quantum mechanically entangled. Alternatively or additionally, the first, second, third, or fourth atoms may not be in a superposition state when quantum mechanically entangled. The first, second, third, and fourth atoms may be quantum mechanically entangled by one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions.

[0066] The entanglement unit may include one or more Rydberg units. The Rydberg units may be configured to electronically excite at least a first atom into a Rydberg state or into 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 atom or dressed Rydberg atom and at least a second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (μm), 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9 μm, 10 μm, or greater from the Rydberg atom or dressed Rydberg atom. The second atom can be located at a distance of at most about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atom or dressed Rydberg atom. The second atom can be located at a distance from the Rydberg atom or dressed Rydberg atom that is within a range defined by any two of the aforementioned values. The Rydberg unit can be configured to allow the Rydberg atom or dressed Rydberg atom to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg unit can be configured to induce the Rydberg atom or dressed Rydberg atom to relax to a lower-energy atomic state. The Rydberg unit may be configured to drive the Rydberg atom or the dressed Rydberg atom to a lower energy atomic state.For example, the Rydberg unit can be configured to apply electromagnetic radiation (such as RF or optical radiation) to drive the Rydberg atom or the dressed Rydberg atom into a lower energy atomic state. The Rydberg unit can be configured to induce any number of quantum mechanical entanglements between any number of the plurality of atoms.

[0067] The Rydberg unit may include 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 wavelengths that form Rydberg atoms or dressed Rydberg atoms. For example, the light may include 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 longer. The light may include one or more wavelengths up to 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values. For example, the light may include one or more wavelengths within the range of 300 nm to 400 nm.

[0068] The Rydberg unit can be configured to induce a two-photon transition to generate entanglement. The Rydberg unit can be configured to induce a two-photon transition to generate entanglement between two atoms. The Rydberg unit can be configured to selectively induce a two-photon transition to selectively generate entanglement between two atoms. For example, the Rydberg unit can be configured to direct electromagnetic energy (e.g., light energy) to specific optical trapping sites to selectively induce a two-photon transition to selectively generate entanglement between two atoms. Two atoms can be trapped in nearby optical trapping sites. For example, two atoms can be trapped in adjacent optical trapping sites. The two-photon transition can be induced using first and second light from first and second light sources, respectively. The first and second light sources can each include any light source described herein (e.g., any laser described herein). The first light source can be the same as or similar to the light source used to perform the single-qubit operations described herein. Alternatively, different light sources can be used to perform single-qubit operations and induce two-photon transitions to generate entanglement. The first light source can emit light containing one or more wavelengths in the visible region of the optical spectrum (e.g., within the range of 400 nm to 800 nm or 650 nm to 700 nm). The second light source can emit light containing one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., within the range of 200 nm to 400 nm or 300 nm to 350 nm). The first and second light sources can emit light with substantially equal and opposite spatially dependent frequency shifts.

[0069] A Rydberg atom or dressed Rydberg atom may contain 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 state may contain a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or greater. The Rydberg state may contain a principal quantum number of up to about 100, 90, 80, 70, 60, 50, or less. The Rydberg state may contain a principal quantum number within a range defined by any two of the aforementioned values. The Rydberg state may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift the atomic energy levels of the atom.

[0070] State-selective excitation of atoms to Rydberg levels can enable the implementation of multi-qubit operations. Multi-qubit operations can include 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 can also drive atoms to the ground state ( 1 S0 ground state) to the Rydberg state (n 3 A two-photon transition can be implemented using first and second laser sources, as described herein. The first laser source can emit pie-polarized light, which does not change the projection of atomic angular momentum along the magnetic field. The second laser can emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. This polarization can be used to excite the first and second qubit levels to Rydberg levels. However, because the Rydberg levels can be more sensitive to magnetic fields than the ground state, large splittings (e.g., on the order of hundreds of MHz) can be easily obtained. This spectral selectivity can enable state-selective excitation to the Rydberg levels.

[0071] Multiqubit operations (e.g., two-qubit, three-qubit, and four-qubit operations) can rely on energy shifts of levels due to van der Waals interactions as described herein. Such shifts may prevent excitation of one atom conditional on the state of the other, or may alter the coherent dynamics of excitations in a two-atom system to perform two-qubit operations. In some cases, "dressing states" can be generated under continuous drive to perform two-qubit operations without requiring full excitation to the Rydberg levels (e.g., as described at www.arxiv.org / abs / 1605.05207, which is incorporated by reference in its entirety for all purposes).

[0072] System 200 may include one or more second electromagnetic delivery units (not shown in FIG. 2). The second electromagnetic delivery units may include any electromagnetic delivery units described herein, such as the electromagnetic delivery units 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 unit may be configured to apply second electromagnetic energy to one or more multi-qubit units. The second electromagnetic energy may include one or more pulse sequences. The first electromagnetic energy may precede, be simultaneous with, or follow the second electromagnetic energy.

[0073] A pulse sequence may include any number of pulses. For example, a pulse sequence may include 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. A pulse sequence may include up to 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 pulse. A pulse sequence may include a number of pulses within a range defined by any two of the aforementioned values. Each pulse in the pulse sequence may include any pulse shape, including any pulse shape described herein.

[0074] Pulse sequences can be configured to reduce the duration required to implement multi-qubit operations, as described herein (e.g., with respect to Example 3). For example, pulse sequences can include durations 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 longer. A pulse sequence can include durations of up to 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. A pulse sequence can include durations that fall within a range defined by any two of the foregoing values.

[0075] Pulse sequences, as described herein, can be configured to enhance the fidelity of multi-qubit operations. For example, pulse sequences can be configured to provide pulse widths 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, 0.999 8, 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, 0.9 ...96, 0.999997, 0.999998, 0.999999, or higher fidelity. The pulse sequence is approximately 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. The pulse sequence may enable multi-qubit operations with a fidelity of 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 lower. The pulse sequence may enable multi-qubit operations with a fidelity within a range defined by any two of the aforementioned values.

[0076] Pulse sequences may enable the implementation of multiqubit operations on nonadiabatic timescales while effectively preserving adiabatic dynamics. For example, pulse sequences may include one or more of the following: shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, inverse adiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences.For example, pulse sequences are 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 Molmer-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.

[0077] The pulse sequence may further include one or more optimal control pulse sequences, which may be derived from one or more procedures including gradient ascent pulse engineering (GRAPE), Krotov, chopped basis, chopped random basis (CRAB), Nelder-Mead, gradient optimization using parameterization (GROUP), genetic algorithm, and gradient optimization of analytic controls (GOAT). For example, the pulse sequence can 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. The pulse sequence can further include a composite pulse.Examples of composite pulses include, but are not limited to, broadband (e.g., BB1), narrowband, passband, Compensation for Off-Resonance with a Pulse Sequence (CORPSE), Short Composite Rotation for Undoing Length Over and Under Shoot (SCROFULOUS), Solovay-Kitaev (SK) sequences, and Knill pulses. Composite pulses can be used in systems with single-qubit gates.

[0078] Cloud Computing System 200 may be operably coupled to a digital computer described herein (such as the digital computer described herein with respect to FIG. 1) via a network described herein (such as the network described herein with respect to FIG. 1). The network may include a cloud computing network.

[0079] Light Capture Unit FIG. 3A illustrates an example of a light trapping unit 210. The light trapping unit can be configured to generate a plurality of spatially distinct light trapping sites 211, as described herein. For example, as shown in FIG. 3B, the light trapping unit can be configured to generate a first light trapping site 211a, a second light trapping site 211b, a third light trapping site 211c, a fourth light trapping site 211d, a fifth light trapping site 211e, a sixth light trapping site 211f, a seventh light trapping site 211g, an eighth light trapping site 211h, and a ninth light trapping site 211i, as depicted in FIG. 3A. The plurality of spatially distinct light trapping sites can be configured to trap a plurality of atoms, such as a first atom 212a, a second atom 212b, a third atom 212c, and a fourth atom 212d, as depicted in FIG. 3A. As depicted in FIG. 3B, each light trapping site can be configured to trap a single atom. As depicted in Figure 3B, some of the light trapping sites may be empty (i.e., do not trap atoms).

[0080] As shown in Figure 3B, the plurality of light capture sites may comprise a two-dimensional (2D) array. The 2D array may be perpendicular to the optical axis of the optical components of the light capture unit depicted in Figure 3A. Alternatively, the plurality of light capture sites may comprise a one-dimensional (1D) array or a three-dimensional (3D) array.

[0081] Although depicted in FIG. 3B as including nine light trapping sites filled with four atoms, the light trapping unit 210 can be configured to generate any number of spatially distinct light trapping sites as described herein and can be configured to trap any number of atoms as described herein.

[0082] Each light capture site of the plurality of light capture sites can be spatially separated from each other light capture site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 μm, 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9 μm, 10 μm, or more. Each light capture site can be spatially separated from each other light capture site by a distance of at most about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each light trapping site can be spatially separated from each other light trapping site by a distance within a range defined by any two of the aforementioned values.

[0083] The optical trapping site may include one or more optical tweezers. The optical tweezers may include one or more focused laser beams to provide attractive or repulsive forces for holding or moving one or more atoms. The beam waist of the focused laser beam may include a strong electric field gradient. Atoms may be attracted or repelled along the electric field gradient toward the center of the laser beam, which may include the strongest electric field. The optical trapping site may include one or more optical lattice sites of one or more optical lattices. The optical trapping site may include one or more optical lattice sites of one or more one-dimensional (1D) optical lattices, two-dimensional (2D) optical lattices, or three-dimensional (3D) optical lattices. For example, the optical trapping site may include one or more optical lattice sites of a 2D optical lattice, as depicted in Figure 3B.

[0084] Optical lattices can be generated by interfering counter-propagating light (such as counter-propagating laser light) to generate a standing wave pattern with a periodic series of intensity minima and maxima along a particular direction. 1D optical lattices can be generated by interfering a pair of counter-propagating light beams. 2D optical lattices can be generated by interfering two pairs of counter-propagating light beams. 3D optical lattices can be generated by interfering three pairs of counter-propagating light beams. The light beams can be generated by different light sources or by the same light source. Thus, optical lattices can be generated by at least about 1, 2, 3, 4, 5, 6, or more light sources, or up to about 6, 5, 4, 3, 2, or 1 light source.

[0085] Returning to the description of Figure 3A, the light capture unit may include one or more light sources configured to emit light to generate the multiple light capture sites described herein. For example, the light capture unit may include a single light source 213, as depicted in Figure 3A. While depicted as including a single light source in Figure 3A, the light capture unit may include 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 up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light source. The light source may include one or more lasers.

[0086] The laser may include one or more continuous wave lasers. The laser may include one or more pulsed lasers. The laser may include 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 example, the laser may include one or more of an argon dimer (Ar2) excimer laser, a krypton dimer (Kr2) excimer laser, a fluorine dimer (F2) excimer laser, a xenon dimer (Xe2) excimer laser, an argon fluoride (ArF) excimer laser, a krypton chloride (KrCl) excimer laser, a krypton fluoride (KrF) excimer laser, a xenon bromide (XeBr) excimer laser, a xenon chloride (XeCl) excimer laser, or a xenon fluoride (XeF) excimer laser. The laser may include one or more dye lasers.

[0087] The laser may include one or more metal vapor lasers, such as one or more helium cadmium (HeCd) metal vapor lasers, helium mercury (HeHg) metal vapor lasers, helium-selenium (HeSe) metal vapor lasers, helium silver (HeAg) metal vapor lasers, strontium (Sr) metal vapor lasers, neon copper (NeCu) metal vapor lasers, copper (Cu) metal vapor lasers, gold (Au) metal vapor lasers, manganese (Mn) metal vapor lasers, or manganese chloride (MnCl) metal vapor lasers.

[0088] The laser may include one or more solid-state lasers, such as one or more ruby ​​lasers, metal-doped crystal lasers, or metal-doped fiber lasers. For example, the laser may include 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 yttrium aluminum garnet (Tm:YAG) lasers, ytterbium The laser may include a chromium-doped yttrium aluminum garnet (Yb:YAG) laser, a ytterbium-doped glass (Yt:glass) laser, a holmium-yttrium aluminum garnet (Ho:YAG) laser, a chromium-doped zinc selenide (Cr:ZnSe) laser, a cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) laser, a cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) laser, an erbium-doped glass (Er:glass) laser, an erbium and ytterbium co-doped glass (Er / Yt:glass) laser, a uranium-doped calcium fluoride (U:CaF2) laser, or a samarium-doped calcium fluoride (Sm:CaF2) laser.

[0089] The laser may include one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGaInP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenide phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.

[0090] The laser may emit continuous wave laser light. The laser may emit pulsed laser light. The laser may emit pulsed laser light. The laser may emit light of at least about 1 femtosecond (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, The pulse length may be 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 longer. Lasers have a maximum of approximately 1,000ns, 900ns, 800ns, 700ns, 600ns, 500ns, 400ns, 300ns, 200ns, 100ns, 90ns, 80ns, 70ns, 60ns, 50ns, 40ns, 30ns, 20ns, 10ns, 9ns, 8ns, 7ns, 6ns, 5ns, 4ns, 3ns, 2ns, 1ns, 900ps, 800ps, 700ps, 600ps, 500ps, 400ps, 300ps, 200ps, 100ps, 90ps, 80ps, 70ps , 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 shorter. The laser may have a pulse length within a range defined by any two of the foregoing values.

[0091] The laser may be configured to operate at frequencies 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 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz ... The repetition rate may be 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.Lasers are available in frequencies up to approximately 1,000MHz, 900MHz, 800MHz, 700MHz, 600MHz, 500MHz, 400MHz, 300MHz, 200MHz, 100MHz, 90MHz, 80MHz, 70MHz, 60MHz, 50MHz, 40MHz, 30MHz, 20MHz, 10MHz, 9MHz, 8MHz, 7MHz, 6MHz, 5MHz, 4MHz, 3MHz, 2MHz, 1MHz, 900kHz, 800kHz, 700kHz, 600kHz, 500kHz, 400kHz, 300kHz, 200kHz, 100kHz, 90kHz , 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 laser may have a repetition rate within a range defined by any two of the foregoing values.

[0092] The laser may be at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (μJ), 2 μJ, 3 μJ, 4 μJ, 5 μJ, 6 μJ, 7 μJ, 8 μJ, 9 μJ, 10 μJ, 20 μJ, 30 μJ, 40 μJ, 50 μJ, 60 μJ, 70 μJ, 80 μJ, 90 μJ, 1 The laser may emit light having a pulse energy of 00 μJ, 200 μJ, 300 μJ, 400 μJ, 500 μJ, 600 μJ, 700 μJ, 800 μJ, 900 μJ, at least 1 millijoule (mJ) or more, 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, at least 1 joule (J), or more. The laser produces up to approximately 1J, 900mJ, 800mJ, 700mJ, 600mJ, 500mJ, 400mJ, 300mJ, 200mJ, 100mJ, 90mJ, 80mJ, 70mJ, 60mJ, 50mJ, 40mJ, 30mJ, 20mJ, 10mJ, 9mJ, 8mJ, 7mJ, 6mJ, 5mJ, 4mJ, 3mJ, 2mJ, 1mJ, 900μJ, 800μJ, 700μJ, 600μJ, 500μJ, 400μJ, 300μJ, 200μJ, 100μJ, 90μJ, 80μJ, 70μJ, 60μJ, 50 The laser may emit light having a pulse energy of 900 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, 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 laser may emit light having a pulse energy within a range defined by any two of the aforementioned values.

[0093] The laser may be at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, Light having an average power of 60mW, 70mW, 80mW, 90mW, 100mW, 200mW, 300mW, 400mW, 500mW, 600mW, 700mW, 800mW, 900mW, 1 Watt (W), 2W, 3W, 4W, 5W, 6W, 7W, 8W, 9W, 10W, 20W, 30W, 40W, 50W, 60W, 70W, 80W, 90W, 100W, 200W, 300W, 400W, 500W, 600W, 700W, 800W, 900W, 1,000W, or more can be emitted. Lasers are available in a range of powers up to approximately 1,000W, 900W, 800W, 700W, 600W, 500W, 400W, 300W, 200W, 100W, 90W, 80W, 70W, 60W, 50W, 40W, 30W, 20W, 10W, 9W, 8W, 7W, 6W, 5W, 4W, 3W, 2W, 1W, 900mW, 800mW, 700mW, 600mW, 500mW, 400mW, 300mW, 200mW, 100mW, 90mW, 80mW, 70mW, 60mW, 50mW, 40mW, and 30mW. , 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. The laser may emit light having a power within a range defined by any two of the foregoing values.

[0094] Lasers can emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. Lasers can emit light comprising 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 0nm, 540nm, 550nm, 560nm, 570nm, 580nm, 590nm, 600nm, 610nm, 620nm, 630nm, 640nm, 650nm, 660nm, 670nm, 680nm, 690nm, 700nm, 710nm, 720nm, 730nm, 740nm, 750nm, 760nm, 770nm, 780nm, 790nm, 800nm, 810nm, 820nm, 830nm, 840nm, 850nm, 860nm, 870nm, 880nm , 890nm, 900nm, 910nm, 920nm, 930nm, 940nm, 950nm, 960nm, 970nm, 980nm, 990nm, 1,000nm, 1,010nm, 1,020nm, 1,030nm, 1,040nm , 1,050nm, 1,060nm, 1,070nm, 1,080nm, 1,090nm, 1,100nm, 1,110nm, 1,120nm, 1,130nm, 1,140nm, 1,150nm, 1,160nm, 1,170nm, 1 The optical fiber may emit light comprising one or more wavelengths of 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 longer.Lasers are available in wavelengths up to approximately 1,400nm, 1,390nm, 1,380nm, 1,370nm, 1,360nm, 1,350nm, 1,340nm, 1,330nm, 1,320nm, 1,310nm, 1,300nm, 1,290nm, 1,280nm, 1,270nm, 1,260nm, 1,250nm, 1,240nm, 1,230nm, 1,220nm, 1,210nm, 1,200nm, 1,190nm, 1,180nm, 1,170nm, 1,160nm, 1,180nm, 1,190nm, 1,210nm, 1,220nm, 1,230nm, 1,240nm, 1,250nm, 1,260nm, 1,270nm, 1,280nm, 1,290nm, 1,300nm, 1,310nm, 1,320nm, 1,330nm, 1,340nm, 1,350nm, 1,360nm, 1,370nm, 1,380nm, 1,390nm, 1,400nm, 1,410nm, 1,420nm, 1,430nm, 1,440nm, 1,450nm, 1,460nm, 1,470nm, 1,480nm, 1,490nm, 1,500nm, 1,510nm, 1,520nm, 1,530nm, 1,540nm, 1,550nm, 1,56 ,150nm, 1,140nm, 1,130nm, 1,120nm, 1,110nm, 1,100nm, 1,090nm, 1,080nm, 1,070nm, 1,060nm, 1,050nm, 1,040nm, 1,030nm, 1 ,020nm, 1,010nm, 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860n m, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, 680nm, 670nm, 660nm, 650nm, 640nm, 630nm, 620nm, 610nm, 600nm, 590nm, 580nm, 570nm, 560nm, 550nm, 540nm, 530nm, 520nm, 510 The laser may emit light comprising one or more wavelengths of: 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 laser may emit light comprising one or more wavelengths within a range defined by any two of the foregoing values.

[0095] The laser has a power of at least about 1 x 10 -15 nm, 2 × 10 -15 nm, 3 × 10 -15nm, 4×10 -15 nm, 5×10 -15 nm, 6×10 -15 nm, 7×10 -15 nm, 8×10 -15 nm, 9×10 -15 nm, 1×10 -14 nm, 2×10 -14 nm, 3×10 -14 nm, 4×10 -14 nm, 5×10 -14 nm, 6×10 -14 nm, 7×10 -14 nm, 8×10 -14 nm, 9×10 -14 nm, 1×10 -13 nm, 2×10 -13 nm, 3×10 -13 nm, 4×10 -13 nm, 5×10 -13 nm, 6×10 -13 nm, 7×10 -13 nm, 8×10 -13 nm, 9×10 -13 nm, 1×10 -12 nm, 2×10 -12 nm, 3×10 -12 nm, 4×10 -12 nm, 5×10 -12 nm, 6×10 -12 nm, 7×10 -12 nm, 8×10 -12 nm, 9×10 -12 nm, 1×10 -11 nm, 2×10 -11 nm, 3×10 -11 nm, 4×10 -11 nm, 5×10 -11 nm, 6×10 -11 nm, 7×10 -11 nm, 8×10 -11 nm, 9×10 -11 nm, 1×10 -10 nm, 2×10 -10 nm, 3×10 -10 nm, 4×10 -10 nm, 5×10 -10 nm, 6×10 -10 nm, 7×10 -10 nm, 8×10 -10nm, 9×10 -10 nm, 1×10 -9 nm, 2×10 -9 nm, 3×10 -9 nm, 4×10 -9 nm, 5×10 -9 nm, 6×10 -9 nm, 7×10 -9 nm, 8×10 -9 nm, 9×10 -9 nm, 1×10 -8 nm, 2×10 -8 nm, 3×10 -8 nm, 4×10 -8 nm, 5×10 -8 nm, 6×10 -8 nm, 7×10 -8 nm, 8×10 -8 nm, 9×10 -8 nm, 1×10 -7 nm, 2×10 -7 nm, 3×10 -7 nm, 4×10 -7 nm, 5×10 -7 nm, 6×10 -7 nm, 7×10 -7 nm, 8×10 -7 nm, 9×10 -7 nm, 1×10 -6 nm, 2×10 -6 nm、3×10 -6 nm, 4×10 -6 nm, 5×10 -6 nm, 6×10 -6 nm, 7×10 -6 nm, 8×10 -6 nm, 9×10 -6 nm, 1×10 -5 nm, 2×10 -5 nm, 3×10 -5 nm, 4×10 -5 nm, 5×10 -5 nm, 6×10 -5 nm, 7×10 -5 nm, 8×10 -5 nm, 9×10 -5 nm, 1×10 -4 nm, 2×10 -4 nm、3×10 -4 nm, 4×10 -4nm, 5 × 10 -4 nm, 6 × 10 -4 nm, 7 × 10 -4 nm, 8 × 10 -4 nm, 9 × 10 -4 nm, 1 × 10 -3 The laser can emit light with a bandwidth of up to about 1×10 nm or longer. -3 nm, 9 × 10 -4 nm, 8 × 10 -4 nm, 7 × 10 -4 nm, 6 × 10 -4 nm, 5 × 10 -4 nm, 4 × 10 -4 nm, 3 × 10 -4 nm, 2 × 10 -4 nm, 1 × 10 -4 nm, 9 × 10 -5 nm, 8 × 10 -5 nm, 7 × 10 -5 nm, 6 × 10 -5 nm, 5 × 10 -5 nm, 4 × 10 -5 nm, 3 × 10 -5 nm, 2 × 10 -5 nm, 1 × 10 -5 nm, 9 × 10 -6 nm, 8 × 10 -6 nm, 7 × 10 -6 nm, 6 × 10 -6 nm, 5 × 10 -6 nm, 4 × 10 -6 nm, 3 × 10 -6 nm, 2 × 10 -6 nm, 1 × 10 -6 nm, 9 × 10 -7 nm, 8 × 10 -7 nm, 7 × 10 -7 nm, 6 × 10 -7 nm, 5 × 10 -7 nm, 4 × 10 -7 nm, 3 × 10 -7 nm, 2 × 10 -7 nm, 1 × 10 -7 nm, 9 × 10 -8 nm, 8 × 10 -8 nm, 7 × 10 -8 nm, 6 × 10 -8 nm, 5 × 10 -8 nm, 4 × 10-8 nm, 3×10 -8 nm, 2×10 -8 nm, 1×10 -8 nm, 9×10 -9 nm, 8×10 -9 nm, 7×10 -9 nm, 6×10 -9 nm, 5×10 -9 nm, 4×10 -9 nm, 3×10 -9 nm, 2×10 -9 nm, 1×10 -9 nm, 9×10 -10 nm, 8×10 -10 nm, 7×10 -10 nm, 6×10 -10 nm, 5×10 -10 nm, 4×10 -10 nm, 3×10 -10 nm, 2×10 -10 nm, 1×10 -10 nm, 9×10 -11 nm, 8×10 -11 nm, 7×10 -11 nm, 6×10 -11 nm, 5×10 -11 nm, 4×10 -11 nm, 3×10 -11 nm, 2×10 -11 nm, 1×10 -11 nm, 9×10 -12 nm, 8×10 -12 nm, 7×10 -12 nm, 6×10 -12 nm, 5×10 -12 nm, 4×10 -12 nm, 3×10 -12 nm, 2×10 -12 nm, 1×10 -12 nm, 9×10 -13 nm, 8×10 -13 nm, 7×10 -13 nm, 6×10 -13 nm, 5×10 -13 nm, 4×10 -13 nm, 3×10 -13 nm, 2×10 -13 nm, 1×10 -13 nm, 9×10 -14 nm, 8×10-14 nm, 7 × 10 -14 nm, 6 × 10 -14 nm, 5 × 10 -14 nm, 4 × 10 -14 nm, 3 × 10 -14 nm, 2 × 10 -14 nm, 1 × 10 -14 nm, 9 × 10 -15 nm, 8 × 10 -15 nm, 7 × 10 -15 nm, 6 × 10 -15 nm, 5 × 10 -15 nm, 4 × 10 -15 nm, 3 × 10 -15 nm, 2 × 10 -15 nm, 1 × 10 -15 A laser may emit light having a bandwidth of 100 .ANG. nm or shorter. A laser may emit light having a bandwidth within a range defined by any two of the aforementioned values.

[0096] The light source can be configured to emit light tuned to one or more magic wavelengths corresponding to a plurality of atoms. The magic wavelengths corresponding to the atoms can include any wavelength of light that induces equal or nearly equal polarizabilities in the first and second atomic states. The magic wavelengths for the transition between the first and second atomic states can be determined by calculating the wavelength-dependent polarizabilities of the first and second atomic states and finding the intersection point. Light tuned to such magic wavelengths can induce equal or nearly equal differential optical shifts in the first and second atomic states, regardless of the intensity of the light emitted from the light source. This effectively decouples the first and second atomic states from the atomic motion. The magic wavelengths can utilize one or more scalar or tensor optical shifts. The scalar or tensor optical shifts can depend on magnetic sublevels within the first and second atomic states.

[0097] For example, group III atoms and metastable states of alkaline earth or alkaline earth-like atoms may have relatively large tensor shifts, the angle with respect to the applied magnetic field can be adjusted to create a situation where the scalar and tensor shifts are balanced, giving a zero or near-zero differential optical shift between the first and second atomic states. The angle θ can be adjusted by choosing the polarization of the emitted light. For example, if the emitted light is linearly polarized, the total polarizability α can be calculated by dividing the scalar component α scalar and tensor component α tensor It can be written as the sum of α=α scalar +(3θ-1)α tensor

[0098] By appropriately choosing θ, the polarizabilities of the first and second atomic states can be chosen to be equal or nearly equal, corresponding to a zero or nearly zero differential optical shift, allowing the atomic motions to be decoupled.

[0099] The light source can be configured to direct light to one or more optical modulators (OMs) configured to generate multiple light trapping sites. For example, the light trapping unit may include an OM 214 configured to generate multiple light trapping sites. While depicted in FIG. 3A as including one OM, the light trapping unit may include any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OM may include one or more digital micromirror devices (DMDs). The OM may include one or more liquid crystal devices, such as one or more liquid crystal on silicon (LCoS) devices. The OM may include one or more spatial light modulators (SLMs). The OM may include one or more acousto-optic deflectors (AODs) or acousto-optic modulators (AOMs). The OM may include one or more electro-optic deflectors (EODs) or electro-optic modulators (EOMs).

[0100] The OM may be optically coupled to one or more optical elements to generate a regular array of light trapping sites. For example, the OM may be optically coupled to optical element 219, as shown in FIG. 3A. The optical element may include a lens or microscope objective configured to redirect light from the OM to form a regular rectangular grid of light trapping sites.

[0101] For example, as shown in Figure 3A, the OM may include an SLM, DMD, or LCoS device, which can be imaged into the back focal plane of a microscope objective, potentially enabling the generation of arbitrary configurations of optical trapping sites in two or three dimensions.

[0102] Alternatively or additionally, the OM may include a first and a second AOD. The active regions of the first and second AODs may be imaged onto the back focal plane of a microscope objective. The output of the first AOD may be optically coupled to the input of the second AOD. In this way, the second AOD can create a copy of the optical output of the first AOD. This may enable the creation of optical trapping sites in two or three dimensions.

[0103] Alternatively or additionally, the OM may include one or more static optical elements, such as microlens arrays or holographic optical elements, which can be imaged onto the back focal plane of the microscope objective, potentially allowing for the generation of arbitrary configurations of optical trapping sites in two or three dimensions.

[0104] The optical trapping unit may include one or more imaging units configured to acquire one or more images of the spatial configuration of the atoms trapped within the optical trapping site. For example, the optical trapping unit may include imaging unit 215. While depicted in FIG. 3A as including a single imaging unit, the optical trapping unit may include 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 up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 imaging units. The imaging unit may include one or more lenses or objective lenses. The imaging unit may include one or more PMTs, photodiodes, avalanche photodiodes, phototransistors, reverse-biased LEDs, CCDs, or CMOS cameras. The imaging unit may include one or more fluorescence detectors. The images may include one or more fluorescence images, single-atom fluorescence images, absorption images, single-atom absorption images, phase-contrast images, or single-atom phase-contrast images.

[0105] The optical capture unit may include one or more artificial intelligence (AI) units configured to perform one or more AI operations to determine the spatial configuration of the plurality of atoms captured within the optical capture site based on the image obtained by the imaging unit. For example, the optical capture unit may include spatial configuration AI unit 216. While depicted in FIG. 3A as including a single spatial configuration AI unit, the optical capture unit may include any number of spatial configuration AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration AI units. The AI ​​operations may include any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0106] The light trapping unit may include one or more atom rearrangement units configured to impart an altered spatial arrangement of the atoms trapped at the light trapping site based on one or more images obtained by the imaging unit. For example, the light trapping unit may include atom rearrangement unit 217. Although depicted in FIG. 3A as including a single atom rearrangement unit, the light trapping unit may include 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 up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units.

[0107] The optical capture unit may include one or more spatial arrangement AI units configured to perform one or more AI operations to determine an altered spatial arrangement of the atoms captured within the optical capture site based on the image obtained by the imaging unit. For example, the optical capture unit may include a spatial arrangement AI unit 218. While depicted in FIG. 3A as including a single spatial arrangement AI unit, the optical capture unit may include any number of spatial arrangement AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial arrangement AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial arrangement AI unit. The AI ​​operations may include any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0108] In some cases, the spatial configuration AI unit and the spatial arrangement AI unit may be integrated into an integrated AI unit. The light capture unit may include any number of integrated AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated AI units.

[0109] The atom rearrangement unit may be configured to change the spatial arrangement of the plurality of light trapping sites to obtain an increased fill factor. The fill factor may be defined as the ratio of the number of computationally active light trapping sites occupied by one or more atoms to the total number of computationally active light trapping sites available in the light trapping unit or a portion of the light trapping unit. For example, the initial loading of atoms in the computationally active light trapping sites may result in a fill factor of 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy less than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active light trapping sites. It may be desirable to rearrange atoms to achieve a fill factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging section, the atomic rearrangement unit may achieve a filling rate 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 greater. The atomic rearrangement unit may achieve a fill factor of up to 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 atomic rearrangement unit may achieve a fill factor within a range defined by any two of the foregoing values.

[0110] As an example, Figure 3C shows an example of a light trapping unit that is partially filled with atoms. As depicted in Figure 3C, the initial loading of atoms within the light trapping sites can result in a fill factor of 44.4% (four atoms filling nine available light trapping sites). By transferring atoms from a different region of the light trapping unit (not shown in Figure 3C) to unoccupied light trapping sites or by transferring atoms from the atom reservoir described herein, much higher fill factors can be obtained, as shown in Figure 3D.

[0111] Figure 3D illustrates an example of a light trapping unit fully filled with atoms. As depicted in Figure 3D, the fifth atom 212e, the sixth atom 212f, the seventh atom 212g, the eighth atom 212h, and the ninth atom 212i can be moved to fill the unoccupied light trapping sites. The fifth, sixth, seventh, eighth, and ninth atoms can be moved by transferring atoms from different regions of the light trapping unit (not shown in Figure 3C) or from the atom reservoir described herein. Thus, the fill factor can be significantly improved following the rearrangement of atoms within the light trapping sites. For example, a fill factor of up to 100% (9 atoms filling the nine available light trapping sites, as shown in Figure 3D) can be achieved.

[0112] Atom rearrangement can be performed by (i) acquiring an image of the optical trapping unit and identifying filled and unfilled optical trapping sites, (ii) determining a sequence of moves to move atoms from filled to unfilled optical trapping sites, and (iii) moving atoms from filled to unfilled optical trapping sites. Operations (i), (ii), and (iii) can be performed iteratively until a large fill factor is achieved. Operation (iii) can include converting the moves identified in operation (ii) into a waveform that can be sent to an arbitrary waveform generator (AWG) and using the AWG to drive the AOD to move the atoms. The sequence of moves can 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.

[0113] Electromagnetic Delivery Unit FIG. 4 illustrates an example of an electromagnetic delivery unit 220. The electromagnetic delivery unit can be configured to apply electromagnetic energy to one or more atoms of the plurality of atoms as described herein. The electromagnetic delivery unit can include one or more light sources, such as any of the light sources described herein. The electromagnetic energy can include optical energy. The optical energy can include any repetition rate, pulse energy, average power, wavelength, or bandwidth described herein.

[0114] The electromagnetic delivery unit may include one or more microwave or radio-frequency (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 (IMPATT) diodes, or masers. The electromagnetic energy may include microwave energy or RF energy. The RF energy may be 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, It may include one or more wavelengths of 4m, 5m, 6m, 7m, 8m, 9m, 10m, 20m, 30m, 40m, 50m, 60m, 70m, 80m, 90m, 100m, 200m, 300m, 400m, 500m, 600m, 700m, 800m, 900m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or longer. RF energy up to approximately 10km, 9km, 8km, 7km, 6km, 5km, 4km, 3km, 2km, 1km, 900m, 800m, 700m, 600m, 500m, 4 00m, 300m, 200m, 100m, 90m, 80m, 70m, 60m, 50m, 40m, 30m, 20m, 10m, 9m, 8m, 7m, 6m, 5m, 4m, 3m, 2m, 1m , 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 shorter. The RF energy may include one or more wavelengths within a range defined by any two of the foregoing values.

[0115] The RF energy may be at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 ... The average power may include W, 50mW, 60mW, 70mW, 80mW, 90mW, 100mW, 200mW, 300mW, 400mW, 500mW, 600mW, 700mW, 800mW, 900mW, 1 Watt (W), 2W, 3W, 4W, 5W, 6W, 7W, 8W, 9W, 10W, 20W, 30W, 40W, 50W, 60W, 70W, 80W, 90W, 100W, 200W, 300W, 400W, 500W, 600W, 700W, 800W, 900W, 1,000W, or more. RF energy can be up to approximately 1,000W, 900W, 800W, 700W, 600W, 500W, 400W, 300W, 200W, 100W, 90W, 80W, 70W, 60W, 50W, 40W, 30W, 20W, 10W, 9W, 8W, 7W, 6W, 5W, 4W, 3W, 2W, 1W, 900mW, 800mW, 700mW, 600mW, 500mW, 400mW, 300mW, 200mW, 100mW, 90mW, 80mW, 70mW, 60mW, 50mW, 40 The RF energy may include an average power of 1000 MHz, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. The RF energy may include an average power within a range defined by any two of the foregoing values.

[0116] The electromagnetic delivery unit may include one or more light sources, such as any of the light sources described herein. For example, the electromagnetic delivery unit may include light source 221. Although depicted in FIG. 4 as including a single light source, the electromagnetic delivery unit may include 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 up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light source.

[0117] The light source can be configured to direct light to one or more OMs configured to selectively apply electromagnetic energy to one or more atoms of the plurality of atoms. For example, the electromagnetic delivery unit can include OM 222. While depicted in FIG. 4 as including a single OM, the electromagnetic delivery unit can include any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OM. The OM can include one or more SLMs, AODs, or AOMs. The OM can include one or more DMDs. The OM can comprise one or more liquid crystal devices, such as one or more LCoS devices.

[0118] The electromagnetic delivery unit may include one or more electromagnetic energy artificial intelligence (AI) units configured to perform one or more AI operations to selectively apply electromagnetic energy to atoms. For example, the electromagnetic delivery unit may include AI unit 223. Although depicted in FIG. 4 as including a single AI unit, the electromagnetic delivery unit may include any number of AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 AI unit. The AI ​​operations may include any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0119] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) to a qubit 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) to a two-qubit unit described herein. Each single-qubit or two-qubit operation may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100 μs, or longer. Each single-qubit or two-qubit operation may include a duration of at most about 100 μs, 90 μs, 80 μs, 70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4 μs, 3 μs, 2 μs, 1 μs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. Each single-qubit or two-qubit operation may include a period within a range defined by any two of the foregoing values. Single-qubit or two-qubit operations may be applied at repetition rates 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 longer.Single-qubit or two-qubit operations may be applied at repetition rates up to 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 shorter. Single-qubit or two-qubit operations may be applied at repetition rates within a range defined by any two of the foregoing values.

[0120] 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 as described herein. 3 P0 or 3For example, 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 up to about 1 GHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The Raman transition may be detuned by a value within a range defined by any two of the aforementioned values.

[0121] Raman transitions may be induced in individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle and / or frequency shift to a light beam based on an applied radio frequency (RF) signal. The SLM or AOD may be combined with an optical alignment system that images the active area of ​​the SLM or AOD onto the back focal plane of a microscope objective. The microscope objective may perform a spatial Fourier transform on the optical field at the location of the SLM or AOD. In this way, angle (which may be proportional to the RF frequency) may be converted to position. For example, applying a radio frequency comb to the AOD may generate a linear array of spots at the focal plane of the objective, each with a finite extent determined by the characteristics of the optical alignment system (such as the point spread function of the optical alignment system).

[0122] To perform Raman transitions on a single atom using a single SLM or AOD, a pair of frequencies may be applied simultaneously to the SLM or AOD, with the two frequencies having a frequency difference that matches or nearly matches the splitting energy between the first and second qubit states. For example, the frequency difference may differ from the split energy by up to 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 can differ from the split 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 can differ from the splitting energy by approximately 0 Hz. The frequency difference can differ from the splitting energy by a value within a range defined by any two of the aforementioned values. The optical system can be configured such that the position interval corresponding to the frequency difference is not resolved and light of both frequencies interacts with a single atom.

[0123] Integrated light capture and electromagnetic delivery unit The optical capture unit and the electromagnetic delivery unit described herein may be integrated into a single optical system. A microscope objective lens may be used to deliver the electromagnetic radiation generated by the electromagnetic delivery unit described herein and to deliver the light for capturing atoms generated by the optical capture unit described herein. Alternatively or additionally, different objective lenses may be used to deliver the electromagnetic radiation generated by the electromagnetic delivery unit and to deliver the light from the captured atoms generated by the optical capture unit.

[0124] A single SLM or AOD can enable the implementation of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or additionally, two separate SLMs or AODs may be configured, each processing light of orthogonal polarizations. Orthogonally polarized light may be superimposed in front of a microscope objective. In such a scheme, each photon used in a two-photon transition described herein is passed to the objective by a separate SLM or AOD, allowing for improved polarization control. Qubit operations can be performed on a two-dimensional array of atoms by transmitting light from a first SLM or AOD via an optical relay to a second SLM or AOD oriented substantially perpendicular to the first. Alternatively or additionally, qubit operations can be performed on a two-dimensional array of atoms using a one-dimensional array of SLMs or AODs.

[0125] The stability of qubit gate fidelity can be improved by maintaining overlap of light from the various light sources described herein (such as the light sources associated with the light trapping or electromagnetic delivery units described herein). Such overlap is maintained by an optical subsystem that measures the direction of light emitted from the various light sources, allowing closed-loop control of the direction of light emission. The optical subsystem can include a pick-off mirror positioned in front of the microscope objective. The pick-off mirror can be configured to direct a small amount of light toward a lens that can focus the collimated beam and convert angular deviations into positional deviations. Position-sensitive photodetectors, such as transverse effect position sensors or quadrant photodiodes, can convert the positional deviations into electronic signals and provide information about the deviations to adaptive optics systems, such as active mirrors.

[0126] The stability of qubit gate operation can be improved by controlling the intensity of light from the various light sources described herein (such as the light sources associated with the optical trapping or electromagnetic delivery units described herein). Such intensity control is maintained by an optical subsystem that measures the intensity of light emitted by the various light sources, allowing for closed-loop control of intensity. Each light source can be coupled to an intensity actuator, such as an intensity servo control. The actuator may comprise an acousto-optic modulator (AOM) or an electro-optic modulator (EOM). Intensity can be measured using a photodetector, such as a photodiode or any other photodetector described herein. Information regarding the intensity can be integrated into a feedback loop to stabilize the intensity.

[0127] State Readiness Unit 5 is a diagram illustrating an example of a state preparation unit 250. The state preparation unit may be configured to prepare states of a plurality of atoms as described herein. The state preparation unit may be coupled to the optical trapping unit and may direct atoms 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 before trapping the plurality of atoms in the plurality of optical trapping sites.

[0128] The state preparation unit may include one or more Zeeman decelerators. For example, the state preparation unit may include Zeeman decelerator 251. While depicted in FIG. 5 as including a single Zeeman decelerator, the state preparation may include any number of Zeeman decelerators, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Zeeman decelerators, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Zeeman decelerators. The Zeeman decelerator may be configured to cool one or more atoms of the plurality of atoms from a first velocity or velocity distribution (e.g., an emission velocity from the atom source, room temperature, liquid nitrogen temperature, or other temperature) to a second velocity that is lower than the first velocity or velocity distribution.

[0129] The first velocity or velocity distribution may be associated with a temperature of at least about 50 Kelvin (K), 60K, 70K, 80K, 90K, 100K, 200K, 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1,000K, or higher. The first velocity or velocity distribution may be associated with a temperature of up to about 1,000K, 900K, 800K, 700K, 600K, 500K, 400K, 300K, 200K, 100K, 90K, 80K, 70K, 60K, 50K, or lower. The first velocity or velocity distribution may be associated with a temperature within a range defined by any two of the foregoing values. The second velocity may be at least about 1 meter per second (m / s), 2 m / s, 3 m / s, 4 m / s, 5 m / s, 6 m / s, 7 m / s, 8 m / s, 9 m / s, 10 m / s, or more. The second velocity may be at most about 10 m / s, 9 m / s, 8 m / s, 7 m / s, 6 m / s, 5 m / s, 4 m / s, 3 m / s, 2 m / s, 1 m / s, or less. The second velocity may be within a range defined by any two of the foregoing values. The Zeeman reducer may include a 1D Zeeman reducer.

[0130] The state preparation unit may include a first magnetic-optical trap (MOT) 252. The first MOT may be configured to cool the atoms to a first temperature, which 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 lower. 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 higher. The first temperature may be within a range defined by any two of the foregoing values. The first MOT may comprise a 1D, 2D, or 3D MOT.

[0131] The first MOT may comprise one or more light sources (such as any light source described herein) configured to emit light 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, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, 1160 nm, 1170 The wavelengths may include one or more wavelengths of 20 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 longer. The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, The light may include one or more wavelengths below 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 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.

[0132] The state preparation unit may include a second MOT 253. The second MOT may be configured to cool the atoms from a first temperature to a second temperature lower than the first temperature. The second temperature may be at most about 100 microkelvin (μK), 90 μK, 80 μK, 70 μK, 60 μK, 50 μK, 40 μK, 30 μK, 20 μK, 10 μK, 9 μK, 8 μK, 7 μK, 6 μK, 5 μK, 4 μK, 3 μK, 2 μK, 1 μK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, or lower. The second temperature may be at least about 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 μK, 2 μK, 3 μK, 4 μK, 5 μK, 6 μK, 7 μK, 8 μK, 9 μK, 10 μK, 20 μK, 30 μK, 40 μK, 50 μK, 60 μK, 70 μK, 80 μK, 90 μK, 100 μK, or higher. The second temperature may be within a range defined by any two of the aforementioned values. The second MOT may comprise a 1D, 2D, or 3D MOT.

[0133] The second MOT may comprise one or more light sources (such as any light source described herein) configured to emit light 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, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, 1160 nm, 1170 The wavelengths may include one or more wavelengths of 20 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 longer. The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, The light may include one or more wavelengths below 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 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.

[0134] Although depicted in FIG. 5 as including two MOTs, the condition preparation unit may include any number of MOTs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more MOTs, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 MOTs.

[0135] The condition preparation unit may comprise one or more sideband or Sisyphus cooling units (such as the sideband cooling unit described at www.arxiv.org / abs / 1810.06626 or the Sisyphus cooling unit described at www.arxiv.org / abs / 1811.06014, each of which is incorporated by reference in its entirety for all purposes). For example, the condition preparation unit may comprise a sideband or Sisyphus cooling unit 254. 5 as including a single sideband or Sisyphus cooling unit, the state preparation may include any number of sideband or Sisyphus cooling units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more sideband or Sisyphus cooling units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband or Sisyphus cooling units. The sideband or Sisyphus cooling unit may be configured to cool atoms from a second temperature to a third temperature lower than the second temperature using sideband cooling. The third temperature may be at most about 10 μK, 9 μK, 8 μK, 7 μK, 6 μK, 5 μK, 4 μK, 3 μK, 2 μK, 1 μK, 900 nK, 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, 90 nK, 80 nK, 70 nK, 60 nK, 50 nK, 40 nK, 30 nK, 20 nK, 10 nK, or lower. The third temperature may be at most about 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80 nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 μK, 2 μK, 3 μK, 4 μK, 5 μK, 6 μK, 7 μK, 8 μK, 9 μK, 10 μK, or more. The third temperature may be within a range defined by any two of the foregoing values.

[0136] The sideband or Sisyphus cooling unit may include one or more light sources (such as any of the light sources described herein) configured to emit light at 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, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, The wavelengths may include one or more wavelengths of 20 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 longer. The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, The light may include one or more wavelengths below 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 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.

[0137] The state preparation unit may include one or more optical pumping units. For example, the state preparation unit may include optical pumping unit 255. While depicted in FIG. 5 as including a single optical pumping unit, the state preparation may include 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 up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 optical pumping unit. The optical pumping unit may be configured to emit light to optically pump atoms from an equilibrium distribution of atomic states to a non-equilibrium atomic state. For example, the optical pumping unit may be configured to emit light to optically pump atoms from an equilibrium distribution of atomic states to a single pure atomic state. The optical pumping unit may be configured to emit light to optically pump atoms to the ground atomic state or any other atomic state. The optical pumping unit may be configured to optically pump atoms between any two atomic states. The optical pumping unit may include one or more light sources (such as any of the light sources described herein) configured to emit light at 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, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, 1160 The wavelengths may include one or more wavelengths of 20 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 longer.The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, The light may include one or more wavelengths below 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values. For example, the light may include one or more wavelengths in the ranges of 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.

[0138] The state preparation unit may include one or more coherent drive units. For example, the state preparation unit may include coherent drive unit 256. While depicted in FIG. 5 as including a coherent drive unit, the state preparation may include any number of coherent drive units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more coherent drive units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 coherent drive units. The coherent drive units may be configured to coherently drive atoms from a non-equilibrium state to a first or second atomic state described herein. Thus, atoms may be optically pumped to a convenient atomic state to access (e.g., based on the availability of a light source emitting a particular wavelength or other factors) and then coherently driven to an atomic state described herein useful for performing quantum computation. The coherent drive unit may be configured to induce a single-photon transition between the non-equilibrium state and the first or second atomic state. The coherent drive unit can be configured to induce a two-photon transition between the non-equilibrium state and the first or second atomic state. The two-photon transition can be induced using light from two light sources described herein (such as two lasers described herein).

[0139] The coherent driving unit may include one or more light sources (such as any of the light sources described herein) configured to emit light at least approximately 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, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, 1160 nm, The wavelengths may include one or more wavelengths of 20 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 longer. The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, The light may include one or more wavelengths below 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 shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 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.

[0140] The coherent drive unit may be configured to induce an RF transition between the non-equilibrium state and the first or second atomic state. The coherent drive unit may include one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. For example, the coherent drive unit may include one or more RF sources (such as any RF source described herein) configured to emit RF radiation. The RF radiation may include 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 longer. The RF radiation may include one or more wavelengths up to 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 shorter. The RF radiation may include one or more wavelengths within a range defined by any two of the foregoing values. Alternatively or additionally, the coherent driving unit may include one or more light sources (such as any light sources described herein) configured to induce two-photon transitions corresponding to the RF transitions.

[0141] controller The optical capture unit, electromagnetic delivery unit, entanglement unit, readout optical unit, vacuum unit, imaging unit, spatial configuration AI unit, spatial arrangement AI unit, atom rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atom reservoir, atom transfer unit, or Rydberg excitation unit may include one or more circuits or controllers (e.g., one or more electronic circuits or controllers) connected to the optical capture unit, electromagnetic delivery unit, entanglement unit, readout optical unit, vacuum unit, imaging unit, spatial configuration AI unit, spatial arrangement AI unit, atom rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atom reservoir, atom transfer unit, or Rydberg excitation unit (e.g., by one or more electronic connections). The circuitry or controller may be configured to control an optical capture unit, an electromagnetic delivery unit, an entanglement unit, a readout optical unit, a vacuum unit, an imaging unit, a spatial configuration AI unit, a spatial arrangement AI unit, an atomic rearrangement unit, a state preparation unit, a sideband cooling unit, an optical pumping unit, a coherent drive unit, an electromagnetic energy AI unit, an atomic reservoir, an atomic transfer unit, or a Rydberg excitation unit.

[0142] Non-classical computers In one aspect, the present disclosure provides a non-classical computer comprising: a plurality of qubits comprising more than 60 atoms, each atom trapped within one optical trapping site of a plurality of spatially distinct optical trapping sites, the plurality of qubits comprising at least a first qubit state and a second qubit state, the first qubit state comprising a first atomic state and the second qubit state comprising 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, the non-classical operation comprising 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 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.

[0143] In one aspect, the present disclosure provides a non-classical computer including a plurality of qubits, each including more than 60 atoms trapped within one of a plurality of spatially distinct optical trapping sites.

[0144] How to perform non-classical computations In one aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (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 including more 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 different from the first atomic state; (c) quantum mechanically entangle 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 states to obtain a non-classical computation.

[0145] FIG. 6 is a flowchart illustrating an example of a first method 600 for performing non-classical computation.

[0146] In a first operation 610, the method 600 may include 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 include more than 60 atoms. The optical trapping sites may include any optical trapping site described herein. The atoms may include any atoms described herein.

[0147] In a second operation 620, the method 600 may include 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 superpositions of a first atomic state and at least a second atomic state different from the first atomic state. The electromagnetic energy may include any electromagnetic energy described herein. The first atomic state may include any first atomic state described herein. The second atomic state may include any second atomic state described herein.

[0148] In a third operation 630, the method 600 may include quantum mechanically entangled at least a subset of the one or more atoms in 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 (e.g., as described herein with respect to FIG. 2).

[0149] In a fourth operation 640, the method 600 may include performing one or more optical measurements of one or more superposition states to obtain a non-classical calculation. The optical measurements may include any optical measurements described herein.

[0150] In one aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) providing a plurality of qubits including more than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, the plurality of qubits including at least a first qubit state and a second qubit state, the first qubit state including a first atomic state and the second qubit state including 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, the non-classical operation including a superposition between at least the first qubit state and the second qubit state; (c) 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 (d) performing one or more optical measurements of the one or more qubits, thereby obtaining a classical computation.

[0151] FIG. 7 is a flowchart illustrating an example of a second method 700 for performing non-classical computation.

[0152] In a first operation 710, the method 700 may include providing a plurality of qubits including more than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, the plurality of qubits including at least a first qubit state and a second qubit state, the first qubit state including a first atomic state and the second qubit state including a second atomic state. The optical trapping site may include any optical trapping site described herein. The qubits may include any qubits described herein. The atoms may include any atoms described herein. The first qubit state may include any first qubit state described herein. The second qubit state may include any second qubit state described herein. The first atomic state may include any first atomic state described herein. The second atomic state may include any second atomic state described herein.

[0153] In a second operation 720, the method 700 may include 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 that includes a superposition between at least a first qubit state and a second qubit state. The electromagnetic energy may include any electromagnetic energy described herein.

[0154] In a third operation 730, the method 700 may include quantum mechanically entangled 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 (e.g., as described herein with respect to FIG. 2).

[0155] In a fourth operation 740, the method 700 may include performing one or more optical measurements of one or more qubits, thereby obtaining a non-classical computation. The optical measurements may include any optical measurements described herein.

[0156] In one aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) providing a plurality of qubits, each qubit comprising more than 60 atoms trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites; and (b) performing a non-classical computation using at least a subset of the plurality of qubits.

[0157] FIG. 8 is a flowchart illustrating an example of a third method 800 for performing non-classical computation.

[0158] In a first operation 810, the method 800 may include providing a plurality of qubits each including more than 60 atoms trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites. The qubits may include any qubit described herein. The atoms may include any atom described herein. The optical trapping site may include any optical trapping site described herein.

[0159] In a second operation 820, the method 800 may include performing a non-classical computation using at least a subset of the plurality of qubits.

[0160] Computer Systems 1 illustrates a computer system 101 programmed or configured to operate any of the methods or systems described herein (such as the systems or methods for performing non-classical computation described herein). The computer system 101 may govern various aspects of the present disclosure. The computer system 101 may be a user's electronic device or a computer system located remotely relative to the electronic device. The electronic device may be a mobile electronic device.

[0161] The computer system 101 includes a central processing unit (CPU, also referred to herein as a "processor" and "computer processor") 105, which may be a single-core or multi-core processor, or multiple processors for parallel processing. The computer system 101 also includes memory or storage 110 (e.g., random access memory, read-only memory, flash memory), electronic storage 115 (e.g., a hard disk), a communication interface 120 (e.g., a network adapter) for communicating with one or more other systems, and peripheral devices 125, such as cache, other memory, data storage, and / or electronic display adapters. The memory 110, storage 115, interface 120, and peripheral devices 125 communicate with the CPU 105 via a communication bus (solid lines), such as a motherboard. The storage unit 115 may also be a data storage unit (or data repository) for storing data. Computer system 101 may be operatively coupled to a computer network ("network") 130 using communication interface 120. Network 130 may be the Internet, an Internet and / or extranet, or an intranet and / or extranet in communication with the Internet. Network 130 may, in some cases, be a telecommunications and / or data network. Network 130 may include one or more computer servers that enable distributed computing, such as cloud computing. Network 130 may, in some cases, implement a peer-to-peer network, allowing devices coupled to computer system 101 to operate as clients or servers, with the assistance of computer system 101.

[0162] CPU 105 may execute a series of machine-readable instructions, which may be embodied in a program or software. The instructions may be stored in a memory location, such as memory 110. The instructions may be directed to CPU 105, which may then program or configure CPU 105 to implement the methods of the present disclosure. Examples of operations performed by CPU 105 may include fetch, decode, execute, and writeback.

[0163] The CPU 105 may be part of a circuit, such as an integrated circuit. One or more other components of the system 101 may be included in the circuit. In some cases, the circuit is an application specific integrated circuit (ASIC).

[0164] The storage device 115 may store files such as drivers, libraries, and saved programs. The storage unit 115 may store user data, such as user preferences and user programs. The computer system 101 may optionally include one or more additional data storage units external to the computer system 101, such as located on a remote server that communicates with the computer system 101 through an intranet or the Internet.

[0165] Computer system 101 can communicate with one or more remote computer systems via network 130. For example, computer system 101 can communicate with a user's remote computer system. Examples of remote computer systems include a personal computer (e.g., a portable PC), a slate or tablet PC (e.g., an Apple® iPad, a Samsung® Galaxy Tab), a telephone, a smartphone (an Apple® iPhone, an Android-enabled device, a Blackberry®), or a personal digital assistant. A user can access computer system 101 via network 130.

[0166] The methods described herein may be implemented by machine (e.g., computer processor) executable code stored in an electronic storage location of computer system 101, such as memory 110 or electronic storage unit 115. The machine-executable or machine-readable code may be provided in the form of software. During use, the code may be executed by processor 105. In some cases, the code may be retrieved from storage unit 115 and stored in memory 110 for immediate access by processor 105. In some situations, electronic storage unit 115 may be omitted, and machine-executable instructions may be stored in memory 110.

[0167] The code may be pre-compiled and configured for use on a machine with a processor adapted to execute the code, or it may be compiled at run time. The code may be provided in a programming language that can be selected to allow the code to be executed in a pre-compiled or co-compiled manner.

[0168] Aspects of the systems and methods provided herein, such as computer system 101, may be embodied in programming. Various aspects of the technology can be considered "products" or "articles of manufacture," typically in the form of machine (or processor) executable code and / or associated data, which are held or embodied in some type of machine-readable medium. The machine-executable code may be stored in electronic storage, such as memory (e.g., read-only memory, random-access memory, flash memory) or a hard disk. "Storage"-type media may include any or all of the tangible memory of a computer, processor, etc., or associated modules, such as various semiconductor memories, tape drives, disk drives, etc., that may provide non-transitory storage for software programming at any time. All or portions of the software may be communicated via the Internet or various other telecommunications networks. Such communication may, for example, enable loading of the software from one computer or processor to another, such as from an administrative server or host computer to an application server computer platform. Thus, other types of media that may hold software elements include the light waves, radio waves, and electromagnetic waves used across physical interfaces between local devices over wired and optical landline networks, as well as various air links. The physical elements that carry such waves, such as wired or wireless links, optical links, etc., may also be considered media that carry software. As used herein, unless limited to non-transitory, tangible "storage" media, terms such as computer or machine "readable medium" refer to any medium that participates in providing instructions to a processor for execution.

[0169] Thus, machine-readable media, such as computer-executable code, may take many forms, including, but not limited to, tangible storage media, carrier wave media, or physical transmission media. Non-volatile storage media include optical or magnetic disks, such as any of the storage devices of any computer, such as those that may be used to implement the databases shown in the figures. Volatile storage media include dynamic memory, such as the main memory of such a computer platform. Tangible transmission media include copper wire and fiber optics, including coaxial cables, i.e., the wires that comprise a bus within a computer system. Carrier-wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Thus, common forms of computer readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tape, any other magnetic media, CD-ROMs, DVDs, or DVD-ROMs, other optical media, punched cards, paper tape, other physical storage media with patterns of holes, RAM, ROM, PROMs and EPROMs, FLASH-EPROMs, other memory chips or cartridges, carrier waves carrying data or instructions, cables or links carrying such carrier waves, or any other medium from which a computer can read programming code and / or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution.

[0170] The computer system 101 may include or communicate with an electronic display 135 that provides a user interface (UI) 140. Examples of a UI include, but are not limited to, a graphical user interface (GUI) and a web-based user interface.

[0171] The methods and systems of the present disclosure may be implemented by one or more algorithms. The algorithms may be implemented by software when executed by the central processing unit 105. The algorithms may, for example, implement the methods for performing non-classical computations described herein. [Example]

[0172] Example 1: Modeling the Strontium-87 Nuclear Spin Levels In the following example, we model the 10 nuclear spin levels (I = 9 / 2) of strontium-87 to demonstrate a two-level system (i.e., a qubit). To achieve spectral separation of the qubit transitions, we employ a Stark shift scheme to shift the unwanted transitions away from the qubit frequency. The separation scheme may improve the effective separation with respect to the achievable Rabi frequency, reduce the influence of shifts or residual scattering on the actual qubit state, may not require full polarization control, and may be accessible with a reasonable amount of optical power. 1 From S0 3 The properties of the resonance to P1 were characterized.

[0173] In Figure 10A, we use a toy model to show the three relevant nuclear spin states, i.e., the m that make up the qubit subspace. F The shifts in the 9 / 2 and 7 / 2 levels and the leakage level 5 / 2 were shown. Here, we simulated the behavior of a single circularly polarized global ac Stark beam addressing an array of atoms in a 700 Gauss magnetic field. Furthermore, we assumed a polarization purity of 100:1 for the intended circular polarization. The ac Stark beam 1 From S0 3 Each detuning to the P1 resonance resulted in a shift in the nuclear spin level. For further clarification, we consider the qubit frequency (m F =9 / 2 and m F = 7 / 2 dressed energy difference) and the leakage transition frequency (m F =7 / 2 and m FBoth the difference between the dressed and undressed states (=5 / 2) were plotted.

[0174] Figure 10B shows that the Stark shift significantly shifted the leakage transition with minimal effect on the qubit frequency, compared to the level splitting at high magnetic fields. 3 This is possible due to the narrow linewidth of the P1 resonance. While we plotted the frequency as a signed quantity, subtleties related to the quantization axis and optical transport make the absolute value of this frequency relevant, and such features appear where the Stark shift brings the leaky state closer to the qubit frequency. For each detuning, taking into account frequency crowding, we can define the maximum achievable usable Rabi frequency. This two-photon Rabi frequency can be used to infer the π pulse duration and to investigate the number of scattering events generated by the nonresonant interaction of the AC Stark beam (Figure 10A).

[0175] Since we did not distinguish between Raman and Rayleigh scattering, we assume a worst-case scenario of AC Stark scattering error per gate. To perform single-qubit gates, we coherently control the light to 3 Two beams detuned from the P1 resonance were used to operate the two-photon transition. 3 Residual scattering from any of the P1 manifold states may be inherently low due to the 7 kHz linewidth of the transition. Including the effects of AC Stark-shifted beams, 3The broadening of the P1 hyperfine magnetic sublevel allows us to separate the energy scale between an AC Stark beam detuned from the F = 1½ manifold and a multiphoton 1Q light detuned from the F = 7½ manifold. A simple toy model including two ground states and several excited states was sufficient to gain insight into the scaling of power, spot size, and achievable Rabi fraction. However, because it includes countless levels (1S0 (F = 9½), 3P1 (F = 7½, 9½, 1½)), including all magnetic sublevels, it may be necessary to perform a full-scale simulation including all relevant levels. To verify the complete operation, we constructed a numerical model utilizing all 40 levels with multiple optical fields to represent both the desired and undesired polarizations. Using a simple square pulse, we can see that transitions to other nuclear spin states can be suppressed with the AC Stark beam (Figures 11A and 11B).

[0176] Example 2: Light trapping array 12A and 12B show arrays of captured light generated by an SLM, such as a square array and an arbitrary array. The hologram is generated by emitting 813 nm light ( 1 S0→ 3The phase shift was generated by reflecting light (the magic wavelength of the P0 transition). The active area of ​​the SLM was an array of 1920 x 1152 square pixels, each approximately 9 microns on a side. Each pixel contained a large volume of liquid crystal that imparted a phase shift to incident light. This phase shift could be controlled by the voltage applied to the pixel, thus creating an arbitrary pixelated phase mask that could be applied to any unstructured light incident on the SLM's surface. The SLM was positioned so that a large collimated beam was incident on it and phase-shifted it, and the light reflected from the SLM was directed toward the microscope objective. This configuration connected the plane of the SLM to the plane below the lens (where the atomic cloud formed) by Fourier conjugation. The complex-valued in-plane electric field at the SLM was the Fourier transform of a similar electric field at the plane below the microscope objective in the volume of the glass cell. The atoms experienced a trapping potential proportional to the strength of the electric field, and thus experienced lateral confinement. Longitudinal confinement arises from the structured light passing through a focal point, the position of which is partly determined (and therefore controllable) by the SLM.

[0177] The light was generated by a titanium-sapphire laser, producing approximately 4 W of optical power at 813 nm. For imaging and other purposes, 2000 traps were generated, each at a depth of 500 microkelvin, far exceeding 1000 times the recoil energy imparted by photon scattering. This means that the device should be in a regime where atoms can be measured hundreds of times without loss due to heating, even without additional cooling. Once cooled to their ground state of motion, the positions of the atoms are known to within 20 nm, allowing a significant separation in scale between the positions of the atoms and the size of the laser beams used to drive the single- and two-qubit gates or the Rydberg interaction length scale. The laser beams driving the gate operations have a spatial extent on the order of 1 micron, and therefore an intensity of 10 -5 Therefore, a fidelity of 0.9999 is expected to be easily achievable. In this way, the gate fidelity is less sensitive to the atomic position.

[0178] Example 3: Ultra-high vacuum A quartz cuvette cell constructed with Spectrosil® 2000 quartz glass was utilized as the vacuum cell. Unlike borosilicate glass, this glass does not fluoresce under UV illumination. The cell featured a glass-to-metal transition from quartz to stainless steel, connecting the cell to the vacuum pump and atom source. The cell dimensions were chosen to avoid clipping of the laser cooling beam and reduce the numerical aperture of the microscope objective. The cell was assembled by Starna Scientific Ltd. using optical contact bonding. The four largest outer surfaces of the cell were coated with a broadband multilayer antireflection coating to minimize reflections from 300 nm to 850 nm for both S- and P-polarized light at normal incidence. A magnesium fluoride coating was applied to the small square window of the cell. The vacuum system was operated for several months at 8 × 10 -12 Torr (1.07 × 10 -9 The pressure was maintained at 100 Pa.

[0179] Example 4: Microscope Objective Lens A microscope objective lens positioned directly above the vacuum cell enables individual trapping, imaging, and addressing of atomic qubits. Due to its high numerical aperture (NA), the objective lens efficiently collects fluorescence from atoms during imaging and transforms a collimated input beam into a tightly focused spot for trapping atoms at the focal plane. The objective lens was manufactured by Special Optics Inc. and has a high NA (0.65), a diffraction-limited field of view (FOV) of 300 μm, and 90% transmission at 461 nm and 813 nm. The end of the objective lens facing the vacuum cell is tapered to prevent clipping of two of the six laser cooling beams. Furthermore, the diameter of the objective barrel was limited to fit between the large magnetic coils used for laser cooling, because the power consumption of these coils is highly proportional to their size and spacing. The mechanical housing of the objective lens was made of Ultem, which is nonmagnetic and nonconductive.

[0180] The objective lens performance was characterized by placing the objective lens and one glass cell window in one arm of a Michelson interferometer. In this arm, the focused beam was retroreflected using a precision ball bearing at the center of the beam focus. The other arm of the Michelson held a reference reflector. Zernike surfaces were reconstructed by fitting the resulting spatial interference pattern. The objective lens was mounted directly on the glass cell to eliminate tilt drift between the cell window and the objective lens. Such tilt, on the order of 1 milliradian (mrad), would otherwise cause fluctuations in wavefront quality. The objective lens was epoxied to a machined McCall mount that contacted the top window of the cell via five brass ball bearings. During this assembly process, the objective lens was interferometrically aligned so that its optical axis remained perpendicular to the cell.

[0181] Three custom dichroic mirrors manufactured by Perkins were used to process four significantly different wavelengths (813 nm, 689 nm, 461 nm, and 319 nm) at the objective. Figure 13 shows the optical system for delivering the four different wavelengths. The three dichroic mirrors are designated DM01, DM02, and DM03. Note that the 319 nm light enters from the bottom of the cell. Custom coatings on the three dichroic mirrors work in concert to maintain the arbitrary polarization states of the 813 nm and 689 nm light, enabling single-qubit or multiqubit gates and magic wavelength and / or magnetic angle trapping.

[0182] Example 5: Trapping and Cooling of Atoms Figure 14 shows the trapping and cooling of strontium-87 and strontium-88 atoms using the red MOT.

[0183] Example 6: Imaging To perform projection measurements, strontium-87 1 S0→ 1Light resonant with the P1 transition is applied to the entire atomic array, and the resulting atomic fluorescence is collected and imaged. 1 For a qubit whose S0 ground state manifold contains two nuclear spin states (both resonant with the imaging light), one of the two states can be considered metastable. 3 The atoms can then be moved to the P0 manifold and measured. This procedure, which is state-selective, is equivalent to the operation of optical lattice clocks and is described in Covey et al., "2000 Times Repeated Imaging of Strontium Atoms in Clock-Magic Tweezer Arrays," Physical Review Letters 122(17):173201 (2019), which is incorporated by reference in its entirety for all purposes. This has the added benefit of reducing readout crosstalk from nearby atoms. Each 1 Fluorescence from the S0 atoms is collected through a microscope objective. This light is then imaged onto a scientific CMOS camera, producing an image of the qubit array that is processed to determine the state of each atom. Such images also help determine whether atoms have been lost from the array. Because the microscope objective is diffraction-limited across the entire atomic array, atoms several microns apart are well resolved.

[0184] Example 7: Single-qubit gate phototransduction The single-qubit method was specifically designed to enable single-site addressability. In particular, the two laser beams used to drive single-qubit operations are transmitted through the same high-numerical-aperture objective lens used to project the optical tweezers trapping potential. As described herein, three dichroic mirrors combine all relevant beams at the back focal plane of the objective. These beams are then generated, manipulated, and modulated to perform site-selective single-qubit operations. The two beams used to drive single-qubit operations have orthogonal linear polarizations (one is pi-polarized because it is aligned with the atomic quantization axis, and the other beam is sigma-polarized). Full control of single-qubit operations requires control of the amplitude, frequency, and phase of each beam at each individual trapping site. This control is obtained through a combination of electro-optic modulators (EOMs), acousto-optic deflectors (AODs), and RF control electronics.

[0185] The light used to drive the single-qubit gates is provided by a common amplified laser source phase-locked to an optical frequency comb. While the overall phase of this light cannot be controlled for each experiment, the laser is a stable local oscillator source that can be modulated with a well-controlled RF source to generate a control field. This global phase sets the global phase of the qubit array, which can only be measured by comparison with an independent qubit array. For maximum flexibility, an electro-optic modulator (EOM) is used to globally phase-modulate the red MOT light, the 689 nm light used for optical pumping, sideband cooling, and single-qubit operations, since these four operations are typically not performed simultaneously. The phase modulation generates symmetric sidebands around a central laser frequency. 3 Detuning the laser from the P1 manifold results in a narrow +1st order sideband. 3 It is chosen to be close enough to the P1 transition to drive the transition. By varying the frequency of this modulation between 5 GHz and 13 GHz, it is possible to use this light to split the excited state manifold, even when large bias fields are used.3 All transitions on the P1 manifold can be treated as resonant.

[0186] The key advantage of this method for generating 689 nm light is that the same beam path is used to generate light for all four of the above beam paths. Furthermore, the overall frequency, amplitude, and phase of these resonant beams are controlled using an advanced microwave RF source. The RF driving the EOM is generated by an arbitrary waveform generator and an IQ mixer, which controls the complex pulse shape of the laser. For qubit manipulation, this global control is used to generate pulses of arbitrary shape with desired spectral characteristics.

[0187] Example 8: Parallel addressing of a single qubit Acousto-optic deflectors (AODs) are used to generate beams that can be directed to different sites within a qubit array by driving the AODs at different frequencies. This introduces a position-dependent frequency and phase matching condition. For single-qubit operations, this complication is overcome by using the same AOD path for both beams, allowing the driven two-photon process to remain resonant while the detuning of the intermediate states varies. In other words, the four AOD frequencies are fully constrained by selecting the specific site to address. The two frequencies select the position of the first beam, and the frequency matching condition ensures that the two frequencies of the second beam are identical, up to the qubit frequency offset (the splitting between the two nuclear spin states, approximately 150 kHz). Using an AOD to generate beams for single-qubit operations allows arbitrary addressing of atoms within a single row (or column) at any given time. This is necessary to maintain full control of their respective amplitudes and phases. This partially serializes the operation. However, the speed at which patterns can be changed with an AOD is significantly improved compared to an SLM and is far more efficient than a DMD. The AOD also allows for full phase control of each beam. This not only allows us to track the phase of each qubit (all rotations can be applied within the local qubit frame), but it can also be used to perform more complex pulse sequences on each qubit. By controlling the amplitude of the RF on each qubit, we can locally scale the pulse area of ​​each qubit operation. Combining both the RF phase and amplitude allows for full control of the operation performed on each qubit during a single pulse from the EOM.

[0188] Example 9: Parallel addressing of multiqubit units Direct excitation of strontium-87 from the ground state to the Rydberg level requires a laser with a wavelength of approximately 218 nm. Alternatively, Rydberg excitation can be performed by using intermediate 3 This can be done using two-photon excitation combining 689 nm and 319 nm light, which is detuned from the P1 state. 3The approximately 7 kHz width of the P1 state is the two-photon effective Rabi rate 3 This provides an effective balance between scattering due to spontaneous decay from P1 and scattering due to spontaneous decay from P1. Figure 15A shows the energy level structure for single and multi-qubit operations in strontium-87.

[0189] The optical system for single-qubit operations is designed to also work well with multiqubit gates. One of the single-qubit beams is used as one leg of a two-photon excitation scheme to drive the transition to the Rydberg electron manifold. To satisfy the spatially dependent frequency and phase matching condition, the AOD is also used with UV light. Importantly, the optical system is tuned so that the frequency shift of the UV light from one site to the other is the same as the frequency shift of the 689 nm light. As a result of this constraint, the performance of a state-of-the-art UV AOD determines the accessible field of view (FOV) for multiqubit operations. Furthermore, because one single-qubit beam is used for multiqubit operations (and the two single-qubit beams are coincident), the FOV for single-qubit operations is the same. The figure of merit for a UV AOD is the product of the active numerical aperture and the device's RF bandwidth. For a fixed beam size in the back focal plane of the objective lens, increasing either of these quantities increases the beam's scanning angle and therefore the FOV in the plane of the qubit array. A FOV of approximately 100 μm × 100 μm was achieved, sufficient to accommodate an array of approximately 1,000 atoms with a 3 μm capture site spacing.

[0190] Figure 15B shows an optical system for transmitting light to perform single-qubit and multi-qubit operations on multiple trapped atoms in parallel. First light for performing a single-qubit operation on a first qubit (qubit 1) is directed to a first two-dimensional AOD (2D AOD) to enable parallel addressing of a first subset of trapped atoms. Second light for performing a single-qubit operation on a second qubit (qubit 2) is directed to a second 2D AOD to enable parallel addressing of a second subset of trapped atoms. Third light for inducing Rydberg interactions in either the first or second subset is delivered through a third 2D AOD to generate multiple entanglements between atoms in the first subset and neighboring atoms in the second subset.

[0191] The third light is generated by an ultraviolet (UV) laser emitting 319 nm light. The UV laser is phase-locked to a frequency comb, providing a narrow-linewidth UV laser beam. Amplitude control is provided by an acousto-optic modulator (AOM). Global phase control is achieved using optical phase stabilization techniques. The stabilized global phase of the 319 nm light is combined with active phase modulation of the 689 nm light to provide phase control. A free-space beam is sent to a third 2D AOD, but from the opposite direction from the first and second 2D AODs. The light is directed toward the trapped atoms through a customized microscope objective. Counter-propagating beam paths are used to monitor the spot position and the light's effect on the atoms (e.g., via pump loss spectroscopy) and optimize alignment. These quantitative effects can also be used to implement an automated alignment scheme, enabling improved autonomous operation of the system.

[0192] Figure 15C shows an optical system configured to dynamically generate and control beams using a single electro-optic modulator (EOM) and two acousto-optic deflectors (AODs) per beam, each driven by an RF signal from an arbitrary waveform generator. The AODs are oriented such that the frequency difference between the beams remains constant whenever they overlap at the qubit array. The frequency difference prevents unwanted operations from being driven, but is easily overcome by the RF drive of the two EOMs. Combining the AODs with an agile RF synthesizer allows for full site-by-site control of operations, a key advantage for performing sequences of quantum operations on an array of atomic qubits, which can be performed in parallel (one row at a time).

[0193] Example 10: Inverse adiabatic drive Without the pulse sequences described herein, multi-qubit operations can be performed by adiabatically varying the Hamiltonian such that adiabatic transitions to Rydberg states are minimized, transferring atoms from the ground state to the dressed state and back to the ground state. The adiabatic condition imposes limitations that make multi-qubit operations relatively slow. However, faster gates are necessary for overall speed and minimization of decoherence effects. The pulse sequences described herein can achieve faster gates while effectively maintaining adiabatic dynamics.

[0194] For example, inverse adiabatic driving can shorten gate times while minimizing errors resulting from transitions to Rydberg states. Counteradiabatic driving involves adding one or more driving fields to cancel Hamiltonian terms that cause undesired adiabatic transitions. Inverse adiabatic driving achieves effective adiabatic dynamics on time scales shorter than those allowed by adiabatic conditions. One example is the "transitionless quantum driving" (TQD) described herein. TQD is achieved by transforming the entire Hamiltonian of the system into a reference frame defined by the instantaneous eigenstates of the Hamiltonian. The Hamiltonian is split into a diagonal part (which does not cause adiabatic transitions between instantaneous eigenstates) and an off-diagonal part (which causes adiabatic transitions). TQD is realized by adding additional control fields that cancel the off-diagonal adiabatic Hamiltonian. Using this technique, effective adiabatic dynamics can be achieved without satisfying the usual slow adiabatic condition. Below is the derivation of the TQD condition for a general two-level system with a single shaft drive that uses a TQD to counteract the adiabatic transition of the Rydberg dressing gate.

[0195] The general problem is to transform a two-level system in the ground state |1〉 into a dressed state that is a mixture of |1〉 and excited states |R〉, and then return to the ground state as quickly as possible without leaving behind a population of excited states. In a rotating frame, the total Hamiltonian (in units of frequency) of the driven two-level system is: (1) H0=Ω(t)δ x +Δ(t)δ z

[0196] where Ω is the Rabi modulus, Δ is the detuning from resonance, and σ x and σ z is the Pauli operator for the two-level system. It is convenient to write the Hamiltonian in a tilted coordinate system. (2) H'0=Ω eff (t)δ z’

[0197]

number

[0198] In the original basis, the instantaneous eigenstates of H0 are (6)|Φ1>=cos(θ)|1>+sin(θ)|R> (7)|Φ2>=-sin(θ)|1>+cos(θ)|R>

[0199] Next, we transform into an "adiabatic frame" described by these instantaneous eigenstates. The unitary operator corresponding to this transformation is

[0200]

number

[0201] where |Φ ad,k > is the instantaneous eigenstate of the adiabatic frame. The transformed Hamiltonian is

[0202]

number

[0203] The second term (W(t)) contains an off-diagonal element that triggers a transition when the adiabatic condition is not satisfied. The adiabatic condition is satisfied if U(t) changes slowly enough to make W(t) small enough. To achieve effective adiabatic dynamics when this term is not small, an additional control field H c (t) to the original Hamiltonian to cancel the effect of W(t), which is

[0204]

number

[0205] Solving for U(t), we get

[0206]

number

[0207] Using the definition of U(t) given above, it can be written in matrix form:

[0208]

number

[0209] Again, simplifying the expression.

[0210]

number

[0211] This result shows that an inverse adiabatic Hamillcian can be achieved by driving with a field that is 90 degrees out of phase with the original driving field. The form of H(t) can generally be found for the desired H0(t).

[0212] To demonstrate the effectiveness of the Rydberg-Dressing gate's transitionless quantum drive, we simulated a two-atom system. Each atom consisted of two ground (qubit) states and a Rydberg state. The two-atom state could be a qubit state |00>, |01>, |10>, or |11>; a singly excited Rydberg state |0R>, |R0>, |R1>, or |1R>; or a doubly excited Rydberg state |RR>. In the presence of Rydberg blockade, the transition to |RR> can be suppressed. The Rydberg-Dressing gate can be configured to apply one or more drive fields to convert an atom pair (e.g., an atom pair starting in a superposition of qubit states) into a mixture of the qubit and Rydberg states. The atom pair can acquire a conditional phase where the two-atom state depends on the initial state. At the gate's edge, the atoms can return to the qubit state space, while any atoms remaining in the Rydberg level can be a source of error. Figure 16A shows a simulation of two atoms initially in the two-atom state |00>. By driving the transition from |0> to |r> in each atom and sweeping the detuning toward and away from resonance, the instantaneous eigenstates of the Hamiltonian transform from bare to dressed and back to bare. As shown in Figure 16A, if the ramp is performed too quickly and the adiabatic condition is violated, a significant population will remain in the Rydberg state |r0>.

[0213] Figure 16B shows a simulation of two atoms in the initial two-atom state |00> with the addition of an inverse adiabatic driving field applied to implement a non-transition quantum driving gate. The population remaining in the Rydberg state is significantly reduced. Figure 16C shows a numerical simulation of a two-atom system with Rydberg blockade passing through a Rydberg dressing gate with (with) and without (without) the addition of an inverse adiabatic driving field configured to achieve a TQD. In this example, for the initial state |01>, the remaining population in the Rydberg state can be reduced by the application of the TQD. In this example, the gate parameters can be selected to emphasize violations of the adiabatic condition to demonstrate the effect of the TQD. In some cases with more than about two levels (e.g., a nine-level system as described elsewhere herein), it may be possible to achieve substantially transition-free quantum driving across multiple transitions. Such driving can be observed even in the presence of different Rabi fractions.

[0214] Inverse adiabatic driving can also be used to suppress unwanted transitions at frequencies other than the driving frequency. This is useful for driving transitions on-resonance while avoiding driving nearby unwanted transitions. Alternatively, off-resonant driving can be used to create dressed states while avoiding excitation to excited states (i.e., adiabatic transitions). An example of inverse adiabatic driving that suppresses unwanted transitions is the "differential removal by adiabatic gating" (DRAG) described herein. Figure 16D shows an example of a DRAG pulse in the time domain (a) and frequency domain (b). Off-resonant DRAG pulses can be used to generate controlled phase gates on Rydberg atoms. Pulses with smooth amplitude profiles can be applied at off-resonant frequencies from qubit transitions to Rydberg transitions. Application at off-resonant frequencies can drive qubit atoms into or out of the Rydberg dressing state. DRAG pulses can be designed to minimize transitions driven at specific frequency offsets from the pulse carrier frequency. By adjusting the location of the spectral hole in the frequency spectrum of the pulse shape, the gating error due to the population remaining in the Rydberg state can be minimized. Figure 16E shows a simulated example of optimizing the DRAG pulse parameters to minimize the error. The spectral hole location with the smallest error can be used as the spectral hole location, thus improving the overall fidelity of the system.

[0215] Example 11: Atomic rearrangement Simulations were performed to determine the time required to perform atomic rearrangements in a 7 × 7 array of optical trapping sites. The simulations assumed an imaging system with a Hamamatsu Orca-Fusion CMOS digital camera using an external trigger in normal mode. The camera has a region of interest of 2304 (fixed, horizontal) × 256 (vertical) pixels. A 20 ms exposure, 4.6 ms readout (256 vertical lines at 18.65 μs per line), and a data transfer latency of 1.75 ms to 5 ms were assumed.

[0216] The data transferred from the camera can be sliced ​​into a 256 x 256 array of 16-bit integers. To determine the capture sites, a calibration image of a fully captured grid must first be used (by averaging many capture realizations). Figure 17A shows a calibration image of a fully filled 7 x 7 array of optical capture sites. Optical capture sites are indexed by coordinates (i, j). This data was used to map capture sites to pixel locations, as shown in Table 1.

[0217] [Table 1]

[0218] Figure 17B shows the labeling of filled and unfilled optical capture sites in a 7 x 7 array. Binning of pixels around each capture site was performed. Figure 17C shows 25 x 25 pixel binning around each optical capture site in the 7 x 7 array. The pixels in each bin were averaged. The average value was compared to a threshold extracted from the calibration procedure to determine whether each optical capture site was filled or unfilled. Filled sites were identified with a "1" and unfilled sites with a "0." Figure 17D shows the identification of each capture site in the 7 x 7 array as filled or unfilled. Thus, the procedure generated a 7 x 7 array of binary values ​​indicating whether each site was filled or unfilled. The total processing time to assign the array of binary values ​​was less than 0.5 ms.

[0219] Once the filled and unfilled sites were identified, the next step was to determine the moves to fill the uncaptured sites. This is a combinatorial optimization problem classified as bipartite matching. This can be solved by setting up an adjacency matrix that can efficiently find the optimal match using algorithms such as the Hungarian matching algorithm described herein. The adjacency matrix d i,j is constructed, where row i is indexed by the target site within the NxN active area, and columns are indexed by the available sites within the full MxM lattice. For example, for a 7x7 array (M=7), an atom can be moved to a 5x5 computationally active area (N=5). Table 2 shows the entries of the adjacency matrix.

[0220] [Table 2]

[0221] The distance metric is the target (i target ,j target ) and the filled area (i filled ,j filled ), the resulting matching produces a collision-free transfer of atoms from a filled to an unfilled light-trapping site. Figure 17E shows a transfer from a filled to an unfilled light-trapping site that avoids collisions between atoms.

[0222] The movements were split into independent subsets and ordered in time for easy parallelization, as shown in Table 3 below. The process of determining the movements took approximately 8ms.

[0223] [Table 3]

[0224] The time required to transfer data to the AWG is less than 1 ms. A single maximum latency is introduced when mapping a series of movements to a series of waveforms within the AWG. A single movement may require a 0.3 ms ramp-up time, a 0.1 ms / µm movement, and a 0.3 ms ramp-down time. Assuming a 3 µm spacing between optical capture sites and that only movements to adjacent sites are allowed, each movement requires approximately 1 ms. In numerous simulations of a 7 × 7 array, up to 34 movements were made, requiring 34 ms to program the AWG.

[0225] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will occur to those skilled in the art without departing from the invention. It is to be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is intended that the following claims define the scope of the invention, and that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

1. 1. A method for performing non-classical computation, the method comprising: (a) (i) one or more capture units that capture a plurality of atoms, which are a plurality of qubits; (ii) one or more first electromagnetic delivery units; (iii) one or more Rydberg units; and (iv) one or more second electromagnetic delivery units; and (v) activating the non-classical computation unit, including: (b) applying first electromagnetic energy to one or more atoms of the plurality of atoms using the one or more first electromagnetic delivery units, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state; (c) using the one or more Rydberg units to quantum mechanically entangle at least a subset of the one or more atoms in the one or more superposition states with at least one other atom of the plurality of atoms, thereby forming one or more multi-qubit units; (d) applying second electromagnetic energy comprising one or more pulse sequences to the one or more multiqubit units using the one or more second electromagnetic delivery units, thereby implementing one or more multiqubit gate operations on the one or more multiqubit units, wherein the one or more pulse sequences comprise one or more non-adiabatic pulses that maintain adiabatic dynamics; and (e) performing one or more measurements of the one or more superposition states using the one or more readout units, thereby obtaining the non-classical calculation.

2. 10. The method of claim 1, wherein the one or more pulse sequences comprise one or more members selected from the group consisting of a shortcut to adiabaticity (STA) pulse sequence, a non-transition quantum drive (TQD) pulse sequence, a superadiabatic pulse sequence, an inverse adiabatic drive pulse sequence, a derivative removal by adiabatic gate (DRAG) pulse sequence, and a weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequence.

3. The method of claim 1 , wherein the first electromagnetic delivery unit and the second electromagnetic delivery unit are the same.

4. The method of claim 1 , wherein the first electromagnetic delivery unit and the second electromagnetic delivery unit are different.

5. The method of claim 1 , wherein the one or more pulse sequences further comprise one or more optimal control pulse sequences.

6. 6. The method of claim 5, wherein the one or more optimal control pulse sequences are obtained from one or more procedures selected from the group consisting of gradient ascent pulse engineering (GRAPE), Krotov, chopped basis, chopped random basis (CRAB), Nelder-Mead, gradient optimization using parameterization (GROUP), genetic algorithm, and gradient optimization of analytical controls (GOAT).

7. The method of any one of claims 1 to 6, wherein the one or more pulse sequences comprise a duration of at least 10 nanoseconds (ns).

8. The method of any one of claims 1 to 7, wherein the one or more pulse sequences comprise a duration of up to 100 microseconds (μs).

9. The method of any one of claims 1 to 8, wherein the one or more multi-qubit gate operations comprise a fidelity of at least 0.

9.

10. 10. The method of claim 1, wherein the one or more multi-qubit gating operations comprise a fidelity of at most 0.999999.

11. 2. The method of claim 1, further comprising: using the one or more Rydberg units to electronically excite at least one atom of the one or more atoms in the one or more superposition states to a Rydberg state, thereby forming one or more Rydberg atoms; or using the one or more Rydberg units to sequentially drive the one or more atoms in the one or more superposition states through transitions to a Rydberg state, thereby forming one or more dressed Rydberg atoms.

12. 12. The method of claim 11, further comprising using the one or more Rydberg units to induce one or more quantum mechanical entanglements between the one or more Rydberg atoms or dressed Rydberg atoms and the at least one other atom, wherein the at least one other atom is located at a distance of 10 micrometers (μm) or less from the one or more Rydberg atoms or dressed Rydberg atoms.

13. 13. The method of claim 12, further comprising using the one or more Rydberg units to drive the one or more Rydberg atoms or dressed Rydberg atoms into a lower energy atomic state, thereby forming the one or more multi-qubit units.

14. The method of any one of claims 1 to 13, wherein the one or more capture units comprise one or more light capture units.

15. The method of claim 14 , further comprising using the one or more light trapping units to generate a plurality of spatially distinct light trapping sites.

16. 16. The method of claim 15, further comprising using each optical trapping site of the plurality of spatially distinct optical trapping sites to trap at most one atom of the plurality of atoms.

17. A method according to any preceding claim, wherein one or more of the multi-qubit units comprises a two-qubit unit.

18. 20. The method of claim 17, wherein one or more of the multi-qubit gate operations comprises a two-qubit gate operation.

19. 10. The method of claim 1, wherein (c) further comprises implementing the one or more multi-qubit gate operations in the one or more multi-qubit units.

20. 20. The method of claim 19, wherein in (c), quantum mechanically entangles at least a subset of the one or more atoms using the one or more Rydberg units comprises using one or more pulse sequences.

21. 21. The method of claim 20, wherein the one or more pulse sequences include one or more non-adiabatic pulses that maintain adiabatic dynamics.

22. 1. A system for performing non-classical computation, the system comprising: one or more capture units; one or more first electromagnetic delivery units; one or more Rydberg units; one or more second electromagnetic delivery units; one or more readout units; and one or more controllers coupled to the one or more capture units, the one or more first electromagnetic delivery units, the one or more Rydberg units, the one or more second electromagnetic delivery units, and the one or more readout units, wherein the one or more controllers: (i) directing the one or more units to capture a plurality of atoms representing a plurality of qubits; (ii) directing the one or more first electromagnetic delivery units to apply first electromagnetic energy to one or more atoms of the plurality of atoms, whereby the one or more atoms adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state; (iii) inducing the one or more Rydberg units to direct at least a subset of the one or more atoms in the one or more superposition states to quantum mechanically entangle with at least another atom of the plurality of atoms to form one or more multiqubit units; (iv) directing the one or more second electromagnetic delivery units to apply second electromagnetic energy comprising one or more pulse sequences to the one or more multiqubit units, thereby implementing one or more multiqubit gate operations on the one or more multiqubit units, the one or more pulse sequences comprising one or more non-adiabatic pulses that maintain adiabatic dynamics; and (v) instructing the one or more readout units to perform one or more measurements of the one or more superposition states, thereby obtaining the non-classical calculation.

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