Scalable Neutral-Atom-Based Quantum Computing

Abstract

The present disclosure provides methods and systems for performing non-classical computations that collectively use a plurality of spatially distinct optical trapping sites for trapping a plurality of atoms, one or more electromagnetic delivery units for applying electromagnetic energy to one or more atoms of the plurality to induce the atoms to adopt one or more superpositions of a first atomic state and a second atomic state, one or more entanglement units for quantum mechanically entangle at least a subset of the one or more atoms in the one or more superpositions with at least another atom of the plurality, and one or more readout optical units for performing measurements of the superpositions to obtain non-classical computations.

Description

Technical Field

[0001] Cross - Reference to Related Applications This application claims the benefit of U.S. Provisional Application No. 62 / 984,174, filed Mar. 2, 2020, which is hereby incorporated by reference in its entirety. Description of Research Funded by the Federal Government

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

Background Art

[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 based on transistors. For example, in a digital computer, data needs to be encoded into binary numbers (bits) where each bit is always in one of two distinct states (0 or 1), while in quantum computing, qubits (quantum bits) that can be in a superposition of states are used.

Summary of the Invention

Problems to be Solved by the Invention

[0004] The need for methods and systems for performing non - classical computing is recognized herein.

[0005] The present disclosure provides systems and methods for performing non - classical or non - charged atoms using atoms (such as neutral atoms or uncharged atoms). The atoms may be optically trapped in a large array. The quantum mechanical state of the atoms (such as the hyperfine state or nuclear spin state of the atoms) may be configured to function as qubit (quantum bit) basis states. The qubit state may be manipulated by interaction with light, radio frequency, or other electromagnetic radiation, thereby performing non-classical or quantum computing.

Means for Solving the Problems

[0006] In one aspect, the present disclosure is a method for performing non-classical computing, comprising: (a) activating a non-classical computing unit including (i) one or more trapping units for trapping a plurality of atoms that are a plurality of 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 wave energy to one or more of the plurality of atoms by using one or more first electromagnetic delivery units, thereby inducing one or more of the 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 entangling at least a subset of one or more atoms in one or more superposition states with at least one other atom of the plurality of atoms by using one or more Rydberg units, thereby forming one or more multi-qubit units; (d) applying second electromagnetic energy including one or more pulse sequences to one or more multi-qubit units by using one or more second electromagnetic delivery units, thereby implementing one or more multi-qubit gate operations on one or more multi-qubit units, wherein one or more of the pulse sequences include one or more non-adiabatic pulses that maintain adiabatic dynamics; and (e) performing one or more measurements of one or more superposition states by using one or more readout units, thereby obtaining non-classical computing.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 non-transition quantum drive (TQD) pulse sequence, a super-adiabatic pulse sequence, a reverse 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 transmission unit and the second electromagnetic transmission unit may be the same. The first electromagnetic transmission unit and the second electromagnetic transmission unit may be different. One or more pulse sequences may further include one or more optimal control pulse sequences. One or more optimal control pulse sequences may be derived from one or more procedures selected from the group consisting of the gradient ascent pulse engineering (GRAPE) method, the Krotov method, the chopped-based method, the chopped random basis (CRAB) method, the Nelder-Mead method, the gradient optimization using parametrization (GROUP) method, the genetic algorithm method, and the gradient optimization of analytic controls (GOAT) method. One or more pulse sequences may include a duration of at least 10 nanoseconds (ns). One or more pulse sequences may include a duration of up to 100 microseconds (μs). One or more multiqubit gate operations may include a fidelity of at least 0.9. One or more multiqubit gate operations may include a fidelity of up to 0.999999.The above method may further include a step of electronically exciting at least one atom out of one or more atoms in one or more superposed states to a Rydberg state using one or more Rydberg units, thereby forming one or more Rydberg atoms, or a step of continuously driving one or more atoms in one or more superposed states to a Rydberg state through transitions using one or more Rydberg units, thereby forming one or more dressed Rydberg atoms. The above method may further include a step of inducing one or more quantum mechanical entanglements between one or more Rydberg atoms or dressed Rydberg atoms and at least another one atom by using one or more Rydberg units, wherein at least another one atom is located at a distance of 10 micrometers (μm) or less from one or more Rydberg atoms or dressed Rydberg atoms. The above method may further include a step of driving one or more Rydberg atoms or dressed Rydberg atoms to a low-energy atomic state by using one or more Rydberg units, thereby forming one or more multiqubit units. One or more trapping units may include one or more optical trapping units. The above method may further include a step of generating a plurality of spatially different optical trapping sites by using one or more optical trapping units. The above method may further include a step of trapping at most one atom out of a plurality of atoms by using each of the plurality of spatially different optical trapping sites. One or more of the multiqubit units may include two-qubit units. One or more of the multiqubit gate operations may include two-qubit gate operations. In some embodiments, (c) may further include a step of implementing one or more multiqubit gate operations on one or more multiqubit units.In some embodiments, in (c), the step of quantum mechanically entangling at least one subset of one or more atoms by using one or more Rydberg units may include using one or more pulse sequences. In some embodiments, the one or more pulse sequences may include one or more non-adiabatic pulses that maintain adiabatic dynamics. In some embodiments, the generation of entanglement by the Rydberg unit may be a gate operation. In some embodiments, the pulse shape or pulse sequence may be applied by the Rydberg unit.

[0007] In another aspect, the present disclosure provides a system for performing non-classical computing, 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, the one or more controllers configured to: (i) instruct the one or more units to capture a plurality of atoms that are a plurality of qubits; (ii) instruct the one or more first electromagnetic delivery units to apply a first electromagnetic energy to one or more of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states between a first atomic state and at least a second atomic state different from the first atomic state; (iii) instruct 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 of the plurality of atoms to form one or more multiqubit units; (iv) instruct the one or more second electromagnetic delivery units to apply a second electromagnetic energy including one or more pulse sequences to the one or more multiqubit units to implement one or more multiqubit gate operations on the one or more multiqubit units, the one or more pulse sequences including one or more non-adiabatic pulses that maintain adiabatic dynamics; and (v) instruct the one or more readout units to perform one or more measurements of the one or more superposition states to obtain non-classical computing.

[0008] Although described herein with respect to a multi - qubit Rydberg gate system, the methods and systems of the present disclosure can be used in a plurality of different gate systems. For example, the methods and systems described herein can be applied to single - qubit gates. In another example, 87 In a system encoded in two nuclear spin states of Sr (e.g., m F = - 9 / 2, and m F = - 7 / 2) of the 1 S0 state, e.g., as shown in FIG. 15A), the remaining nuclear spin states can exist outside the qubit subspace, and the population of these states can constitute leakage errors. Two - photon transitions driving a single - qubit gate can further 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 transitions. As described elsewhere herein, the use of DRAG pulses, executed on two - photon transitions, suppresses leakage transitions, which enables fast and 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, which shows and describes only exemplary embodiments of the present disclosure. As will be understood, the present disclosure is capable of other and different embodiments, and some of the details thereof are capable of various obvious modifications without departing from the present disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature and not restrictive.

[0010] Incorporation by reference All publications, patents, and patent applications mentioned in this specification are hereby incorporated by reference into this specification 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 incorporated publications and patents or patent applications conflict with the disclosure contained in this specification, this specification is intended to supersede and / or prevail over 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 invention will be obtained from the following detailed description which illustrates exemplary embodiments in which the principles of the invention are utilized, and from the appended drawings (also referred to herein as "FIGures" and "FIGs").

Brief Description of the Drawings

[0012]

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DETAILED DESCRIPTION OF THE INVENTION

[0013] Although various embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. Those skilled in the art can conceive of numerous variations, modifications, and substitutions without departing from the present invention. It should be understood that various alternatives to the embodiments of the present invention described herein may be employed.

[0014] Unless otherwise specified, all technical terms used in this specification shall have the same meaning as commonly understood by those skilled in the art to which the present invention pertains. As used in this specification and the appended claims, the singular forms "a", "an", and "the" shall include plural references unless the context clearly dictates otherwise. References to "or" in this specification are intended to include "and / or" unless specifically stated otherwise.

[0015] When the terms "at least", "greater than", or "above" are before or after the first numerical value of a series of two or more numerical values, the terms "at least", "greater than", or "above" shall always apply to each numerical value of that series of numerical values. For example, 1, 2, or 3 or more is equivalent to 1 or more, 2 or more, or 3 or more.

[0016] When the terms "no more than", "less than", "less than or equal to", or "at most" are before or after the first numerical value of a series of two or more numerical values, the terms "no more than", "less than", "less than or equal to", or "at most" shall always apply to each numerical value of that series of numerical values. For example, 3, 2, or 1 or less is equivalent to 3 or less, 2 or less, or 1 or less.

[0017] When a value is described as a range, such disclosure shall be understood to include all possible sub-ranges within such range, as well as the disclosure of specific numerical values falling within such range, regardless of whether such specific numerical values or specific sub-ranges are explicitly stated.

[0018] As used in this specification, like characters 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 increase 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 the computer performance of a task. Machine learning may include machine learning algorithms. 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., where various parameters are determined as weights or scaling factors). 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 may include, but is not limited to, k-means, k-means clustering, k-nearest neighbor, learning vector quantization, linear regression, non-linear 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 operator, 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, t-distributed stochastic neighbor embedding, AdaBoosting, boosting, gradient boosting, bagging, ensemble averaging, decision trees, conditional decision trees, boosted decision trees, gradient boosted decision trees, random forests, stacked generalization, Bayesian networks, Bayesian belief networks, naive Bayes, Gaussian naive Bayes, polynomial naive Bayes, hidden Markov models, hierarchical hidden Markov models, support vector machines, encoders, decoders, autoencoders, stacked autoencoders, perceptrons, multilayer perceptrons, artificial neural networks, feedforward neural networks, convolutional neural networks, recurrent neural networks, long short-term memory, deep belief networks, deep Boltzmann machines, deep convolutional neural networks, deep recurrent neural networks, or adversarial generative networks.

[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 executes one or more actions to reinforce or maximize a notion of cumulative reward for interaction with an environment. An agent that executes a reinforcement learning (RL) procedure may execute one or more actions within an environment, thereby putting itself and the environment into various new states, and may receive positive or negative reinforcement called "instantaneous reward".

[0022] The goal of the agent may be to reinforce or maximize the concept of cumulative reward. For example, the goal of the agent may be to reinforce or maximize a "discounted reward function" or an "average reward function". The "Q-function" may represent the maximum cumulative reward that can be obtained from a state and the action executed in that state. The "value function" and the "generalized advantage estimator" may represent the maximum cumulative reward that can be obtained from a state given the selection of an optimal or best action. RL may utilize any one or more of such concepts of cumulative reward. As used herein, such a function may sometimes be referred to as a "cumulative reward function". Therefore, calculating the best or optimal cumulative reward function may be equivalent to finding the best or optimal policy of the agent.

[0023] The interaction between the agent and its environment can be formulated as one or more Markov Decision Processes (MDPs). The RL procedure may not require knowledge of the exact mathematical model of the MDP. The MDP may be completely unknown, partially known, or completely known to the agent. The RL procedure may lie within a spectrum between two ranges, "model-based" or "model-free", with respect to the prior knowledge of the MDP. Therefore, the RL procedure may be applicable to large-scale MDPs where an exact method may be infeasible or unavailable due to the unknown or probabilistic nature of the MDP.

[0024] The RL procedure can be implemented using one or more computer processors described herein. The digital processing unit may utilize an agent that trains, saves, and later deploys a "policy" to reinforce or maximize the cumulative reward. The policy can be explored (such as searched) as much as possible or for a desired period. Such optimization problems can be solved by storing an approximation of the optimal policy, storing an approximation of the cumulative reward function, or both. In some cases, the RL procedure may store one or more tables of approximations of such functions. In other cases, the RL procedure may utilize one or more "function approximations".

[0025] Examples of function approximations can include neural networks (such as deep neural networks) and probabilistic graphical models (such as Boltzmann machines, Helmholtz machines, and Hopfield networks). The function approximator can create a parameterization of the approximation of the cumulative reward function. The optimization of the function approximation with respect to the parameterization can be composed of perturbing the parameters in a direction that reinforces or maximizes the cumulative reward and thus reinforces or optimizes the policy (such as policy gradient methods), or by perturbing the function approximator to approach satisfying the Bellman optimality criterion (such as by temporal difference methods).

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

[0027] As used herein, the terms "non-classical computing", "non-classical procedure", "non-classical operation", and any "non-classical computer" generally refer to any method or system for performing computational procedures outside the paradigm of classical computing. Non-classical computing, non-classical procedures, non-classical operations, or non-classical computers can include quantum computing, quantum procedures, quantum operations, or quantum computers.

[0028] As used herein, the terms "quantum computing," "quantum procedure," "quantum operation," and "quantum computer" generally refer to any method or system for performing computations using quantum mechanical operations (such as unitary transformations in a quantum channel or completely positive trace-preserving (CPTP) maps) in a Hilbert space represented by a quantum device. Thus, quantum computing and classical (or digital) computing may be similar in the following aspect: both computations can involve a series of instructions that are executed on input information and provide an output. Various paradigms of quantum computing can decompose quantum operations into a sequence of basic quantum operations that simultaneously affect a subset of the qubits of a quantum device. The quantum operations can be selected, for example, based on their locality or ease of physical implementation. A quantum procedure or computation can be composed of a series of such instructions that can represent different quantum evolutions on a quantum device for various applications. For example, a procedure for computing or simulating quantum chemistry can represent quantum states and the annihilation and creation operators of electron spin orbits by using qubits (such as two-level quantum systems) and a universal set of quantum gates (such as Hadamard, controlled-not (CNOT), and π / 8 rotations) through so-called Jordan-Wigner transformation or Bravyi-Kitaev transformation.

[0029] Additional examples of quantum procedures or computations can include optimization procedures such as the quantum approximate optimization algorithm (QAOA) and finding the quantum minimum. QAOA can involve the execution of single-qubit rotations and multi-qubit entanglement gates. In quantum adiabatic computing, the instructions can carry a probabilistic or non-probabilistic evolution path from an initial quantum system to a final quantum system.

[0030] Procedures inspired by quantum can include simulated annealing, parallel tempering, master equation solvers, Monte Carlo procedures, etc. Quantum classical or hybrid algorithms or procedures can include procedures such as variational quantum eigensolver (VQE) and variational and adiabatically navigated quantum eigensolver (VanQver).

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

[0032] As used herein, the term "adiabatic" refers to any process executed in a quantum mechanical system where the parameters of the Hamiltonian change slowly compared to the natural time scale of the evolution of the system.

[0033] As used herein, the term "non-adiabatic" refers to any process-executing quantum mechanical system where the parameters of the Hamiltonian change rapidly compared to the natural time scale of the evolution of the system or on a time scale similar to the natural time scale of the evolution of the system.

[0034] Systems for performing non-classical computations In one aspect, the present disclosure provides a system for performing non-classical computing. The system includes one or more optical trapping units configured to generate a plurality of spatially distinct optical trapping sites, where the plurality of optical trapping sites are 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 of the plurality of atoms, thereby inducing one or more of the plurality of 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 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 non-classical computing.

[0035] FIG. 2 is a diagram showing an example of a system 200 for performing non-classical computing. Non-classical computing may include quantum computing. Quantum computing may include gate-model quantum computing.

[0036] System 200 may include one or more capture units 210. The capture unit may include one or more light capture units. The light capture unit may include any light capture unit described herein, such as the light capture unit described herein with respect to FIG. 3A. The light capture unit may be configured to generate a plurality of light capture sites. The light capture unit may be configured to generate a plurality of spatially distinct light capture sites. For example, the light capture unit may be configured to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more light capture sites. The light capture unit may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer light capture sites. The light capture unit may be configured to capture some light capture sites 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 be configured to trap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. The light trapping unit can be configured to trap at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The light trapping unit can be configured to trap some atoms within a range defined by any two of the foregoing values.

[0038] Each light trapping site of the light trapping unit can be configured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. Each light trapping site can be configured to trap at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each light trapping site can be configured to trap some atoms within a range defined by any two of the foregoing values. Each light trapping site can be configured to trap a single atom.

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

[0040] One or more atoms may include alkali atoms. 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. 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. One or more atoms may include alkaline earth atoms. 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. One or more atoms may include a beryllium-9 atom, a magnesium-24 atom, a magnesium-25 atom, a magnesium-26 atom, a calcium-40 atom, a calcium-42 atom, a calcium-43 atom, a calcium-44 atom, a calcium-46 atom, a calcium-48 atom, a strontium-84 atom, a strontium-86 atom, a strontium-87 atom, a strontium-88 atom, a barium-130 atom, a barium-132 atom, a barium-134 atom, a barium-135 atom, a barium-136 atom, a barium-137 atom, or a barium-138 atom. One or more atoms may include rare earth atoms. 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, a ytterbium (Yb) atom, or a lutetium (Lu) atom.One or more atoms may include scandium 45 atoms, yttrium 89 atoms, lanthanum 139 atoms, cerium 136 atoms, cerium 138 atoms, cerium 140 atoms, cerium 142 atoms, praseodymium 141 atoms, neodymium-142 atoms, neodymium 143 atoms, neodymium 145 atoms, neodymium 146 atoms, neodymium 148 atoms, samarium 144 atoms, samarium 149 atoms, samarium 150 atoms, samarium 152 atoms, samarium 154 atoms, europium 151 atoms, europium 153 atoms, gadolinium 154 atoms, gadolinium 155 atoms, gadolinium 156 atoms, gadolinium 157 atoms, gadolinium 158 atoms, gadolinium 160 atoms, terbium 159 atoms, dysprosium 156 atoms, dysprosium 158 atoms, dysprosium 160 atoms, dysprosium 161 atoms, dysprosium 162 atoms, dysprosium 163 atoms, dysprosium 164 atoms, erbium 162 atoms, erbium 164 atoms, erbium 166 atoms, erbium 167 atoms, erbium 168 atoms, erbium 170 atoms, holmium 165 atoms, thulium 169 atoms, ytterbium 168 atoms, ytterbium 170 atoms, ytterbium 171 atoms, ytterbium 172 atoms, ytterbium 173 atoms, ytterbium 174 atoms, ytterbium 176 atoms, lutetium 175 atoms, or lutetium 176 atoms.

[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 isotope 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 isotope-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 isotope 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 include an isotope-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 include rare earth atoms. For example, the plurality of atoms may include 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, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, enriched to an isotope 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.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 may be included. The plurality of atoms may be 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, concentrated to an isotope abundance 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.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 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms may be included. The plurality of atoms are 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, enriched in the isotope abundance within the range defined by any two of the aforementioned values.Barium 135 atoms, barium 136 atoms, barium 137 atoms, barium 138 atoms, scandium 45 atoms, yttrium 89 atoms, lanthanum 139 atoms, cerium 136 atoms, cerium 138 atoms, cerium 140 atoms, cerium 142 atoms, praseodymium 141 atoms, neodymium 142 atoms, neodymium 143 atoms, neodymium 145 atoms, neodymium 146 atoms, neodymium 148 atoms, samarium 144 atoms, samarium 149 atoms, samarium 150 atoms, samarium 152 atoms, samarium 154 atoms, europium 151 atoms, europium 153 atoms, gadolinium 154 atoms, gadolinium 155 atoms, gadolinium 156 atoms, gadolinium 157 atoms, gadolinium 158 atoms, gadolinium 160 atoms, terbium 159 atoms, dysprosium 156 atoms, dysprosium 158 atoms, dysprosium 160 atoms, dysprosium 161 atoms, dysprosium 162 atoms, dysprosium 163 atoms, dysprosium 164 atoms, erbium 162 atoms, erbium 164 atoms, erbium 166 atoms, erbium 167 atoms, erbium 168 atoms, erbium 170 atoms, holmium 165 atoms, thulium 169 atoms, ytterbium 168 atoms, ytterbium 170 atoms, ytterbium 171 atoms, ytterbium 172 atoms, ytterbium 173 atoms, ytterbium 174 atoms, ytterbium 176 atoms, lutetium 175 atoms, or, may contain lutetium 176 atoms.,

[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 a first electromagnetic energy to one or more of a plurality of atoms. Applying the first electromagnetic energy can induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state different from the first atomic state.,

[0043] The first atomic state may include a first single qubit state. The second atomic state may include a second single qubit state. The first atomic state or the second atomic state may have an increased energy relative to the ground atomic state of the atom. The first atomic state or the second atomic state may have an equal energy relative 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 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 3 include first and second hyperfine states on P1 or 3 P2 manifolds, respectively. The first and second atomic states may be strontium 87 3 on the P1 manifold or strontium-87 3 on the P2 manifold, etc., and may include first and second hyperfine states on the P1 or 3 P2 manifold of any atom described herein. 3 include first and second hyperfine states on the P1 or

[0045] FIG. 9 is a diagram showing an example of a qubit including the P2 state of strontium 87. The left panel of FIG. 9 shows the rich energy level structure of the P2 state of strontium 87. The right panel of FIG. 9 shows potential qubit transitions within the P2 state of strontium 87 that are (up to first order) unaffected by changes in the magnetic field near 70 gauss. 3 The left panel of FIG. 9 shows the rich energy level structure of the P2 state of strontium 87. The right panel of FIG. 9 shows potential qubit transitions within the P2 state of strontium 87 that are (up to first order) unaffected by changes in the magnetic field near 70 gauss. 3 The left panel of FIG. 9 shows the rich energy level structure of the P2 state of strontium 87. The right panel of FIG. 9 shows potential qubit transitions within the P2 state of strontium 87 that are (up to first order) unaffected by changes in the magnetic field near 70 gauss. 3 The left panel of FIG. 9 shows the rich energy level structure of the P2 state of strontium 87. The right panel of FIG. 9 shows potential qubit transitions within the P2 state of strontium 87 that are (up to first order) unaffected by changes in the magnetic field near 70 gauss.

[0046] In some cases, the first and second atomic states are the 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 can excite the first hyperfine state and / or the second hyperfine state to the second electronic state. A 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, respectively, can be applied to drive the two-photon transition. In some cases, the first and second hyperfine states are the hyperfine states of the ground electronic state. The ground electronic state may not decay to a lower electronic state by spontaneous emission or stimulated emission. The hyperfine state may include a nuclear spin state. In some cases, the hyperfine state includes the nuclear spin state of strontium 87 1 and includes the nuclear spin states of the S0 manifold, and the qubit transition drives one or both of the two nuclear spin states of strontium 87 1 S0 to a state detuned from, or within, the 3 P2 or 3 P1 manifold. In some cases, a one qubit transition is a two-photon Raman transition between the nuclear spin states of strontium 87 1 S0 and involves a state detuned from, or within, the 3 P2 or 3 P1 manifold. In some cases, the nuclear spin state may be a Stark-shifted nuclear spin state. The Stark shift can be optically driven. The optical Stark shift can be detuned from resonance with any, all, or a combination of single qubit transitions, two qubit transitions, shelving transitions, imaging transitions, etc.

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

[0048] In the case of the first and second nuclear spin states associated with a nucleus containing a spin greater than 1 / 2 (such as a spin 1, spin 3 / 2, spin 2, spin 5 / 2, spin 3, spin 7 / 2, spin 4, or spin 9 / 2 nucleus), the transition between the first nuclear spin state and the second nuclear spin state may involve transitions between other spin states on the nuclear spin manifold. For example, in the case of a spin 9 / 2 nucleus in the presence of a homogeneous magnetic field, all nuclear spin levels may be separated by equal energies. Thus, for example, a transition (such as a Raman transition) designed to transition an atom from the m N = 9 / 2 spin state to the m N = 7 / 2 spin state may also drive from m 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 = -9 / 2, where m N is the nuclear spin state. Similarly, for example, from the m N = 9 / 2 spin state to the mN Transitions (such as Raman transitions) designed to transition atoms to a 5 / 2 spin state can also drive from m N = 7 / 2 to m N = 3 / 2, from m N = 5 / 2 to m N = 1 / 2, from m N = 3 / 2 to m N = -1 / 2, from m N = 1 / 2 to m N = -3 / 2, from m N = -1 / 2 to m N = -5 / 2, from m N = -3 / 2 to m N = -7 / 2, and from m N = -5 / 2 to m N = -9 / 2. Thus, such transitions may not be selective in inducing transitions between specific spin states on the nuclear spin manifold.

[0049] Instead, it may be desirable to implement a selective transition between a specific first spin state and a second spin state 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 transitions that accompany transitions between the desired first and second nuclear spin states. For example, if transitions from first and second nuclear spin states having m N = -9 / 2 and m N = -7 / 2 are desired, the light can provide an AC Stark shift to the m N = -5 / 2 spin state, thereby significantly reducing the transition between m N = -7 / 2 and m N = -5 / 2 states. Similarly, if transitions from first and second nuclear spin states having m N = -9 / 2 and m N = -5 / 2 are desired, the light can provide an AC Stark shift to the m N = -1 / 2 spin state, thereby reducing m N = -5 / 2 and m N- Significantly reduce transitions between the -1 / 2 states. This can effectively create a two-level subsystem within a nuclear spin manifold separated from the rest of the nuclear spin manifold, significantly simplifying the dynamics of the qubit system. To have only one necessary AC Stark shift, it may be advantageous to use nuclear spin states near the edges of 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 = 9 / 2). Alternatively, nuclear spin states far from the edges 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, and two AC Stark shifts can be implemented (e.g., m N = -7 / 2 and m N = -1 / 2 or m N = -9 / 2 and m N = 3 / 2).

[0050] The Stark shift of the nuclear spin manifold can shift adjacent nuclear spin states from resonance with a desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. The Stark shift can reduce leakage from the first and second nuclear spin states of the nuclear spin manifold to other states. The Stark shift can potentially be achieved up to several hundred kHz with a beam output of less than 10 mW. Due to the frequency selectivity of the upper state, scattering due to imperfect polarization control can be reduced. 3The separation of different angular momentum states in the P1 manifold can be on the order of gigahertz, which is larger than the numerical values for single- and two-qubit gate light. Leakage to other states of the nuclear spin manifold can lead to decoherence. The Rabi frequency of the two-qubit transition (e.g., how fast the transition can be driven) may 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 can enable the use of long-lived metastable excited electronic states (such as the 3 P0 state of strontium 87) for qubit storage. Atoms can be selectively transferred to such states to reduce crosstalk or improve the fidelity of gates or detection. Such a storage process or shelving process can be atom-selective using the SLM or AOD described herein. The shelving transition can include a transition from the 1 S0 state of strontium 87 to the 3 P0 or 3 P2 state of strontium 87.

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

[0053] The shelving clock light may or may not be atom-selective. In some cases, the clock transition is applied globally (e.g., not atom-selective). A globally applied clock transition may involve directing light without passing through the microscope objective or structuring the light. In some cases, the clock transition is atom-selective. Atom-selective clock transitions may potentially improve gate fidelity by minimizing crosstalk. For example, to reduce atomic crosstalk, atoms may be shelved into clock states that are not affected by light. This may reduce crosstalk between adjacent qubits during transitions. 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, digital micromirror device, acousto-optic deflector, etc.

[0054] System 200 may include one or more reading units 230. The reading unit may include one or more reading optical units. The reading optical unit may be configured to perform one or more measurements of one or more superposition states in order to obtain non-classical calculations. The reading optical unit may include one or more photodetectors. The detector 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 photodetector may include one or more fluorescence detectors. The reading optical 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 lens may have an NA 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 lens may have an NA within a range defined by any two of the aforementioned values.

[0055] One or more reading optical units 230 can perform measurements, such as projective measurements, by applying light that resonates with the imaging transition. Fluorescence may be generated by the imaging transition. The imaging transition may include a transition from the S0 state of strontium 87 to the 1 P1 state of strontium 87. The imaging transition of strontium 87 may include a transition from the 1 S0 state to the1 The P1 state can emit fluorescence. The lower state of the qubit transition can include 1 two nuclear spin states of the S0 manifold. One or more states can resonate with the imaging transition. The measurement can include two excitations. In the first excitation, one of the two lower states can be excited to a shelving state (e.g., the 3 P0 state of strontium 87). In the second excitation, the imaging transition can be excited. The first transition can reduce crosstalk between adjacent 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 whether one or more atoms have been lost from the capture. The imaging unit can be used to observe an array of atoms within the capture.

[0057] 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 can include any combination of the 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 pump can be configured to discharge gas from the system 200 to achieve a low vacuum pressure state. For example, the roughing pump can be configured to pump gas out of the system 200 to achieve a low vacuum pressure of up to about 10 3 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 pumping or ultra-high vacuum pumping. The high vacuum pump can be configured to discharge gas from the system 200 to achieve a high vacuum pressure of up to about 10 -3 Pa or an ultra-high vacuum pressure of up to about 10 -6 Pa when the system 200 reaches the low vacuum pressure state provided by one or more roughing pumps.

[0059] The vacuum unit can evacuate the system 200 to up to about 10 -6 Pa, 9×10 -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, 9×10 -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 Pa, or can be configured to maintain at a pressure equal to or lower than that. The vacuum unit maintains the system 200 at at least about 10 -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 Pa, or can be configured to maintain at a pressure equal to or higher than that. The vacuum unit may be configured to maintain the system 200 at a pressure within the range defined by any two of the aforementioned values.

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

[0061] System 200 may include one or more atomic reservoirs 260. The atomic reservoir may be configured to supply one or more replacement atoms to replace one or more atoms at one or more light trapping sites when the atoms are lost from the light trapping sites. The atomic reservoir may be spatially separated from the light trapping unit. For example, the atomic reservoir may be disposed at a position away from the light trapping unit.

[0062] Alternatively or additionally, the atomic reservoir may include a portion of the light trapping sites of the light trapping unit. A first subset of the light trapping sites can be utilized to perform quantum computing and can be referred to as the set of computationally active light trapping sites, and a second subset of the light trapping sites can function as an atomic reservoir. For example, the first subset of the light trapping sites may include an internal array of the light trapping sites, and the second subset of the light trapping sites may include an external array of the light trapping sites surrounding the internal array. The internal array may include an array of rectangles, squares, rectangular prisms, or cubes of the light trapping sites.

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

[0064] 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 among a plurality of atoms with at least a second atom among 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. The multiqubit unit may include a two-qubit unit, a three-qubit unit, a four-qubit unit, or an n-qubit unit, where n may be 5, 6, 7, 8, 9, 10, or greater. 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 second and third atoms, a four-qubit unit may include a first atom quantum mechanically entangled with second, third, and fourth atoms, 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 atom may not be in a superposition state at the time of quantum mechanical entanglement. 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 unit may be configured to electronically excite at least a first atom to a Rydberg state or a superposition of a Rydberg state and a low-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg unit may be configured to induce one or more quantum mechanical entanglements between the Rydberg atom or the 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 longer from the Rydberg atom or the dressed Rydberg atom. The second atom may be located at a distance of up to 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 shorter from the Rydberg atom or the dressed Rydberg atom. The second atom may be arranged at a distance from the Rydberg atom or the dressed Rydberg atom within a range defined by any two of the aforementioned values. The Rydberg unit can be configured such that the Rydberg atom or the dressed Rydberg atom can relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg unit may be configured to induce the relaxation of the Rydberg atom or the dressed Rydberg atom 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, a Rydberg unit may be configured to apply electromagnetic radiation (such as RF radiation or optical radiation) to drive a Rydberg atom or a dressed Rydberg atom to a lower energy atomic state. The Rydberg unit may be configured to induce any number of quantum mechanical entanglements between any number of atoms among a 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 wavelength may be selected to correspond to a wavelength that forms a Rydberg atom or a dressed Rydberg atom. 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 of 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 two-photon transitions to generate entanglement. The Rydberg unit can be configured to induce two-photon transitions to generate entanglement between two atoms. The Rydberg unit can be configured to selectively induce two-photon transitions to selectively generate entanglement between two atoms. For example, the Rydberg unit can be configured to direct electromagnetic energy (such as light energy) towards a specific light-capturing site to selectively induce two-photon transitions to selectively generate entanglement between two atoms. The two atoms may be captured by nearby light-capturing sites. For example, the two atoms can be captured by adjacent light-capturing sites. The two-photon transition can be induced using the first and second light from the first and second light sources, respectively. The first and second light sources may each include any light source described herein (such as 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 operation 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., in the range from 400 nm to 800 nm or from 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., in the range from 200 nm to 400 nm or from 300 nm to 350 nm). The first and second light sources can emit light having substantially equal and opposite spatial-dependent frequency shifts.

[0069] A Rydberg atom or a dressed Rydberg atom can include a Rydberg state that has a sufficiently strong interatomic interaction with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) to enable the implementation of multiqubit operations. The Rydberg state can include a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or greater. The Rydberg state can include a principal quantum number of at most about 100, 90, 80, 70, 60, 50 or less. The Rydberg state can include a principal quantum number within a range defined by any two of the foregoing values. The Rydberg state can interact with nearby atoms through van der Waals interactions. The van der Waals interactions can shift the atomic energy levels of the atoms.

[0070] State-selective excitation of an atom to a Rydberg level can enable the implementation of multiqubit operations. The multiqubit operations can include two-qubit operations, three-qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or greater. Two-photon transitions can be used to excite an atom from a ground state ( 1 such as the S0 ground state) to a Rydberg state (n 3 such as the S1 state, where n is the principal quantum number described herein). State selectivity can be achieved by a combination of laser polarization and spectral selectivity. The two-photon transitions can be implemented using a first and a second laser source as described herein. The first laser source can emit π-polarized light that does not change the projection of the atomic angular momentum along the magnetic field. The second laser can emit circular polarization and can change the projection of the atomic angular momentum along the magnetic field by one unit. This polarization can be used to excite the first and second qubit levels to the Rydberg levels. However, since the Rydberg levels can be more sensitive to the magnetic field than the ground state, a large splitting (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] Multi - qubit operations (such as two - qubit operations, three - qubit operations, four - qubit operations, etc.) may depend on the energy shifts of levels due to the van der Waals interactions described herein. Such shifts can prevent the excitation of one atom conditional on the state of the other atom, or can modify the coherent dynamics of the excitation of a two - atom system to perform a two - qubit operation. In some cases, a “dressed state” can be generated under continuous driving to perform a two - qubit operation without the need for complete excitation to a Rydberg level (for example, as described in www.arxiv.org / abs / 1605.05207, which is hereby incorporated by reference in its entirety for all purposes).

[0072] System 200 may comprise one or more second electromagnetic delivery units (not shown in FIG. 2). The second 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 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 a 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 can precede, be simultaneous with, or follow the second electromagnetic energy.

[0073] The pulse sequence may include any number of pulses. For example, the 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. The pulse sequence may include at most about 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 pulse. The pulse sequence may include some pulses within a range defined by any two of the foregoing values. Each pulse of the pulse sequence may include any pulse shape, such as any of the pulse shapes described herein.

[0074] The pulse sequence may be configured to reduce the duration required to implement multiqubit operations, as described herein (e.g., with respect to Example 3). For example, the pulse sequence may include 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. The pulse sequence 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 shorter. The pulse sequence may include a duration within a range defined by any two of the foregoing values.

[0075] The pulse sequence can be configured to increase the fidelity of multiqubit operations, as described herein. For example, the pulse sequence can enable multiqubit operations with a fidelity of at least about 0.5, 0.6, 0.7, 0.8, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 0.991, 0.992, 0.993, 0.994, 0.995, 0.996, 0.997, 0.998, 0.999, 0.9991, 0.9992, 0.9993, 0.9994, 0.9995, 0.9996, 0.9997, 0.9998, 0.9999, 0.99991, 0.99992, 0.99993, 0.99994, 0.99995, 0.99996, 0.99997, 0.99998, 0.99999, 0.999991, 0.999992, 0.999993, 0.999994, 0.999995, 0.999996, 0.999997, 0.999998, 0.999999, or higher. The pulse sequence can enable multiqubit operations with a fidelity of up to about 0.999999, 0.999998, 0.999997, 0.999996, 0.999995, 0.999994, 0.999993, 0.999992, 0.999991, 0.99999, 0.99998, 0.99997, 0.99996, 0.99995, 0.99994, 0.99993, 0.99992, 0.99991, 0.9999, 0.9998, 0.9997, 0.9996, 0.9995, 0.9994, 0.9993, 0.9992, 0.9991, 0.999, 0.998, 0.997, 0.996, 0.995, 0.994, 0.993, 0.992, 0.991, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.9, 0.8, 0.7, 0.6, 0.5, or lower. The pulse sequence can enable multiqubit operations with a fidelity within a range defined by any two of the foregoing values.

[0076] The pulse sequence may enable the implementation of multiqubit operations on a nonadiabatic time scale while effectively maintaining adiabatic dynamics. For example, the pulse sequence may include one or more of a shortcut to adiabaticity (STA) pulse sequence, a transitionless quantum driving (TQD) pulse sequence, a superadiabatic pulse sequence, an inverse adiabatic driving pulse sequence, a derivative removal by adiabatic gate (DRAG) pulse sequence, and a weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequence.For example, the pulse sequence may be similar to those described in M. V. Berry, "Transitionless Quantum Driving", Journal of Physics A: Mathematical and Theoretical 42(36), 365303(2009), www.doi.org / 10.1088 / 1751-8113 / 42 / 36 / 365303; Y.-Y. Jau et al., "Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction", Nature Physics 12(1), 71-74(2016); T. Keating et al., "Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing", Physical Review A 91, 012337(2015); A. Mitra et al., "Robust 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 hereby incorporated by reference in its entirety for all purposes.

[0077] The pulse sequence may further include one or more optimal control pulse sequences. The optimal control pulse sequence can be derived from one or more procedures including gradient ascent pulse engineering (GRAPE) method, Krotov method, chopped-based method, chopped random basis (CRAB) method, Nelder-Mead method, gradient optimization using parametrization (GROUP) method, genetic algorithm method, and gradient optimization of analytic controls (GOAT) method. 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 hereby incorporated by reference in its entirety for all purposes. The pulse sequence may further include composite pulses.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 a system having single qubit gates.

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

[0079] Light trapping unit FIG. 3A shows an example of the light trapping unit 210. The light trapping unit can be configured to generate a plurality of spatially different 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 different 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 FIG. 3B, some of the light trapping sites may be empty (i.e., do not trap atoms).

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

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

[0082] Each light - trapping site of the plurality of light - trapping sites can be spatially separated from each other's light - trapping sites by 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 a longer distance. Each light - trapping site can be spatially separated from each other's light - trapping sites by 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 a shorter distance. Each light - trapping site can be spatially separated from each other's light - trapping sites by a distance within the range defined by any two of the aforementioned values.

[0083] The light - trapping site can include one or more optical tweezers. The optical tweezers can include one or more focused laser beams to provide an attractive or repulsive force for holding or moving one or more atoms. The beam waist of the focused laser beam can include a strong electric - field gradient. Atoms can be attracted or repelled along the electric - field gradient towards the center of the laser beam, and the center of the laser beam can include the strongest electric field. The light - trapping site can include one or more grating sites of one or more optical gratings. The light - trapping site may include one or more grating sites of one or more one - dimensional (1D), two - dimensional (2D), or three - dimensional (3D) optical gratings. For example, the light - trapping site can include one or more grating sites of a 2D optical grating as depicted in FIG. 3B.

[0084] An optical lattice can be generated by interfering light propagating in opposite directions (such as laser light propagating in opposite directions) to generate a standing wave pattern having a periodic succession of minimum and maximum intensities along a specific direction. A 1D optical lattice can be generated by interfering a pair of light beams propagating in opposite directions. A 2D optical lattice can be generated by interfering two pairs of light beams propagating in opposite directions. A 3D optical lattice can be generated by interfering three pairs of light beams propagating in opposite directions. The light beams may be generated by different light sources or the same light source. Thus, the optical lattice can be generated by at least about 1, 2, 3, 4, 5, 6, or more light sources, or at most about 6, 5, 4, 3, 2, or 1 light source.

[0085] Returning to the description of FIG. 3A, the optical trapping unit may comprise one or more light sources configured to emit light to generate a plurality of optical trapping sites as described herein. For example, the optical trapping unit may comprise a single light source 213 as depicted in FIG. 3A. Although depicted as including a single light source in FIG. 3A, the optical trapping unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light 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 argon dimer (Ar2) excimer lasers, krypton dimer (Kr2) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe2) excimer lasers, argon fluoride (ArF) excimer lasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF) excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride (XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The laser may 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 (MnCl2) 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-doped yttrium aluminum garnet (Yb:YAG) lasers, ytterbium-doped glass (Yb:glass) lasers, holmium yttrium aluminum garnet (Ho:YAG) lasers, chromium-doped zinc selenide (Cr:ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) lasers, erbium-doped glass (Er:glass) lasers, erbium and ytterbium co-doped glass (Er / Yb:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers.

[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 have a pulse length 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, 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. The laser may have a pulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3 ns, 2 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800 fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80 fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6 fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or shorter. The laser may have a pulse length within a range defined by any two of the aforementioned values.

[0091] The laser can have a repetition rate of at least about 1 Hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more.The laser can have a repetition rate of up to approximately 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The laser may have a repetition rate within the range defined by any two of the aforementioned values.

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

[0093] The laser can emit light having an average output of at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800 W, 900 W, 1,000 W, or more. The laser can emit light having an average output of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. The laser can emit light having an output within a range defined by any two of the foregoing values.

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

[0095] The laser is at least about 1×10 -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 nm, or can emit light having a bandwidth equal to or longer than that. The laser has a maximum of about 1×10 -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 nm, or can emit light having a bandwidth equal to or shorter than that. The laser can emit light having a bandwidth within the 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 wavelength corresponding to an atom can include any wavelength of light that causes equal or nearly equal polarizabilities of a first atomic state and a second atomic state. The magic wavelength of the transition between the first atomic state and the second atomic state can be determined by calculating the wavelength-dependent polarizabilities of the first atomic state and the second atomic state and finding the intersection. Light tuned to such a magic wavelength can cause equal or nearly equal differential optical shifts in the first atomic state and the second atomic state regardless of the intensity of the light emitted from the light source. Thereby, the first atomic state and the second atomic state can be effectively separated from the motion of the atoms. The magic wavelength may utilize one or more scalar or tensor optical shifts. The scalar optical shift or the tensor optical shift can depend on the magnetic sub-levels within the first atomic state and the second atomic state.

[0097] For example, the metastable states of atoms such as those of group III and alkaline earth or alkaline earth can have relatively large tensor shifts such that the angle with respect to the applied magnetic field is adjusted to cause a situation where the balance of the scalar shift and the tensor shift gives a differential optical shift that is zero or close to zero between the first atomic state and the second atomic state. The angle θ can be adjusted by selecting the polarization of the emitted light. For example, when the emitted light is linearly polarized, the total polarization rate α can be described as the sum of the scalar component α scalar and the tensor component α tensor . α = α scalar + (3θ - 1)α tensor

[0098] By appropriately selecting θ, the polarization rates of the first atomic state and the second atomic state can be selected to be equal or approximately equal corresponding to a differential optical shift of zero or approximately zero, and the motion of the atoms can be separated.

[0099] The light source can be configured to direct light at one or more optical modulators (OMs) configured to generate a plurality of light trapping sites. For example, the light trapping unit may include an OM214 configured to generate a plurality of light trapping sites. Although depicted as including one OM in FIG. 3A, 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 OM. 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 an optical element 219, as shown in FIG. 3A. The optical element may include a lens or a 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 FIG. 3A, the OM may include an SLM, a DMD, or an LCoS device. The SLM, DMD, or LCoS device may be imaged on the rear focal plane of the microscope objective lens. This may enable the generation of any configuration of the light trapping sites in two or three dimensions.

[0102] Alternatively or additionally, the OM may include first and second AODs. The active regions of the first and second AODs may be imaged on the rear focal plane of the microscope objective lens. 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 generation of the light trapping sites in two or three dimensions.

[0103] Alternatively or additionally, the OM may comprise static optical elements such as one or more microlens arrays or holographic optical elements. The static optical elements may be imaged on the rear focal plane of the microscope objective lens. This may enable the generation of any configuration of the light trapping sites in two or three dimensions.

[0104] The light trapping unit may include one or more imaging units configured to acquire one or more images of the spatial configuration of a plurality of atoms trapped within the light trapping site. For example, the light trapping unit may include an imaging unit 215. Although depicted in FIG. 3A as including a single imaging unit, the light 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 unit. 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 image 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 light trapping unit may include one or more artificial intelligence (AI) units configured to perform one or more AI operations to determine the spatial configuration of a plurality of atoms trapped within the light trapping site based on an image obtained by the imaging unit. For example, the light trapping unit may include a spatial configuration AI unit 216. Although depicted in FIG. 3A as including a single spatial configuration AI unit, the light trapping 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 unit. The AI operations may include any machine learning (ML) or reinforcement learning (RL) operations described herein.

[0106] The light capture unit may include one or more atom rearrangement units configured to provide an altered spatial arrangement of a plurality of atoms captured at the light capture site based on one or more images obtained by the imaging unit. For example, the light capture unit may include an atom rearrangement unit 217. Although depicted in FIG. 3A as including a single atom rearrangement unit, the light capture 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 unit.

[0107] The light capture unit may include one or more spatial arrangement AI units configured to perform one or more artificial intelligence (AI) operations to determine an altered spatial arrangement of a plurality of atoms captured within the light capture site based on an image obtained by the imaging unit. For example, the light capture unit may include a spatial arrangement AI unit 218. Although depicted in FIG. 3A as including a single spatial arrangement AI unit, the light 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 unit.

[0109] The atomic rearrangement unit may be configured to change the spatial arrangement in order to obtain an increase in the filling rate of a plurality of light trapping sites. The filling rate 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 part of the light trapping unit. For example, the initial loading of atoms within the computationally active light trapping sites may cause the filling rate to be 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy 100%, 90%, 70%, 60%, less than 50%, or a lower percentage of the available computationally active light trapping sites. It may be desirable to rearrange the atoms to achieve a filling rate of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, 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 filling rate 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 filling rate within a range defined by any two of the foregoing values.

[0110] As an example, FIG. 3C shows an example of a light trapping unit that is partially filled with atoms. As depicted in FIG. 3C, the initial loading of atoms within the light trapping sites can result in a filling rate of 44.4% (4 atoms filling 9 available light trapping sites). By moving atoms from different regions of the light trapping unit (not shown in FIG. 3C) to the unoccupied light trapping sites, or by moving atoms from the atomic reservoir described herein, a much higher filling rate can be obtained, as shown in FIG. 3D.

[0111] FIG. 3D is a diagram showing an example of a light trapping unit that is completely filled with atoms. As depicted in FIG. 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 moving atoms from different regions of the light trapping unit (not shown in FIG. 3C) or from the atomic reservoir described herein. Thus, the filling rate can be significantly improved following the rearrangement of atoms within the light trapping sites. For example, a maximum filling rate of 100% (9 atoms filling 9 available light trapping sites, as shown in FIG. 3D) can be achieved.

[0112] The rearrangement of atoms can be performed by (i) acquiring an image of the light trapping unit, identifying filled and unfilled light trapping sites, (ii) determining a series of movements to move atoms from the filled light trapping sites to the unfilled light trapping sites, and (iii) moving atoms from the filled light trapping sites to the unfilled light trapping sites. Operations (i), (ii), and (iii) can be repeatedly performed until a large filling rate is achieved. Operation (iii) can include converting the movements identified in operation (ii) into waveforms that can be sent to an arbitrary waveform generator (AWG) and using the AWG to drive an AOD to move the atoms. The series of movements 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 hereby incorporated by reference in its entirety for all purposes.

[0113] Electromagnetic delivery unit Figure 4 shows an example of the electromagnetic delivery unit 220. The electromagnetic delivery unit can be configured to apply electromagnetic energy to one or more of a plurality of atoms, as described herein. The electromagnetic delivery unit may include one or more light sources, such as any of the light sources described herein. The electromagnetic energy can include light energy. The light energy can include any repetition rate, pulse energy, average output, wavelength, or bandwidth described herein.

[0114] The electromagnetic delivery unit can 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 lasers. The electromagnetic energy can include microwave energy or RF energy. The RF energy can include one or more wavelengths that are at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or longer. The RF energy can include one or more wavelengths that are at most about 10 km, 9 km, 8 km, 7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m, 500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m, 30 m, 20 m, 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm, 800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm, 80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or shorter. The RF energy can include one or more wavelengths that are within a range defined by any two of the aforementioned values.

[0115] RF energy can include an average power of at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800 W, 900 W, 1,000 W, or greater. RF energy can include an average power of up to about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. RF energy can 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 comprise a light source 221. Although depicted as including a single light source in FIG. 4, the electromagnetic delivery unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light source.

[0117] The light source may be configured to direct light to one or more OMs configured to selectively apply electromagnetic energy to one or more of a plurality of atoms. For example, the electromagnetic delivery unit may include an OM 222. Although depicted as including a single OM in FIG. 4, the electromagnetic delivery 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 OM. The OM may include one or more SLMs, AODs, or AOMs. The OM may include one or more DMDs. The OM may 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 an AI unit 223. Although depicted as including a single AI unit in FIG. 4, 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 of the machine learning (ML) or reinforcement learning (RL) operations described herein.

[0119] The electromagnetic delivery unit can be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) to the qubits described herein. The electromagnetic delivery unit can be configured to apply one or more two-qubit operations (such as one or more two-qubit gate operations) to the two-qubit units described herein. Each single-qubit or two-qubit operation can include 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 can 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 shorter. Each single-qubit or two-qubit operation can include a period within the range defined by any two of the foregoing values. The single-qubit or two-qubit operation can be applied at a repetition frequency of at least 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or longer.Single qubit or two qubit operations can be applied at a repetition frequency of 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. The single qubit or two qubit operations may be applied at a repetition frequency within the range defined by any two of the foregoing values.

[0120] The electromagnetic delivery unit can be configured to apply one or more single qubit operations by inducing one or more Raman transitions between the first qubit state and the second qubit state described herein. The Raman transitions are those described herein 3 P0 or 3It may be detuned from the P1 line. For example, the Raman transition 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 transition may be detuned by at most about 1 GHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The Raman 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 an optical 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 region of the SLM or AOD onto the rear focal plane of the microscope objective lens. The microscope objective lens may perform a spatial Fourier transform on the optical field at the position of the SLM or AOD. In this way, an angle (which may be proportional to the RF frequency) may be converted to a position. For example, applying a comb of radio frequencies to an AOD may generate a linear array of spots with a finite extent determined by the characteristics of the optical alignment system (such as the point spread function of the optical alignment system) at the focal plane of the objective lens.

[0122] To perform Raman transitions on a single atom using a single SLM or AOD, a pair of frequencies may be applied to the SLM or AOD simultaneously. The two frequencies of the pair may have a frequency difference that matches or approximately matches the splitting energy between the first qubit state and the second qubit state. For example, the frequency difference may have a difference of at most about 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less from the splitting energy. The frequency difference may differ from the splitting energy by at least about 1 Hz, 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, or more. The frequency difference may differ from the splitting energy by about 0 Hz. The difference in frequencies may differ from the splitting energy by only values within the range defined by any two of the foregoing values. The optical system may be configured such that the position spacing corresponding to the frequency difference is not resolved and the light of both frequencies interacts with a single atom.

[0123] Integrated Light Trapping Unit and Electromagnetic Delivery Unit The light trapping unit and 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 light for trapping atoms generated by the light trapping 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 light from the trapped atoms generated by the light trapping 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 polarization. The light of orthogonal polarization may be superimposed in front of the microscope objective lens. In such a scheme, each photon used in the two-photon transitions described herein is passed to the objective lens by an individual SLM or AOD, enabling improved polarization control. By bringing the light from the first SLM or AOD to a second SLM or AOD oriented substantially perpendicular to the first SLM or AOD via an optical relay, qubit operations can be performed on a two-dimensional array of atoms. Alternatively or additionally, qubit operations can be performed on a two-dimensional array of atoms by using a one-dimensional array of SLMs or AODs.

[0125] The stability of qubit - gate fidelity can be improved by maintaining the overlap of light from various light sources described herein (such as light sources associated with the light - trapping units or electromagnetic - delivery units described herein). Such overlap is maintained by an optical subsystem that measures the directions of light emitted from various light sources, enabling closed - loop control of the direction of light emission. The optical subsystem can include a pick - off mirror disposed in front of the microscope objective lens. The pick - off mirror can be configured to direct a small amount of light towards a lens that can focus the collimated beam and convert an angular deviation into a position deviation. A position - sensitive photodetector, such as a lateral - effect position sensor or a quadrant photodiode, can convert the position deviation into an electronic signal and supply information about the deviation to a compensating optical system, such as an active mirror.

[0126] The stability of qubit - gate operations can be improved by controlling the intensity of light from various light sources described herein (such as light sources associated with the light - trapping units or electromagnetic - delivery units described herein). Such intensity control is maintained by an optical subsystem that measures the intensity of light emitted by various light sources, enabling 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). The intensity can be measured using a photodetector, such as a photodiode, or any other photodetector described herein. Information about the intensity can be integrated into a feedback loop to stabilize the intensity.

[0127] State preparation unit FIG. 5 is a diagram showing an example of the state preparation unit 250. The state preparation unit can be configured to prepare the states of a plurality of atoms as described herein. The state preparation unit is coupled to the optical trapping unit and can direct the atoms prepared by the state preparation unit towards the optical trapping unit. The state preparation unit may be configured to cool a plurality of atoms. The state preparation unit can be configured to cool a plurality of atoms before trapping the plurality of atoms at a 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 a Zeeman decelerator 251. Although 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 decelerator. The Zeeman decelerator can be configured to cool one or more of the plurality of atoms from a first velocity or velocity distribution (such as the emission velocity from an atomic 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 related to a temperature of at least about 50 Kelvin (K), 60 K, 70 K, 80 K, 90 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1,000 K, or higher. The first velocity or velocity distribution may be related to a temperature of up to about 1,000 K, 900 K, 800 K, 700 K, 600 K, 500 K, 400 K, 300 K, 200 K, 100 K, 90 K, 80 K, 70 K, 60 K, 50 K, or 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 faster. 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 slower. The second velocity may be within a range defined by any two of the foregoing values. The Zeeman decelerator may include a 1D Zeeman decelerator.

[0130] The state preparation unit may include a first magneto-optical trap (MOT) 252. The first MOT may be configured to cool atoms to a first temperature. The first temperature may be up to 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 include a 1D, 2D, or 3D MOT.

[0131] The first MOT may comprise one or more light sources configured to emit light (such as any of the light sources described herein). The light may include one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or longer. The light may include one or more wavelengths of up to about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, light may include one or more wavelengths within 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 MOT253. 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 include a 1D, 2D, or 3D MOT.

[0133] The second MOT may comprise one or more light sources (such as any of the light sources described herein) configured to emit light. The light may include one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or longer. The light may include one or more wavelengths of up to about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, light may include one or more wavelengths within the range 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 FIG. 5 depicts the inclusion of two MOTs, the state 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 MOT.

[0135] The state preparation unit may comprise one or more sideband cooling units or Sisyphus cooling units (such as the sideband cooling units described in www.arxiv.org / abs / 1810.06626, or the Sisyphus cooling units described in www.arxiv.org / abs / 1811.06014. Each of these is hereby incorporated by reference in its entirety for all purposes). For example, the state preparation unit may comprise a sideband cooling unit or a Sisyphus cooling unit 254. In FIG. 5, it is depicted as including a single sideband cooling unit or Sisyphus cooling unit, but state preparation may include at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more sideband cooling units or Sisyphus cooling units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband cooling unit or Sisyphus cooling unit, i.e., any number of sideband cooling units or Sisyphus cooling units. The sideband cooling unit or Sisyphus cooling unit can 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 higher. The third temperature may be within a range defined by any two of the aforementioned values.

[0136] The sideband cooling unit or the sheeshpos cooling unit may comprise one or more light sources configured to emit light (such as any of the light sources described herein). The light may include one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or longer. The light may include one or more wavelengths of up to about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or shorter. The light may include one or more wavelengths within a range defined by any two of the aforementioned values.For example, light may include one or more wavelengths within the range 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 comprise one or more optical pumping units. For example, the state preparation unit may comprise an optical pumping unit 255. Although depicted in FIG. 5 as including a single optical pumping unit, 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 a 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 comprise one or more light sources (such as any light source described herein) configured to emit light. The light may include one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or longer.The light may include one or more wavelengths up to approximately 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or shorter. The light may include one or more wavelengths within a range defined by any two of the aforementioned values. For example, the light may include one or more wavelengths within the range of 400 nm to 1,000 nm, 500 nm - 1,000 nm, 600 nm - 1,000 nm, 650 nm - 1,000 nm, 400 nm - 900 nm, 400 nm - 800 nm, 400 nm - 700 nm, 400 nm - 600 nm, 400 nm - 500 nm, 500 nm - 700 nm, or 650 nm - 700 nm.

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

[0139] The coherent drive unit may comprise one or more light sources configured to emit light (such as any of the light sources described herein). The light may include one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or longer. The light may include one or more wavelengths of up to about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or shorter. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, light may include one or more wavelengths within the range 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 a non-equilibrium state and a 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 an RF transition. For example, the coherent drive unit may include one or more RF sources (such as any of the RF sources 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 of 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 drive unit may include one or more light sources (such as any of the light sources described herein) configured to induce a two-photon transition corresponding to the RF transition.

[0141] Controller The optical trapping unit, electromagnetic transmission unit, entanglement unit, readout light unit, vacuum unit, imaging unit, spatial configuration AI unit, spatial arrangement AI unit, atomic rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atomic reservoir, atomic transfer unit, or Rydberg excitation unit may include one or more circuits or controllers (such as one or more electronic circuits or controllers) connected (e.g., by one or more electrical connections) to the optical trapping unit, electromagnetic transmission unit, entanglement unit, readout light unit, vacuum unit, imaging unit, spatial configuration AI unit, spatial arrangement AI unit, atomic rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atomic reservoir, atomic transfer unit, or Rydberg excitation unit. The circuit or controller may be configured to control the optical trapping unit, electromagnetic transmission unit, entanglement unit, readout optical unit, vacuum unit, imaging unit, spatial configuration AI unit, spatial arrangement AI unit, atomic rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atomic reservoir, atomic transfer unit, or Rydberg excitation unit.

[0142] Non-classical computer In one aspect, the present disclosure provides a non-classical computer comprising a plurality of qubits comprising more than 60 atoms, each atom being trapped within one of a plurality of spatially distinct light-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 of the plurality of qubits, thereby imparting a non-classical operation to one or more of the qubits, the non-classical operation comprising at least a superposition between 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 a 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 one or more of the plurality of qubits, thereby obtaining a non-classical computation.

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

[0144] Method for performing non-classical computation In one aspect, the present disclosure is a method for performing non-classical computing, comprising: (a) generating a plurality of spatially distinct light-trapping sites, the plurality of light-trapping sites being configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms; (b) applying electromagnetic energy to one or more of the plurality of atoms, thereby inducing one or more of the 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 entangling at least a subset of the one or more atoms in the one or more superposition states with at least another of the plurality of atoms; and (d) performing one or more optical measurements of the one or more superposition states to obtain non-classical computing.

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

[0146] In a first operation 610, the method 600 may include generating a plurality of spatially distinct light-trapping sites. The plurality of light-trapping sites may be configured to trap a plurality of atoms. The plurality of atoms may include more than 60 atoms. The light-trapping sites may include any of the light-trapping sites described herein. The atoms may include any of the atoms described herein.

[0147] In a second operation 620, the method 600 may include applying electromagnetic energy to one or more of the plurality of atoms, thereby inducing one or more of the 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. The electromagnetic energy may include any of the electromagnetic energies described herein. The first atomic state may include any of the first atomic states described herein. The second atomic state may include any of the second atomic states described herein.

[0148] In a third operation 630, method 600 may include entangling at least a subset of one or more atoms in one or more superposed states with at least another atom among the plurality of atoms. The atoms may be quantum mechanically entangled in any way described herein (e.g., as described herein with respect to FIG. 2).

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

[0150] In one aspect, the present disclosure provides a method for performing a non-classical computation, the method comprising: (a) providing a plurality of qubits including more than 60 atoms, each atom being trapped within one 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 of the plurality of qubits, thereby imparting a non-classical operation including a superposition between at least the first qubit state and the second qubit state to the one or more qubits; (c) entangling at least a subset of the plurality of qubits in the superposition with at least another qubit among the plurality of qubits; and (d) performing one or more optical measurements of one or more of the qubits, thereby obtaining a classical computation.

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

[0152] In a first operation 710, method 700 may include providing a plurality of qubits that include more than 60 atoms, where each atom is trapped within one of a plurality of spatially distinct light-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 light-trapping sites may include any of the light-trapping sites described herein. The qubits may include any of the qubits described herein. The atoms may include any of the atoms described herein. The first qubit state may include any of the first qubit states described herein. The second qubit state may include any of the second qubit states described herein. The first atomic state may include any of the first atomic states described herein. The second atomic state may include any of the second atomic states described herein.

[0153] In a second operation 720, method 700 may include applying electromagnetic energy to one or more of the plurality of qubits, thereby imparting to the one or more qubits a non-classical operation that includes at least an overlap between the first qubit state and the second qubit state. The electromagnetic energy may include any of the electromagnetic energy described herein.

[0154] In a third operation 730, method 700 may include quantum mechanically entangling at least a subset of the plurality of qubits in the overlap with at least another of the plurality of qubits. The qubits may be quantum mechanically entangled in any of the ways described herein (e.g., as described herein with respect to FIG. 2).

[0155] In a fourth operation 740, method 700 may include performing one or more optical measurements of one or more qubits, thereby obtaining a non-classical computation. The optical measurement may include any optical measurement 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 including more than 60 atoms trapped within one of a plurality of spatially distinct light trapping sites; and (b) performing a non-classical computation using at least a subset of the plurality of qubits.

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

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

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

[0160] Computer system FIG. 1 is a diagram showing a computer system 101 programmed or configured to operate any method or system described herein (such as a system or method for performing non-classical computing described herein). The computer system 101 may regulate various aspects of the present disclosure. The computer system 101 may be a user's electronic device or a computer system remotely located with respect to the electronic device. The electronic device may be a mobile electronic device.

[0161] Computer system 101 includes a central processing unit (CPU, also referred to herein as "processor" and "computer processor") 105, which can be a single-core or multi-core processor, or multiple processors for parallel processing. Computer system 101 also includes a memory or storage location 110 (e.g., random access memory, read-only memory, flash memory), an electronic storage device 115 (e.g., hard disk), a communication interface 120 (e.g., network adapter) for communicating with one or more other systems, and peripheral devices 125 such as a cache, other memory, data storage, and / or an electronic display adapter. Memory 110, storage device 115, interface 120, and peripheral device 125 communicate with CPU 105 via a communication bus (solid line) such as a motherboard. Storage unit 115 may be a data storage unit (or data repository) for storing data. Computer system 101 may be operably coupled to a computer network ("network") 130 using communication interface 120. Network 130 may be the Internet, the Internet and / or an extranet, or an intranet and / or an extranet that communicates 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 that enables devices coupled to computer system 101 to operate as clients or servers with the assistance of computer system 101.

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

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

[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 personal computers (e.g., portable PCs), slate or tablet PCs (e.g., Apple® iPad, Samsung® Galaxy Tab), telephones, smartphones (Apple® iPhone, Android-enabled devices, Blackberry®), or personal digital assistants. 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 memory location of computer system 101, such as, for example, memory 110 or electronic storage unit 115. The machine executable or machine readable code may be provided in the form of software. In use, the code may be executed by processor 105. Optionally, for immediate access by processor 105, the code may be retrieved from storage device 115 and stored in memory 110. In some circumstances, electronic storage unit 115 may be excluded and machine executable instructions may be stored in memory 110.

[0167] The code can be pre-compiled and configured for use in a machine with a processor adapted to execute the code, or can be compiled at runtime. The code can be provided in a programming language selected to enable 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 technology can typically be thought of as a "product" or "article of manufacture" in the form of machine (or processor) executable code and / or associated data, which is carried on or incorporated in a type of machine-readable medium. The machine executable code can be stored in an electronic memory device such as a memory (e.g., read-only memory, random access memory, flash memory) or a hard disk. A "storage" type of medium can include any or all of various semiconductor memories, tape drives, disk drives, etc. that can provide non-transitory storage for software programming at any time for a computer, processor, or tangible memory associated therewith. All or part of the software may be communicated via the Internet or other various electrical communication networks. Such communication can enable, for example, the loading of software from one computer or processor to another, such as from an administrative server or host computer to an application server computer platform. Thus, another type of medium that can hold software elements includes light waves, radio waves, and electromagnetic waves used throughout the physical interface between local devices via wired and optical fixed telephone networks as well as various air links. Physical elements that carry such waves, such as wired or wireless links, optical links, etc., can also be considered media that carry software. As used herein, the term "machine-readable medium" such as a computer or machine, unless limited to non-transitory and tangible "storage" media, refers to any medium involved in providing instructions to a processor for execution.

[0169] Accordingly, machine-readable media such as computer-executable code can take many forms including, but not limited to, tangible storage media, carrier wave media, or physical transmission media. Non-volatile storage media includes, for example, optical disks or magnetic disks such as any computer storage device that can be used to implement a database shown in the drawings. Volatile storage media includes dynamic memory such as the main memory of such a computer platform. Tangible transmission media includes coaxial cables, i.e., copper wires and optical fibers including the wires that make up the buses within a computer system. Carrier wave transmission media can take the form of electrical or electromagnetic signals, or acoustic or light waves such as those generated during radio frequency (RF) and infrared (IR) data communications. Accordingly, common forms of computer-readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tape, any other magnetic media, CD-ROM, DVD, or DVD-ROM, any other optical media, punched card paper tape, any other physical storage media with patterns of holes, RAM, ROM, PROM, and EPROM, FLASH-EPROM, any other memory chip or cartridge, carrier waves carrying data or instructions, cables or links carrying such carrier waves, or any other media that a computer can read programming code and / or data from. Many of these forms of computer-readable media can be involved in carrying one or more sequences of one or more instructions to a processor for execution.

[0170] Computer system 101 includes, or can communicate with, an electronic display 135 having a user interface (UI) 140. Examples of UIs include, but are not limited to, graphical user interfaces (GUIs) and web-based user interfaces.

[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 a central processing unit 105. The algorithms may implement, for example, the methods for performing non-classical computations described herein.

Example

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

[0173] In FIG. 10A, a toy model was used to show the shifts of three related nuclear spin states, namely, the m that makes up the qubit subspace F = 9 / 2 and 7 / 2 levels, and the leakage level 5 / 2. Here, the operation of a single circularly polarized global ac Stark beam was simulated for an array of atoms in a 700 Gauss magnetic field. Further, a polarization purity of 100:1 was assumed for the intended circular polarization. Each time the AC Stark beam 1 From S0 3 detuned from the S0 to F P1 resonance, a shift occurred for each nuclear spin level. For greater clarity, the qubit frequency (the difference in dressed energies between m F = 9 / 2 and m F = 7 / 2) and the leakage transition frequency (between m FBoth (the difference in the dressed state of =5 / 2) were plotted.

[0174] Figure 10B shows that the Stark shift has significantly shifted the leakage transition while minimizing the impact on the qubit frequency. This can be made possible by the narrow linewidth of the P1 resonance compared to the level splitting in a high magnetic field. Although the frequency was plotted as a signed quantity, due to subtle points related to the quantization axis and light transmission, the absolute value of this frequency becomes appropriate, and such features appear where the Stark shift brings the leakage state closer to the qubit frequency. For each detuning, the achievable maximum available Rabi frequency can be defined considering frequency congestion. Using this two-photon Rabi frequency, the π-pulse time can be inferred, and the number of scattering events generated by the off-resonant interaction of the AC Stark beam can be examined (Figure 10A). 3 Here, since Raman scattering and Rayleigh scattering were not distinguished, it is assumed to be the worst-case scenario of scattering errors due to AC Stark per gate. To execute a single-qubit gate, light was coherently controlled to

[0175] operate two-photon transitions using two beams detuned from the P1 resonance. 3 Residual scattering from any of the P1 manifold states may be inherently low because the linewidth of the transition is 7 kHz. Including the effect of the AC Stark shift beam, 3 3 ​By utilizing the broadening of the P1 ultrafine magnetic sublevels, the energy scale between the AC Stark beam detuned from the F = 11 / 2 manifold and the multi-photon 1Q light detuned from the F = 7 / 2 manifold can be separated. A simple toy model containing two ground states and several excited states was sufficient to gain insights into the scaling of power, spot size, and achievable Rabi rates. However, since there are numerous levels (1S0(F = 9 / 2), 3P1(F = 7 / 2, 9 / 2, 11 / 2)) including all magnetic sublevels, it may be necessary to perform a full simulation including all relevant levels. To verify the complete operation, a numerical model representing both desired and undesired polarizations was constructed using all 40 levels with multiple optical fields. Using a simple square pulse, it was found that transitions to other nuclear spin states can be suppressed by the AC Stark beam (Figs. 11A and 11B).

[0176] Example 2: Light trapping array Figs. 12A and 12B show arrays of trapped light generated by an SLM such as a square array and an arbitrary array. The hologram is light at 813 nm from a spatial light modulator (SLM) ([[]] 1 S0→ 3It was generated by reflecting the magic wavelength of the P0 transition. The active area of the SLM was an array of 1920×1152 square pixels with a side length of approximately 9 microns. Each pixel contained a large amount of liquid crystal that imparted a phase shift to the incident light. This phase shift could be controlled by the voltage applied to the pixel, and in this way, an arbitrary pixelated phase mask could be generated and applied to any unstructured light incident on the surface of the SLM. The SLM was arranged such that a large collimated beam was incident and phase-shifted, and the light reflected from the SLM was directed towards the objective lens of the microscope. This configuration connected the plane of the SLM to the plane under the lens (where the atomic cloud was formed) by Fourier conjugation. The complex in-plane electric field at the SLM was the Fourier transform of a similar electric field in the volume of the glass cell and in the plane under the microscope objective lens. The atoms experienced a trapping potential proportional to the intensity of the electric field and thus experienced lateral confinement. The longitudinal confinement occurred because the structured light passed through the focus, and the position of the focus was partially determined (and thus controllable) by the SLM.

[0177] The light was generated by a titanium-sapphire laser that produced an optical output of approximately 4 W at 813 nm. For imaging and other purposes, 2000 traps were generated at a depth of 500 microkelvins each, far exceeding 1000 times the recoil energy imparted by photon scattering. This means that the device needs to be in a regime where it can measure the atoms hundreds of times without being lost due to heating without additional cooling. When cooled to the ground state of motion, the position of the atoms is known within 20 nm, which allows for a large separation of scales between the position of the atoms and the size of the laser beam used to drive the single and two-qubit gate or Rydberg interaction length scale. Since the laser beam driving the gate operation has a spatial spread on the order of 1 micron, the intensity varies at a level of 10 -5 and thus it is expected that a fidelity of 0.9999 can be easily achieved. In this way, the fidelity of the gate is less affected by the position of the atoms.

[0178] Example 3: Ultra-high vacuum A quartz cuvette cell made of Spectrosil (registered trademark) 2000 quartz glass was used as the vacuum cell. Unlike borosilicate glass, this glass does not fluoresce under UV illumination. The cell was characterized by the transition from glass to metal from quartz to stainless steel connecting the cell to the vacuum pump and the atomic source. The dimensions of the cell were selected to avoid clipping of the laser cooling beam and to reduce the numerical aperture of the microscope objective lens. 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 reflection for both S-polarized and P-polarized light at normal incidence angles from 300 nm to 850 nm. The small square window of the cell was coated with magnesium fluoride. The vacuum system maintained a pressure of 8×10 -12 Torr (1.07×10 -9 Pa) for several months.

[0179] Example 4: Microscope objective lens A microscope objective lens placed 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 the atoms during imaging and converts the collimated input beam into a well-focused spot for trapping the atoms on the focal plane. The objective lens was manufactured by Special Optics Inc. and has a high NA (0.65) and a diffraction-limited field of view (FOV) of 300 μm, and shows a transmittance of 90% at 461 nm and 813 nm. The tip of the objective lens facing the vacuum cell is tapered so that two of the six laser cooling beams are not clipped. 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 because it is non-magnetic and non-conductive.

[0180] The performance of the objective lens was characterized by placing the objective lens and a glass cell window in one arm of a Michelson interferometer. In this arm, the focused beam was retroreflected using a precision ball bearing centered at the beam focus. A reference reflector was held in the other arm of the Michelson. The Zernike surface was reconstructed by fitting the resulting spatial interference pattern. The objective lens was directly attached to the glass cell, eliminating tilt drift between the cell window and the objective lens. Such a tilt of about 1 milliradian (mrad) would otherwise cause fluctuations in the wavefront quality. The objective lens was adhesively bonded with epoxy resin to a machined Macor mount that contacted the upper window of the cell via five brass ball bearings. During this assembly process, the objective lens was aligned interferometrically so that the optical axis was kept perpendicular to the cell.

[0181] Three custom dichroic mirrors from Perkins were used to process four significantly different wavelengths (813 nm, 689 nm, 461 nm, and 319 nm) with the objective lens. FIG. 13 shows the optical system for delivering the four different wavelengths. The three dichroic mirrors are shown as DM01, DM02, and DM03. Note that light at 319 nm enters from the bottom of the cell. The custom coatings of the three dichroic mirrors operate in concert to maintain any polarization state of the light at 813 nm and 689 nm, performing single qubit or multiqubit gates and magic wavelength and / or magnetic angle trapping.

[0182] Example 5: Atom Trapping and Cooling FIG. 14 shows the trapping and cooling of strontium 87 and strontium 88 atoms using a red MOT.

[0183] Example 6: Imaging To perform projective measurements, strontium 87 1 S0→ 1Apply light resonant with the P1 transition to the entire atomic array and collect and image the resulting atomic fluorescence. 1 In the case of a qubit containing two nuclear spin states (both resonant with the imaging light) in the S0 ground state manifold, one of the two states can be metastable 3 and moved to the P0 manifold and then measured. This procedure is state-selective and identical to the operation of an optical lattice clock 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 hereby incorporated by reference in its entirety for all purposes. This provides the additional advantage of reduced 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 to generate an image of the qubit array that is processed to determine the state of each atom. Such an image also serves to determine whether an atom has been lost from the array. Since the microscope objective is diffraction-limited across the entire atomic array, atoms separated by several microns are well resolved.

[0184] Example 7: Single-Qubit Gate Optical Delivery The single qubit scheme is specifically designed to enable addressability of single sites. 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 tweezer trapping potential. As described herein, three dichroic mirrors couple all the relevant beams at the back focal plane of the objective lens. These beams are 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 π-polarized as it is aligned with the atomic quantization axis and the other beam is σ-polarized). To fully control the single qubit operations, it is necessary to control the amplitude, frequency, and phase of each beam at the individual trapping sites. This control is obtained by a combination of electro-optic modulators (EOMs), acousto-optic deflectors (AODs), and RF control electronics.

[0185] The light used to drive single qubit gates is provided by a common amplified laser source phase-locked to an optical frequency comb. Although the overall phase of this light cannot be controlled in each experiment, the laser is a stable local oscillator source and can be modulated with a properly controlled RF source to generate the control fields. This global phase sets the global phase of the qubit array, which cannot be measured without comparison to an independent qubit array. To obtain maximum flexibility, an electro-optic modulator (EOM) is used to globally phase-modulate the red MOT light, optical pumping, sideband cooling, and 689 nm light used for single qubit operations. This is because these four operations are not normally performed simultaneously. The phase modulation generates symmetric sidebands around the central laser frequency. 3 Detuning of the laser from the P1 manifold is such that only the +1st order sideband is narrow 3 enough to the P1 transition and is selected to drive the transition. By changing the frequency of this modulation between 5 GHz and 13 GHz, this light can be used even when a large bias field is used to split the excited state manifold.3 All transitions of the P1 manifold can be treated resonantly.

[0186] The main advantage of this method of generating 689 nm light is that it uses the same beam path to generate light in all four of the above beam paths. Further, 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 and controls the complex pulse shape of the laser. In the case of qubit operations, this global control is used to generate pulses of any shape with favorable spectral characteristics.

[0187] Example 8: Parallel addressing of single qubits Using an acousto-optic deflector (AOD), by driving the AOD at different frequencies, a beam can be generated that can be directed to different sites within the qubit array. This introduces position-dependent frequency and phase matching conditions. In the case of single qubit operations, this complexity is overcome by using the same AOD path for two beams, while the detuning of the intermediate state changes, the driven two-photon process remains resonant. In other words, by selecting the specific site to be addressed, the four AOD frequencies are completely restricted. Two frequencies select the position of the first beam, and due to the frequency matching condition, the two frequencies of the second beam become the same up to the offset of the qubit frequency (the splitting between two nuclear spin states, about 150 kHz). When using the AOD to generate a beam for single qubit operations, atoms within an arbitrary single row (or column) can be arbitrarily addressed at any given time. This is necessary to maintain full control of the respective amplitudes and phases. This partially serializes the operations. However, the speed at which the pattern can be changed with the AOD is significantly improved compared to the SLM and is much more efficient than the DMD. Using the AOD also enables full phase control of each beam. This allows not only tracking the phase of each qubit (applying all rotations within the local qubit frame), but also using it to execute more complex pulse sequences on each qubit. By controlling the amplitude of the RF for each qubit, the pulse area of each qubit operation can be locally scaled. By combining both the phase and amplitude of the RF, the operations performed on each qubit during a single pulse from the EOM can be fully controlled.

[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 about 218 nm. Alternatively, the Rydberg excitation operation can be performed using two-photon excitation that combines light at 689 nm and 319 nm, each detuned from the intermediate 3 P1 state. 3The approximately 7 kHz width in the P1 state provides an effective balance between the two-photon effective Rabi rate and 3 scattering due to spontaneous decay from P1. FIG. 15A shows the energy level structure for single qubit and multiqubit operations in strontium 87.

[0189] The optical system for single qubit operations is designed to also function well for multiqubit gates. One of the single qubit beams is used as one leg of a two-photon excitation scheme that drives transitions to the Rydberg electron manifold. The AOD is also used for UV light to satisfy the spatial-dependent frequency and phase matching conditions. Importantly, the optical system is adjusted such that the frequency shift of the UV light from one site to another is the same as that of the 689 nm light. As a result of this constraint, the performance of state-of-the-art UV AODs determines the accessible field of view (FOV) for multiqubit operations. Furthermore, since one of the single qubit beams is used for multiqubit operations (and the two single qubit beams are coincident), the FOVs for single qubit operations are the same. The figure of merit of the UV AOD is the product of the active numerical aperture and the RF bandwidth of the device. For a fixed beam size at the back focal plane of the objective lens, increasing either of these quantities increases the beam scan angle and the FOV in the plane of the qubit array. A FOV of approximately 100 μm × 100 μm was achieved, which is sufficient to accommodate an array of approximately 1,000 atoms with a trapping site spacing of 3 μm.

[0190] Figure 15B shows an optical system for transmitting light and performing single-qubit and multi-qubit operations in parallel on a plurality of trapped atoms. A first light for performing a single-qubit operation on a first qubit (qubit 1) is directed to a first two-dimensional acousto-optic deflector (2D AOD) to enable parallel addressing of a first subset of the trapped atoms. A 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 the trapped atoms. A third light for inducing a Rydberg interaction in either the first subset or the second subset is delivered via a third 2D AOD to create a plurality of entanglements between the atoms of the first subset and the adjacent atoms of the second subset.

[0191] The third light is generated by an ultraviolet (UV) laser that emits light at 319 nm. The UV laser is phase-locked to a frequency comb to provide a narrow-linewidth UV laser beam. Amplitude control is provided by an acousto-optic modulator (AOM). Global phase control is achieved by an optical phase stabilization technique. The stabilized global phase of the 319-nm light is combined with an active phase modulation of the 689-nm light to provide phase control. The free-space beam is transmitted to the third 2D AOD but in the opposite direction from the first and second 2D AODs. The light is directed at the trapped atoms via a customized microscope objective. The beam paths propagating in opposite directions are used to monitor the position of the spots and the effect of the light on the atoms (e.g., via excitation-loss spectroscopy) and optimize the alignment. These quantitative effects can also be used in the implementation of an automatic alignment scheme that enables improvement of the self-automated operation of the system.

[0192] FIG. 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 is kept constant each time the beam overlaps with the qubit array. The difference in frequency prevents driving of unwanted operations, but is easily overcome by the RF drives of the two EOMs. Combining the AODs with an agile RF synthesizer provides the key advantage of being able to fully control, site by site, operations for performing a series of quantum operations on an array of atomic qubits that can be executed in parallel (one row at a time).

[0193] Example 10: Inverse adiabatic drive In the absence of the pulse sequences described herein, multiqubit operations can be performed by adiabatically changing the Hamiltonian so as to minimize the adiabatic transition to the Rydberg state, transferring the ground state atoms to the dressed state, and then back to the ground state. The adiabatic conditions impose limitations and the multiqubit operations are relatively slow. However, faster gates are needed for overall speed and minimization of the decoherence effect. The pulse sequences described herein can achieve faster gates while effectively maintaining adiabatic dynamics.

[0194] For example, inverse adiabatic driving can shorten the gate time while minimizing errors arising from the transition to the Rydberg state. Counterdiabatic driving is to add one or more driving fields to cancel out the terms of the Hamiltonian that cause unwanted adiabatic transitions. Inverse adiabatic driving achieves effective adiabatic dynamics on a time scale shorter than the time scale allowed by adiabatic conditions. As an example, there is the "transitionless quantum driving" (TQD) described in this specification. TQD is achieved by transforming the total Hamiltonian of the system into a reference frame defined by the instantaneous eigenstates of the Hamiltonian. The Hamiltonian is divided 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 an additional control field that cancels the off-diagonal adiabatic Hamiltonian. Using this method, effective thermodynamics can be realized without satisfying the normal slow adiabatic conditions. The following is the derivation of the TQD conditions for a general two-level system with uniaxial driving that uses TQD to cancel the adiabatic transition of the Rydberg dressing gate.

[0195] A general problem is to convert a two-level system in the ground state |1> into a dressed state that is a mixture of |1> and the excited state |R>, and to return to the ground state as quickly as possible without leaving a population in the excited state. In the rotating coordinate system, the total Hamiltonian (in units of frequency) of the two-level system during driving is as follows. (1) H0 = Ω(t)δ x +Δ(t)δ z

[0196] Here, Ω is the Rabi rate, Δ is the detuning from resonance, and σ x and σ z are the Pauli operators of the two-level system. It is convenient to write the Hamiltonian in an "tilted coordinate system". (2) H’0 = Ω eff (t)δ z’

[0197]

Number

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

[0199] Next, convert to the "adiabatic frame" described by these instantaneous eigenstates. The unitary operator corresponding to this conversion is

[0200]

Number

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

[0202]

Number

[0203] The second term (W(t)) contains off-diagonal elements that cause transitions when the adiabatic condition is not satisfied. When the change in U(t) is slow enough for W(t) to be made sufficiently small, the adiabatic condition is satisfied. To achieve effective adiabatic dynamics when this term is not small, an additional control field H c (t) is added to the original Hamiltonian to cancel out the effect of W(t). This can be achieved by

[0204]

Number

[0205] Solving for U(t),

[0206]

Mathematics

[0207] Using the above definition of U(t), it can be described in matrix form.

[0208]

Mathematics

[0209] Simplify the expression again.

[0210]

Mathematics

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

[0212] To demonstrate the effectiveness of the non-transition quantum drive of the Rydberg dressing gate, a two-atom system was simulated. Each atom was composed of two basis (qubit) states and the Rydberg state. The two-atom states could be the qubit states |00>, |01>, |10>, or |11>, the single-excitation Rydberg states |0R>, |R0>, |R1>, or |1R>, or the double-excitation Rydberg state |RR>. In the presence of the 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 from a superposition of qubit states) into a mixture of qubit states and Rydberg states. The atom pair can acquire a conditional phase depending on the initial state such that the two-atom state. At the end of the gate, 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 in the initial two-atom state |00>. By driving the transition from |0> to |r> in each atom, sweeping the detuning from resonance, and moving away from resonance, the instantaneous eigenstate of the Hamiltonian is converted from the bare state to the dressed state and back to the bare state. As shown in Figure 16A, if the ramp is executed too quickly and violates the adiabatic condition, a significant population remains in the Rydberg state |r0>.

[0213] Figure 16B shows a simulation of two atoms in the initial two-atom state |00>, with an additional inverse adiabatic drive field applied to execute a non-transition quantum drive gate. The population remaining in the Lieb state decreases significantly. Figure 16C is a diagram showing a numerical simulation of a two-atom system with a Lieb blockade passing through a Lieb dressing gate, with the case where an inverse adiabatic drive field configured to achieve TQD is added (with TQD) and the case where it is not added (without TQD). In this example, for the initial state |01>, the remaining population in the Lieb state can be reduced by the application of TQD. In this example, the gate parameters can be selected to emphasize the violation of adiabatic conditions in order to demonstrate the effect of TQD. In some cases with more than about two levels (e.g., a nine-level system as described elsewhere in this specification), it may be possible to achieve a quantum drive with substantially no transitions in multiple transitions. Such a drive can be observed even in the presence of different Rabi rates.

[0214] Counter-adiabatic driving can also be used to suppress unwanted transitions at frequencies other than the driving frequency. This helps drive transitions on resonance while avoiding driving nearby unwanted transitions. Alternatively, off-resonant driving can be used to create dressed states while avoiding excitation to the excited state (i.e., adiabatic transitions). An example of counter-adiabatic driving to suppress unwanted transitions is "differential removal by adiabatic gates" (DRAG) described herein. FIG. 16D is a diagram showing an example of a DRAG pulse in the time domain (a) and the frequency domain (b). The off-resonant DRAG pulse can be used to generate a controlled phase gate for a Rydberg atom. A pulse with a smooth amplitude profile can be applied at an off-resonant frequency from the qubit transition to the Rydberg transition. Application at an off-resonant frequency can put the qubit atoms into a Rydberg dressing state or remove them from the Rydberg dressing state. The DRAG pulse can be designed to minimize transitions driven at a specific frequency offset from the pulse carrier frequency. By adjusting the position of the spectral hole in the frequency spectrum of the pulse shape, the gate error due to the population left in the Rydberg state can be minimized. FIG. 16E is a diagram showing a simulated example of optimizing the DRAG pulse parameters to minimize errors. The spectral hole position with the minimum error can be used as the spectral hole position, thus improving the overall fidelity of the system.

[0215] Example 11: Rearrangement of Atoms To determine the time required to perform the atomic rearrangement at the light-capturing sites of a 7×7 array, simulations were run. The simulations assumed an imaging system equipped with a Hamamatsu Orca-Fusion CMOS digital camera that uses an external trigger in normal mode. This camera has a region of interest of 2304 (fixed, horizontal) × 256 (vertical) pixels. An exposure of 20 ms, a readout of 4.6 ms (18.65 μs per line for 256 vertical lines), 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 16-bit integer 256×256 array. To determine the capture sites, it is first necessary to use a calibrated image of the fully captured lattice (by averaging over many capture realizations). Figure 17A shows a calibrated image of a fully filled 7×7 array of light-capturing sites. The light-capturing sites are indexed by coordinates (i,j). This data was used for the mapping from the capture sites to the pixel positions as shown in Table 1.

[0217] [Table 1]

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

[0219] Once the filled and unfilled sites were identified, the next step was to determine the movements to fill the un-captured sites. This is a combinatorial optimization problem classified as bipartite matching. This can be solved by setting up an adjacency matrix that allows efficient finding of an optimal matching using an algorithm such as the Hungarian matching algorithm described herein. The adjacency matrix d i,j is constructed, where row i is indexed by the target sites within the N×N active area and column is indexed by the available sites within the complete M×M grid. For example, in the case of a 7×7 array (M = 7), the atoms can move within a 5×5 computationally active area (N = 5). Table 2 shows the entries of the adjacency matrix.

[0220]

Table 2

[0221] If the distance metric is the square of the distance between the target (i target , j target ) and the filled site (i filled , j filled ), the resulting matching generates a collision-free movement of the atoms from the filled light trapping sites to the unfilled light trapping sites. Figure 17E shows the movement of the atoms from the filled light trapping sites to the unfilled light trapping sites that avoids collisions between the atoms.

[0222] As shown in Table 3 below, the movements were split into independent subsets and ordered chronologically so that they could be easily parallelized. The process of determining the movements took approximately 8 ms.

[0223]

Table 3

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

[0225] Preferred embodiments of the invention have been shown and described herein. It will be apparent to those skilled in the art that such embodiments are provided by way of example only. Those skilled in the art will envision numerous variations, modifications, and substitutions without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. The following claims define the scope of the invention, and it is intended that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

1. A system for performing non-classical calculations, 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, and one or more controllers communicatively coupled to the one or more readout units, the one or more controllers being configured to: (i) instruct the one or more capture units to capture a plurality of atoms, which are a plurality of qubits; (ii) instruct the one or more first electromagnetic delivery units to apply first electromagnetic energy to one or more of the plurality of atoms, whereby the one or more atoms adopt one or more superposition states between a first atomic state and at least a second atomic state different from the first atomic state; (iii) instruct 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 multiqubit units comprising two or more qubits, wherein the instruction in (iii) comprises implementing one or more multiqubit gate operations on the one or more multiqubit units by parallel addressing each qubit of the two or more qubits with a second electromagnetic energy comprising one or more pulse sequences for 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 (iv) instruct the one or more readout units to perform one or more measurements of the one or more superposition states to obtain the non-classical calculation. A system configured to perform the above.

2. The system according to claim 1, wherein the one or more pulse sequences include one or more members selected from the group consisting of a shortcut to adiabaticity (STA) pulse sequence, a non-adiabatic quantum drive (TQD) pulse sequence, a super-adiabatic pulse sequence, a reverse adiabatic drive pulse sequence, and a derivative removal by adiabatic gate (DRAG) pulse sequence.

3. The system according to claim 1, wherein the first electromagnetic delivery unit and the second electromagnetic delivery unit are the same.

4. The system according to claim 1, wherein the first electromagnetic delivery unit and the second electromagnetic delivery unit are different.

5. The system according to claim 1, wherein the one or more pulse sequences further include one or more optimal control pulse sequences.

6. The system according to 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) method, Krotov method, chopped base method, chopped random basis (CRAB) method, Nelder-Mead method, gradient optimization using parametrization (GROUP) method, genetic algorithm method, and gradient optimization of analytic controls (GOAT) method.

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

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

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

9.

10. The system according to any one of claims 1 to 9, wherein the one or more multiqubit gate operations include a fidelity of up to 0.9999.

11. The system according to claim 1, wherein the one or more Rydberg units are configured to electronically excite at least one of the one or more atoms in the one or more superposed states to a Rydberg state, thereby forming one or more Rydberg atoms, or the one or more Rydberg units are configured to continuously drive the one or more atoms in the one or more superposed states to a Rydberg state via a transition, thereby forming one or more dressed Rydberg atoms.

12. The system according to claim 11, wherein the one or more Rydberg units are configured to induce one or more quantum mechanical entanglements between the one or more Rydberg atoms or the one or more dressed Rydberg atoms and the at least another atom, and the at least another atom is located at a distance of 10 micrometers (μm) or less from the one or more Rydberg atoms or the one or more dressed Rydberg atoms.

13. The system according to claim 12, wherein the one or more Rydberg units are configured to drive the one or more Rydberg atoms or the one or more dressed Rydberg atoms to a low-energy atomic state, thereby forming the one or more multiqubit units.

14. The system according to any one of claims 1 to 13, wherein the one or more trapping units include one or more optical trapping units.

15. The system according to claim 14, wherein the one or more optical trapping units are configured to generate a plurality of spatially different optical trapping sites.

16. The system according to claim 15, wherein each optical trapping site of the plurality of spatially different optical trapping sites is configured to trap at most one of the plurality of atoms.

17. The system according to any one of claims 1 to 16, wherein one or more of the multiqubit units include two-qubit units.

18. The system according to claim 17, wherein one or more of the multiqubit gate operations include two-qubit gate operations.

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