Scalable neutral atom based quantum computation

Abstract

To provide a method and system for executing non-classical computation.SOLUTION: This method includes: generating a plurality of spatially distinct optical trapping sites for trapping a plurality of atoms; using one or more electromagnetic delivery units to apply electromagnetic energy to one or more atoms of the plurality, inducing the atoms to adopt one or more superposition states of a first atomic state and a second atomic state; using one or more entanglement units to quantum-mechanically entangle at least a subset of one or more atoms in one or more superposition states with at least another atom of the plurality; and using one or more readout optical units to perform measurements of the superposition states to obtain non-classical computation.SELECTED DRAWING: Figure 6

Description

[Technical field]

[0001] CROSS-REFERENCE This application is a continuation-in-part of U.S. Provisional Patent Application No. 62 / 760,781, filed November 13, 2018, entitled "Scalable neutral atom based quantum computing," and U.S. Provisional Patent Application No. 62 / 815,985, filed March 8, 2019, entitled "Scalable neutral atom based quantum computing," which claims the benefit of U.S. Provisional Patent Application No. 16 / 405,877, filed May 7, 2019, entitled "Scalable neutral atom based quantum computing," which are hereby incorporated by reference in their entireties for all purposes.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH This invention was made with United States Government support under the Small Business Innovation Research Program Grant No. 1843926 awarded by the National Science Foundation. The United States Government has certain rights in this invention. [Background technology]

[0003] Quantum computers typically use quantum-mechanical phenomena such as superposition and quantum entanglement to perform operations on data. Quantum computers may differ from digital electronic computers that are based on transistors. For example, digital computers require data to be coded into binary digits (bits), each of which can always be in one of two finite states (0 or 1), whereas quantum computing uses quantum bits (qubits), which can be in a superposition of states. Summary of the Invention

[0004] SUMMARY OF THE DISCLOSURE There is a need herein for methods and systems for performing non-classical computations.

[0005] The present disclosure provides systems and methods for utilizing atoms (such as neutral or uncharged atoms) to perform non-classical or quantum computations. The atoms may be optically trapped in large arrays. The quantum mechanical states of the atoms (such as the atomic hyperfine or nuclear spin states) may be configured to serve as quantum bit (qubit) basis states. The qubit states may be manipulated through interaction with optical, radio frequency, or other electromagnetic radiation to perform non-classical or quantum computations.

[0006] In one aspect, the present disclosure provides a system for performing non-classical computation, the system including one or more optical trapping units configured to generate a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms; one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state; one or more entanglement units configured to quantum entangle at least a subset of the one or more atoms of the superposition states with at least another atom of the plurality of atoms; and one or more readout optical units configured to perform one or more measurements of the one or more superposition states to obtain a non-classical computation. The non-classical computation may include quantum computation. The quantum computation may include gate model quantum computation. The one or more atoms of the plurality of atoms may include a qubit. The first atomic state may include a first single qubit state and the second atomic state may include a second single qubit state. The first atomic state or the second atomic state may be elevated in energy relative to a ground atomic state of the atom. 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. 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 plurality of atoms may include at least 100 atoms. The plurality of atoms may include neutral atoms. The plurality of atoms may include rare earth atoms. The plurality of atoms may include alkali atoms. The plurality of atoms may include alkaline earth atoms. The alkaline earth atoms may include strontium atoms. The strontium atoms may include strontium-87 atoms. The first and second atomic states are strontium-87 3The P1 manifold may include a first hyperfine state and a second hyperfine state. The first atomic state and the second atomic state may be strontium-87 3The first and second atomic states may include a first hyperfine state and a second hyperfine state on a P2 manifold. The first and second atomic states may include a first hyperfine state and a second hyperfine state on a multiplet manifold. The first and second atomic states may include a first hyperfine state and a second hyperfine state on a triplet manifold. The first and second atomic states may include a first and a second nuclear spin state of a quadrupolar nucleus. The first and second atomic states may include a first and a second nuclear spin state of nuclear spin 9 / 2. The first and second atomic states may include a first and a second nuclear spin state of strontium-87. The subset of one or more atoms in the one or more superpositions and another atom may be quantum entangled with a coherence lifetime of at least 1 second. The plurality of atoms may include a temperature of up to 10 microkelvin (μK). The system may further include one or more vacuum units configured to maintain the system at a pressure of up to 10-6 Pascal (Pa). Each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by at least 200 nanometers (nm). Each optical trapping site of the plurality of optical trapping sites may be configured to trap a single atom of the plurality of atoms. The one or more optical trapping sites may include one or more optical tweezers. The one or more optical trapping sites may include one or more optical lattice sites of one or more optical lattices. The one or more optical lattices may include one or more members selected from the group consisting of a one-dimensional (1D) optical lattice, a two-dimensional (2D) optical lattice, and a three-dimensional (3D) optical lattice. The one or more optical trapping units may include one or more spatial light modulators (SLMs) configured to generate the plurality of optical trapping sites. The one or more SLMs may include one or more Digital Micromirror Devices (DMDs) or one or more Liquid Crystal On-Site (LCoS) devices. The one or more optical trapping units may include one or more light sources configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms.The one or more optical trapping units may include one or more imaging units configured to obtain one or more images of a spatial shape of the plurality of atoms trapped within the optical trapping site. The one or more images may include one or more members selected from the group consisting of a fluorescence image, a single atom fluorescence image, an absorption image, a single atom absorption image, a phase contrast image, and a single atom phase contrast image. The system may further include one or more spatial shape artificial intelligence (AI) units configured to perform one or more AI operations to determine a spatial shape of the plurality of atoms trapped within the optical trapping site based on the one or more images. The one or more AI operations may include one or more machine learning (ML) operations. The one or more AI operations may include one or more reinforcement learning (RL) operations. The one or more optical trapping units may include one or more atom relocation units configured to impart an altered spatial arrangement of the plurality of atoms trapped at the optical trapping site based on the one or more images. The system may further include one or more spatial arrangement artificial intelligence (AI) units configured to perform one or more AI operations to determine an altered spatial arrangement of the plurality of atoms trapped within the optical trapping site based on the one or more images. The one or more AI operations may include one or more machine learning (ML) operations. The one or more AI operations may include one or more reinforcement learning (RL) operations. The one or more atom rearrangement units may be configured to alter the spatial arrangement to obtain an increased fill factor of the plurality of optical trapping sites. The fill factor may include a value of at least 70%. The system may further include one or more state preparation units configured to prepare a state of the plurality of atoms. One or more of the state preparation units may be configured to cool the plurality of atoms. The one or more state preparation units may be configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites. One or more of the conditioning units may include a Zeeman reducer configured to reduce the one or more atoms from a first velocity or distribution of velocities to a second velocity that is lower than the first velocity or distribution of velocities. The Zeeman reducer may include a one-dimensional (1D) Zeeman reducer. The second velocity may be up to 10 meters per second (m / s).One or more of the condition preparation units may further include a first magneto-optical trap (MOT) configured to cool one or more atoms to a first temperature. The first MOT may include a three-dimensional (3D) MOT. The first MOT may include one or more light sources configured to emit light having one or more wavelengths in a range of 400 nm to 500 nm. The first temperature may be at most 10 millikelvin (mK). One or more of the condition preparation units may further include a second MOT configured to cool one or more atoms from the first temperature to a second temperature lower than the first temperature. The second MOT may include one or more light sources configured to emit light having one or more wavelengths in a range of 400 nm to 1,000 nm. The second temperature may be at most 100 microkelvin (mK). One or more of the condition preparation units may further include a sideband cooling unit. The sideband cooling unit may be configured to use sideband cooling to cool the one or more atoms from the second temperature to a third temperature lower than the second temperature. The sideband cooling unit may include one or more light sources configured to emit light having one or more wavelengths in a range of 400 nm to 1,000 nm. The third temperature may be at most 10 microkelvin (mK). One or more of the state preparation units may include an optical pumping unit configured to emit light to optically pump one or more atoms of the plurality of atoms from an equilibrium atomic state to a non-equilibrium atomic state. The optical pumping unit may include one or more light sources configured to emit light including one or more wavelengths in a range of 400 nanometers (nm) to 1,000 nm. The light may include one or more wavelengths in a range of 650 nm to 700 nm. One or more of the state preparation units may include a coherent drive unit configured to coherently drive one or more atoms from a non-equilibrium atomic state to a first atomic state or a second atomic state. The coherent drive unit may be configured to induce a two-photon transition between the non-equilibrium state and the first atomic state or the second atomic state.The coherent driving unit may include one or more light sources configured to emit light having one or more wavelengths in a range of 400 nm to 1000 nm. The coherent driving unit may be configured to induce a one-photon transition between a non-equilibrium state and the first atomic state or the second atomic state. The coherent driving unit may include one or more light sources configured to emit light having one or more wavelengths in a range of 400 nm to 1000 nm. The coherent driving unit may be configured to induce a radio frequency (RF) transition between a non-equilibrium state and the first atomic state or the second atomic state. The coherent driving unit may include one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. The one or more electromagnetic delivery units may include one or more spatial light modulators (SLMs), acousto-optical devices (AODs), or acousto-optical modulators (AOMs) configured to selectively apply electromagnetic energy to one or more atoms of the plurality of atoms. The one or more electromagnetic delivery units may include one or more digital micromirror devices (DMDs), or one or more Elcos (LCoS) devices. The system may further include one or more electromagnetic energy artificial intelligence (AI) units configured to perform one or more AI operations to selectively apply electromagnetic energy to the one or more atoms. The one or more AI operations may include one or more machine learning (ML) operations. The one or more AI operations may include one or more reinforcement learning (RL) operations. The electromagnetic energy may include optical energy. The electromagnetic energy may include microwave energy. The electromagnetic energy may include radio frequency (RF) energy. The RF energy may include one or more wavelengths of at least 30 millimeters (mm). The RF energy may include an average power of up to 10 watts (W). The one or more electromagnetic delivery units may be configured to perform one or more single qubit gate operations on the one or more qubits. The one or more readout optical units may include one or more optical detectors. The one or more optical detectors may include one or more cameras. The one or more optical detectors may include one or more fluorescence detectors.The system may further include one or more atom reservoirs configured to provide one or more replacement atoms to replace one or more atoms at one or more of the plurality of optical trapping sites when one or more atoms are lost from the one or more optical trapping sites. The system may further include one or more atom movement units configured to move the one or more replacement atoms to the one or more optical trapping sites. The one or more atom movement units may include one or more electrical. The one or more entanglement units may include a lens, an acousto-optical deflector (AOD), or a spatial light modulator (SLM) that can be adjusted to move the one or more atoms in the one or more superposition states. The one or more subsets of atoms in the one or more superposition states and another atom may be quantum entangled through magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The one or more entanglement units may include one or more Rydberg excitation units configured to electronically excite the one or more subsets of atoms in the one or more superposition states into a Rydberg state or into a superposition of a Rydberg state and a lower energy atomic state, thereby forming one or more Rydberg or Rydberg-less atoms. The one or more Rydberg excitation units may be configured to induce one or more quantum entanglements between the one or more Rydberg or Rydberg-less atoms and another atom, the other atom being located at a distance of at most 10 micrometers (μm) from the one or more Rydberg or Rydberg-less atoms. The one or more Rydberg units may be configured to drive one or more Rydberg atoms or Rydbergless atoms to a lower energy atomic state, thereby forming one or more two-qubit units. The one or more electromagnetic delivery units may be configured to perform one or more two-qubit gate operations on the one or more qubits. The one or more Rydberg excitation units may include one or more light sources configured to emit light having one or more ultraviolet (UV) wavelengths. The light may include one or more wavelengths in a range from 300 nm to 400 nm. The system may be operably coupled to a digital computer over a network. The network may include a cloud network.

[0007] In another aspect, the disclosure provides a non-classical computer comprising a plurality of qubits including more than 60 atoms, each atom being trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, where the plurality of qubits includes at least a first qubit state and a second qubit state, where the first qubit state includes a first atomic state and the second qubit state includes a second atomic state, and one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more qubits of the plurality of qubits. the one or more electromagnetic delivery units, whereby a non-classical operation is imparted to one or more quantum bits, the non-classical operation comprising a superposition between at least a first quantum bit state and a second quantum bit state; one or more entanglement units configured to quantum entangle at least a subset of the plurality of quantum bits in the superposition state with at least another quantum bit of the plurality of quantum bits; and one or more readout optical units configured to perform one or more measurements of the one or more quantum bits, thereby obtaining a non-classical computation.

[0008] In another aspect, the present disclosure provides a non-classical computer, the computer comprising a plurality of qubits, each atom comprising more than 60 atoms, each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites.

[0009] In another aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more optical measurements of the one or more superposition states to obtain a non-classical computation.

[0010] In another aspect, the disclosure provides a method for performing a non-classical computation, the method comprising: (a) providing a plurality of qubits comprising more than 60 atoms, each atom being trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, where the plurality of qubits comprises at least a first qubit state and a second qubit state, where the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; (b) applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, the non-classical operation comprising a superposition between at least the first qubit state and the second qubit state; (c) quantum entanglement of at least a subset of the plurality of qubits in the superposition state with at least another qubit of the plurality of qubits; and (d) performing one or more optical measurements of the one or more qubits, thereby obtaining a non-classical computation.

[0011] In another aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) providing a plurality of qubits including more than 60 atoms each trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites; and (b) using at least a subset of the plurality of qubits to perform a non-classical computation.

[0012] In another aspect, the disclosure provides a method for performing non-classical computation, the method including: (a) providing a plurality of optical trapping sites including a plurality of atoms, the plurality of atoms being a plurality of quantum bits; (b) moving one or more of the plurality of atoms from an occupied trapping site to an unoccupied trapping site, thereby changing a spatial arrangement of the plurality of atoms; (c) applying electromagnetic energy to one or more atoms of the plurality of atoms to induce the one or more atoms to enter one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state, where an atom of the one or more atoms of the one or more superposition states is quantum entangled with another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superposition states.

[0013] In another aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, where the plurality of atoms are quantum bits, where an atom of the plurality of atoms is trapped at an optical trapping site of the plurality of optical trapping sites by an attractive force; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superposition states to obtain a non-classical computation.

[0014] In another aspect, the present disclosure provides a method for performing a non-classical computation, the method comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, where the plurality of atoms are qubits; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state, where the applying comprises modulating the electromagnetic energy with at least two optical modulators; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superposition states to obtain a non-classical computation.

[0015] Another aspect of the present disclosure provides a non-transitory computer-readable medium containing machine-executable code that, when executed by one or more computer processors, performs any of the methods described above or anywhere herein.

[0016] Another aspect of the present disclosure provides a system including one or more computer processors and a computer memory coupled thereto, the computer memory including machine-executable code that, when executed by the one or more computer processors, performs any of the methods described above or anywhere herein.

[0017] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in the art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its various details can be modified in various obvious respects, all without departing from the present disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.

[0018] INCORPORATION BY REFERENCE All publications, patents, and patent applications mentioned in this specification are incorporated herein by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent that a publication, patent, or patent application incorporated by reference conflicts with the disclosure contained herein, the specification is intended to supersede and / or take precedence over such conflicting material. [Brief description of the drawings]

[0019] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also referred to herein as "Figures" and "FIGs.") of the following drawings. [Figure 1] 1 illustrates a computer control system programmed or configured to carry out the methods provided herein. [Diagram 2]1 illustrates an example of a system for performing non-classical computation. [Figure 3A] 1 shows an example of an optical trapping unit. [Figure 3B] 1 shows an example of multiple optical trapping sites. [Figure 3C] 1 shows an example of an optical trapping unit that is partially filled with atoms. [Figure 3D] An example of an optical trapping unit that is completely filled with atoms is shown. [Figure 4] 1 shows an example of an electromagnetic delivery unit. [Diagram 5] 1 shows an example of a state preparation unit. [Figure 6] 1 shows a flowchart of a first example method for performing non-classical computation. [Figure 7] 1 shows a flowchart of a second example method for performing non-classical computation. [Figure 8] 13 shows a flowchart of a third example method for performing non-classical computation. [Figure 9] An example qubit containing the 3P2 state of strontium-87 is shown. [Figure 10A] Stark shift simulation of the 1S0 hyperfine state of strontium-87 is shown. [Figure 10B] Stark shift simulation of the 1S0 hyperfine state of strontium-87 is shown. [Figure 11A] We present a simulation of single-qubit control using the Stark shift. [Figure 11B] We present a simulation of single-qubit control using the Stark shift. [Figure 12A] 1 shows an exemplary arrangement of trapped light generated by an SLM. [Figure 12B] 1 shows an exemplary arrangement of trapped light generated by an SLM. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0020] While 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. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the invention described herein may be utilized.

[0021] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. As used within the specification and the appended claims, the singular forms "a," "an," and "the" include plural references unless the context clearly dictates otherwise. Any reference to "or" is intended to encompass "and / or" unless otherwise specified.

[0022] Whenever the terms "at least," "greater than," or "greater than or equal to" precede the first number in a series of two or more numbers, the terms "at least," "greater than," or "greater than or equal to" apply to each number in the series. For example, 1, 2, or 3 or more means greater than or equal to 1, greater than or equal to 2, greater than or equal to 3, etc.

[0023] Whenever the terms "no more than," "less than," "less than or equal to," or "at most" precede the first number in a series of two or more numbers, the terms "up to," "less than," "less than" and "at most" apply to each number in the series. For example, 3, 2, or 1 or less is less than or equal to 3, less than or equal to 2, and less than or equal to 1.

[0024] When values ​​are listed as ranges, such disclosure will be understood to include disclosure of all possible subranges within such ranges, as well as specific numerical values ​​falling within such ranges, whether or not a specific numerical value or subrange is explicitly stated.

[0025] As used herein, like letters refer to like elements.

[0026] 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 enhance or maximize the chances of successfully achieving a goal. The term "artificial intelligence" may include "generative modeling," "machine learning" (ML), and / or "reinforcement learning" (RL).

[0027] 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 incrementally improves a computer's performance on a task. Machine learning may include a machine learning algorithm. A machine learning algorithm may be a trained algorithm. Machine learning (ML) may include one or more supervised, semi-supervised, or unsupervised machine learning techniques. For example, an ML algorithm may be a trained algorithm that is trained through supervised learning (e.g., where various parameters are determined as weighting or scaling factors). ML may include one or more of regression analysis, regularization, classification, dimensionality reduction, ensemble learning, meta-learning, combination rule learning, cluster analysis, anomaly detection, deep learning, or ultra-deep learning. ML includes k-means, k-means clustering, k-nearest neighbor, learning vector quantization, linear regression, nonlinear regression, least squares regression, partial least squares regression, logistic regression, stepwise regression, multivariate adaptive regression spline, ridge regression, principal component regression, least absolute shrinkage and selection operation, least angle regression, canonical correlation analysis, factor analysis, independent component analysis, linear discriminant analysis, multidimensional scaling, nonnegative matrix factorization, principal component analysis, principal coordinate analysis, projection pursuit, Sammon mapping, t-distributed stochastic neighbor embedding, AdaBoosting, boosting, gradient boosting, bootstrap aggregation, ensemble averaging, decision trees, and conditional decision trees. boosted decision treetree, gradient boosted decision tree, random forest, stacked generalization, Bayesian network, Bayesian belief network, naive Bayes, Gaussian naive Bayes, multinomial naive Bayes, hidden Markov model, hierarchical hidden Markov model, support vector machine, encoder, decoder, autoencoder, stacking autoencoder, perceptron, multi-layer perceptron, artificial neural network, feedforward neural network, convolutional neural network, recurrent neural network, long short-term memory, deep belief network, deep Boltzmann machine These include, but are not limited to, deep neural networks, deep convolutional neural networks, deep recurrent neural networks, or generative adversarial networks.

[0028] As used herein, "reinforcement learning," "reinforcement learning procedure," "reinforcement learning operation," and "reinforcement learning algorithm" generally refer to a system or computational procedure that takes one or more actions to reinforce or maximize some notion of cumulative reward for interacting with an environment. An agent executing a reinforcement learning (RL) procedure can receive positive or negative reinforcement, called an "immediate reward," from taking one or more actions in an environment, thereby placing itself and the environment in various new states.

[0029] An agent's goal may be to enhance or maximize some notion of cumulative reward. For example, an agent's goal may be to enhance or maximize a "discounted reward function" or an "average reward function." A "Q-function" may represent the maximum cumulative reward obtainable from a state and an action taken in that state. A "value function" and a "generalized advantage estimator" may represent the maximum cumulative reward obtainable from a state given an optimal or best choice of action. RL may utilize any one or more of such notions of cumulative reward. As used herein, any such function may be referred to as a "cumulative reward function." Thus, computing the best or optimal cumulative reward function may be equivalent to finding the best or optimal policy for the agent.

[0030] The agent, and its interactions with the environment, may be formulated as one or more Markov Decision Processes (MDPs). The RL procedure may not assume 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 exist in a spectrum between two ranges: "model-based" or "model-free" with respect to the prior knowledge of the MDP. Thus, the RL procedure may target large MDPs, where exact methods may be infeasible or unavailable due to unknown or stochastic nature of the MDP.

[0031] The RL procedure may be implemented using one or more computer processors as described herein. The digital processing device may utilize an agent that trains, stores, and later deploys "policies" to enhance or maximize cumulative reward. Policies may be sought (e.g., searched) over as long a time period as possible, or over a desired time period. Such optimization problems may be solved by storing approximations of optimal policies, storing approximations of cumulative reward functions, or both. In some cases, the RL procedure may store one or more tables of approximations for such functions. In other cases, the RL procedure may utilize one or more "function approximators."

[0032] Examples of functional approximators may include neural networks (such as deep neural networks) and probabilistic graphical models (e.g., Boltzmann machines, Helmholtz machines, and Hopfield networks). The functional approximator may produce an approximate parameterization of the cumulative reward function. Optimization of the functional approximator with respect to the parameterization may be configured to move the parameters in a direction that enhances or maximizes the cumulative reward and thus enhance or optimize the policy (such as by a policy gradient method) or move the functional approximator to satisfy Bellman's optimality criteria (such as by a time difference method).

[0033] During training, the agent may perform actions in the environment to obtain further information about the environment and about the appropriate or best choice of policy for survival or better utility. The agent's actions may be generated randomly (e.g., especially in the early stages of training) or may be prescribed by another machine learning paradigm (such as supervised learning, imitation learning, or other machine learning procedures described herein). The agent's actions may be improved by selecting an action that is closer to the agent's perception of what the augmented or optimal policy is. Various training strategies may lie between two ranges, off-policy and on-policy methods, with regard to the choice between vetting and exploitation.

[0034] As used herein, the terms "nonclassical computation," "nonclassical procedure," "nonclassical operation," and "nonclassical computer" generally refer to any method or system for performing a computational procedure outside the paradigm of classical computing. A nonclassical computation, nonclassical procedure, nonclassical operation, or nonclassical computer may include quantum computation, quantum procedure, quantum operation, or quantum computer.

[0035] As used herein, the terms "quantum computation", "quantum procedure", "quantum operation", and "quantum computer" generally refer to any method or system for performing computations using quantum mechanical operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) in a Hilbert space represented by a quantum device. Thus, quantum computation and classical (or digital) computation may be similar in the following respects: both computations may involve a sequence of instructions performed on input information and subsequently provide an output. Various paradigms of quantum computation may decompose quantum operations into a sequence of elementary quantum operations that simultaneously affect a subset of qubits of a quantum device. Quantum operations may be selected, for example, based on their locality, or on their ease of physical implementation. A quantum procedure or quantum computation may then consist of a sequence of instructions that can represent different quantum evolutions of a quantum device in various applications. For example, a procedure for calculating or simulating quantum chemistry may represent quantum states and annihilation and creation operators of electron spin-orbitals through the so-called Jordan-Wigner or Bravyi-Kitaev transformations by using qubits (such as 2-level quantum systems) and a set of universal quantum gates (such as Hadamard, controlled-not (CNOT), π / 8 rotation, etc.).

[0036] Additional examples of quantum procedures or computations may include procedures for optimization, such as quantum approximate optimization algorithms (QAOA) or quantum minimum confirmation. QAOA may include performing single qubit rotations and multi-qubit entanglement gates. In quantum adiabatic computation, instructions may be passed down a probabilistic or non-probabilistic path of evolution from an initial quantum system to a final quantum system.

[0037] Quantum-inspired procedures may include simulated annealing, parallel tempering, master equation solver, Monte Carlo procedures, etc. Quantum-classical or hybrid algorithms or procedures may include procedures such as the variational quantum eigensolver (VQE) and the variational and adiabatically navigated quantum eigensolver (VanQver).

[0038] The quantum computer may include one or more of an adiabatic quantum computer, a quantum gate array, a one-way quantum computer, a topological quantum computer, a quantum Turing machine, a quantum annealer, an Ising solver, or a gate model of quantum computing.

[0039] A system for performing non-classical computations In one aspect, the present disclosure provides a system for performing a non-classical computation, the system including one or more optical trapping units configured to generate a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms, one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state, one or more entanglement units configured to quantum entangle at least a subset of the one or more atoms of the one or more superposition states with at least another atom of the plurality of atoms, and one or more readout optical units configured to perform one or more measurements of the one or more superposition states to obtain a non-classical computation.

[0040] 2 illustrates an example of a system 200 for performing non-classical computing. The non-classical computing may include quantum computing. The quantum computing may include gate model quantum computing.

[0041] The system (200) may include one or more optical trapping units (210). The optical trapping units may include any optical trapping units described herein, such as the optical trapping units described herein with reference to FIG. 3A. The optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites. For example, the optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites, such as at least 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, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 50,000, 60,000, 70,000, 80,000, 90,000, 10,000, 20,000, 30,000, 50,000, 80,000, 90,000, 10,000, 120,000, 140,000, 150,000, 160,000, 170,000, 180,000, 190,000, 210,000, 220, , 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000 or more optical trapping sites. Optical trap units are available in sizes up to 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10 ,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer optical trap sites. The optical trapping unit may be configured to trap a plurality of optical trap sites within a range defined by any two of the aforementioned values.

[0042] The optical trapping unit may be configured to trap a plurality of atoms. For example, the optical trapping unit may be configured to trap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30, In one embodiment, the nanotube may be configured to trap 1,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. The optical trap unit can accommodate up to approximately 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000 , 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The optical trapping unit may be configured to trap a plurality of atoms within a range defined by any two of the aforementioned values.

[0043] Each optical trapping site of the optical trapping unit may be configured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. Each optical trapping site may be configured to trap up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each optical trapping site may be configured to trap multiple atoms within a range defined by any two of the aforementioned values. Each optical trapping site may be configured to trap a single atom.

[0044] As described herein (e.g., with reference to FIG. 4), one or more of the plurality of atoms may comprise a qubit. The two or more atoms may be quantum entangled. The two or more atoms may be quantum entangled for at least about 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100 μs, 200 μs, 300 μs, 400 μs, 500 μs, 600 μs, 700 μs, 800 μs, 900 μs, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, It may be quantum entangled with a coherence lifetime of 6 ms, 7 ms, 8 ms, 9 ms, 10 ms, 20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms, 300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2 s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or longer. For atoms of 2 or more, the maximum is about 10s, 9s, 8s, 7s, 6s, 5s, 4s, 3s, 2s, 1s, 900ms, 800ms, 700ms, 600ms, 500ms, 400ms, 300ms, 200ms, 100ms, 90ms, 80ms, 70ms, 60ms, 50ms, 40ms, 30ms, 20ms, 10ms, 9ms, 8ms, 7ms, 6ms, 5ms, 4ms, 3ms, 2m s, 1 ms, 900 μs, 800 μs, 700 μs, 600 μs, 500 μs, 400 μs, 300 μs, 200 μs, 100 μs, 90 μs, 80 μs, 70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4 μs, 3 μs, 2 μs, 1 μs, or less. Two or more atoms may be quantum entangled with a coherence lifetime within a range defined by any two of the preceding values. One or more atoms may include neutral atoms. One or more atoms may include uncharged atoms.

[0045] The one or more atoms may include an alkali atom. The one or more atoms may include a lithium (Li) atom, a sodium (Na) atom, a potassium (K) atom, a rubidium (Rb) atom, or a cesium (Cs) atom. The one or more atoms may include a lithium-6 atom, a lithium-7 atom, a sodium-23 atom, a potassium-39 atom, a potassium-40 atom, a potassium-41 atom, a rubidium-85 atom, a rubidium-87 atom, or a cesium-133 atom. The one or more atoms may include an alkaline earth atom. The one or more atoms may include a beryllium (Be) atom, a magnesium (Mg) atom, a calcium (Ca) atom, a strontium (Sr) atom, or a barium (Ba) atom. The one or more atoms may include 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. The one or more atoms may include a rare earth atom. The one or more atoms may include a scandium (Sc) atom, a yttrium (Y) atom, a lanthanum (La) atom, a cerium (Ce) atom, a praseodymium (Pr) atom, a neodymium (Nd) atom, a samarium (Sm) atom, a europium (Eu) atom, a gadolinium (Gd) atom, a terbium (Tb) atom, a dysprosium (Dy) atom, a holmium (Ho) atom, an erbium (Er) atom, a thulium (Tm) atom, an ytterbium (Yb) atom, or a lutetium (Lu) atom.The one or more atoms are: scandium-45, yttrium-89, lanthanum-139, cerium-136, cerium-138, cerium-140, cerium-142, praseodymium-141, neodymium-142, neodymium-143, neodymium-145, neodymium-146, neodymium-148, samarium-144, samarium-149, samarium-150, samarium-152, samarium-154, europium-151, europium-153, gadolinium-154, gadolinium-155, gadolinium-156, gadolinium-157, gadolinium-158, gadolinium-160, 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 The cations may include tungsten-containing atoms such as tungsten-containing fluorine-containing atoms, tungsten-containing atoms, fluorine ...

[0046] The plurality of atoms may include a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may include an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The atoms may 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-8 8 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 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 and may comprise at least about 50%, 60% or more of tungsten-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms. Enriched to 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 higher isotopic abundance. The multiple atoms are lithium-6 atom, lithium-7 atom, sodium-23 atom, potassium-39 atom, potassium-40 atom, potassium-41 atom, rubidium-85 atom, rubidium-87 atom, cesium-133 atom, beryllium-9 atom, magnesium-24 atom, magnesium-25 atom, magnesium-26 atom, calcium-40 atom, calcium-42 atom, calcium-43 atom, calcium-44 atom, calcium-46 atom, calcium-48 atom, strontium-84 atom, strontium-86 atom, strontium-87 atom, strontium-88 atom, barium-130 atom, barium-132 atom, barium-134 atom, barium-135 atom, barium-136 atom, barium-137 atom,Barium-138 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 child, 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, Rubium-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 and atoms enriched to an isotopic 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. The multiple atoms are: lithium-6 atom, lithium-7 atom, sodium-23 atom, potassium-39 atom, potassium-40 atom, potassium-41 atom, rubidium-85 atom, rubidium-87 atom, cesium-133 atom, beryllium-9 atom, magnesium-24 atom, magnesium-25 atom,Magnesium-26 atoms, Calcium-40 atoms, Calcium-42 atoms, Calcium-43 atoms, Calcium-44 atoms, Calcium-46 atoms, Calcium-48 atoms, Strontium-84 atoms, Strontium-86 atoms, Strontium-87 atoms, Strontium-88 atoms, Barium-130 atoms, Barium-132 atoms, Barium-134 atoms, Barium-135 atoms, Barium-136 atoms, Barium-137 atoms, Barium-138 atoms, Scandium-45 atoms, Yttrium -89 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 child, 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-169 atom, erbium-170 atom, erbium-171 atom, erbium-172 atom, erbium-173 atom, erbium-174 atom, erbium-175 atom, erbium-176 atom, erbium-177 atom, erbium-178 atom, erbium-179 ...9 atom, erbium-170 atom, erbium-171 atom, erbium-172 atom, erbium-173 atom, erbium-174 atom, erbium- The isotopic abundance may include rubium-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, which are enriched to an isotopic abundance within a range defined by any two of the preceding values.

[0047] The system (200) may include one or more electromagnetic delivery units (220). The electromagnetic delivery unit may include any electromagnetic delivery unit described herein, such as the electromagnetic delivery unit described herein with reference to FIG. 4. The electromagnetic delivery unit may be configured to apply electromagnetic energy to one or more atoms of the plurality of atoms. The applying of electromagnetic energy may induce the atoms to adopt one or more superpositions of a first atomic state and a second atomic state different from the first atomic state.

[0048] The first atomic state may comprise a first single qubit state. The second atomic state may comprise a second single qubit state. The first atomic state or the second atomic state may be elevated in energy relative to a ground atomic state of the atom. The first atomic state or the second atomic state may be equivalent in energy to a ground atomic state of the atom.

[0049] 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 atomic state and the second atomic state may include the first hyperfine state and the second hyperfine state on a multiplet manifold, such as a triplet manifold. The first atomic state and the second atomic state may include 3 P1 or 3 The P2 manifold may include a first hyperfine state and a second hyperfine state, respectively. The first atomic state and the second atomic state may be strontium-87 3 P1 Manifold or Strontium-87 3 of any atom described herein, such as the P2 manifold. 3 P1 or 3 A first hyperfine state and a second hyperfine state may be included on the P2 manifold.

[0050] Figure 9 shows the 3 The left panel of Figure 9 shows an example of a qubit containing the P2 state. 3The right panel of Figure 9 shows the high energy level structure of the P2 state of strontium-87, which has a low sensitivity (to first order) to magnetic field changes of approximately 70 Gauss. 3 Illustrates potential qubit transitions within the P2 state.

[0051] In some cases, the first and second atomic states are first and second hyperfine states of the first electronic state. Optical excitation may be applied between the first and second electronic states. The optical excitation may excite the first and / or second hyperfine states to the second electronic state. The single qubit transition may include a two-photon transition between two hyperfine states in the first electronic state using the second electronic state as an intermediate state. To drive the single qubit transition, a pair of frequencies may be applied, each detuned from the single-photon transition to the intermediate state, to drive the two-photon transition. In some cases, the first and second hyperfine states are hyperfine states of the ground electronic state. The ground electronic state may not decay by spontaneous or stimulated emission to a lower electronic state. The hyperfine state may include a nuclear spin state. In some cases, the hyperfine state may be a strontium-87 1 The S0 manifold contains nuclear spin states, and the qubit transition is strontium-87 1 One or both of the two nuclear spin states of S0 can be 3 P2 manifold or 3 From the P1 manifold, or 3 P2 manifold or 3 In some cases, the single qubit transition is 3 P2 manifold or 3 From the P1 manifold, or 3 P2 manifold or 3 Strontium-87 via detuned states in the P1 manifold 1The two-photon Raman transition between the nuclear spin states of S0 and S1. In some cases, the nuclear spin states are Stark-shifted nuclear spin states. The Stark shift may be optically driven. The optical Stark shift may be driven off-resonant with any, all, or combination of single qubit transitions, two-qubit transitions, shelving transitions, imaging transitions.

[0052] 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 atomic state and the second atomic state may include a first nuclear spin state and a second nuclear spin state, respectively, of a quadrupolar nucleus. The first atomic state and the second atomic state may include a first nuclear spin state and a second nuclear spin state, respectively, of nuclear spin-1, nuclear spin-3 / 2, nuclear spin-2, nuclear spin-5 / 2, nuclear spin-3, nuclear spin-7 / 2, nuclear spin-4, or nuclear spin-9 / 2. The first atomic state and the second atomic state may include a first nuclear spin state and a second nuclear spin state, respectively, of any atom described herein, such as the first spin state and the second spin state of strontium-87.

[0053] For first and second nuclear spin states associated with nuclei containing spins greater than 1 / 2 (such as nuclear spin-1, nuclear spin-3 / 2, nuclear spin-2, nuclear spin-5 / 2, nuclear spin-3, nuclear spin-7 / 2, nuclear spin-4, or nuclear spin-9 / 2), the transition between the first and second nuclear spin states may involve transitions between other spin states on the nuclear spin manifold. For example, for a 9 / 2 nuclear spin in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, an atom may be separated into, for example, m N = 9 / 2 spin state to m N Transitions (such as Raman transitions) designed to transfer the m = 7 / 2 spin state also N =7 / 2~m N =5 / 2, mN =5 / 2~m N =3 / 2, m N =3 / 2~m N = 1 / 2, m N =1 / 2~m N =-1 / 2, m N =-1 / 2~m N =-3 / 2, m N =-3 / 2~m N =-5 / 2, m N =-5 / 2~m N = -7 / 2mN, and m N =-7 / 2~m N =-9 / 2, where mN is the nuclear spin state. Similarly, an atom may be driven to, for example, m N = 9 / 2 spin state to m N Transitions (such as Raman transitions) designed to transfer the m = 5 / 2 spin state also N =7 / 2~m N =3 / 2, m N =5 / 2~m N = 1 / 2, m N =3 / 2~m N =-1 / 2, m N =1 / 2~m N =-3 / 2, m N =-1 / 2~m N =-5 / 2, m N =-3 / 2~m N = -7 / 2, and m N =-5 / 2~m N =-9 / 2. As such, such transitions may not be selective for inducing transitions between specific spin states on the nuclear spin manifold. Instead, it may be desirable to implement selective transitions between specific first and second spin states on the nuclear spin manifold. This may be accomplished by directing light from a light source that provides an AC Stark shift and pushes adjacent nuclear spin states out of resonance with the transition between the desired transition between the first and second nuclear spin states. For example, m N =-9 / 2 and m N If transitions from the first and second nuclear spin states with m = -7 / 2 are desired, the light will N=-5 / 2 spin state, which provides an AC Stark shift to the m N =-7 / 2 state and m N =-5 / 2 state. Similarly, m N =-9 / 2 and m N If transitions from the first and second nuclear spin states with m = -5 / 2 are desired, the light will N =-1 / 2 spin state, which provides an AC Stark shift to N =-5 / 2 state and m N =-1 / 2 state. This effectively creates a two-level subsystem within the nuclear spin manifold that is decoupled from the rest of the nuclear spin manifold, which can greatly simplify the dynamics of the qubit system. Nuclear spin states near the edges of the nuclear spin manifold (e.g., for nuclear spin -9 / 2, 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) may be advantageous. Alternatively, it may be advantageous to use nuclear spin states far from the edge of the nuclear spin manifold (e.g., m N =-5 / 2 and m N =-3 / 2, or m N =-5 / 2 and m N =-1 / 2) may be used, and (e.g., m N =-7 / 2 and m N =-1 / 2, or m N =-9 / 2 and m N =3 / 2) two AC Stark shifts may be performed.

[0054] The Stark shift of the nuclear spin manifold may shift adjacent nuclear spin states out of resonance with a desired transition between the first and second nuclear spin states and the second electronic state or states detuned therefrom. The Stark shift may reduce leakage from the first and second nuclear spin states into other states of the nuclear spin manifold. The Stark shift may be achievable up to several hundred kHz with beam powers of less than 10 mW. The upper state frequency selectivity may reduce scattering from imperfect polarization control. 3 Separation of different angular momentum states in the P1 manifold may be multi-gigahertz from single and two-qubit gated light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for the two-qubit transition (e.g., how fast the transition can be driven) may be faster than the decoherence rate. Scattering from an intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve the fidelity of the two-qubit transition.

[0055] Qubits based on the nuclear spin state of the electronic ground state can be used for quantum bit storage (strontium-87 3 This may allow for the exploitation of long-lived metastable excited electronic states (such as the P0 state). Atoms may be selectively transferred to such states to reduce cross-talk or improve gating or detection fidelity. Such storage or shelving processes may be atom-selective using the SLM or AOD described herein. Shelving transitions may be used to detect the presence of strontium-87 in the presence of strontium-87. 1 S0 state and strontium-87 3 P0 state or 3 It may include transitions to and from the P2 state.

[0056] The clock transition (also referred to herein as a "shelving transition" or "storage transition") may be qubit-state selective. The top state of the clock transition may have a very long natural lifetime, e.g., greater than 1 second. The linewidth of the clock transition may be much narrower than the qubit energy spacing. This may allow for direct spectral resolution. The ensemble may be transferred from one of the qubit states to the clock state. This may allow for separate readout of individual qubit states by first transferring the ensemble from the one-qubit state to the clock state and performing imaging on the qubit, and then transferring the ensemble from the clock state to the ground state and imaging again. In some cases, magic wavelength transitions are used to drive the clock transitions.

[0057] The clock light for shelving may or may not be atom-selective. In some cases, the clock transition is applied globally (e.g., not atom-selective). Globally applied clock transitions may include directing the light without passing through a microscope objective or structuring the light. In some cases, the clock transition is atom-selective. Clock transitions that are atom-selective may potentially improve gate fidelity by minimizing crosstalk. For example, to reduce crosstalk at an atom, the atom may be shelved to a clock state where it may not be affected by the light. This may reduce crosstalk between adjacent qubits undergoing a transition. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives and / or be structured with one or more spatial light modulators, digital micromirror devices, crossed acousto-optic deflectors, etc.

[0058] The system (200) may include one or more readout optical units (230). The readout optical unit may be configured to perform one or more measurements of one or more superposition states to obtain a non-classical calculation. The readout optical unit may include one or more optical detectors. The detectors may include one or more photomultiplier tubes (PMTs), photodiodes, avalanche diodes, single-photon avalanche diodes, single-photon avalanche diode arrays, phototransistors, reverse-biased light-emitting diodes (LEDs), charge-coupled devices (CCDs), or complementary metal-oxide semiconductor (CMOS) cameras. The optical detectors may include one or more fluorescence detectors. The readout 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 lenses may have an NA of up to about 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, or less. The objective lenses may have an NA that is within a range defined by any two of the aforementioned values.

[0059] One or more readout optical units (230) may perform measurements, such as projection measurements, by applying light that is resonant with the imaging transition. The imaging transition may cause fluorescence. 1 S0 state and strontium-87 1 The transition between the P1 state and the P2 state may include 1 The P1 state may be fluorescent. The lower states of the qubit transition 1The S0 manifold may include two nuclear spin states. One or more of the states may be resonant with an imaging transition. The measurement may include two excitations. In the first excitation, the two lower states are shelving states (e.g., strontium-87). 3 In a second excitation, an imaging transition may be excited. The first transition may reduce crosstalk between neighboring atoms during calculations. Fluorescence generated from the imaging transition may be collected in one or more readout optical units (230).

[0060] The imaging unit may be used to determine whether one or more atoms have been lost from the trap.The imaging unit may be used to observe the arrangement of the atoms within the trap.

[0061] The system (200) may include one or more vacuum units (240). The one or more vacuum units may include one or more vacuum pumps. The vacuum units 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 units may include one or more high vacuum pumps, such as one or more cryosorption pumps, diffusion pumps, turbomolecular pumps, molecular drag pumps, turbo-drag hybrid pumps, cryogenic pumps, ion pumps, or getter pumps.

[0062] The vacuum unit may include any combination of vacuum pumps described herein. For example, the vacuum unit may include one or more roughing pumps (such as scroll-type pumps) configured to provide a first stage of roughing vacuum pumping. The roughing vacuum pumps may be configured to evacuate gases from the system (200) to achieve a low vacuum pressure condition. For example, the roughing pumps may be configured to provide a first stage of roughing vacuum pumping. 3 The system (200) may be configured to evacuate gases to achieve a rough vacuum pressure of 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 pumps may be configured to evacuate gases from the system (200) to achieve a rough vacuum pressure condition provided by the one or more roughing pumps, up to about 10 -3 High vacuum pressure of up to 10 Pa, or up to 10 -6 The system (200) may be configured to evacuate gas to achieve ultra-high vacuum pressures of up to 100 Pa.

[0063] The vacuum unit can be used for up to 10 -6 Pa, 9x10 -7 Pa, 8x10 -7 Pa, 7x10 -7 Pa, 6x10 -7 Pa, 5x10 -7 Pa, 4x10 -7 Pa, 3x10 -7 Pa, 2x10 -7 Pa, 10 -7 Pa, 9x10 -8 Pa, 810 -8 Pa, 7x10 -8 Pa, 6x10 -8 Pa, 5x10 -8 Pa, 4x10 -8 Pa, 3x10 -8 Pa, 2x10 -8 Pa, 10 -8 Pa, 9x10 -9 Pa, 8x10 -9 Pa, 7x10 -9 Pa, 6x10 -9Pa, 5x10 -9 Pa, 4x10 -9 Pa, 3x10 -9 Pa, 2x10 -9 Pa, 10 -9 Pa, 9x10 -10 Pa, 8x10 -10 Pa, 7x10 -10 Pa, 6x10 -10 Pa, 5x10 -10 Pa, 4x10 -10 Pa, 3x10 -10 Pa, 2x10 -10 Pa, 10 -10 Pa, 9x10 -11 Pa, 8x10 -11 Pa, 7x10 -11 Pa, 6x10 -11 Pa, 5x10 -11 Pa, 4x10 -11 Pa, 3x10 -11 Pa, 2x10 -11 Pa, 10 -11 Pa, 9x10 -12 Pa, 8x10 -12 Pa, 7x10 -12 Pa, 6x10 -12 Pa, 5x10 -12 Pa, 4x10 -12 Pa, 3x10 -12 Pa, 2x10 -12 Pa, 10 -12 The vacuum unit may be configured to maintain the system (200) at a pressure of at least about 10 Pa or less. -12 Pa, 2x10 -12 Pa, 3x10 -12 Pa, 4x10 -12 Pa, 5x10 -12 Pa, 6x10 -12 Pa, 7x10 -12 Pa, 8x10 -12 Pa, 9x10 -12 Pa, 10 -11 Pa, 2x10 -11 Pa, 3x10 -11 Pa, 4x10 -11 Pa, 5x10 -11 Pa, 6x10 -11 Pa, 7x10 -11Pa, 8x10 -11 Pa, 9x10 -11 Pa, 10 -10 Pa, 2x10 -10 Pa, 3x10 -10 Pa, 4x10 -10 Pa, 5x10 -10 Pa, 6x10 -10 Pa, 7x10 -10 Pa, 8x10 -10 Pa, 9x10 -10 Pa, 10 -9 Pa, 2x10 -9 Pa, 3x10 -9 Pa, 4x10 -9 Pa, 5x10 -9 Pa, 6x10 -9 Pa, 7x10 -9 Pa, 8x10 -9 Pa, 9x10 -9 Pa, 10 -8 Pa, 2x10 -8 Pa, 3x10 -8 Pa, 4x10 -8 Pa, 5x10 -8 Pa, 6x10 -8 xPa, 7x10 -8 Pa, 8x10 -8 Pa, 9x10 -8 Pa, 10 -7 Pa, 2x10 -7 Pa, 3x10 -7 Pa, 4x10 -7 Pa, 5x10 -7 Pa, 6x10 -7 Pa, 7x10 -7 Pa, 8x10 -7 Pa, 9x10 -7 Pa, 10 -6 The vacuum device may be configured to maintain the system (200) at a pressure of 100 Pa or greater. The vacuum device may be configured to maintain the system (200) at a pressure within a range defined by any two of the aforementioned values.

[0064] The system (200) may include one or more state preparation units (250). The state preparation units may include any state preparation units described herein, such as the state preparation units described herein with reference to Figure 5. The state preparation units may be configured to prepare states of a plurality of atoms.

[0065] The system 200 may include one or more atom reservoirs 260. The atom reservoirs may be configured to provide one or more replacement atoms to replace one or more atoms at one or more optical trapping sites upon loss of atoms from the optical trapping sites.

[0066] The system (200) may include one or more atom movement units (270). The atom movement units may be configured to move one or more displaced atoms to one or more optical trapping sites. For example, the one or more atom movement units may include one or more electrically tunable lenses, an acousto-optical deflector (AOD), or a spatial light modulator (SLM).

[0067] The system (200) may include one or more entanglement units (280). The entanglement unit may be configured to quantum entangle at least a first atom of the plurality of atoms with at least a second atom of the plurality of atoms. The first atom and the second atom may be in a superposition state when quantum entangled. Alternatively, or in combination, the first atom and the second atom may not be in a superposition state when quantum entangled. The first atom and the second atom may be quantum entangled through one or more of magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The entanglement unit may be configured to quantum entangle any number of atoms described herein.

[0068] The entanglement unit may include one or more Rydberg excitation units. The Rydberg excitation units may be configured to electronically excite at least a first atom into a Rydberg state or into a superposition of a Rydberg state and a lower energy atomic state, thereby forming one or more Rydberg or Rydbergless atoms. The Rydberg excitation units may be configured to induce one or more quantum entanglements between the Rydberg or Rydbergless atoms 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 more from the Rydberg or Rydbergless atom. The second atom may be located at a distance of at most about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg or Rydberg-dressed atom. The second atom may be located at a distance from the Rydberg or Rydberg-dressed atom that is within a range defined by any two of the aforementioned values. The Rydberg excitation unit may be configured to allow the Rydberg or Rydberg-dressed atom to relax to a lower energy atomic state, thereby forming one or more two-qubit units. The Rydberg excitation unit may be configured to induce the Rydberg or Rydberg-dressed atom to relax to a lower energy atomic state. The Rydberg excitation unit may be configured to drive the Rydberg or Rydberg-dress atoms to a lower energy atomic state. For example, the Rydberg excitation unit may be configured to apply electromagnetic radiation (such as RF radiation or optical radiation) to drive the Rydberg or Rydberg-dress atoms to a lower energy atomic state.The Rydberg excitation unit may be configured to induce any number of quantum entanglements between any number of atoms of the plurality of atoms.

[0069] The Rydberg excitation unit may include one or more light sources (such as any light sources described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths may be selected to correspond to wavelengths at which Rydberg or Rydbergless atoms form. 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 more. The light may include one or more wavelengths up to about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. 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 a range of 300 nm to 400 nm.

[0070] The Rydberg excitation unit may be configured to induce a two-photon transition to generate quantum entanglement. The Rydberg excitation unit may be configured to induce a two-photon transition to generate quantum entanglement between two atoms. The Rydberg excitation unit may be configured to selectively induce a two-photon transition to selectively generate quantum entanglement between two atoms. For example, the Rydberg excitation unit may be configured to direct electromagnetic energy (such as optical energy) to a specific optical trapping site to selectively induce a two-photon transition to selectively generate quantum entanglement between two atoms. The two atoms may be trapped in nearby optical trapping sites. For example, the two atoms may be trapped in adjacent optical trapping sites. The two-photon transition may be induced using first and second light from a first and second light source, 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 may be the same as or similar to the light source used to perform the single qubit operations described herein. Alternatively, a different light source may be used to perform the single qubit operations and induce two-photon transitions to generate quantum entanglement. The first light source may emit light including one or more wavelengths in the visible region of the optical spectrum (e.g., within the range of 400 nm to 800 nm, or 650 nm to 700 nm). The second light source may emit light including one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., within the range of 200 nm to 400 nm, or 300 nm to 350 nm). The first light source and the second light source may emit light having substantially equal and opposite spatially dependent frequency shifts.

[0071] A Rydberg atom or Rydbergless atom may include a Rydberg state that may have sufficiently strong interatomic interactions with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) to allow for the implementation of two-qubit operations. The Rydberg state may include a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or more. The Rydberg state may include a principal quantum number of up to about 100, 90, 80, 70, 60, 50, or less. The Rydberg state may include a principal quantum number that is within a range defined by any two of the aforementioned values. The Rydberg state may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift the atomic energy levels of the atoms.

[0072] State-selective excitation of atoms into Rydberg levels may enable the implementation of two-qubit operations. Two-photon transitions are 1 from a ground state (such as the ground state of S0), where n is the principal quantum number as described herein, 3 The polarization may be used to excite the atoms into a Rydberg state (such as the S1 state). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transition may be performed using a first laser source and a second laser source as described herein. The first laser source may emit pi-polarized light, which may not change the angular momentum of the atoms along the magnetic field. The second laser may emit circularly polarized light, which may change the angular momentum of the atoms along the magnetic field by one unit. The first and second qubit levels may be excited into the Rydberg level using this polarization. However, the Rydberg level may be more sensitive to the magnetic field than the ground state, such that large splitting (e.g., on the order of hundreds of MHz) may be easily obtained. This spectral selectivity may allow state-selective excitation into the Rydberg level.

[0073] Two-qubit operations may rely on energy shifts of levels due to van der Waals interactions as described herein. Such shifts may prevent excitation of one atom conditioned by the state of the other, or may alter the coherent dynamics of excitations of the two-atom system to effect two-qubit operations. In some cases, a "dressed state" may be created under continuous drive to effect two-qubit operations without requiring full excitation to the Rydberg levels (see, for example, "Dressed States," which are incorporated by reference in their entirety for all purposes).<https: / / arxiv.org / abs / 1605.05207> ").

[0074] Cloud Computing The system (200) may be operably coupled to a digital computer as described herein (such as the digital computer as described herein with respect to FIG. 1) on a network as described herein (such as the network as described herein with respect to FIG. 1). The network may include a cloud computing network.

[0075] Optical Trap Unit FIG. 3A shows an example of an optical trapping unit (210). The optical trapping unit may be configured to generate a plurality of spatially distinct optical trapping sites (211) as described herein. For example, as shown in FIG. 3B, the optical trapping unit may be configured to generate a first optical trapping site (211a), a second optical trapping site (211b), a third optical trapping site (211c), a fourth optical trapping site (211d), a fifth optical trapping site (211e), a sixth optical trapping site (211f), a seventh optical trapping site (211g), an eighth optical trapping site (211h), and a ninth optical trapping site (211i) as depicted in FIG. 3A. The multiple spatially distinct optical trapping sites may be configured to trap multiple atoms, such as a first atom (212a), a second atom (212b), a third atom (212c), and a fourth atom (212d) as depicted in Figure 3A. As depicted in Figure 3B, each optical trapping site may be configured to trap a single atom. As depicted in Figure 3B, some of the optical trapping sites may be empty (i.e., not trapping any atoms).

[0076] As shown in Figure 3B, the plurality of optical trapping sites may comprise a two-dimensional (2D) array. The 2D array may be perpendicular to the optical axis of the optical components of the optical trapping unit depicted in Figure 3A. Alternatively, the plurality of optical trapping sites may comprise a one-dimensional (1D) array, or a three-dimensional (3D) array.

[0077] Although depicted in FIG. 3B as including nine optical trapping sites filled with four atoms, the optical trapping unit (210) may be configured to generate any number of spatially distinct optical trapping sites as described herein, and may be configured to trap any number of atoms as described herein.

[0078] Each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 μm, 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9 μm, 10 μm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site may be spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the aforementioned values.

[0079] The optical trapping site may include one or more optical tweezers. The optical tweezers may include one or more focused laser beams to provide an attractive or repulsive force to hold or move one or more atoms. The beam waist of the focused laser beam may include a strong electric field gradient. Atoms may be attracted or repelled along the electric field gradient to the center of the laser beam, which may include the strongest electric field. The optical trapping site may include one or more optical lattice sites of one or more optical lattices. The optical trapping site may include one or more optical lattice sites of one or more one-dimensional (1D), two-dimensional (2D), or three-dimensional (3D) optical lattices. For example, the optical trapping site may include one or more optical lattice sites of a 2D optical lattice, as depicted in FIG. 3B.

[0080] An optical grating may be generated by interfering counter-propagating light (such as counter-propagating laser light) to generate a standing wave pattern with periodic and successive intensity minima and maxima along a particular direction. A 1D optical grating may be generated by a pair of interfering counter-propagating light beams. A 2D optical grating may be generated by two pairs of interfering counter-propagating light beams. A 3D optical grating may be generated by three pairs of interfering counter-propagating light beams. The light beams may be generated by different light sources or by the same light source. Thus, an optical grating may be generated by at least about 1, 2, 3, 4, 5, 6, or more, or up to 6, 5, 4, 3, 2, or 1 light sources.

[0081] Returning to the description of Figure 3A, the optical trapping unit may include one or more light sources configured to emit light for generating a plurality of optical trapping sites as described herein. For example, as depicted in Figure 3A, the optical trapping unit may include a single light source (213). Although depicted in Figure 3A as including a single light source, the optical trapping unit may include any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light source. The light source may include one or more lasers.

[0082] The laser may include one or more continuous wave lasers. The laser may include one or more pulsed lasers. The laser may include one or more gas lasers, such as one or more helium neon (HeNe) lasers, argon (Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N2) lasers, carbon dioxide (CO2) lasers, carbon monoxide (CO) lasers, transversely excited atmospheric (TEA) lasers, or excimer lasers. For example, the laser may include one or more of an argon dimer (Ar2) excimer laser, a krypton dimer (Kr2) excimer laser, a fluorine dimer (F2) excimer laser, a xenon dimer (Xe2) excimer laser, an argon fluoride (ArF) excimer laser, a krypton chloride (KrCl) excimer laser, a krypton fluoride (KrF) excimer laser, a xenon bromide (XeBr) excimer laser, a xenon chloride (XeCl) excimer laser, or a xenon fluoride (XeF) excimer laser. The laser may include one or more dye lasers.

[0083] 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.

[0084] 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 be one or more of a neodymium-doped yttrium aluminum garnet (Nd:YAG) laser, a neodymium / chromium-doped yttrium aluminum garnet (Nd / Cr:YAG) laser, an erbium-doped yttrium aluminum garnet (Er:YAG) laser, a neodymium-doped yttrium lithium fluoride (Nd:YLF) laser, a neodymium-doped yttrium orthovanadate (ND:YVO4) laser, a neodymium-doped yttrium calcium oxoborate (Nd:YCOB) laser, a neodymium-glass (Nd:glass) laser, a titanium-sapphire (Ti:sapphire) laser, a thulium-doped yttrium aluminum garnet (Tm:YAG) laser. The laser may include a ytterbium-doped yttrium aluminum garnet (Yb:YAG) laser, a ytterbium-doped glass (Yt:glass) laser, a holmium-yttrium aluminum garnet (Ho:YAG) laser, a chromium-doped zinc selenide (Cr:ZnSe) laser, a cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) laser, a cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) laser, an erbium-doped glass (Er:glass) laser, an erbium-ytterbium co-doped glass (Er / Yt:glass) laser, a uran-doped calcium fluoride (U:CaF2) laser, or a samarium-doped calcium fluoride (Sm:CaF2) laser.

[0085] 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.

[0086] The laser may emit continuous wave laser light. The laser may emit pulsed laser light. The laser may emit pulsed laser light. The laser may emit pulsed laser light. The laser may emit pulsed laser light. The laser may emit pulsed laser light. The pulse length may be ps, 90ps, 100ps, 200ps, 300ps, 400ps, 500ps, 600ps, 700ps, 800ps, 900ps, 1 nanosecond (ns), 2ns, 3ns, 4ns, 5ns, 6ns, 7ns, 8ns, 9ns, 10ns, 20ns, 30ns, 40ns, 50ns, 60ns, 70ns, 80ns, 90ns, 100ns, 200ns, 300ns, 400ns, 500ns, 600ns, 700ns, 800ns, 900ns, 1,000ns, or longer. Lasers have a maximum of about 1,000ns, 900ns, 800ns, 700ns, 600ns, 500ns, 400ns, 300ns, 200ns, 100ns, 90ns, 80ns, 70ns, 60ns, 50ns, 40ns, 30ns, 20ns, 10ns, 9ns 8ns, 7ns, 6ns 5ns, 4ns, 3ns, 2ns, 1ns, 900ps, 800ps, 700ps, 600ps, 500ps, 400ps, 300ps, 200ps, 100ps, 90ps, 80ps, 70ps, 60ps, 50ps, 40ps, 30ps, 20ps, 10ps, 9ps, 8ps, 7ps, 6ps, 5ps, 4ps, 3ps, 2ps, 1ps, 9 In some embodiments, the pulse length may be 00fs, 800fs, 700fs, 600fs, 500fs, 400fs, 300fs, 200fs, 100fs, 90fs, 80fs, 70fs, 60fs, 50fs, 40fs, 30fs, 20fs, 10fs, 9fs, 8fs, 7fs, 6fs, 5fs, 4fs, 3fs, 2fs, 1fs, or less.The laser may have a pulse length that is within the range defined by any two of the aforementioned values.

[0087] The laser may be 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 The optical fiber may have a repetition rate of 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 operates at frequencies up to approximately 1,000MHz, 900MHz, 800MHz, 700MHz, 600MHz, 500MHz, 400MHz, 300MHz, 200MHz, 100MHz, 90MHz, 80MHz, 70MHz, 60MHz, 50MHz, 40MHz, 30MHz, 20MHz, 10MHz, 9MHz, 8MHz, 7MHz, 6MHz, 5MHz, 4MHz, 3MHz, 2MHz, 1MHz, 900kHz, 800kHz, 700kHz, 600kHz, 500kHz, 400kHz, 300kHz, 200kHz, 100kHz, 90kHz, The laser may have a repetition rate of 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 that is within a range defined by any two of the aforementioned values.

[0088] The laser may provide 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, The light may be emitted having a pulse energy of 100 μJ, 200 μJ, 300 μJ, 400 μJ, 500 μJ, 600 μJ, 700 μJ, 800 μJ, 900 μJ, at least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, at least 1 Joule (J), or more. The laser has a maximum output of about 1J, 900mJ, 800mJ, 700mJ, 600mJ, 500mJ, 400mJ, 300mJ, 200mJ, 100mJ, 90mJ, 80mJ, 70mJ, 60mJ, 50mJ, 40mJ, 30mJ, 20mJ, 10mJ, 9mJ, 8mJ 7mJ, 6mJ, 5mJ, 4mJ, 3mJ, 2mJ, 1mJ, 900μJ, 800μJ, 700μJ, 600μJ, 500μJ, 400μJ, 300μJ, 200μJ, 100μJ, 90μJ, 80μJ, 70μJ, 60μJ, 50μJ, 40μJ, 30μJ, 20μJ, 10μJ, 9μJ, 8μJ, 7μJ, 6μJ, 5μJ, 4μJ, 3μJ, 2μJ, 1μJ, 9 The laser may emit light having a pulse energy of 00nJ, 800nJ, 700nJ, 600nJ, 500nJ, 400nJ, 300nJ, 200nJ, 100nJ, 90nJ, 80nJ, 70nJ, 60nJ, 50nJ, 40nJ, 30nJ, 20nJ, 10nJ, 9nJ, 8nJ, 7nJ, 6nJ, 5nJ, 4nJ, 3nJ, 2nJ, 1nJ, or less. The laser may emit light having a pulse energy within a range defined by any two of the aforementioned values.

[0089] The laser may be at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, In some embodiments, the light source may emit light having an average power of 100W, 60mW, 70mW, 80mW, 90mW, 100mW, 200mW, 300mW, 400mW, 500mW, 600mW, 700mW, 800mW, 900mW, 1 Watt (W), 2W, 3W, 4W, 5W, 6W, 7W, 8W, 9W, 10W, 20W, 30W, 40W, 50W, 60W, 70W, 80W, 90W, 100W, 200W, 300W, 400W, 500W, 600W, 700W, 800W, 900W, 1,000W, or more. Lasers are available in a range of powers up to approximately 1,000W, 900W, 800W, 700W, 600W, 500W, 400W, 300W, 200W, 100W, 90W, 80W, 70W, 60W, 50W, 40W, 30W, 20W, 10W, 9W, 8W, 7W, 6W, 5W, 4W, 3W, 2W, 1W, 900mW, 800mW, 700mW, 600mW, 500mW, 400mW, 300mW, 200mW, 100mW, 90mW, 80mW, 70mW, 60mW, 50mW, 40mW, 3 The laser may emit light having an average power of 0 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or more. The laser may emit light having a power that is within a range defined by any two of the aforementioned values.

[0090] The laser may emit light that includes one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The laser may emit light that includes 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, 9 20nm, 530nm, 540nm, 550nm, 560nm, 570nm, 580nm, 590nm, 600nm, 610nm, 620nm, 630nm, 640nm, 650nm, 660nm, 670nm, 680nm, 69 0nm, 700nm, 710nm, 720nm, 730nm, 740nm, 750nm, 760nm, 770nm, 780nm, 790nm, 800nm, 810nm, 820nm, 830nm, 840nm, 850nm, 860 nm, 870nm, 880nm, 890nm, 900nm, 910nm, 920nm, 930nm, 940nm, 950nm, 960nm, 970nm, 980nm, 990nm, 1,000nm, 1,010nm, 1,020n m, 1,030nm, 1,040nm, 1,050nm, 1,060nm, 1,070nm, 1,080nm, 1,090nm, 1,100nm, 1,110nm, 1,120nm, 1,130nm, 1,140nm, 1,150 The optical fiber may emit light comprising one or more wavelengths, such as 1160 nm, 1170 nm, 1180 nm, 1190 nm, 1200 nm, 1210 nm, 1220 nm, 1230 nm, 1240 nm, 1250 nm, 1260 nm, 1270 nm, 1280 nm, 1290 nm, 1300 nm, 1310 nm, 1320 nm, 1330 nm, 1340 nm, 1350 nm, 1360 nm, 1370 nm, 1380 nm, 1390 nm, 1400 nm, or more.Lasers are available in a range of wavelengths up to approximately 1,400nm, 1,390nm, 1,380nm, 1,370nm, 1,360nm, 1,350nm, 1,340nm, 1,330nm, 1,320nm, 1,310nm, 1,300nm, 1,290nm, 1,280nm, 1,270nm, 1,260nm, 1,250nm, 1,240nm, 1,230nm, 1,220nm, 1,210nm, 1,200nm, 1,190nm, 1,180nm, 1,170nm, 1,160nm, 1,150nm, 1,140nm, 1,130nm, 1,120nm, 1,110nm, 1,100nm, 1,090nm, 1,080nm, 1,070nm, 1,060nm, 1,050nm, 1,040nm, 1,030nm, 1 ,020nm, 1,010nm, 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860n m, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm, 6 80nm, 670nm, 660nm, 650nm, 640nm, 630nm, 620nm, 610nm, 600nm, 590nm, 580nm, 570nm, 560nm, 550nm, 540nm, 530nm, 520nm, 510n The laser may emit light including one or more wavelengths of: 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, or 200 nm. The laser may emit light including one or more wavelengths within a range defined by any two of the foregoing values.

[0091] The laser has a resolution of at least about 1x10 -15 nm, 2x10 -15 nm, 3x10 -15nm、4x10 -15 nm、5x10 -15 nm、6x10 -15 nm、7x10 -15 nm、8x10 -15 nm、9x10 -15 nm、1x10 -14 nm、2x10 -14 nm、3x10 -14 nm、4x10 -14 nm、5x10 -14 nm、6x10 -14 nm、7x10 -14 nm、8x10 -14 nm、9x10 -14 nm、1x10 -13 nm、2x10 -13 nm、3x10 -13 nm、4x10 -13 nm、5x10 -13 nm、6x10 -13 nm、7x10 -13 nm、8x10 -13 nm、9x10 -13 nm、1x10 -12 nm、2x10 -12 nm、3x10 -12 nm、4x10 -12 nm、5x10 -12 nm、6x10 -12 nm、7x10 -12 nm、8x10 -12 nm、9x10 -12 nm、1x10 -11 nm、2x10 -11 nm、3x10 -11 nm、4x10 -11 nm、5x10 -11 nm、6x10 -11 nm、7x10 -11 nm、8x10 -11 nm、9x10 -11 nm、1x10 -10 nm、2x10 -10 nm、3x10 -10 nm、4x10 -10 nm、5x10 -10 nm、6x10 -10 nm、7x10 -10 nm、8x10 -10nm、9x10 -10 nm、1x10 -9 nm、2x10 -9 nm、3x10 -9 nm、4x10 -9 nm、5x10 -9 nm、6x10 -9 nm、7x10 -9 nm、8x10 -9 nm、9x10 -9 nm、1x10 -8 nm、2x10 -8 nm、3x10 -8 nm、4x10 -8 nm、5x10 -8 nm、6x10 -8 nm、7x10 -8 nm、8x10 -8 nm、9x10 -8 nm、1x10 -7 nm、2x10 -7 nm、3x10 -7 nm、4x10 -7 nm、5x10 -7 nm、6x10 -7 nm、7x10 -7 nm、8x10 -7 nm、9x10 -7 nm、1x10 -6 nm、2x10 -6 nm、3x10 -6 nm、4x10 -6 nm、5x10 -6 nm、6x10 -6 nm、7x10 -6 nm、8x10 -6 nm、9x10 -6 nm、1x10 -5 nm、2x10 -5 nm、3x10 -5 nm、4x10 -5 nm、5x10 -5 nm、6x10 -5 nm、7x10 -5 nm、8x10 -5 nm、9x10 -5 nm、1x10 -4 nm、2x10 -4 nm、3x10 -4 nm、4x10 -4nm, 5x10 -4 nm, 6x10 -4 nm, 7x10 -4 nm, 8x10 -4 nm, 9x10 -4 nm, 1x10 -3 The laser may emit light having a bandwidth of up to about 1x10 nm or even larger. -3 nm, 9x10 -4 nm, 8x10 -4 nm, 7x10 -4 nm, 6x10 -4 nm, 5x10 -4 nm, 4x10 -4 nm, 3x10 -4 nm, 2x10 -4 nm, 1x10 -4 nm, 9x10 -5 nm, 8x10 -5 nm, 7x10 -5 nm, 6x10 -5 nm, 5x10 -5 nm, 4x10 -5 nm, 3x10 -5 nm, 2x10 -5 nm, 1x10 -5 nm, 9x10 -6 nm, 8x10 -6 nm, 7x10 -6 nm, 6x10 -6 nm, 5x10 -6 nm, 4x10 -6 nm, 3x10 -6 nm, 2x10 -6 nm, 1x10 -6 nm, 9x10 -7 nm, 8x10 -7 nm, 7x10 -7 nm, 6x10 -7 nm, 5x10 -7 nm, 4x10 -7 nm, 3x10 -7 nm, 2x10 -7 nm, 1x10 -7 nm, 9x10 -8 nm, 8x10 -8 nm, 7x10 -8 nm, 6x10 -8 nm, 5x10 -8nm、4x10 -8 nm、3x10 -8 nm、2x10 -8 nm、1x10 -8 nm、9x10 -9 nm、8x10 -9 nm、7x10 -9 nm、6x10 -9 nm、5x10 -9 nm、4x10 -9 nm、3x10 -9 nm、2x10 -9 nm、1x10 -9 nm、9x10 -10 nm、8x10 -10 nm、7x10 -10 nm、6x10 -10 nm、5x10 -10 nm、4x10 -10 nm、3x10 -10 nm、2x10 -10 nm、1x10 -10 nm、9x10 -11 nm、8x10 -11 nm、7x10 -11 nm、6x10 -11 nm、5x10 -11 nm、4x10 -11 nm、3x10 -11 nm、2x10 -11 nm、1x10 -11 nm、9x10 -12 nm、8x10 -12 nm、7x10 -12 nm、6x10 -12 nm、5x10 -12 nm、4x10 -12 nm、3x10 -12 nm、2x10 -12 nm、1x10 -12 nm、9x10 -13 nm、8x10 -13 nm、7x10 -13 nm、6x10 -13 nm、5x10 -13 nm、4x10 -13 nm、3x10 -13 nm、2x10 -13 nm、1x10 -13 nm、9x10 -14nm, 8x10 -14 nm, 7x10 -14 nm, 6x10 -14 nm, 5x10 -14 nm, 4x10 -14 nm, 3x10 -14 nm, 2x10 -14 nm, 1x10 -14 nm, 9x10 -15 nm, 8x10 -15 nm, 7x10 -15 nm, 6x10 -15 nm, 5x10 -15 nm, 4x10 -15 nm, 3x10 -15 nm, 2x10 -15 nm, 1x10 -15 The laser may emit light having a bandwidth of 100 nm, or less. The laser may emit light having a bandwidth that is within the range defined by any two of the preceding values.

[0092] The light source may be configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms. The magic wavelengths corresponding to the atoms may include any wavelength of light that produces equal or nearly equal polarizations in the first and second atomic states. The magic wavelengths for the transition between the first and second atomic states may be determined by calculating the wavelength-dependent polarizabilities of the first and second atomic states and finding the crossover point. Such magic wavelength-tuned light may produce equal or nearly equal differential optical shifts in the first and second atomic states regardless of the intensity of the light emitted by the light source. This may effectively decouple the first and second atomic states from the motion of the atoms. The magic wavelengths may utilize one or more scalar or tensor optical shifts. The scalar or tensor optical shifts may depend on magnetic sublevels within the first and second atomic states.

[0093] For example, the metastable states of group III atoms and alkaline earth or alkaline earth-like atoms may have relatively large tensor shifts, and the angle θ may be adjusted with respect to the applied magnetic field to create a situation where the scalar and tensor shifts balance and give a zero or near-zero differential optical shift between the first and second atomic states. The angle θ may be adjusted by the choice of polarization of the emitted light. For example, if the emitted light is linearly polarized, the total polarizability α is calculated by dividing the scalar component α scalar and the tensor component α tensor may be written as the sum of: α=α scalar +(3cos 2 θ-1)α tensor

[0094] By appropriately choosing θ, the polarizabilities of the first and second atomic states may be chosen to be equal or nearly equal, corresponding to a zero or near-zero differential optical shift, and the motion of the atoms may be decoupled.

[0095] The light source may be configured to direct light to one or more optical modulators (OMs) configured to generate a plurality of optical trapping sites. For example, the optical trapping unit may include an OM (214) configured to generate a plurality of optical trapping sites. Although depicted in FIG. 3A as including one OM, the optical trapping unit may include any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OM may include one or more digital micromirror devices (DMDs). The OM may include one or more liquid crystal devices, such as one or more liquid crystal on silicon (LCoS) devices. The OM may include one or more spatial light modulators (SLMs). The OM may include one or more acousto-optic deflectors (AODs) or acousto-optic modulators (AOMs). The OM may include one or more electro-optic deflectors (EODs) or electro-optic modulators (EOMs).

[0096] The OM may be optically coupled to one or more optical elements to generate a regular array of optical trapping sites. For example, as shown in FIG. 3A, the OM may be optically coupled to optical element (219). 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 optical trapping sites.

[0097] For example, as shown in Figure 3A, the OM may include an SLM, DMD, or LCoS device. The SLM, DMD, or LCoS device may be imaged onto the back focal plane of a microscope objective. This may allow the creation of any configuration of two- or three-dimensional optical trapping sites.

[0098] Alternatively or in combination, the OM may include a first AOD and a second AOD. The active areas of the first and second AOD may be imaged onto the back focal plane of the microscope objective. The output of the first AOD may be optically coupled to the input of the second AOD. In this way, the second AOD may create a copy of the optical output of the first AOD. This may enable the creation of two-dimensional or three-dimensional optical trapping sites.

[0099] Alternatively or in combination, the OM may include static optical elements, such as one or more microlens arrays or holographic optical elements, which may be imaged onto the back focal plane of the microscope objective. This may allow the creation of any configuration of two- or three-dimensional optical trapping sites.

[0100] The optical trapping unit may include one or more imaging units configured to obtain one or more images of the spatial shape of the atoms trapped within the optical trapping site. For example, the optical trapping unit may include an imaging unit (215). Although depicted in FIG. 3A as including a single imaging unit, the optical trapping unit may include any number of imaging units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more imaging units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 imaging units. The imaging unit may include one or more lenses, or objective lenses. The imaging unit may include one or more PMTs, photodiodes, avalanche photodiodes, phototransistors, reverse-biased LEDs, CCDs, or CMOS cameras. The imaging unit may include one or more fluorescence detectors. The images may include one or more fluorescence images, single atom fluorescence images, absorption images, single atom absorption images, phase contrast images, and single atom phase contrast images.

[0101] The optical trapping unit may include one or more spatial shape artificial intelligence (AI) units configured to perform one or more AI operations to determine the spatial shape of the plurality of atoms trapped within the optical trapping site based on the image obtained by the imaging unit. For example, the optical trapping unit may include a spatial shape AI unit (216). Although depicted in FIG. 3A as including a single spatial shape AI unit, the optical trapping unit may include any number of spatial shape AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial shape AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial shape AI units. The AI ​​operations may include any machine learning (ML) operations or reinforcement learning (RL) operations described herein.

[0102] The optical trapping unit may include one or more atom rearrangement units configured to impart an altered spatial arrangement of the trapped atoms at the optical trapping site based on one or more images obtained by the imaging unit. For example, the optical trapping unit may include an atom rearrangement unit (217). Although depicted in FIG. 3A as including a single atom rearrangement unit, the optical trapping unit may include any number of atom rearrangement units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atom rearrangement units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units.

[0103] The optical trapping unit may include one or more spatial configuration artificial intelligence (AI) units adapted to perform one or more AI operations to determine an altered spatial configuration of the plurality of atoms trapped within the optical trapping site based on the image obtained by the imaging unit. For example, the optical trapping unit may include a spatial configuration AI unit (218). Although depicted in FIG. 3A as including a single spatial configuration AI unit, the optical 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 units. The AI ​​operations may include any machine learning (ML) operations or reinforcement learning (RL) operations described herein.

[0104] In some cases, the spatial shape AI unit and the spatial configuration AI unit may be integrated into an integrated AI unit. The optical trapping unit may include any number of integrated AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated AI units.

[0105] The atom rearrangement unit may be configured to change the spatial arrangement to obtain an increased filling rate of the plurality of optical trapping sites. The filling rate may be defined as a ratio of the number of optical trapping sites occupied by one or more atoms to the total number of optical trapping sites available in the optical trapping unit or in a region of the optical trapping unit. For example, an initial loading of atoms in the optical trapping sites may result in a filling rate of less than 100%, less than 90%, less than 80%, less than 70%, less than 60%, less than 50%, or less, such that the atoms occupy less than 100%, less than 90%, less than 70%, less than 60%, less than 50%, or less of the available optical trapping sites, respectively. 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 atom rearrangement unit may reach a fill factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99% or more. The atomic rearrangement unit may achieve a fill factor of up to about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atomic rearrangement unit may achieve a fill factor within a range defined by any two of the aforementioned values.

[0106] As an example, Figure 3C shows an example of an optical trapping unit that is partially filled with atoms. As depicted in Figure 3C, the initial loading of atoms in the optical trapping sites may rise to a fill factor of 44.4% (4 atoms filling the 9 available optical trapping sites). By moving atoms from a different region of the optical trapping unit (not shown in Figure 3C) to the unoccupied optical trapping sites, or by moving atoms from an atom reservoir as described herein, much higher fill factors may be obtained, as shown in Figure 3D.

[0107] FIG. 3D shows an example of an optical 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) may be moved to fill the unoccupied optical trapping sites. The fifth, sixth, seventh, eighth, and ninth atoms may be moved from different regions of the optical trapping unit (not shown in FIG. 3C) or by moving atoms from an atom reservoir as described herein. Thus, the filling factor may be substantially improved after the rearrangement of the atoms in the optical trapping sites. For example, a filling factor of up to 100% may be reached (such as 9 atoms filling the 9 available optical trapping sites as shown in FIG. 3D).

[0108] Electromagnetic Delivery Unit 4 shows an example of an electromagnetic delivery unit (220). The electromagnetic delivery unit may be configured to apply electromagnetic energy to one or more atoms of the plurality of atoms as described herein. The electromagnetic delivery unit may include one or more light sources, such as any of the light sources described herein.

[0109] The electromagnetic energy may include optical energy. The optical energy may include any repetition rate, pulse energy, average power, wavelength, or bandwidth described herein. The electromagnetic delivery unit may include one or more microwave energy sources, or radio frequency (RF) energy sources, such as one or more magnetrons, klystrons, traveling wave tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMPATT) diodes, or masers. The electromagnetic energy may include microwave energy, or RF energy. The RF energy may be at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 The wavelength may include one or more wavelengths of 1 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or more.RF energy up to approximately 10km, 9km, 8km, 7km, 6km, 5km, 4km, 3km, 2km, 1km, 900m, 800m, 700m, 600m, 500m, 400m, 300m, 200m, 100m, 90m, 80m, 70m, 60m, 50m, 40m, 30m, 20m, 10m, 9m, 8m, 7m, 6m, 5m, 4m, 3m, 2m, The RF energy may include one or more wavelengths that are within a range defined by any two of the preceding values.

[0110] The RF energy may be at least about 1 microwatt (μW), 2 μW, 3 μW, 4 μW, 5 μW, 6 μW, 7 μW, 8 μW, 9 μW, 10 μW, 20 μW, 30 μW, 40 μW, 50 μW, 60 μW, 70 μW, 80 μW, 90 μW, 100 μW, 200 μW, 300 μW, 400 μW, 500 μW, 600 μW, 700 μW, 800 μW, 900 μW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 90 ... The average power may include, for example, 1 W, 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. RF energy can be up to approximately 1,000W, 900W, 800W, 700W, 600W, 500W, 400W, 300W, 200W, 100W, 90W, 80W, 70W, 60W, 50W, 40W, 30W, 20W, 10W, 9W, 8W, 7W, 6W, 5W, 4W, 3W, 2W, 1W, 900mW, 800mW, 700mW, 600mW, 500mW, 400mW, 300mW, 200mW, 100mW, 90mW, 80mW, 70mW, 60mW, 50mW, 40 The RF energy may include an average power of 1000 uW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 μW, 800 μW, 700 μW, 600 μW, 500 μW, 400 μW, 300 μW, 200 μW, 100 μW, 90 μW, 80 μW, 70 μW, 60 μW, 50 μW, 40 μW, 30 μW, 20 μW, 10 μW, 9 μW, 8 μW, 7 μW, 6 μW, 5 μW, 4 μW, 3 μW, 2 μW, 1 μW, or less. The RF energy may include an average power that is within a range defined by any two of the aforementioned values.

[0111] The electromagnetic delivery unit may include one or more light sources, such as any of the light sources described herein. For example, the electromagnetic delivery unit may include a light source (221). Although depicted in FIG. 4 as including a single light source, the electromagnetic delivery unit may include any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light source.

[0112] The light source may be configured to direct light to one or more OMs configured to selectively apply electromagnetic energy to one or more atoms of the plurality of atoms. For example, the electromagnetic delivery unit may include an OM (222). Although depicted in FIG. 4 as including a single OM, 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 OMs. The OM may include one or more SLMs, AODs, or AOMs. The OM may include one or more DMDs. The OM may include one or more liquid crystal devices, such as one or more LCoS devices.

[0113] 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 in FIG. 4 as including a single AI unit, the electromagnetic delivery unit may include any number of AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more AI units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 AI units. The AI ​​operations may include any machine learning (ML) operations or reinforcement learning (RL) operations described herein.

[0114] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) on a qubit as described herein. The electromagnetic delivery unit may be configured to apply one or more two-qubit operations (such as one or more two-qubit gate operations) on a two-qubit unit as described herein. Each single-qubit operation or two-qubit operation may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100 μs, or more. Each single qubit or two-qubit operation may include a duration of at most about 100 μs, 90 μs, 80 μs, 70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4 μs, 3 μs, 2 μs, 1 μs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. Each single qubit or two-qubit operation may include a duration that is within a range defined by any two of the foregoing values. Single qubit or two-qubit operations may be applied at a repetition rate of at least 1 kilohertz (kHz), 2 kHz 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or more.Single or two-qubit operations may be applied at a repetition rate of up to about 1,000 kHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. Single or two-qubit operations may be applied at a repetition rate that is within a range defined by any two of the foregoing values.

[0115] The electromagnetic delivery unit may be configured to apply one or more single qubit operations by inducing one or more Raman transitions between a first qubit state and a second qubit state as described herein. 3 P0 or 3For example, the Raman 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 transitions may be detuned by up to about 1 GHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz 30 kHz 20 kHz 10 kHz 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The Raman transition may be detuned by a value that is within a range defined by any two of the aforementioned values.

[0116] Raman transitions may be induced on individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle and / or frequency shift to the light beam based on an applied radio frequency (RF) signal. The SLM or AOD may be coupled to an optical alignment system that images the SLM or AOD active area onto the back focal plane of a microscope objective. The microscope objective may perform a spatial Fourier transform on the optical field of the position of the SLM or AOD. In that way, the angle (which may be proportional to the RF frequency) may be converted to a position. For example, applying a radio frequency comb to the AOD may generate a linear array of spots in the focal plane of the objective, with each spot having a finite extent determined by a property of the optical alignment system (such as the point spread function of the optical alignment system).

[0117] To perform a Raman transition on a single atom with a single SLM or AOD, a pair of frequencies may be applied simultaneously to the SLM or AOD, and the two frequencies in the pair of frequencies may have a frequency difference that matches or nearly matches the splitting energy between the first qubit state and the second qubit state. For example, the frequency difference may differ from the splitting energy by up to about 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. For example, the frequency difference may differ from the splitting energy by up to 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 be different from the splitting energy by about 0 Hz. The frequency difference may be different from the splitting energy by a value that is within a range defined by any two of the aforementioned values. The optical system may be configured such that the position interval corresponding to the frequency difference is not resolved and such that the light interacts with a single atom at both of the two frequencies.

[0118] Integrated optical trapping unit and electromagnetic delivery unit The optical trapping unit and the electromagnetic delivery unit described herein may be integrated into a single optical system. A microscope objective lens may be used to deliver the electromagnetic radiation generated by the electromagnetic delivery unit described herein and to deliver light for trapping atoms generated by the optical trapping unit described herein. Alternatively or in combination, 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 optical trapping unit.

[0119] A single SLM or AOD may enable the performance of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or in combination, two separate SLMs or AODs may be configured to process light, each with orthogonal polarizations. The light with orthogonal polarizations may overlap in front of a microscope objective. In such a scheme, each photon used in a two-photon transition described herein may be passed to the objective by a separate SLM or AOD, which may allow for increased polarization control. Qubit operations may be performed on a two-dimensional arrangement of atoms by delivering light from a first SLM or AOD to a second SLM or AOD that is oriented substantially perpendicular to the first SLM or AOD via an optical relay. Alternatively or in combination, qubit operations may be performed on a two-dimensional arrangement of atoms by using a one-dimensional array of SLMs or AODs.

[0120] The stability of the qubit gate fidelity may be improved by maintaining an overlap of light from the various light sources described herein (such as the light sources associated with the optical trapping unit or electromagnetic delivery unit described herein). Such overlap may be maintained by an optical subsystem that measures the direction of light emitted by the various light sources, allowing closed-loop control of the direction of the light emission. The optical subsystem may include a pickoff mirror located in front of the microscope objective lens. The pickoff mirror may be configured to direct a small amount of light to a lens that may focus the collimated beam and convert the angular deviation into a position deviation. A position-sensitive optical detector, such as a lateral-effect position sensor or a quadrant photodiode, may convert the position deviation into an electronic signal, and information about the deviation may be fed to a corrective optical device, such as an active mirror.

[0121] The stability of the qubit gate operation may be improved by controlling the intensity of light from the various light sources described herein (such as the light sources associated with the optical trapping unit or electromagnetic delivery unit described herein). Such intensity control may be maintained by an optical subsystem that measures the intensity of the light emitted by the various light sources, allowing for closed-loop control of the intensity. Each light source may be coupled to an intensity actuator, such as an intensity servo control. The actuator may include an acousto-optical modulator (AOM) or an electro-optical modulator (EOM). The intensity may be measured using an optical detector, such as a photodiode or any other optical detector described herein. Information regarding the intensity may be integrated into a feedback loop to stabilize the intensity.

[0122] Conditioning Unit 5 shows an example of a condition preparation unit (250). The condition preparation unit may be configured to prepare the condition of the plurality of atoms as described herein. The condition preparation unit may be coupled to the optical trapping unit and may direct the atoms prepared by the condition preparation unit to the optical trapping unit. The condition preparation unit may be configured to cool the plurality of atoms. The condition preparation unit may be configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites.

[0123] The condition preparation unit may include one or more Zeeman reducers. For example, the condition preparation unit may include a Zeeman reducer (251). Although depicted in FIG. 5 as including a single Zeeman reducer, the condition preparation unit may include any number of Zeeman reducers, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Zeeman reducers, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Zeeman reducers. The Zeeman reducer may be configured to reduce one or more atoms of the plurality of atoms from a first velocity or distribution of velocities (e.g., an emission velocity from an atom source, room temperature, liquid nitrogen temperature, or any other temperature, etc.) to a second velocity that is lower than the first velocity or distribution of velocities.

[0124] The first velocity or velocity distribution may be associated with a temperature of at least about 50 Kelvin (K), 60K, 70K, 80K, 90K, 100K, 200K, 300K, 400K, 500K, 600K, 700K, 800K, 900K, 1,000K, or more. The first velocity or velocity distribution may be associated with a temperature of up to about 1,000K, 900K, 800K, 700K, 600K, 500K, 400K, 300K, 200K, 100K, 90K, 80K, 70K, 60K, 50K, or less. The first velocity or velocity distribution may be associated with a temperature that is within a range defined by any two of the foregoing values. The second velocity may be at least about 1 meter per second (m / s), 2 m / s, 3 m / s, 4 m / s, 5 m / s, 6 m / s, 7 m / s, 8 m / s, 9 m / s, 10 m / s, or more. The second velocity may be up to about 10 m / s, 9 m / s, 8 m / s, 7 m / s, 6 m / s, 5 m / s, 4 m / s, 3 m / s, 2 m / s, 1 m / s, or less. The second velocity may be within a range defined by any two of the aforementioned values. The Zeeman reducer may include a 1D Zeeman reducer.

[0125] The condition preparation unit may include a first magneto-optical trap (MOT) (252). The first MOT may be configured to cool the atoms to a first temperature. The first temperature may be 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 less. The first temperature may be at least about 0.1 mK, 0.2 mK, 0.3 mK, 0.4 mK, 0.5 mK, 0.6 mK, 0.7 mK, 0.8 mK, 0.9 mK, 1 mK, 2 mK, 3 mK, 4 mK, 5 mK, 6 mK, 7 mK, 8 mK, 9 mK, 10 mK, or more. The first temperature may be within a range defined by any two of the aforementioned values. The first MOT may include a 1D MOT, a 2D MOT, or a 3D MOT.

[0126] The first MOT may include one or more light sources (such as any light sources described herein) configured to emit light, the light being 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, m, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light can range up to approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 6 ...90nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790 The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0127] The condition preparation unit may include a second MOT (253). The second MOT may be configured to cool the atoms from a first temperature to a second temperature lower than the first temperature. The second temperature may be up to about 100 microkelvin (μK), 90 μK, 80 μK, 70 μK, 60 μK, 50 μK, 40 μK, 30 μK, 20 μK, 10 μK, 9 μK, 8 μK, 7 μK, 6 μK, 5 μK, 4 μK, 3 μK, 2 μK, 1 μK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, or less. The second temperature may be at least about 100nK, 200nK, 300nK, 400nK, 500nK, 600nK, 700nK, 800nK, 900nK, 1μK, 2μK, 3μK, 4μK, 5μK, 6μK, 7μK, 8μK, 9μK, 10μK, 20μK, 30μK, 40μK, 50μK, 60μK, 70μK, 80μK, 90μK, 100μK, or more. The second temperature may be within a range defined by any two of the aforementioned values. The second MOT may include a 1D MOT, a 2D MOT, or a 3D MOT.

[0128] The second MOT may include one or more light sources (such as any light sources described herein) configured to emit light, the light being 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, m, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light can range up to approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 6 ...90nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790 The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0129] Although depicted in FIG. 5 as including two MOTs, the conditioning unit may include any number of MOTs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 or more MOTs, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 MOTs.

[0130] The condition adjustment unit may include any of the following:<https: / / arxiv.org / abs / 1810.06626> " or "<https: / / arxiv.org / abs / 1811.06014> "). For example, the condition preparation unit may include a sideband or Sisyphus cooling unit (254). Although depicted in FIG. 5 as including a single sideband or Sisyphus cooling unit, the condition preparation unit may include any number of sideband or Sisyphus cooling units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more sideband or Sisyphus cooling units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband or Sisyphus cooling units. The sideband or Sisyphus cooling unit may be configured to use sideband cooling to cool atoms from a second temperature to a third temperature that is lower than the second temperature. The third temperature may be up to about 10 μK, 9 μK, 8 μK, 7 μK, 6 μK, 5 μK, 4 μK, 3 μK, 2 μK, 1 μK, 900nK, 800nK, 700nK, 600nK, 500nK, 400nK, 300nK, 200nK, 100nK, 90nK, 80nK, 70nK, 60nK, 50nK, 40nK, 30nK, 20nK, 10nK, or less. The third temperature may be up to 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80 nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 μK, 2 μK, 3 μK, 4 μK, 5 μK, 6 μK, 7 μK, 8 μK, 9 μK, 10 μK, or more. The third temperature may be within a range defined by any two of the aforementioned values.

[0131] The sideband or Sisyphus cooling unit may include one or more light sources (such as any light sources described herein) configured to emit light, the light being 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, m, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light can range up to approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 6 ...90nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790 The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0132] The state preparation unit may include one or more optical pumping units. For example, the state preparation unit may include an optical pumping unit (255). Although depicted in FIG. 5 as including a single optical pumping unit, the state preparation unit 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 units. 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 include one or more light sources (such as any of the light sources described herein) configured to emit light, the light being 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, m, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more.The light can range up to approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 6 ...90nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790 The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may include one or more wavelengths within a range defined by any two of the foregoing values. For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0133] The state preparation unit may include one or more coherent driving units. For example, the state preparation unit may include a coherent driving unit (256). Although depicted in FIG. 5 as including a single coherent driving unit, the state preparation unit may include any number of coherent driving units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more coherent driving units, or up to about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 coherent driving units. The coherent driving units may be configured to coherently drive atoms from a non-equilibrium state to a first atomic state or a second atomic state described herein. Thus, the atoms may be coherently driven to an atomic state described herein useful for performing quantum computation after being optically pumped to an atomic state convenient to access (e.g., based on the availability of a light source emitting a particular wavelength, or based on other factors). The coherent drive unit may be configured to induce a one-photon transition between the non-equilibrium state and the first atomic state or the second atomic state. The coherent drive unit may be configured to induce a two-photon transition between the non-equilibrium state and the first atomic state or the second atomic state. The two-photon transition may be induced using light from two light sources described herein (such as two lasers described herein).

[0134] The coherent driving unit may include one or more light sources (such as any of the light sources described herein) configured to emit light, the light being 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, m, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light can range up to approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 6 ...90nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 780nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790nm, 790 The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0135] The coherent driving unit may be configured to induce an RF transition between the non-equilibrium state and the first atomic state or the second atomic state. The coherent driving unit may include one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. The coherent driving unit may include one or more RF sources (such as any RF source described herein) configured to emit RF radiation. The RF radiation may include one or more wavelengths of at least about 10 centimeters (cm), 20 cm, 30 cm, 40 cm, 50 cm, 60 cm, 70 cm, 80 cm, 90 cm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, or more. The RF radiation may include one or more wavelengths up to about 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 90 cm, 80 cm, 70 cm, 60 cm, 50 cm, 40 cm, 30 cm, 20 cm, 10 cm, or less. The RF radiation may include one or more wavelengths within a range defined by any two of the foregoing values. Alternatively or in combination, the coherent driving unit may include one or more light sources (such as any of the light sources described herein) configured to induce two-photon transitions corresponding to the RF transitions.

[0136] controller The optical trapping unit, electromagnetic delivery unit, entanglement unit, readout optical unit, vacuum unit, imaging unit, spatial shape AI unit, spatial configuration AI unit, atomic rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atomic reservoir, atomic motion 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 electronic connections) to the one or more optical trapping units, electromagnetic delivery unit, entanglement unit, readout optical unit, vacuum unit, imaging unit, spatial shape AI unit, spatial configuration AI unit, atomic rearrangement unit, state preparation unit, sideband cooling unit, optical pumping unit, coherent drive unit, electromagnetic energy AI unit, atomic reservoir, atomic motion unit, or Rydberg excitation unit. The circuit or controller may be configured to control one or more of an optical trapping unit, an electromagnetic delivery unit, an entanglement unit, a readout optical unit, a vacuum unit, an imaging unit, a spatial shape AI unit, a spatial configuration AI unit, an atomic rearrangement unit, a state preparation unit, a sideband cooling unit, an optical pumping unit, a coherent drive unit, an electromagnetic energy AI unit, an atomic reservoir, an atomic motion unit, or a Rydberg excitation unit.

[0137] Non-Classical Computers In one aspect, the disclosure provides a non-classical computer comprising a plurality of qubits comprising more than 60 atoms, each atom being trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, where the plurality of qubits comprises at least a first qubit state and a second qubit state, where the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; and one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more qubits of the plurality of qubits. The system includes one or more electromagnetic delivery units configured to impart a non-classical operation to one or more quantum bits, the non-classical operation comprising a superposition between at least a first quantum bit state and a second quantum bit state, one or more entanglement units configured to quantum entangle at least a subset of the plurality of quantum bits in the superposition state with at least another quantum bit of the plurality of quantum bits, and one or more readout optical units configured to perform one or more measurements of the one or more quantum bits, thereby obtaining a non-classical computation.

[0138] In one aspect, the present disclosure provides a non-classical computer that includes a plurality of qubits, each atom including more than 60 atoms trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites.

[0139] Methods for performing non-classical computations In one aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superposition states of a first atomic state and at least a second atomic state different from the first atomic state; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superposition states with at least another atom of the plurality of atoms; and (d) performing one or more optical measurements of the one or more superposition states to obtain a non-classical computation.

[0140] FIG. 6 shows a flow chart of an example of a first method (600) for performing non-classical computation.

[0141] In a first operation (610), the method (600) may include generating a plurality of spatially distinct optical trapping sites. The plurality of optical trapping units may be configured to trap a plurality of atoms. The plurality of atoms may include more than 60 atoms. The optical trapping sites may include any optical trapping site described herein. The atoms may include any atoms described herein.

[0142] In a second operation (620), the method (600) may include applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state. The electromagnetic energy may include any electromagnetic energy described herein. The first atomic state may include any first atomic state described herein. The second atomic state may include any second atomic state described herein.

[0143] In a second operation (630), the method (600) may include quantum entanglement of at least a subset of the one or more atoms in the one or more superposition states with at least another atom of the plurality of atoms. The atoms may be quantum entangled in any manner described herein (e.g., as described herein with respect to FIG. 2).

[0144] In a fourth operation (640), the method (600) may include performing one or more optical measurements of the one or more superposition states to obtain the non-classical computation. The optical measurements may include any optical measurements described herein.

[0145] In one aspect, the disclosure provides a method for performing a non-classical computation, the method including: (a) providing a plurality of qubits including more than 60 atoms, each atom being trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, where the plurality of qubits includes at least a first qubit state and a second qubit state, where the first qubit state includes a first atomic state and the second qubit state includes a second atomic state; (b) applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, the non-classical operation including a superposition between at least the first qubit state and the second qubit state; (c) quantum entanglement of at least a subset of the plurality of qubits in the superposition state with at least another qubit of the plurality of qubits; and (d) performing one or more optical measurements of the one or more qubits, thereby obtaining a non-classical computation.

[0146] FIG. 7 shows a flow chart of an example of a second method (700) for performing non-classical computation.

[0147] In a first operation (710), the method (700) may include providing a plurality of qubits including more than 60 atoms, each atom being trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, where the plurality of qubits includes at least a first qubit state and a second qubit state, where the first qubit state includes a first atomic state and the second qubit state includes a second atomic state. The optical trapping site may include any optical trapping site described herein. The qubit may include any qubit described herein. The atom may include any atom described herein. The first qubit state may include any first qubit state described herein. The second qubit state may include any second qubit state described herein. The first atomic state may include any first atomic state described herein. The second atomic state may include any second atomic state described herein.

[0148] In a second operation (720), the method (700) may include applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, the non-classical operation comprising a superposition between at least a first qubit state and a second qubit state. The electromagnetic energy may include any electromagnetic energy described herein.

[0149] In a third operation (730), the method (700) may include quantum entanglement of at least a subset of the plurality of qubits in a superposition state with at least another qubit of the plurality of qubits. The qubits may be quantum entangled in any manner described herein (e.g., as described herein with respect to FIG. 2).

[0150] In a fourth operation (740), the method (700) may include performing one or more optical measurements of one or more qubits, thereby obtaining a non-classical computation. The optical measurements may include any optical measurements described herein.

[0151] In one aspect, the present disclosure provides a method for performing a non-classical computation, the method including: (a) providing a plurality of qubits including more than 60 atoms, each atom trapped in an optical trapping site of a plurality of spatially distinct optical trapping sites; and (b) using at least a subset of the plurality of qubits to perform a non-classical computation.

[0152] FIG. 8 shows a flow chart for an example of a third method (800) for performing non-classical computation.

[0153] In a first operation (810), the method (800) may include providing a plurality of qubits including more than 60 atoms, each atom trapped in an optical trapping site of a plurality of spatially distinct optical trapping sites. The qubits may include any qubit described herein. The atoms may include any atom described herein. The optical trapping sites may include any optical trapping sites described herein.

[0154] In a second operation (820), the method (800) includes using at least a subset of the plurality of qubits to perform a non-classical computation.

[0155] Computer Systems 1 shows a computer system (101) programmed or otherwise configured to operate any of the methods or systems described herein (such as the systems or methods for performing non-classical computation described herein). The computer system (101) can accommodate various aspects of the present disclosure. The computer system (101) can be a user's electronic device or a computer system located remotely relative to the electronic device. The electronic device can be a mobile electronic device.

[0156] The computer system (101) includes a central processing unit (CPU, also referred to herein as "processor" and "computer processor") (105), which may be a single-core or multi-core processor, or multiple processors for parallel processing. The computer system (101) also includes memory or storage locations (110) (e.g., random access memory, read-only memory, flash memory), electronic storage (115) (e.g., hard disk), communication interfaces (120) (e.g., network adapters) for communicating with one or more other systems, and peripherals (125), such as cache, other memory, data storage devices, and / or electronic display adapters. The memory (110), storage (115), interface (120), and peripherals (125) communicate with the CPU (105) through a communication bus (solid lines), such as a motherboard. The storage (115) may be a data storage device (or data repository) for storing data. The computer system (101) may be operably coupled to a computer network ("network") (130) with the aid of a communication interface (120). The network (130) may be the Internet and / or an extranet, an intranet and / or an extranet in communication with the Internet. In some cases, the network (130) is a telecommunications and / or data network. The network (130) may include one or more computer servers, which may enable distributed computing, such as cloud computing. The network (130), in some cases, with the aid of the computer system (101), may implement a peer-to-peer network, which may enable devices coupled to the computer system (101) to act as clients or servers.

[0157] The CPU (105) can execute a series of machine-readable instructions, which may be embodied in a program or software. The instructions may be stored in a memory location, such as the memory (110). The instructions may be directed to the CPU (105), which may then program or configure the CPU (105) to perform the methods of the present disclosure. Examples of operations performed by the CPU (105) may include fetch, decode, execute, and writeback.

[0158] 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).

[0159] The storage device (115) can store files such as drivers, libraries, and saved programs. The storage device (115) can store user data, such as user preferences and user programs. The computer system (101) may optionally include one or more additional data storage devices external to the computer system (101), such as located on a remote server in communication with the computer system (101) over an intranet or the Internet.

[0160] The computer system (101) can communicate with one or more remote computer systems through the network (130). For example, the computer system (101) can communicate with a user's remote computer system. Examples of remote computer systems include a personal computer (e.g., a portable PC), a slate or tablet PC (e.g., Apple® iPad®, Samsung® Galaxy Tab), a phone, a smart phone (e.g., Apple® iPhone®, Android-enabled device, Blackberry®), or a personal digital assistant. A user can access the computer system (101) through the network (130).

[0161] Methods as described herein may be implemented by machine (e.g., computer processor) executable code stored in an electronic storage location of the computer system (101), such as on memory (110) or electronic storage (115). The machine executable or machine readable code may be provided in the form of software. In use, the code may be executed by the processor (105). In some cases, the code may be retrieved from storage (115) and stored in memory (110) for immediate access by the processor (105). In some situations, the electronic storage (115) may be omitted and machine executable instructions are stored in memory (110).

[0162] The code may be pre-compiled and configured for use with a machine having a suitable processor to execute the code, or may be compiled during run-time. The code may be provided in a programming language that may be selected to enable the code to execute in a pre-compiled or as-compiled fashion.

[0163] Aspects of the systems and methods provided herein, such as the computer system (101), may be integrated in programming. Various aspects of the technology may be considered as an "article of manufacture" or "article of manufacture" in the form of machine (or processor) executable code and / or associated data, typically carried on or embedded in a type of machine-readable medium. The machine executable code may be stored in electronic storage, such as memory (e.g., read-only memory, random access memory, flash memory) or a hard disk. A "storage" type medium may include any or all of the tangible memory of a computer or processor, or its associated modules, such as various semiconductor memories, tape drives, disk drives, etc., which may provide non-transitory storage at any time for the programming of the software. All or a portion of the software may be communicated from time to time over the Internet or various other telecommunications networks. Such communication may enable, for example, loading of the software from one computer or processor to another, for example, from a management server or host computer to the computer platform of an application server. Thus, another type of medium that may bear software elements includes light waves, radio waves, and electromagnetic waves, such as those used across physical interfaces between local devices, through wired and optical landline networks, and over various air-links. The physical elements that carry such waves, such as wired or wireless links, optical links, etc., may also be considered as media bearing software. As used herein, unless limited to non-transitory, tangible "storage" media, terms such as computer or machine "readable medium" refer to any medium that participates in providing instructions to a processor for execution.

[0164] Thus, a machine-readable medium such as a computer executable code may take many forms, including but not limited to a tangible storage medium, a carrier wave medium, or a physical transmission medium. Non-volatile storage media include optical or magnetic disks, such as any of the storage devices in a computer, such as those that may be used to implement the databases, etc., shown in the figures. Volatile storage media include dynamic memory, such as the main memory of such a computer platform. Tangible transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise a bus within a computer system. Carrier wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Thus, common forms of computer readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tape, other magnetic media, CD-ROMs, DVDs, or DVD-ROMs, other optical media, punch cards, paper tape, other physical storage media with patterns of holes, RAM, ROM, PROMs, and EPROMs, FLASH-EPROMs, other memory chips or cartridges, carrier waves carrying data or instructions, cables or links which transmit such carrier waves, or other media from which a computer may read programming code and / or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution.

[0165] The computer system 101 may include or be in communication with an electronic display 135 that includes a user interface (UI) 140. Examples of UIs include, but are not limited to, graphical user interfaces (GUIs) and web-based user interfaces.

[0166] The methods and systems of the present disclosure may be implemented by one or more algorithms. The algorithms may be implemented by software when executed by the central processing unit (105). The algorithms may, for example, implement the methods for performing non-classical computations described herein. EXAMPLES

[0167] In the following example, the 10 nuclear spin levels of strontium-87 (I=9 / 2) were modeled to demonstrate a two-level system (i.e., qubit). To achieve spectral separation of the qubit transitions, a Stark-shift scheme was utilized, which shifts the undesired transitions away from the qubit frequency. The separation scheme may improve the effective separation for achievable Rabi frequencies, reduce effects on the actual qubit state via shifts or residual scattering, do not require full polarization control, and can be accessed with a reasonable amount of optical power. 1 S0- 3 The properties of the P1 resonance were characterized.

[0168] In Figure 10A, a toy model is utilized to demonstrate the shift of three relevant nuclear spin states: the mF=9 / 2 and mF=7 / 2 levels that create the qubit subspace, and the leaky mF=5 / 2 level. Here, we simulate the behavior of a single, circularly polarized, wide-area ac Stark beam directed at a set of atoms in a magnetic field of 700 Gauss. In addition, a polarization purity of 100:1 was assumed for the intended circular polarization. 1 S0- 3 At each detuning of the AC Stark beam from the P1 resonance, a shift occurs at each nuclear spin level. For further clarification, we plot the qubit frequency (the difference between the dressed energies of mF=9 / 2 and mF=7 / 2) and the leakage transition frequency (the difference between the dressed states of mF=7 / 2 and mF=5 / 2).

[0169] FIG. 10B shows that the Stark shift significantly shifts the leakage transition while having minimal effect on the qubit frequency. This is indicative of the effect of the high magnetic field on the level splitting. 3 This may be possible due to the narrow linewidth of the P1 resonance. Although the frequency is plotted as a signed quantity, subtleties related to the quantization axis and light delivery make the absolute value of this frequency relevant, so that the Stark shift features moving the leaky state to the close vicinity of the qubit frequency. At each detuning, we can define the maximum usable Rabi frequency achievable assuming frequency crowding. At this two-photon Rabi frequency, we can infer the pi-pulse time and see the number of scattering events that occur due to the off-resonant interaction of the AC Stark beam (Figure 10A).

[0170] Here we do not distinguish between Raman and Rayleigh scattering, which is therefore assumed to be the worst-case scenario of AC Stark-induced scattering errors per gate. To perform single-qubit gates, we coherently control the light, 3 Two beams detuned from the P1 resonance were used to operate the two-photon transition. 3 Residual scattering from any of the P1 manifold states may be essentially low due to the 7 kHz linewidth of the transition. Including the effects of AC Stark shifted beams, 3 The broadening of the P1 hyperfine magnetic sublevels can be used to separate the energy scales between an AC Stark beam detuned from the F=11 / 2 manifold and a multiphoton 1Q light detuned from the F=7 / 2 manifold. A simple toy model including two ground states and a few excited states was sufficient to gain insight into the scaling of powers, spot size, and achievable Rabi rate. However, the expansion of an infinite number of levels including all magnetic sublevels ( 1 S0(F=9 / 2), 3Since the 40 levels (F = 7 / 2, 9 / 2, 11 / 2) are included, it may be necessary to perform a full-scale simulation including all relevant levels. To verify a complete run, a numerical model was constructed using all 40 levels with multiple optical fields, representing both the desired and undesired polarizations. Using a simple square pulse, it can be seen that transitions to other nuclear spin states are suppressed with the AC Stark beam (Figures 11A and 11B).

[0171] 12A and 12B show the arrangement of trapped light generated by the SLM, such as a square array and a random array. 1 S0→ 3 The hologram was generated by reflecting light from the SLM at the magic wavelength of the P0 transition (the magic wavelength of the P0 transition). The active area of ​​the SLM was a 1920x1152 array of square pixels, approximately 9 microns on a side. Each pixel contains a large amount of liquid crystal that imparts a phase shift to the incident light. This phase shift can be controlled with a voltage applied to the pixel, and in this way an arbitrary, pixelated phase mask can be generated and applied to any unstructured light incident on the surface of the SLM. The SLM is positioned such that a large collimated beam is incident on it, and the phase-shifted light reflected from the SLM is later directed through the microscope objective. This configuration connected the plane of the SLM to the plane below the lens (where the atomic cloud is formed) by Fourier conjugacy. The complex-valued in-plane electric field of the SLM is the Fourier transform of a similar electric field in the plane below the microscope objective, in the volume of the glass cell. The atoms experienced a trapping potential proportional to the strength of the electric field, and therefore lateral confinement. Longitudinal confinement results from structured light passing through a focal spot, the position of which is also partly determined (and therefore controllable) by the SLM.

[0172] We create 2000 traps, each 500 microkelvins deep, which is well over 1000 times the recoil energy imparted from scattering photons for imaging and so on. This means that the device should be well within the regime where atoms can be measured hundreds of times without being lost due to heating, without additional cooling. Once cooled to the vibrational ground state, the positions of the atoms are known to be within 20 nm, allowing a large separation in scale between the positions of the atoms and the size of the laser beams used to drive the single-qubit and two-qubit gates, or the length scale of the Rydberg interaction. The laser beams driving the gate operation have a spatial extent of about microns, and therefore the intensity varies at levels 10-5, and therefore it is expected that a fidelity of 0.9999 is easily achievable. Thus, the gate fidelity is not very sensitive to the position of the atoms.

[0173] While preferred 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. Numerous modifications, changes, and substitutions will occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the invention described herein may be utilized in practicing the invention. It is intended that the following claims define the scope of the invention, and that methods and structures within the scope of these claims and their equivalents are covered thereby.

Claims

1. 1. A system for performing non-classical computation, the system comprising: one or more optical trapping units configured to generate a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms including more than 60 atoms; one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superpositions 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 entangle at least a subset of the one or more atoms in the superposition state with at least another atom of the plurality of atoms; one or more readout optical units configured to perform one or more measurements of the one or more superposition states to obtain the non-classical calculation; Including, the system.

2. The system of claim 1 , wherein the non-classical computing comprises quantum computing.

3. The system of claim 2 , wherein the quantum computing comprises gate model quantum computing.

4. The system of claim 1 , wherein one or more atoms of the plurality of atoms comprises a quantum bit.

5. 2. The system of claim 1 , wherein the first atomic state comprises a first single qubit state and the second atomic state comprises a second single qubit state.

6. The system of claim 1 , wherein the first atomic state or the second atomic state is elevated in energy relative to a ground atomic state of the atom.

7. 2. The system of claim 1, wherein the first atomic state comprises a first hyperfine electronic state and the second atomic state comprises a second hyperfine electronic state different from the first hyperfine electronic state.

8. The system of claim 1 , wherein the first atomic state comprises a first nuclear spin state and the second atomic state comprises a second nuclear spin state different from the first nuclear spin state.

9. The system of claim 1 , wherein the plurality of atoms comprises at least 100 atoms.

10. The system of claim 1 , wherein the plurality of atoms comprises neutral atoms.

11. The system of claim 1 , wherein the plurality of atoms comprises rare earth atoms.

12. The system of claim 1 , wherein the plurality of atoms comprises alkali atoms.

13. The system of claim 1 , wherein the plurality of atoms comprises alkaline earth atoms.

14. The system of claim 13 , wherein the alkaline earth atoms include strontium atoms.

15. The system of claim 14, wherein the strontium atoms comprise strontium-87 atoms.

16. The first atomic state and the second atomic state are strontium-87 ３ P １ The system of claim 15 comprising a first hyperfine state and a second hyperfine state on a manifold.

17. The first atomic state and the second atomic state are strontium-87 ３ P ２ The system of claim 15 comprising a first hyperfine state and a second hyperfine state on a manifold.

18. 2. The system of claim 1, wherein the first atomic state and the second atomic state comprise a first hyperfine state and a second hyperfine state on a manifold of multiplets.

19. 2. The system of claim 1, wherein the first atomic state and the second atomic state comprise a first hyperfine state and a second hyperfine state on a triplet manifold.

20. The system of claim 1 , wherein the first and second atomic states comprise first and second nuclear spin states of a quadrupolar nucleus.

21. 2. The system of claim 1, wherein the first atomic state and the second atomic state comprise a first nuclear spin state and a second nuclear spin state with a nuclear spin of 9 / 2.

22. 16. The system of claim 15, wherein the first atomic state and the second atomic state comprise a first nuclear spin state and a second nuclear spin state of strontium-87.

23. 2. The system of claim 1, wherein the subset of the one or more atoms in the one or more superposition states and the other atom are quantum entangled with a coherence lifetime of at least 1 second.

24. The system of claim 1 , wherein the plurality of atoms comprises a temperature of up to 10 microkelvin (μK).

25. The system of claim 1 , further comprising one or more vacuum units configured to maintain the system at a pressure of up to 10 −6 Pascals (Pa).

26. The system of claim 1 , wherein each optical trapping site of the plurality of optical trapping sites is spatially separated from each other optical trapping site by at least 200 nanometers (nm).

27. The system of claim 1 , wherein each optical trapping site of the plurality of optical trapping sites is configured to trap a single atom of the plurality of atoms.

28. The system of claim 1 , wherein the one or more optical trapping sites include one or more optical tweezers.

29. The system of claim 1 , wherein the one or more optical trapping sites comprise one or more optical lattice sites of one or more optical lattices.

30. 30. The system of claim 29, wherein the one or more optical gratings comprise one or more members selected from the group consisting of a one-dimensional (1D) optical grating, a two-dimensional (2D) optical grating, and a three-dimensional (3D) optical grating.

31. The system of claim 1 , wherein the one or more optical trapping units include one or more spatial light modulators (SLMs) configured to generate the plurality of optical trapping sites.

32. 32. The system of claim 31, wherein the one or more SLMs include one or more digital micromirror devices (DMDs) or one or more Light-CoS (LCoS) devices.

33. 2. The system of claim 1, wherein the one or more optical trapping units include one or more light sources configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms.

34. 2. The system of claim 1, wherein the one or more optical trapping units include one or more imaging units configured to obtain one or more images of a spatial shape of the plurality of atoms trapped within the optical trapping sites.

35. 35. The system of claim 34, wherein the one or more images include one or more members selected from the group consisting of a fluorescence image, a single atom fluorescence image, an absorption image, a single atom absorption image, a phase contrast image, and a single atom phase contrast image.

36. 35. The system of claim 34, further comprising one or more spatial shape artificial intelligence (AI) units configured to perform one or more AI operations to determine the spatial shape of the plurality of atoms trapped within the optical trapping site based on the one or more images.

37. 37. The system of claim 36, wherein the one or more AI operations include one or more machine learning (ML) operations.

38. 38. The system of claim 37, wherein the one or more AI operations include one or more reinforcement learning (RL) operations.

39. 35. The system of claim 34, wherein the one or more optical trapping units comprise one or more atom rearrangement units configured to impart an altered spatial arrangement of the plurality of atoms trapped at the optical trapping sites based on the one or more images.

40. 40. The system of claim 39, further comprising one or more spatial configuration artificial intelligence (AI) units configured to perform one or more AI operations to determine the altered spatial configuration of the plurality of atoms trapped within the optical trapping site based on the one or more images.

41. 41. The system of claim 40, wherein the one or more AI operations include one or more machine learning (ML) operations.

42. 41. The system of claim 40, wherein the one or more AI operations include one or more reinforcement learning (RL) operations.

43. 40. The system of claim 39, wherein the one or more atomic rearrangement units are configured to modify the spatial arrangement to obtain an increased filling factor of the plurality of optical trapping sites.

44. 44. The system of claim 43, wherein the fill factor comprises a value of at least 70%.

45. The system of claim 1 , further comprising one or more state preparation units configured to prepare states of the plurality of atoms.

46. 46. ​​The system of claim 45, wherein one or more of the conditioning units are configured to cool the plurality of atoms.

47. 47. The system of claim 46, wherein the one or more conditioning units are configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites.

48. 48. The system of claim 47, wherein one or more of the conditioning units comprises a Zeeman decelerator configured to slow the one or more atoms from a first velocity or distribution of velocities to a second velocity that is lower than the first velocity or distribution of velocities.

49. 49. The system of claim 48, wherein the Zeeman reducer comprises a one-dimensional (1D) Zeeman reducer.

50. 49. The system of claim 48, wherein the second velocity is up to 10 meters per second (m / s).

51. 49. The system of claim 48, wherein one or more of the conditioning units further comprises a first magneto-optical trap (MOT) configured to cool the one or more atoms to a first temperature.

52. 52. The system of claim 51, wherein the first MOT comprises a three-dimensional (3D) MOT.

53. 52. The system of claim 51, wherein the first MOT includes one or more light sources configured to emit light having one or more wavelengths in a range from 400 nm to 500 nm.

54. 52. The system of claim 51, wherein the first temperature is at most 10 milliKelvin (mK).

55. 55. The system of claim 54, wherein one or more of the conditioning units further comprises a second MOT configured to cool the one or more atoms from the first temperature to a second temperature lower than the first temperature.

56. 56. The system of claim 55, wherein the second MOT includes one or more light sources configured to emit light having one or more wavelengths in a range from 400 nm to 1,000 nm.

57. 56. The system of claim 55, wherein the second temperature is up to 100 microKelvin (μK).

58. 56. The system of claim 55, wherein one or more of the conditioning units further comprises a sideband cooling unit.

59. 60. The system of claim 58, wherein the sideband cooling unit is configured to use sideband cooling to cool the one or more atoms from the second temperature to a third temperature that is lower than the second temperature.

60. 60. The system of claim 58, wherein the sideband cooling unit includes one or more light sources configured to emit light having one or more wavelengths in a range from 400 nm to 1,000 nm.

61. 60. The system of claim 58, wherein the third temperature is at most 10 microKelvin (μK).

62. 46. ​​The system of claim 45, wherein one or more of the state preparation units comprises an optical pumping unit configured to emit light to optically pump one or more atoms of the plurality of atoms from an equilibrium atomic state to a non-equilibrium atomic state.

63. 63. The system of claim 62, wherein the optical pumping unit comprises one or more light sources configured to emit light comprising one or more wavelengths in a range from 400 nanometers (nm) to 1,000 nm.

64. 64. The system of claim 63, wherein the light comprises one or more wavelengths in the range of 650 nm to 700 nm.

65. 63. The system of claim 62, wherein one or more of the state preparation units comprises a coherent driving unit configured to coherently drive the one or more atoms from the non-equilibrium atomic state to the first atomic state or the second atomic state.

66. 66. The system of claim 65, wherein the coherent drive unit is configured to induce a two-photon transition between a non-equilibrium state and the first atomic state or the second atomic state.

67. 67. The system of claim 66, wherein the coherent driving unit includes one or more light sources configured to emit light having one or more wavelengths in the range of 400 nm to 1,000 nm.

68. 66. The system of claim 65, wherein the coherent drive unit is configured to induce a one-photon transition between the non-equilibrium state and the first atomic state or the second atomic state.

69. 70. The system of claim 68, wherein the coherent driving unit includes one or more light sources configured to emit light having one or more wavelengths in the range of 400 nm to 1,000 nm.

70. 66. The system of claim 65, wherein the coherent driving unit is configured to induce a radio frequency (RF) transition between the non-equilibrium state and the first atomic state or the second atomic state.

71. 71. The system of claim 70, wherein the coherent driving unit comprises one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transitions.

72. 2. The system of claim 1, wherein the one or more electromagnetic delivery units comprise one or more spatial light modulators (SLMs), acousto-optical devices (AODs), or acousto-optical modulators (AOMs) configured to selectively apply the electromagnetic energy to one or more atoms of the plurality of atoms.

73. 73. The system of claim 72, wherein the one or more electromagnetic delivery units include one or more digital micromirror devices (DMDs) or one or more Light-CoS (LCoS) devices.

74. 73. The system of claim 72, further comprising one or more electromagnetic energy artificial intelligence (AI) units configured to perform one or more AI operations to selectively apply the electromagnetic energy to the one or more atoms.

75. 75. The system of claim 74, wherein the one or more AI operations include one or more machine learning (ML) operations.

76. 75. The system of claim 74, wherein the one or more AI operations include one or more reinforcement learning (RL) operations.

77. The system of claim 1 , wherein the electromagnetic energy comprises optical energy.

78. The system of claim 1 , wherein the electromagnetic energy comprises microwave energy.

79. The system of claim 1 , wherein the electromagnetic energy comprises radio frequency (RF) energy.

80. 80. The system of claim 79, wherein the RF energy comprises one or more wavelengths of at least 30 millimeters (mm).

81. 80. The system of claim 79, wherein the RF energy comprises an average power of up to 10 Watts (W).

82. The one or more electromagnetic delivery units are configured to perform one or more single-qubit gate operations on the one or more qubits. The system of claim 4.

83. The system of claim 1 , wherein the one or more readout optical units include one or more optical detectors.

84. 84. The system of claim 83, wherein the one or more optical detectors include one or more cameras.

85. 84. The system of claim 83, wherein the one or more optical detectors comprise one or more fluorescence detectors.

86. 10. The system of claim 1, further comprising: one or more atom reservoirs configured to provide one or more replacement atoms to replace one or more atoms at one or more optical trapping sites of the plurality of optical trapping sites when the one or more atoms are lost from the one or more optical trapping sites.

87. 87. The system of claim 86, further comprising one or more atom movement units configured to move the one or more replacement atoms to the one or more optical trapping sites.

88. 88. The system of claim 87, wherein the one or more atomic motion units include one or more electrically tunable lenses, acousto-optical deflectors (AODs), or spatial light modulators (SLMs).

89. 2. The system of claim 1, wherein the subset of one or more atoms in the one or more superposition states and the other atom are quantum entangled through a magnetic dipole interaction, an induced magnetic dipole interaction, an electric dipole interaction, or an induced electric dipole interaction.

90. 2. The system of claim 1 , wherein the one or more entanglement units comprise one or more Rydberg excitation units configured to electronically excite the subset of one or more atoms in the one or more superposition states into a Rydberg state or into a superposition of a Rydberg state and a lower energy atomic state, thereby forming one or more Rydberg atoms or Rydberg-less atoms.

91. 91. The system of claim 90, wherein the one or more Rydberg excitation units are configured to induce one or more quantum entanglements between the one or more Rydberg or Rydberg-dress atoms and the other atoms, the other atoms being located at a distance of at most 10 micrometers (μm) from the one or more Rydberg or Rydberg-dress atoms.

92. 92. The system of claim 91 , wherein the one or more Rydberg units are configured to drive one or more Rydberg or Rydberg-less atoms to lower energy atomic states, thereby forming one or more two-qubit units.

93. 93. The system of claim 92, wherein the one or more electromagnetic delivery units are configured to perform one or more two-qubit gate operations on the one or more two-qubit units.

94. 92. The system of claim 91, wherein the one or more Rydberg excitation units include one or more light sources configured to emit light having one or more ultraviolet (UV) wavelengths.

95. 95. The system of claim 94, wherein the light comprises one or more wavelengths in the range of 300 nm to 400 nm.

96. The system of claim 1 , wherein the system is operably linked to a digital computer over a network.

97. The system of claim 1 , wherein the network comprises a cloud computing network.

98. A non-classical computer, the non-classical computer comprising: a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, the plurality of qubits comprising at least a first qubit state and a second qubit state, the first qubit state comprising a first atomic state and the second qubit state comprising a second atomic state; one or more electromagnetic delivery units configured to apply electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, the non-classical operation comprising a superposition between at least the first qubit state and the second qubit state; one or more entanglement units configured to quantum entangle at least a subset of the plurality of qubits in the superposition state with at least another qubit of the plurality of qubits; one or more readout optical units configured to perform one or more measurements of the one or more qubits, thereby obtaining a non-classical computation; and Non-classical computers, including

99. A non-classical computer, comprising a plurality of qubits, each of the qubits comprising more than 60 atoms trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites; Non-classical computers.

100. 1. A method for performing non-classical computation, the method comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms comprising greater than 60 atoms; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state; (c) quantum entanglement of 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; (d) performing one or more optical measurements of the one or more superposition states to obtain the non-classical calculation; A method comprising:

101. 1. A method for performing non-classical computation, the method comprising: (a) providing a plurality of qubits comprising greater than 60 atoms, each atom trapped within an optical trapping site of a plurality of spatially distinct optical trapping sites, wherein the plurality of qubits comprises at least a first qubit state and a second qubit state, wherein the first qubit state comprises a first atomic state and the second qubit state comprises a second atomic state; (b) applying electromagnetic energy to one or more qubits of the plurality of qubits, thereby imparting a non-classical operation to the one or more qubits, the non-classical operation comprising a superposition between at least the first qubit state and the second qubit state; (c) quantum entanglement of at least a subset of the plurality of qubits in the superposition state with at least another qubit of the plurality of qubits; (d) performing one or more optical measurements of the one or more qubits, thereby obtaining the non-classical computation; and A method comprising:

102. 1. A method for performing a non-classical computation, the method comprising: (a) providing a plurality of qubits comprising more than 60 atoms, each atom trapped in an optical trapping site of a plurality of spatially distinct optical trapping sites; and (b) using at least a subset of the plurality of qubits to perform the non-classical computation.

103. 1. A method for performing a non-classical computation, the method comprising: (a) providing a plurality of optical trapping sites comprising a plurality of atoms, the plurality of atoms being a plurality of quantum bits; (b) moving one or more of the plurality of atoms from an occupied trapping site to an unoccupied trapping site, thereby changing a spatial arrangement of the plurality of atoms; (c) applying electromagnetic energy to one or more atoms of the plurality of atoms to induce the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state, where one atom of the one or more atoms of the one or more superpositions is quantum entangled with another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superpositions of the one or more states.

104. 1. A method for performing a non-classical computation, the method comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, the plurality of atoms being quantum bits, and wherein an atom of the plurality of atoms is trapped at an optical trapping site of the plurality of optical trapping sites by attractive forces; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superpositions with at least another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superpositions to obtain the non-classical computation.

105. 1. A method for performing a non-classical computation, the method comprising: (a) generating a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms, wherein the plurality of atoms are qubits; (b) applying electromagnetic energy to one or more atoms of the plurality of atoms, thereby inducing the one or more atoms to adopt one or more superpositions of a first atomic state and at least a second atomic state different from the first atomic state, wherein the applying comprises modulating the electromagnetic energy with at least two optical modulators; (c) quantum entanglement of at least a subset of the one or more atoms of the one or more superpositions with at least another atom of the plurality of atoms; and (d) performing one or more measurements of the one or more superpositions to obtain the non-classical computation.

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