Methods and systems for non-destructive atomic qubit state-resolved imaging for quantum computation

Abstract

Disclosed are systems and methods of performing state detection for non-classical computing. Methods can include obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites, performing one or more qubit gate operations on at least a portion of said first plurality of qubits, performing a measurement operation; and determining that the qubit was in the initial state.

Description

METHODS AND SYSTEMS FOR NON-DESTRUCTIVE ATOMIC QUBIT STATE- RESOLVED IMAGING FOR QUANTUM COMPUTATIONCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 483,160, filed February 3, 2023, which is incorporated by reference herein in its entirety.BACKGROUND

[0002] Quantum computers can make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers may be different from digital electronic computers based on transistors. For instance, whereas digital computers can require data to be encoded into binary digits (bits), each of which is in one of two definite states (0 or 1), quantum computation can use quantum bits (qubits), which can be in superpositions of states.

[0003] In neutral-atom quantum computers or simulation devices, qubits may be encoded in optically trapped atoms. The qubit can be represented by a linear superposition of its two orthonormal basis states. The two orthonormal basis states are usually denoted as |0) = [ ] (the “zero state”) and 11) = (the “one state”). The two orthonormal basis states, {| 0), 11)},together called the computational basis, span the two-dimensional linear vector (Hilbert) space of the qubit. The basis states can also be combined to form product basis states, e.g., 100), 101), 110), 111), each called a quantum register. Generally, n qubits are represented by a superposition state vector in 2ndimensional Hilbert space.

[0004] The ability to reliably detect the quantum state of a qubit may be useful to the operation of a quantum computer. In architectures making use of trapped ions or neutral atoms, the state of the qubit can be read out by collecting photons through imaging systems that spatially or temporally resolve the qubit states.SUMMARY

[0005] Numerous technical challenges can arise in accurately identifying the states of qubits. Systems and methods described herein can provide a technical solution for identifying the states of qubits. For example, systems and methods herein can provide non-lossy qubit state detection by fluorescence imaging in neutral-atom quantum computers. Accordingly, systems and methods herein may avoid reliance on shelving atoms in a metastable clock state, avoid errors caused by imperfect transfer to clock state or scatter out of the clock state, and avoid the complexities of generating moving tune-out tweezers. Moreover, systems and methods herein may non-destructively detect both qubit states and may therefore be compatible with mid-circuit detection and error-correction schemes.

[0006] In an aspect, the present disclosure provides a method of performing state detection for non-classical computing, comprising: (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites; (b) performing one or more qubit gate operations on at least a portion of said first plurality of qubits; (c) performing a measurement operation, wherein said measurement operation comprises exposing a second plurality of qubits to electromagnetic energy, wherein said first plurality of qubits comprises said second plurality of qubits, wherein said electromagnetic energy is configured to selectively drive a qubit of said second plurality of qubits from an initial state to an excited state in a presence of an applied magnetic field, wherein a selectivity of a transition to said excited state is based at least in part on a strength of said applied magnetic field to said first plurality of qubits, said second plurality of qubits, or both; and (d) determining that said qubit was in said initial state, wherein said determining is based at least in part on said qubit returning to said initial state by emission of a photon in response to said electromagnetic energy in (c).

[0007] In some embodiments, said method further comprises repeating (c) to (d) a plurality of times. In some embodiments, said returning to said initial state by emission of said photon is a fluorescence transition. In some embodiments, said returning to said initial state is from a manifold of excited states, wherein said strength of said magnetic field determines a separation between states in said manifold of excited states. In some embodiments, said separation between states determines which states are in resonance with said radiation. In some embodiments, a transition to a single state with said manifold of excited states is allowed by a selection rule for each qubit state. In some embodiments, said (c) comprises exposing a first subset of said second plurality of qubits to a first electromagnetic energy and then exposing a second subset of said second plurality of qubits to a second electromagnetic energy. In some embodiments, said first electromagnetic energy comprises a first polarization and said second electromagnetic energy comprises a second polarization. In some embodiments, said method further comprises: (e) determining that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation. In some embodiments, said method further comprises: (f) determining, based at least in part on determining at (e) that a number (n) of spatially distinct optical trapping sites of said array of spatially distinct optical trapping sites is missing a qubit. In some embodiments, said first plurality of qubits has at most n more qubits than said second plurality of qubits. In some embodiments, said electromagnetic energy is polarized. In some embodiments, one or more of said electromagnetic energy, said first electromagnetic energy, orsaid second electromagnetic energy is circularly polarized. In some embodiments, said qubit comprises a first state |0) and said second state |1). In some embodiments, said initial state is |0). In some embodiments, said initial state is |1). In some embodiments, said first plurality of qubits comprises neutral atoms. In some embodiments, said neutral atoms comprise a Group II element. In some embodiments, said Group II element is strontium. In some embodiments, said neutral atoms comprise rubidium or cesium. In some embodiments, said neutral atoms comprise ytterbium. In some embodiments, said qubits comprise a temperature of at most 10 microkelvin (pK). In some embodiments, said array is two-dimensional. In some embodiments, said array is three-dimensional. In some embodiments, said transition is a cycling transition. In some embodiments, said qubit state comprises a first state |0) and said second state |1), wherein said excited state comprises a manifold of excited states, wherein a transition between said first state and said manifold and said second state in said manifold comprise two closed two-level systems. In some embodiments, a state of said qubit is identified as one or said first state or said state by emission of said photon or by an absence of said emission of said photon. In some embodiments, said returning to said initial state comprises spontaneous emission, wherein said returning to said initial state comprises stimulated emission. In some embodiments, prior to said (c), the method comprises tuning said strength of said applied magnetic field to a selected measurement state. In some embodiments, said (d) comprises imaging said photon at a measurement unit. In some embodiments, said (c) to (d) is configured to perform an atom cooling operation.

[0008] In another aspect, provided is a system for performing state detection for non-classical computing, comprising: one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits; one or more non-classical computation units configured to perform a non-classical computation using at least a portion of said first plurality of qubits; one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and one or more measurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0009] In another aspect, provided is a non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of performing state detection for non-classical computing on a non-classical computer, wherein said non-classical computer is configured to execute said one or more instructions, the method comprising: (a) obtaining a first plurality of qubits in an array of spatially distinct opticaltrapping sites; (b) performing a non-classical computation using at least a portion of said first plurality of qubits; (c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and (d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0010] In another aspect the present disclosure provides a method for non-destructive atomic qubit state-resolved imaging. The method may comprise: (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites; (b) performing a non-classical computation using at least a portion of said first plurality of qubits; (c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and (d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0011] In some embodiments, exposing said second plurality of qubits to said radiation comprises exposing said second plurality of qubits to a first radiation and then exposing said second plurality of qubits to a second radiation. In some embodiments, said first radiation is a first light with a first polarization and said second radiation is a second light with a second polarization. In some embodiments, the method further comprises: (e) determining that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation. In some embodiments, the method further comprises: (f) determining, based at least in part on determining at (e) that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation, that a number (n) of spatially distinct optical trapping sites of said array of spatially distinct optical trapping sites is missing a qubit.

[0012] In some embodiments, said first plurality of qubits has at most n more qubits than said second plurality of qubits. In some embodiments, said radiation is light. In some embodiments, said light is polarized light. In some embodiments, one or more of said polarized light, said first light, or said second light is circularly polarized. In some embodiments, said second plurality of qubits are exposed to said radiation while said second plurality of qubits are in a magnetic field. In some embodiments, said at least one of said second plurality of qubits that fluoresces in response to being exposed to said radiation is said first subset of said second plurality of qubits.In some embodiments, said first state is |0) and said second state is |1). In some embodiments, said first state is 11) and said second state is |0). In some embodiments, said first plurality of qubits comprise neutral atoms. In some embodiments, said neutral atoms comprise a Group II element. In some embodiments, said Group II element is strontium. In some embodiments, said neutral atoms comprise ytterbium. In some embodiments, said qubits comprise a temperature of at most 10 microkelvin ( l<). In some embodiments, said array is two-dimensional. In some embodiments, said array is three-dimensional.

[0013] In another aspect, the present disclosure provides: a system for performing state detection for non-classical computing. The system may comprise: one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits; one or more non-classical computation units configured to perform a non-classical computation using at least a portion of said first plurality of qubits; one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and one or more measurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0014] In another aspect, the present disclosure provides non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of performing state detection for non-classical computing on a non-classical computer, wherein said non-classical computer is configured to execute said one or more instruction. The method may comprise: (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites; (b) performing a non-classical computation using at least a portion of said first plurality of qubits; (c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and (d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0015] Another aspect of the present disclosure provides a system for non-destructive atomic qubit state-resolved imaging comprising one or more computer processors and computer memory coupled thereto. The computer memory comprises machine executable code that, upon execution by the one or more computer processors, implements any of the methods above or elsewhere herein.

[0016] Another aspect of the present disclosure provides a non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of non-destructive atomic qubit state-resolved imaging, wherein said non-classical computer is configured to execute said one or more instructions. The method implemented being the methods above or elsewhere herein.

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

[0018] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and / or take precedence over any such contradictory material.BRIEF DESCRIPTION OF THE DRAWINGS

[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 “Figure” and “FIG.” herein), of which:

[0020] FIG. 1 illustrates an example energy level diagram of a system enabling cycling transition for qubit states;

[0021] FIG. 2 illustrates an example energy spectrum of excited states;

[0022] FIG. 3 illustrates a schematic of a method of performing state detection for non-classical computing;

[0023] FIG. 4 illustrates a schematic of another method of performing state detection for non- classical computing;

[0024] FIG. 5A illustrates an example system configured to implement methods provided herein.

[0025] FIG. 5B illustrates (top) a level diagram for an example atomic system which can be used in the methods and systems disclosed herein and (bottom) experimental data showing twosubsequent single-shot images of a fully filled 10 by 3 array of atoms generated by an example of the methods and systems disclosed herein.

[0026] FIG. 6 illustrates a computer control system that is programmed or otherwise configured to implement methods provided herein;

[0027] FIG. 7 illustrates an example of state-resolved imaging infidelity; and

[0028] FIG. 8 illustrates another example of state-resolved imaging infidelity.DETAILED DESCRIPTION

[0029] While various embodiments of the invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed.

[0030] Systems and methods herein provide improved methods of measuring a state of a qubit, e.g., measurement operations. It may be useful to measure a qubit state selectively. For example, a measurement operation that selectively determines whether the state of the qubit is state |0> or state |1>. It may be useful to measure a qubit state site selectively. For example, a measurement operation that selectively determines a state of a particular qubit in an array may be useful for a quantum computation. It may be useful to measure a qubit without causing atom loss of the qubit from the array. Additionally, because error correcting codes may involve measurements on qubits during an error correcting cycle, measurements which preserve the atom in the array may be helpful for improving the efficiency of error correcting codes.

[0031] Systems and methods herein can improve upon nondestructive state-resolved detection without using a high magnetic field (B field). In some cases, the detection mechanism may rely on (i) the electric dipole selection rules (or angular momentum selection rules) and (ii) purity control of polarization of the imaging beam. For example, one of the qubit states can be bright and the other qubit state dark, which can lead to false-positive atom detection when it comes to the dark state call-out. Additionally, systems and methods herein can improve upon non-strict qubit state resolvers resolving between two hyperfine manifolds (e.g., of87Rb,87Sr,171Yb, etc.). Additionally, systems and methods herein can improve upon destructive (e.g., lossy) ways of state-resolved detection that remove (e.g., blow away) one of the qubit states from the trap to detect the existence of atoms.

[0032] Systems and methods of the present disclosure may approve upon methods of measuring atoms which implement state-selective atom loss followed by state-independent imaging of the remaining atoms (e.g., destructive methods). Destructive methods may not be able to distinguishbetween errors due to atom loss and those that populate the ejected state. Destructive methods may require frequent reloading in order to address these errors. Methods and systems disclosed herein which are non-destructive may be more robust to atom loss and may require fewer reloading cycles. Systems and methods of the present disclosure may improve upon methods of measuring atoms which utilize selective movements of the atoms apart followed by state-independent imaging at least because additional movement steps may be slower, more complicated, or both. Systems and methods of the present disclosure may improve upon methods of measuring atoms which use an additional pulse to selectively shelve atoms to be images at least because additional pulses may be slower, more complicated, or both. Systems and methods may improve upon methods which chose states which have selective scattering rates for the imaging light or which use high-finesse optical cavities because the possible imaging conditions, qubit states, and trapping states may be more flexibly chosen without using these methods. Systems and method so the present disclosure may relieve at least some of the above referenced drawbacks.

[0033] Systems and methods of the present disclosure may employ narrow line-width imaging in combination with a differential Zeeman shift (e.g., a shift in the presence of a magnetic field). Either or both of the qubit states (e.g., |0>, 11>) may be separately imaged by light from an imaging beam. Either or both states may be shifted by an applied magnetic field. The applied magnetic field may determine a selectivity of the imaging transition for either or both qubit states. For example, a strength of the applied magnetic field may selectively move an upper state of the imaging transition, a lower state of the imaging transition or both. In some cases, compared to other methods, methods and systems of the present disclosure can provide improved methods by combining narrow-line imaging with large Zeeman shifts created by operating in a low magnetic field, e.g., about 500 Gauss magnetic field, as described herein.

[0034] In some cases, detecting qubit states may include (A) initializing qubits in some initial state; (B) performing a quantum computation (e.g., which can involve the atoms flipping back and forth, where, at the end of the computation, an atom will either be in the zero state or the one state); (C) and then, in order to determine which states each qubit is in, bathing the qubits in the circularly, polarized, resonant light in the presence of magnetic fields.

[0035] Methods and systems disclosed herein have various applications in the field of quantum computing. For example, measurement-based quantum error correction can rely on the ability to determine the state of a subset of qubits (e.g., ancilla qubits) within a processor without revealing or disturbing the state of the remaining qubits, e.g., data qubits or other ancilla qubits. Measurement-based quantum error correction can include repeated “mid-circuit” measurements (e.g., imaging) in a single-species tweezer-array of neutral171Yb atoms ancilla qubits. Tweezer- confined alkaline earth atoms, such as171Yb, or lanthanide atoms can be used with systems andmethods described herein for quantum computation. Systems and methods herein can perform mid-circuit measurements (MCM) using a narrow-linewidth transition in an array of tweezer- confined171Yb atoms thereby providing nondestructive state-selective detection. In some cases, by applying site-specific light shifts, selected atoms within the array can be hidden from imaging light, which can allow a subset of qubits to be measured while causing only percent-level errors on the remaining qubits.

[0036] The system and methods of the present disclosure may have various useful extensions. For example, the cycling transitions that is utilized for the detection of the qubit states may also provide the cooling mechanisms, e.g., Doppler cooling, preventing the atoms escaping from optical traps.Non-destructive atomic qubit state-resolved measurements

[0037] Examples of non-lossy qubit state detection by fluorescence imaging in neutral-atom quantum computers are presented. In some cases, detecting qubit states may include (A) initializing qubits in some initial state; (B) performing a quantum computation (which involves the atoms flipping back and forth, where, at the end of the computation, an atom will either be in the zero state or the one state); (C) and then, in order to determine which states each qubit is in, exposing the qubits in circularly, polarized, resonant electromagnetic energy (e.g., light at a predetermined wavelength) in the presence of magnetic fields to generate spontaneous or stimulated emissions of photons.

[0038] The present disclosure provides methods for non-destructive atomic qubit state-resolved imaging. In some cases, the method may comprise: (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites; (b) performing a non-classical computation using at least a portion of said first plurality of qubits; (c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and (d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0039] The present disclosure provides methods of performing state detection for non-classical computing. In some cases, the method comprises (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites. In some cases, the method comprises (b) performing one or more qubit gate operations on at least a portion of said first plurality of qubits In some cases, the method comprises (c) performing a measurement operation, wherein said measurement operation comprises exposing a second plurality of qubits to electromagnetic energy, wherein saidfirst plurality of qubits comprises said second plurality of qubits, wherein said electromagnetic energy is configured to selectively drive a qubit of said second plurality of qubits from an initial state to an excited state in a presence of an applied magnetic field, wherein a selectivity of a transition to said excited state is based at least in part on a strength of said applied magnetic field to said first plurality of qubits, said second plurality of qubits, or both. In some cases, the method comprises (d) determining that said qubit was in said initial state, wherein said determining is based at least in part on said qubit returning to said initial state by emission of a photon in response to said electromagnetic energy in (c). In some cases, the method can comprise repeating (c) to (d) a plurality of times. In some cases, the initial state is |0). In some cases, the initial state is | 1). In some cases, the qubit comprises a first state |0) and the second state 11 ).

[0040] In some cases, the qubit state comprises a first state |0) and a second state 11). In some cases, the excited state comprises a manifold of excited states. In some cases, a transition between the first state and the manifold and the second state in the manifold comprise two closed two-level systems. In some cases, a state of the qubit is identified as one or the first state or the state by emission of the photon or by an absence of the emission of the photon.

[0041] For example, in some cases, atomic qubit state-resolved detection may include fluorescence imaging utilizing cycling transitions between qubit states and excited states that fluoresce photons for the imaging. In some cases, the fluorescing of the photons for imaging can be done without losing atoms from optical traps before and after the detection. The cycling transitions in between the qubit states and the excited states can be driven by electromagnetic energy (e.g., lasers) near resonant to the qubit states and the excited states.

[0042] In some cases, the transitions may be closed two-level systems, respective to each one of the qubit states as shown in FIG. 1. For example, the transition from |0 > (| 1 >) to |el > (|e4 >) may be a closed two-level system when » 8, I20 1, f, where r is the spontaneous decay rate, 8 is the detuning between the energy level and the excitation laser light, and Q is the Rabi frequency. Such a two-level system can enable cycling transition for qubit states resolved imaging by collecting spontaneous photon emissions. A spontaneous photon emission may decay the excited population of |el> (|e4>) exclusively to |0> (|1>) due to the electric dipole selection rules.

[0043] In some cases, the applied magnetic field can change the excited states energy levels by Zeeman shift as shown in FIG. 2. In some cases, the magnetic field can energetically separate the transition paths of two different qubit states to an independently closed system. In the example of FIG. 2, the energy spectrum of the excited states is illustrated as a function of the magnetic field by 4 = 2TT x 1.4 MHz / Gauss * B. The electric dipole selection rules (or angular momentum selection rules) may cause the population of excited states to decay to the origin of the excited states. In some cases, the independent closed system for each qubit state formed by the largeZeeman shift does not require the imaging light to have clean polarization, which can enable systems and methods herein to use relatively lower trap depths and cool the atoms while imaging. Accordingly, the state-resolved detection is enabled by time-multiplexed fluorescence imaging of the qubit states, one after the other.

[0044] Referring again to FIG. 1, the energy level diagram shows, in some cases, the ground state of qubits (e.g., Ytterbium atoms that are spin ’A, such as Ytterbium 171), as well as excited states to which the qubits may be driven to fluorescence. For the example, in which the qubits are spin ’A, there may be two possible states for the qubits. In some cases, the qubits may be driven to a manifold of states associated with f=3 / 2, therefore having spin states of -3 / 2, -1 / 2, +1 / 2, and +3 / 2. Further, if a qubit in the zero state is driven (e.g., with circularly polarized light), the qubit, by selection rules, may in some cases, have one possible energy level to which it can be driven. Therefore, in order to have states that can be determined or predicted using imaging, one state (e.g., the zero state) may fluoresce and the other state (e.g., the one state) may not fluoresce. The straight arrows in FIG. 1 illustrate, based on the selection rules, each energy level that each respective qubit state may be able to reach. For example, in the left frame of FIG. 1, a qubit in the zero state may be able to reach the |el> energy level, and a qubit in the one state may be able to reach the |e2> energy level. Continuing with this example, qubits of the zero state reaching the |el> energy level may fluoresce (illustrated with the wavy arrow), while the qubits of the one state reaching the |e2> may not fluoresce, enabling detection (e.g., via an imager) of which qubits are in which state. The right frame of FIG. 1 illustrates a different polarization of light than the polarized light in the left frame. As illustrated, with the polarization of light in the right frame, qubits in the zero state may be driven to reach the |e3> energy level, while qubits in the one state may be driven to reach the |e4> energy level. Continuing with this example, qubits of the one state reaching the |e4> energy level may fluoresce (illustrated with the wavy arrow), while the qubits of the zero state reaching the |e3> may not fluoresce, enabling detection (e.g., via an imager) of which qubits are in which state.

[0045] In some cases, the application of a magnetic field may enable this detection based on differences in fluorescence for different qubit states. In some cases, the magnetic field may cause only one energy level transition for each of the left frame and the right frame of FIG. 1 to be on resonance and accordingly fluoresce. Conversely, if there is zero magnetic field, qubits of both states may fluoresce. The amount that the upper levels shift in the presence of a magnetic field may be about 1 to 2 megahertz per Gauss, whereas the amount the ground state level shifts may be about a few hundred hertz per Gauss. As such, FIG. 1 illustrates that the ground states may shift less than the excited states. It can be understood that it is not strictly necessary for the groundstates to shift less than the excited states, rather it may, in some cases, just be useful that the ground states do not shift by the same amount as the excited states shift.

[0046] In some cases, it may be useful for there to be a differential in the amount or magnitude of shifts between the ground states and the excited states such that, as a stronger magnetic field is applied, one energy level may be in resonance while the other energy level is out of resonance. For the example illustrated in the left frame of FIG. 1, if a first qubit was in the zero state, photons may scatter from it (e.g., producing a bright spot in a camera) and if a second qubit was in the one state, photons may not scatter from it (e.g., leaving a dark spot in the camera). In some cases, there may be additional complication relating to qubit loss. For example, if there is a dark spot in the camera, which can mean, in this example, there is a qubit in the one state in the corresponding position; however, this may alternatively mean that there has been qubit lost from the corresponding position. However, as illustrated in the two frames of FIG. 1, because a qubit in a given state may fluoresce when sending light polarized in one way, but not light polarized in a different way, by performing the light bathing process of both the left frame and the right frame of FIG. 1, qubit loss may be identified to be at positions that do not fluoresce after either the process of the left frame or the right frame of FIG. 1. In some cases, one light polarization may be used, provided qubit loss occurs at a sufficiently low rate. In some cases, a first light polarization may be used primarily, with a second light polarization used at a certain interval (e.g., every 3 cycles, every 10 cycles, every 100 cycles, etc.) to check if qubit loss has occurred. The certain interval may be dependent on qubit loss rates. For example, a lower qubit loss rate may mean that checking to see if qubit loss has occurred using a second light polarization may be done less frequently. If qubit loss is detected, the qubit may be replaced.

[0047] In some examples, methods and systems disclosed herein can selectively image the1Somf = 1 / 2, -1 / 2 qubit states (or alternatively |1) and |0), respectively) of a Group II element or a Group II like element by applying electromagnetic energy (e.g., light at a predetermined wavelength). In some cases, the states may be imaged from either one of two counter-propagating imaging beams, each configured to image one of thexSo m = / i and -1 / 2 qubit states. In some cases, the electromagnetic energy is polarized. In some cases, the transition is a cycling transition. In some cases, the beams can be tuned to address one of the3Pi mf= ±3 / 2 states, which can provide access to narrow linewidth (e.g., about 180 kHz) closed cycling transitions. Scattering from the mf = ±1 / 2 excited states, which may allow population leakage between the qubit states, can be suppressed by the large ratio of Zeeman shifts to transition linewidth. In some cases, the ratios can include 771 MHz, 681 MHz between the -3 / 2 and -1 / 2 states and the 1 / 2 and 3 / 2 states, respectively.

[0048] FIG. 3 illustrates a schematic of a method 300 of performing state detection for non- classical computing. FIG. 4 illustrates a schematic of a method 400 of performing state detection for non-classical computing.

[0049] Plurality of Spatially Distinct Optical Trapping Sites - At an operation 310 of a method 300 the method may comprise obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites. Operation 410 of a method 400 may comprise an example, variation, or embodiment of the method 310 of method 300.

[0050] Trapping - In some examples, the optical traps may be formed by tightly focused light (tweezers) or by standing-wave lattices, or by imaged masks or gratings. Optical trapping may additionally include various methods where atoms are cooled with optical illumination, e.g., a laser, and a spatially varying magnetic field to create a trap. Such optical traps may be called magneto-optical traps (MOTs).

[0051] In some cases, the array is two dimensional. In some cases, the array is three dimensional. In some cases, the plurality of spatially distinct optical traps comprises a ID, 2D, or 3D optical trap. In some examples, the arrays may be linear, two-dimensional, three-dimensional, or may involve synthetic dimensions. A synthetic dimension may include, for example, dimensions consisting of internal atomic states or motional states. The plurality of spatially distinct optical traps may comprise single or multiple reservoir regions. In some examples, the arrays may be of regular or irregular or quasi-regular geometry.

[0052] In some cases, the array is two-dimensional. For example, an array of two-dimensional optical traps can be formed. The two-dimensional array can include a rectangular, square, rectangular prism, or cubic array of optical trapping sites. In some cases, the method further comprises determining, based at least in part on determining that a number of spatially distinct optical trapping sites of the array of spatially distinct optical trapping sites is missing a qubit. For example, each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 pm, 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site may be spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the preceding values. In some cases, the array is three-dimensional. For example, an array of three-dimensional optical traps can be formed.

[0053] Cooling - The array of spatially distinct optical trapping sites may comprise a potion of an atom cooling and trapping system. The atom cooling and trapping system may comprise oneor a plurality of optical lattices. For example, the atom cooling and trapping system may comprise a first optical lattice followed by a second optical lattice. In some cases, the optical lattices may be filled by a reservoir trap. In some cases, the reservoir may comprise an unstructured or semistructured optical trap. In some cases, atoms can be loaded from a pre-cooled atomic beam into a two-stage magneto-optical trap (formed using the 399 nm1Pi transition followed by the 556 nm3Pi narrow-line transition). In some cases, atoms can then be loaded into an optical lattice formed using 532 nm light from an optical trapping system.

[0054] The qubits in the array of spatially distinct traps may be cooled to a temperature. In some cases, the qubits comprise a temperature of at most 10 microkelvin (pK). In some cases, the qubits comprise a temperature of at most 10 microkelvin (pK). In some cases, the one or more atoms disposed within the optical traps can include a temperature of at least about 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 300, 400, 500, or more microkelvin. In some cases, the one or more atoms disposed within the within the optical traps can include a temperature of at most about 500, 400, 300, 200, 190, 180, 170, 160, 150, 140, 130, 120, 110, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 4, 3, 2, 1, or less microkelvin. In some cases, the one or more atoms disposed within the optical traps can include a temperature in a range as defined by any two of the proceeding values.

[0055] Optical tweezers - In some cases, the plurality of spatially distinct optical traps comprises optical tweezers. The optical trapping sites may comprise one or more optical tweezers. Optical tweezers may comprise one or more focused laser beams to provide an attractive or repulsive force to hold or move the one or more atoms. The beam waist of the focused laser beams may comprise a strong electric field gradient. The atoms may be attracted or repelled along the electric field gradient to the center of the laser beam, which may contain the strongest electric field. The optical trapping sites may comprise one or more optical tweezer sites of one or more optical arrays of tweezers. The optical trapping sites may comprise one or more optical tweezer sites of one or more one-dimensional (ID) optical arrays of tweezers, two-dimensional (2D) optical arrays of tweezers, or three-dimensional (3D) optical arrays of tweezers. In some cases, the methods and systems described herein may be applied similarly to optical lattices. Optical tweezers may be useful in moving atoms or arrays of atoms.

[0056] Sites - The optical trapping system may be configured to generate a plurality of optical trapping sites. The optical trapping system may be configured to generate a plurality of spatially distinct optical trapping sites. Each optical trapping system may comprise any number of sites disclosed herein. Each optical trapping system may comprise any number of trapped atoms disclosed herein.

[0057] For instance, each optical trapping system may be configured to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. Each optical trapping system may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer optical trapping sites. The optical trapping system(s) may be configured to trap a number of optical trapping sites that is within a range defined by any two of the preceding values.

[0058] Each optical trapping system may be configured to trap a plurality of atoms. For instance, each optical trapping system may be configured to trap a total number of atoms in the plurality of optical trapping sites of at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. For example, the optical trapping system(s) may be configured to trap a total number of atoms in the plurality of optical trapping sites of at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The optical trapping system(s) may be configured to trap a number of atoms that is within a range defined by any two of the preceding values.

[0059] Trap Electromagnetic Energy - In some cases, method and systems disclosed herein may be configured to form a plurality of optical trapping sites using a trap electromagnetic energy (e.g. , a “trap excitation” herein). The trap excitation may be generated by a trapping optical source.

[0060] The trap excitation may comprise an optical excitation, such as in a magneto-optical trap, an optical tweezer, etc. In some cases, the trap excitation is delivered by one or more optical trapping systems as disclosed herein. In some cases, each optical trapping system comprises its own trap excitation (e.g., trap wavelength, trap power, trap focus, number of spots, etc.). In some cases, a single trap excitation may be split into multiple arrays in order to form a plurality of arrays of traps with similar characteristics.

[0061] Atoms - Systems and methods of the present disclosure may be applied to any atomic system that may be cooled and trapped. In some cases, the first plurality of qubits comprisesneutral atom qubits. In some cases, the plurality of atoms comprises neutral atoms. In some cases, the plurality of atoms comprises a Group II element. In some cases, the plurality of atoms comprises Strontium. In some cases, the plurality of atoms comprises a Group Il-like element. In some cases, the plurality of atoms an atom with two-valence electrons. In some cases, the plurality of atoms comprises Ytterbium. In some cases, the plurality of atoms are qubits.

[0062] The optical trapping system may be configured to trap neutral atoms. In some cases, the optical trapping system may trap alkaline earth or an alkaline earth-like atom. In some cases, an alkaline earth-like atom comprises two valence electrons. In some cases, an alkaline earth or an alkaline earth-like atom comprises strontium or ytterbium.

[0063] In some cases, one or more atoms may comprise alkali atoms. One or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium- 23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium- 9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise Scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms,holmium- 165 atoms, thulium- 169 atoms, ytterbium- 168 atoms, ytterbium- 170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms.

[0064] In some cases, the plurality of atoms may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium -23 atoms, potassium-39 atoms, potassium -40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium- 42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium- 84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium- 142 atoms, praseodymium- 141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium- 165 atoms, thulium- 169 atoms, ytterbium- 168 atoms, ytterbium- 170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%,99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum - 139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium- 142 atoms, praseodymium- 141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium- 165 atoms, thulium- 169 atoms, ytterbium- 168 atoms, ytterbium- 170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium- 137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum - 139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium- 142 atoms, praseodymium- 141 atoms, neodymium- 142 atoms, neodymium- 143 atoms, neodymium- 145 atoms, neodymium- 146 atoms, neodymium- 148 atoms, samarium- 144 atoms, samarium- 149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms,europium-153 atoms, gadolinium- 154 atoms, gadolinium- 155 atoms, gadolinium- 156 atoms, gadolinium- 157 atoms, gadolinium-158 atoms, gadolinium- 160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium- 160 atoms, dysprosium-161 atoms, dysprosium- 162 atoms, dysprosium- 163 atoms, dysprosium- 164 atoms, erbium- 162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium- 165 atoms, thulium- 169 atoms, ytterbium- 168 atoms, ytterbium- 170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance that is within a range defined by any two of the preceding values.

[0065] In some cases, the first plurality of qubits comprises neutral atoms. In some cases, the second plurality of qubits can include neutral atoms. For example, the one or more atoms of the array can include one or more qubits. The one or more atoms can be configured to be usable as one or more qubits. The one or more qubits may be configured to perform a non-classical computation. For example, the one or more qubits can be configured to perform a gate-based quantum computation. In another example, the one or more qubits may be configured to perform a quantum computation. The one or more atoms may comprise at least about 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 300, 400, 500, or more atoms. The one or more atoms may comprise at most about 500, 400, 300, 200, 190, 180, 170, 160, 150, 140, 130, 120, 110, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 4, 3, 2, or fewer atoms. The one or more atoms may comprise a number of atoms as defined by any two of the proceeding values. For example, the one or more atoms may comprise from about 75 to about 150 atoms.

[0066] Gate Operations - At an operation 320 of a method 300, the method may comprise performing a non-classical computation using at least a portion of a first plurality of qubits. At an operation 420 of a method 400, the method may comprise performing one or more qubit gate operations on at least a portion of a first plurality of qubits. The one or more qubit gate operations may comprise sub-potions of a non-classical computation disclosed herein. For example, the one or more qubit gate operations may comprise an error correcting code. In some cases, methods and systems described herein may facilitate mid-circuit measurement operations on qubits (e.g., ancilla qubits) during a non-classical computation. Operation 420 of a method 400 may comprise an example, variation, or embodiment of the method 320 of method 300.

[0067] A non-classical computation may comprise a gate-model quantum computation, a quantum annealing procedure, etc. In some cases, a qubit may comprise a first state |0) and said second state |1). In some cases, the qubit state to be measured is |0). In some cases, the qubit state to be measured is 11).

[0068] In some cases, a qubit state comprises a first state |0) and a second state | 1), wherein the excited state comprises a manifold of excited states, and wherein a transition between said first state and said manifold and said second state in said manifold comprises two closed two-level systems.

[0069] Qubits - In some cases, the neutral atoms are qubits. In some cases, the neutral atoms comprise a Group II element. In some cases, the Group II element is strontium. In some cases, the neutral atoms comprise rubidium or cesium. In some cases, the neutral atoms comprise ytterbium. For example, the first plurality of qubits or the second plurality of qubits can include neutral atoms. The one or more atoms can include atoms that are not ionized (e.g., are in a neutral state). In some cases, each atom of the one or more atoms may be a neutral atom. For example, each atom of an array of atoms can be not ionized. In some cases, the one or more atoms may comprise rare earth atoms (e.g., lanthanide series atoms (e.g., ytterbium, neodymium, lanthanum, erbium, etc.), alkali atoms (e.g., sodium, potassium, rubidium, cesium, etc.), alkali earth atoms (e.g., calcium, strontium (e.g., strontium-87 atoms), etc.), or the like, or any combination thereof.

[0070] In some cases, the qubits described herein may comprise nuclear spin qubits. The qubit states (e.g., |0>, |1>) may comprise nuclear spin states within an electronic state manifold. For example, the qubit states may be nuclear spin states on a ground state manifold. In some cases, the qubit states may comprise nuclear spin states on a ground state manifold of a Group II or a Group II like element. A Group II like element may comprise two valence electrons. In some cases, the ground state manifold is a1So state. In some cases, the ground state manifold is a1So state of87Rb,87Sr,171Yb, etc.

[0071] Qubit States- In some cases, the qubits described herein may comprise a first atomic state and a second atomic state. The first atomic state may comprise a first single-qubit state. The second atomic state may comprise a second single-qubit state. The first atomic state or second atomic state may be elevated in energy with respect to a ground atomic state of the atoms, e.g., within an excited state manifold. The first atomic state or second atomic state may be within a ground state manifold.

[0072] The first atomic state may comprise a first hyperfine electronic state and the second atomic state may comprise a second hyperfine electronic state that is different from the first hyperfine electronic state. For instance, the first and second atomic states may comprise first and second hyperfine states on a multiplet manifold, such as a triplet manifold, a singlet manifold, etc. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi,3P2, ' SO manifold, etc. The first and second atomic states may comprise first and second hyperfine states, respectively, on a3Pi,3P2, manifold of any atom described herein, such as astrontium-873Pi manifold, a strontium-873P2, a strontium-871So, ytterbium-1713Pi manifold, a ytterbium-1713P2, a ytterbium-171xSo manifold.

[0073] In some cases, the first and second atomic states are first and second hyperfine states of a first electronic state. Optical excitation may be applied between a first electronic state and a second electronic state. The optical excitation may excite the first hyperfine state and / or the second hyperfine state to the second electronic state. A single-qubit transition may comprise a two-photon transition between two hyperfine states within the first electronic state using a second electronic state as an intermediate state. To drive a single-qubit transition, a pair of frequencies, each detuned from a single-photon transition to the intermediate state, may be applied to drive a two-photon transition. In some cases, the first and second hyperfine states are hyperfine states of the ground electronic state. The ground electronic state may not decay by spontaneous or stimulated emission to a lower electronic state. The hyperfine states may comprise nuclear spin states.

[0074] In some cases, the hyperfine states comprise nuclear spin states of a strontium-87xSo or a ytterbium-171 ' So manifold and the qubit transition drives one or both of two nuclear spin states of strontium-87xSo or a ytterbium-171JSoto a state detuned from or within the3P2 or3Pi manifold. In some cases, the one-qubit transition is a two photon Raman transition between nuclear spin states of strontium-87 ' So or ytterbium-1711So via a state detuned from or within the3P2 or3Pi manifold. In some cases, the nuclear spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift may be driven off resonance with any, all, or a combination of a single-qubit transition, a two-qubit transition, a shelving transition, an imaging transition, etc.

[0075] In some cases, the hyperfine states comprise nuclear spin states of ytterbium

[0076] The first atomic state may comprise a first nuclear spin state and the second atomic state may comprise a second nuclear spin state that is different from the first nuclear spin state. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a quadrupolar nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin- 9 / 2 nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of any atom described herein, such as first and second spin states of strontium-87.

[0077] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1 / 2 (such as a spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, forinstance, an mN = 9 / 2 spin state to an mN = 7 / 2 spin state, may also drive mN = 7 / 2 to mN = 5 / 2, mN = 5 / 2 to mN = 3 / 2, mN = 3 / 2 to mN = 1 / 2, mN = 1 / 2 to mN = -1 / 2, mN = -1 / 2 to mN = -3 / 2, mN = -3 / 2 to mN = -5 / 2, mN = -5 / 2 to mN = -7 / 2, and mN = -7 / 2 to mN = -9 / 2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9 / 2 spin state to an mN = 5 / 2 spin state, may also drive mN = 7 / 2 to mN = 3 / 2, mN = 5 / 2 to mN = 1 / 2, mN = 3 / 2 to mN = -1 / 2, mN = 1 / 2 to mN = -3 / 2, mN = -1 / 2 to mN = -5 / 2, mN = -3 / 2 to mN = -7 / 2, and mN = -5 / 2 to mN = -9 / 2. Such a transition may thus not be selective for inducing transitions between particular spin states on the nuclear spin manifold.

[0078] It may be desirable to instead implement selective transitions between particular first and second spins states on the nuclear spin manifold. This may be accomplished by providing light from a light source that provides an AC Stark shift and pushes neighboring nuclear spin states out of resonance with a transition between the desired transition between the first and second nuclear spin states. For instance, if a transition from first and second nuclear spin states having mN = -9 / 2 and mN = -7 / 2 is desired, the light may provide an AC Stark shift to the mN = -5 / 2 spin state, thereby greatly reducing transitions between the mN = -7 / 2 and mN = -5 / 2 states. Similarly, if a transition from first and second nuclear spin states having mN = -9 / 2 and mN = -5 / 2 is desired, the light may provide an AC Stark shift to the mN = -1 / 2 spin state, thereby greatly reducing transitions between the mN = -5 / 2 and mN = -1 / 2 states. This may effectively create a two-level subsystem within the nuclear spin manifold that is decoupled from the remainder of the nuclear spin manifold, greatly simplifying the dynamics of the qubit systems. It may be advantageous to use nuclear spin states near the edge of the nuclear spin manifold (e.g., mN = -9 / 2 and mN = -7 / 2, mN = 7 / 2 and mN = 9 / 2, mN = -9 / 2 and mN = -5 / 2, or mN = 5 / 2 and mN = 9 / 2 for a spin-9 / 2 nucleus) such that only one AC Stark shift is required. Alternatively, nuclear spin states farther from the edge of the nuclear spin manifold (e.g., mN = -5 / 2 and mN = -3 / 2 or mN = -5 / 2 and mN = -1 / 2) may be used and two AC Stark shifts may be implemented (e.g., at mN = -7 / 2 and mN = -1 / 2 or mN = - 9 / 2 and mN = 3 / 2).

[0079] Stark shifting of the nuclear spin manifold may shift neighboring nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. Stark shifting may decrease leakage from the first and second nuclear spin state to other states in the nuclear spin manifold. Starks shifts may be achievable up to 100s of kHz for less than 10 mW beam powers. Upper state frequency selectivity may decrease scattering from imperfect polarization control. Separation of different angular momentum states in the3Pi manifold may be many gigahertz from the single and two- qubit gate light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for two-qubit transitions (e.g., how quickly the transition can be driven) may befaster than the decoherence rate. Scattering from the intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve fidelity of two-qubit transitions.

[0080] Qubits based on nuclear spin states in the electronic ground state may allow exploitation of long-lived metastable excited electronic states (such as a3Po state in strontium-87 or ytterbium - 171) for qubit storage. Atoms may be selectively transferred into such a state to reduce cross-talk or to improve gate or detection fidelity. Such a storage or shelving process may be atom-selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the ' So state in strontium-87 or ytterbium-171 to the3Po or3P2 state in strontium-87 or a ytterbium-171.

[0081] The clock transition (also a “shelving transition” or a “storage transition” herein) may be qubit-state selective. The upper state of the clock transition may have a very long natural lifetime, e.g., greater than 1 second. The linewidth of the clock transition may be much narrower than the qubit energy spacing. This may allow direct spectral resolution. Population may be transferred from one of the qubit states into the clock state. This may allow individual qubit states to be read out separately, by first transferring population from one qubit state into the clock state, performing imaging on the qubits, then transferring the population back into the ground state from the clock state and imaging again. In some cases, a magic wavelength transition is used to drive the clock transition.

[0082] The clock light for shelving can be atom -selective or not atom-selective. In some cases, the clock transition is globally applied (e.g., not atom selective). A globally applied clock transition may include directing the light without passing through a microscope objective or structuring the light. In some cases, the clock transition is atom-selective. Clock transition which are atom-selective may potentially allow us to improve gate fidelities by minimizing cross-talk. For example, to reduce cross talk in an atom, the atom may be shelved in the clock state where it may not be affected by the light. This may reduce cross-talk between neighboring qubits undergoing transitions. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives and / or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc.

[0083] Multi-qubit gates - Two-qubit gates and multi-qubit gates may be enabled by entanglement units herein. For example, an entanglement excitation may be configured to implement an entanglement operation between a first qubit and another qubit distinct from the first qubit. The entanglement units may be configured to quantum mechanically entangle at least a first atom of the plurality of atoms with at least a second atom of the plurality of atoms. The first or second atom may be in a superposition state at the time of quantum mechanical entanglement.Alternatively or in addition, the first or second atom may not be in a superposition state at the time of quantum mechanical entanglement. The first atom and the second atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The entanglement units may be configured to quantum mechanically entangle any number of atoms described herein.

[0084] The entanglement units may also be configured to quantum mechanically entangle at least a subset of the atoms with at least another atom to form one or more multi-qubit units. The multiqubit units may comprise two-qubit units, three-qubit units, four-qubit units, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For instance, a two-qubit unit may comprise a first atom quantum mechanically entangled with a second atom, a three-qubit unit may comprise a first atom quantum mechanically entangled with a second and third atom, a four-qubit unit may comprise a first atom quantum mechanically entangled with a second, third, and fourth atom, and so forth. The first, second, third, or fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first, second, third, or fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first, second, third, and fourth atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions.

[0085] The entanglement units may comprise one or more Rydberg units. The Rydberg units may be configured to electronically excite the at least first atom to a Rydberg state or to a superposition of a Rydberg state and a lower-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg units may be configured to induce one or more quantum mechanical entanglements between the Rydberg atoms or dressed Rydberg atoms and the at least second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (pm), 2 pm, 3 pm, 4 pm, 5 pm, 6 pm, 7 pm, 8 pm, 9 pm, 10 pm, or more from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance of at most about 10 pm, 9 pm, 8 pm, 7 pm, 6 pm, 5 pm, 4 pm, 3 pm, 2 pm, 1 pm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance from the Rydberg atoms or dressed Rydberg atoms that is within a range defined by any two of the preceding values. The Rydberg units may be configured to allow the Rydberg atoms or dressed Rydberg atoms to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg units may be configured to induce the Rydberg atoms or dressed Rydberg atoms to relax to a lower-energy atomic state. The Rydberg units may beconfigured to drive the Rydberg atoms or dressed Rydberg atoms to a lower-energy atomic state. For instance, the Rydberg units may be configured to apply electromagnetic radiation (such as RF radiation or optical radiation) to drive the Rydberg atoms or dressed Rydberg atoms to a lower- energy atomic state. The Rydberg units may be configured to induce any number of quantum mechanical entanglements between any number of atoms of the plurality of atoms.

[0086] The Rydberg units may comprise one or more light sources (such as any light source described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths may be selected to correspond to a wavelength that forms the Rydberg atoms or dressed Rydberg atoms. For instance, the light may comprise one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. The light may comprise one or more wavelengths of at most about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 300 nm to 400 nm.

[0087] The Rydberg units may be configured to induce a two-photon transition to generate an entanglement. The Rydberg units may be configured to induce a two-photon transition to generate an entanglement between two atoms. The Rydberg units may be configured to selectively induce a two-photon transition to selectively generate an entanglement between two atoms. For instance, the Rydberg units may be configured to direct electromagnetic energy (such as optical energy) to particular optical trapping sites to selectively induce a two-photon transition to selectively generate the entanglement between the two atoms. The two atoms may be trapped in nearby optical trapping sites. For instance, the two atoms may be trapped in adjacent optical trapping sites. The two-photon transition may be induced using first and second light from first and second light sources, respectively. The first and second light sources may each comprise any light source described herein (such as any laser described herein). The first light source may be the same or similar to a light source used to perform a single-qubit operation described herein. Alternatively, different light sources may be used to perform a single-qubit operation and to induce a two-photon transition to generate an entanglement. The first light source may emit light comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from 650 nm to 700 nm). The second light source may emit light comprising one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., within a range from 200nm to 400 nm or from 300 nm to 350 nm). The first and second light sources may emit light having substantially equal and opposite spatially-dependent frequency shifts.

[0088] The Rydberg atoms or dressed Rydberg atoms may comprise a Rydberg state that may have sufficiently strong interatomic interactions with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) to enable the implementation of multi-qubit operations. The Rydberg states may comprise a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or more. The Rydberg states may comprise a principal quantum number of at most about 100, 90, 80, 70, 60, 50, or less. The Rydberg states may comprise a principal quantum number that is within a range defined by any two of the preceding values. The Rydberg states may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift atomic energy levels of the atoms.

[0089] State selective excitation of atoms to Rydberg levels may enable the implementation of multi-qubit operations. The multi-qubit operations may comprise two-qubit operations, three- qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or more. Two-photon transitions may be used to excite atoms from a ground state (such as a1So ground state) to a Rydberg state (such as an n3Si state, wherein n is a principal quantum number described herein). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transitions may be implemented using first and second laser sources, as described herein. The first laser source may emit pi-polarized light, which may not change the projection of atomic angular momentum along a magnetic field. The second laser may emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. The first and second qubit levels may be excited to Rydberg level using this polarization. However, the Rydberg levels may be more sensitive to magnetic fields than the ground state so that large splittings (for instance, on the order of 100s of MHz) may be readily obtained. This spectral selectivity may allow state selective excitation to Rydberg levels.

[0090] Multi-qubit operations (such as two-qubit operations, three-qubit operations, four-qubit operations, and so forth) may rely on energy shifts of levels due to van der Waals interactions described herein. Such shifts may either prevent the excitation of one atom conditional on the state of the other or change the coherent dynamics of excitation of the two-atom system to enact a two- qubit operation. In some cases, “dressed states” may be generated under continuous driving to enact two-qubit operations without requiring full excitation to a Rydberg level (for instance, as described in www.arxiv.org / abs / 1605.05207, which is incorporated herein by reference in its entirety for all purposes).

[0091] One qubit gates, two qubit gates, and multi qubit gates may be implemented by one or more non-classical computation units. The non-classical computation units may be configured toperform one-qubit gate operations, two-qubit gate operations, multi-qubit gate operations and sequences and combinations thereof to perform a non-classical computation. The non-classical computation units may comprise electromagnetic delivery units. For example, any electromagnetic delivery unit disclosed herein. The electromagnetic delivery units disclosed herein with respect to cooling and trapping may be the same electromagnetic delivery units used for qubit gate operations or different. The electromagnetic energy may comprise one or more pulses, pulse sequences, or optical waveforms. The non-classical computation units may comprise one or more entanglement units disclosed herein, one or more Rydberg units disclosed herein, or both. In some cases, a Rydberg unit disclosed herein is an example of an entanglement unit disclosed herein which uses a Rydberg excitation to generate entanglement and to perform two- qubit gate or multi-qubit gate operations.

[0092] A non-classical computation may be configured to provide pulses, pulse sequences, or optical waveforms to perform the non-classical computation. The pulses, pulse sequences, or optical waveforms may comprise any number of pulses, pulse sequences, or optical waveforms. For instance, pulses, pulse sequences, or optical waveforms may comprise at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, or more pulses or sub-waveforms. The pulses, pulse sequences, or optical waveforms may comprise at most about 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 pulses or sub -waveforms. The pulses, pulse sequences, or optical waveforms may comprise a number of pulses or sub-waveforms that is within a range defined by any two of the preceding values. Each pulse of the pulse sequence may comprise any pulse shape, such as any pulse shape described herein.

[0093] The pulses, pulse sequences, or optical waveforms may be configured to decrease the duration of time required to implement multi -qubit operations, as described herein (for instance, with respect to Example 3). For instance, the pulse sequences may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, or more. The pulse sequences may comprise a duration of at most about 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. The pulse sequences may comprise a duration that is within a range defined by any two of the preceding values.

[0094] The pulses, pulse sequences, or optical waveforms may be configured to increase the fidelity of multi-qubit operations, as described herein. For instance, the pulses, pulse sequences,or optical waveforms may enable multi-qubit operations with a fidelity of at least about 0.5, 0.6, 0.7, 0.8, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 0.991, 0.992, 0.993, 0.994, 0.995, 0.996, 0.997, 0.998, 0.999, 0.9991, 0.9992, 0.9993, 0.9994, 0.9995, 0.9996, 0.9997, 0.9998, 0.9999, 0.99991, 0.99992, 0.99993, 0.99994, 0.99995, 0.99996, 0.99997, 0.99998, 0.99999, 0.999991, 0.999992, 0.999993, 0.999994, 0.999995, 0.999996, 0.999997, 0.999998, 0.999999, or more. The pulse sequences may enable multi-qubit operations with a fidelity of at most about 0.999999, 0.999998, 0.999997, 0.999996, 0.999995, 0.999994, 0.999993, 0.999992, 0.999991, 0.99999, 0.99998, 0.99997, 0.99996, 0.99995, 0.99994, 0.99993, 0.99992, 0.99991, 0.9999, 0.9998, 0.9997, 0.9996, 0.9995, 0.9994, 0.9993, 0.9992, 0.9991, 0.999, 0.998, 0.997, 0.996, 0.995, 0.994, 0.993, 0.992, 0.991, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.9, 0.8, 0.7, 0.6, 0.5, or less. The pulse sequences may enable multi-qubit operations with a fidelity that is within a range defined by any two of the preceding values.

[0095] The pulses, pulse sequences, or optical waveforms may enable the implementation of multi-qubit operations on non-adiabatic timescales while maintaining effectively adiabatic dynamics. For instance, the pulse sequences may comprise one or more of shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, counterdiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences. For instance, the pulse sequences may be similar to those described in M.V. Berry, “Transitionless Quantum Driving,” Journal of Physics A: Mathematical and Theoretical 42(36), 365303 (2009), www.doi.org / 10.1088 / 1751-8113 / 42 / 36 / 365303; Y.-Y. Jau et al., “Entangling Atomic Spins with a Strong Rydberg -Dressed Interaction,” Nature Physics 12(1), 71-74 (2016); T. Keating et al., “Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing,” Physical Review A 91, 012337 (2015); A. Mitra et al., “Robust Mblmer-Sbrenson Gate for Neutral Atoms Using Rapid Adiabatic Rydberg Dressing,” www.arxiv.org / abs / 1911.04045 (2019); or L.S. Theis et al., “Counteracting Systems of Diabaticities Using DRAG Controls: The Status after 10 Years,” Europhysics Letters 123(6), 60001 (2018), each of which is incorporated herein by reference in its entirety for all purposes.

[0096] The pulses, pulse sequences, or optical waveforms may further comprise one or more optimal control pulse sequences. The optimal control pulse sequences may be derived from one or more procedures, including gradient ascent pulse engineering (GRAPE) methods, Krotov’s method, chopped basis methods, chopped random basis (CRAB) methods, Nelder-Mead methods, gradient optimization using parametrization (GROUP) methods, genetic algorithm methods, and gradient optimization of analytic controls (GOAT) methods. For instance, the pulse sequences may be similar to those described in N. Khaneja et al., “Optimal Control of Coupled SpinDynamics: Design of NMR Pulse Sequences by Gradient Ascent Algorithms,” Journal of Magnetic Resonance 172(2), 296-305 (2005); or J.T. Merrill et al., “Progress in Compensating Pulse Sequences for Quantum Computation,” Advances in Chemical Physics 154, 241-294 (2014), each of which is incorporated by reference in its entirety for all purposes.

[0097] Measurement Operations - At an operation 330 of a method 300, the method may comprise exposing a second plurality of qubits to radiation. The first plurality of qubits may comprise a second plurality of qubits. In some cases, the first plurality of qubits is identical to the second plurality of qubits. In some cases, the second plurality is a selected subset of qubits which are to be measured. In some cases, operation 330 is an embodiment, variation, or example of a measurement operation disclosed herein. In some cases, the radiation may comprise an electromagnetic field disclosed herein.

[0098] At an operation 430 of a method 400, the method may comprise performing a measurement operation. The measurement operation may comprise exposing a second plurality of qubits to electromagnetic energy. The first plurality of qubits may comprise a second plurality of qubits. In some cases, the first plurality of qubits is identical to the second plurality of qubits. In some cases, the second plurality is a selected subset of qubits which are to be measured. In some cases, the electromagnetic energy is configured to selectively drive a qubit of a second plurality of qubits from an initial state to an excited state in a presence of an applied magnetic field. In some cases, a selectivity of a transition to an excited state is based at least in part on a strength of an applied magnetic field to a first plurality of qubits, a second plurality of qubits, or both. Operation 430 of a method 400 may comprise an example, variation, or embodiment of the method 330 of method 300.

[0099] In some cases, operation 430 or 330 comprises exposing a first subset of said second plurality of qubits to a first electromagnetic energy and then exposing a second subset of said second plurality of qubits to a second electromagnetic energy. In some cases, the first electromagnetic energy comprises a first polarization and the second electromagnetic energy comprises a second polarization.

[0100] In some cases, method 300 or method 400 further comprises determining that a first subset of a second plurality of qubits fluoresces in response to being exposed to a first radiation and that a second subset of a second plurality of qubits fluoresces in response to being exposed to a second radiation. In some cases, the states may be imaged from either one of two counter-propagating imaging beams, each configured to image one of the1So m / = ’ and -1 / 2 qubit states. In some cases, the electromagnetic energy is polarized. In some cases, one or more of an electromagnetic energy, a first electromagnetic energy, or a second electromagnetic energy is circularly polarized. In some cases, the transition is a cycling transition. In some cases, the beams can be tuned toaddress one of the3Pi mf= ±3 / 2 states, which can provide access to narrow linewidth (e.g., about 180 kHz) closed cycling transitions. Scattering from the mf= ±1 / 2 excited states, which may allow population leakage between the qubit states, can be suppressed by the large ratio of Zeeman shifts to transition linewidth. In some cases, the ratios can include 771 MHz, 681 MHz between the -3 / 2 and -1 / 2 states and the 1 / 2 and 3 / 2 states, respectively.

[0101] In some cases, a method comprises tuning a strength of an applied magnetic field to a selected measurement state. For example, FIG. 2 shows a curve for magnetic field strength such that a particular state may be moved in and out of resonance with a particular measurement state.

[0102] In some cases, the first plurality of qubits has at most n more qubits than the second plurality of qubits. For example, the first plurality and the second plurality of qubits can be configured in a two-dimensional array of optical trapping sits. The first plurality of qubits can include at most n more qubits than the second plurality of qubits. In some cases, the first plurality of qubits can include at least about 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 300, 400, 500, or more qubits than the second plurality of qubits. In some cases, the first plurality of qubits can include at most about 500, 400, 300, 200, 190, 180, 170, 160, 150, 140, 130, 120, 110, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 4, 3, 2, or fewer more quantity than the second plurality of qubits.

[0103] In some cases, the second plurality of qubits has at most n more qubits than the first plurality of qubits. For example, the first plurality and the second plurality of qubits can be configured in a two-dimensional array of optical trapping sits. The second plurality of qubits can include at most n more qubits than the first plurality of qubits. In some cases, the second plurality of qubits can include at least about 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 300, 400, 500, or more qubits than the first plurality of qubits. In some cases, the second plurality of qubits can include at most about 500, 400, 300, 200, 190, 180, 170, 160, 150, 140, 130, 120, 110, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 4, 3, 2, or fewer more quantity than the first plurality of qubits.

[0104] In some cases, performing the measurement operation in the methods and systems disclosed herein comprises exposing a first subset of said second plurality of qubits to a first electromagnetic energy and then exposing a second subset of said second plurality of qubits to a second electromagnetic energy. In some cases, the first electromagnetic energy comprises a first polarization and the second electromagnetic energy comprises a second polarization. In some cases, the method further comprises determining that the first subset of the second plurality of qubits fluoresces in response to being exposed to the first radiation and that the second subset ofthe second plurality of qubits fluoresces in response to being exposed to the second radiation. For example, the states may be imaged from either one of two imaging beams, each configured to image one of the1So m / = / i and —1 / 2 qubit states. The two imaging beams may be counterpropagating.

[0105] The measurement operations may be implemented by one or more electromagnetic delivery units, such as for example, any electromagnetic delivery unit disclosed herein. In some cases, the electromagnetic delivery unit configured to perform a measurement operation is the same as an electromagnetic delivery unit configured to perform a non-classical computation. In some cases, the electromagnetic delivery unit configured to perform a measurement operation is different than an electromagnetic delivery unit configured to perform a non-classical computation.

[0106] In some cases, the magnetic field, e.g., a 500 Gauss magnetic field, can be selected to minimize unwanted scattering, as well as other potential errors described herein elsewhere, while being consistent with operational constraints, e.g., mechanical and thermal constraints. Scattered light can be collected by an objective, e.g., a high-numerical-aperture objective, and imaged onto a camera, e.g., a low-noise digital camera. For each site in the tweezer array, methods herein can include applying a threshold to the counts in an integration region to determine if an atom in the imaged state occupies that site. Methods herein can include operating with an imaging duration, e.g., 5 milliseconds (ms). With a predetermined imaging duration, methods herein can include registering or detecting a number of photons, e.g., approximately 30 photons, from a bright atom. In some cases, imaging duration can be chosen to balance distinguishability, loss, data-rate, and robustness to experimental drifts.

[0107] State Identification - Methods and systems disclosed herein may provide state selective identification of a qubit state. At an operation 340 of a method 300, the method may comprise determining or predicting, based at least in part on at least one of a second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state. In some cases, a state of a qubit is identified as one or a first state or a state by emission of a photon or by an absence of said emission of said photon.

[0108] At an operation 440 of the method 400, the method may comprise determining that a qubit was in an initial state. The determining may be based at least in part on a qubit returning to an initial state by emission of a photon in response to an electromagnetic energy at operation 430. Operation 440 of a method 400 may comprise an example, variation, or embodiment of the method 340 of method 300.

[0109] In some cases, operation 340 and / or operation 440 comprise detecting a photon indicating a transition between the measurement state and the qubit state (e.g., the initial state). In somecases, a photon is created by returning to said initial state. In some cases, the returning to the initial state is from a manifold of excited states. In some cases, the strength of the magnetic field determines a separation between states in the manifold of excited states. For example, the strength of the magnetic field can include a strength of at least about 100, 200, 300, 400, 500, or more Gauss. For example, the strength of the magnetic field can include a strength of at most about 500, 400, 300, 200, 100, or less Gauss. The strength of the magnetic field can include a strength in a range as defined by any two of the proceeding values.

[0110] In some cases, the method comprises tuning the strength of the applied magnetic field to a selected measurement state. In some cases, the separation between states determines which states are in resonance with the radiation. In some cases, a transition to a single state with the manifold of excited states is allowed by a selection rule for each qubit state.[OHl] In some cases, returning to the initial state comprises spontaneous emission. In some cases, returning to the initial state comprises stimulated emission. In some cases, returning to the initial state by emission of the photon is a fluorescence transition. In some cases, the method comprises imaging the photon at a measurement unit.

[0112] In some cases, operation 440 or 340 comprises imaging said photon at a measurement unit. In some cases, the fluorescence associated with emission of the photon can be detected by the measurement unit, e.g., one or more optical detection devices. For example, the optical detection devices may be configured to perform measurements of the states for non-classical computation. The optical detection devices can 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 detection devices can include one or more fluorescence detectors. The optical detection devices can include one or more objectives. For example, the one or more objectives can have a numerical aperture (NA) of at least about 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, or more. The objective may have an NA of at most about 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, or less. The objective may have an NA that is within a range defined by any two of the preceding values.

[0113] In some cases, method 400 further comprises repeating operations 430 to 440 a plurality of times. In some cases, method 400 further comprises determining, based at least in part on determining a florescence at 430 that a number (n) of spatially distinct optical trapping sites of said array of spatially distinct optical trapping sites is missing a qubit. In some cases, a first plurality of qubits has at most n more qubits than said second plurality of qubits.

[0114] In some cases, the method is configured to perform an atom cooling operation. In some cases, operations 430 to 440 are configured to perform an atom cooling operation. For example, achieving low off-resonant scattering while maintaining cooling from the imaging beam can be achieved by operating with low imaging beam power and relatively small red detuning, e.g., of order of the transition linewidth. This can be facilitated by using “magic” traps, e.g., where ground and excited states can experience the same trapping potential. In some cases, the magic wavelength for thexSo to3Pi mf= ±3 / 2 transitions can occur near about 483 nm. In some cases, the trapping electromagnetic energy (e.g., light at a predetermined wavelength) may be polarized perpendicular to the magnetic field. In some cases, magic traps can also reduce requirements for array uniformity. In some cases, a magic wavelength corresponding to an atom may comprise any wavelength of light that gives rise to equal or nearly equal polarizabilities of the first and second states. The magic wavelengths for a transition between the first and second states may be determined by calculating the wavelength-dependent polarizabilities of the first and second states and finding crossing points. Electromagnetic energy, e.g., light, tuned to such a magic wavelength may give rise to equal or nearly equal differential light shifts in the first and second states, regardless of the intensity of the light emitted by the light sources. In some cases, this may effectively decouple the first and second states from motion of the atoms. In some cases, the magic wavelengths may utilize one or more scalar or tensor light shifts.

[0115] Mid Circuit Measurements - The ability to determine the quantum state of an atom without it being lost from an optical trap can be useful for mid-circuit measurements. In some systems, such as neutral-atom systems, the state of single atoms can be determined by introducing state- selective loss followed by state-independent imaging of the remaining atoms. Compared to the present disclosure, such approaches may not distinguish atoms that populate the ejected state from those that already underwent loss. If used in an error-correction protocol, such approaches can require frequent reloading and state preparation of new ancilla qubits.Systems for performing state detection for non-classical computing

[0116] Disclosed herein are systems for performing state detection for non-classical computing. In some cases, the system can include one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits. In some cases, the system can include one or more non-classical computation units configured to perform a non-classical computation using at least a portion of said first plurality of qubits. In some cases, the system can include one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits. In some cases, the system can include one or moremeasurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0117] FIGS. 5A illustrates an example system configured to perform state detection for non- classical computing. As shown in the illustrated example, the system may comprise sites of an optical tweezer array, a magnetic field, laser beams with opposite circular polarization and different frequencies applied along the direction of the magnetic field, high-numerical-aperture objectives, and a movable tweezer. The system may comprise at least sites of an optical tweezer array, shown as the two-dimensional array of spots. The system may comprise a magnetic field shown by the left and right grey coils on each side of the optical tweezer array.

[0118] The system may comprise various forms of electromagnetic radiation directed at the optical tweezer array. The electromagnetic radiation may be generated by the various electromagnetic delivery units disclosed herein. For example, trapping optical excitation may be provided from an optical trapping unit to implement an array of spatially distinct optical traps (e.g., a tweezer array herein). For example, measurement electromagnetic energy may be provided from an electromagnetic delivery unit associated with a measurement unit. In the illustrated example, the system may comprise laser beams with opposite circular polarization and different frequencies applied along the direction of the magnetic field. Each of G- and c may be configured to measure a qubit state as disclosed herein. In some cases, light (e.g., fluorescence, spontaneous emission, stimulated emission, etc.) may be directed to a measurement unit to detect a state as disclosed herein. For example, electromagnetic energy from a non-classical computation unit may be directed to the array to perform various gate operations disclosed herein. For example, electromagnetic energy from a measurement unit may be configured to suppress incoherent scattering on the excited state by a “hiding” light applied to qubits at a hiding wavelength. The hiding procedure may be similar to that disclosed in commonly owned international application no. PCT / US2023 / 026730, which is incorporated herein by reference for all purposes.

[0119] FIG. 5A illustrates a system configured to optically trap individual atoms (e.g.,171Yb atoms) in a plurality of sites of an optical tweezer array in the presence of a magnetic field, e.g., a 500 Gauss magnetic field. In some cases, the system can include one or more high-numerical- aperture objectives (e.g., 2 objectives) configured for site-resolved imaging. In some cases, the one or more objectives can be configured for targeted application of trapping light at a trapping wavelength, e.g., 483 nm wavelength. In some cases, the one or more objectives can be configured for targeted application of hiding light at a hiding wavelength, e.g., 460 nm wavelength. In some cases, the system can include one or more movable or optical tweezers. In some cases, the one ormore movable or optical tweezers can be configured at a trapping wavelength, e.g., 483 nm wavelength. In some cases, the one or more moveable or optical tweezers can be configured to rearrange atoms between trapping sites. In some cases, the system can include one or more electromagnetic energy sources to perform non-classical computations. In some cases, a measurement unit may be configured to direct one or more lasers can be configured at a global Raman transition wavelength, e.g., 556 nm wavelength. In some cases, the one or more lasers can be configured to generate a laser incident through a hole in the imaging objective to drive global Raman transitions between the qubit states.

[0120] FIG. 5B illustrates (top) a level diagram for an example atomic system which can be used in the methods and systems disclosed herein and (bottom) experimental data showing two subsequent single-shot images of a fully filled 10 by 3 array of atoms generated by an example of the methods and systems disclosed herein.

[0121] FIG. 5B (top) illustrates the system configured with one or more laser beams to perform state detection for non-classical computing. In some cases, the one or more laser beams (e.g., two laser beams) can be configured with opposite circular polarization and at different frequencies, e.g., <J_ and <J+. In some cases, the one or more laser beams can be configured to apply along the direction of the magnetic field to selectively image the two qubit states,xSo mf= 1 / 2, -1 / 2, labelled as |1) and 0). In some cases, the two qubit states can couple to either of3Pi mf= 3 / 2 or -3 / 2 at a predetermined wavelength, e.g., 556 nm wavelength.

[0122] In some cases, as illustrated in FIG. 5B (bottom), state-resolved, nondestructive imaging can be shown by performing two subsequent single-shot images of a fully filled array. In some cases, the array is a 10 by 3 array of atoms prepared in an equal superposition of |1) and |0) prior to the first image.

[0123] The system may comprise various forms of electromagnetic radiation directed at the optical tweezer array. The electromagnetic radiation may be generated by the various electromagnetic delivery units disclosed herein. In some cases, the electromagnetic energy can include optical energy. The optical energy can include any repetition rate, pulse energy, average power, wavelength, or bandwidth. In some cases, the electromagnetic energy can be sourced by magnetrons, klystrons, traveling-wave tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMP ATT) diodes, or masers. In some cases, the electromagnetic energy can include one or more wavelengths of at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m,700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or more. In some cases, the electromagnetic energy can include one or more wavelengths of at most about 10 km, 9 km, 8 km, 7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m, 500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m, 30 m, 20 m, 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm, 800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm, 80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or less. In some cases, the electromagnetic energy can include one or more wavelengths that are within a range defined by any two of the preceding values.

[0124] In some cases, the electromagnetic energy can include an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 Watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. In some cases, the electromagnetic energy can include an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or less. In some cases, the electromagnetic energy can include an average power that is within a range defined by any two of the preceding values.

[0125] In some cases, the system can include one or more optical trapping units. The optical trapping units may be configured to generate a plurality of optical trapping sites. In some cases, 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 at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. For example, the optical trapping units may beconfigured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer optical trapping sites. In some cases, the optical trapping units may be configured to trap a number of optical trapping sites that is within a range defined by any two of the preceding values.

[0126] In some cases, the optical trapping units may be configured to trap a plurality of atoms. For example, the optical trapping units may be configured to trap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. For example, the optical trapping units may be configured to trap at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. In some cases, the optical trapping units may be configured to trap a number of atoms that is within a range defined by any two of the preceding values.

[0127] In some cases, each optical trapping site of the optical trapping units may be configured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. In some cases, each optical trapping site may be configured to trap at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. In some cases, each optical trapping site may be configured to trap a number of atoms that is within a range defined by any two of the preceding values. In some cases, each optical trapping site may be configured to trap a single atom.

[0128] In some cases, one or more atoms of the plurality of atoms can include qubits. Two or more atoms may be quantum mechanically entangled. In some cases, two or more atoms may be quantum mechanically entangled with a coherence lifetime of at least about 1 microsecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, 6 ms, 7 ms, 8 ms, 9 ms, 10 ms, 20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms, 300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2 s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or more. In some cases, two or more atoms may be quantum mechanically entangled with a coherence lifetime of at most about 10 s, 9 s, 8 s, 7 s, 6 s, 5 s, 4 s, 3 s, 2 s, 1 s, 900 ms, 800 ms, 700 ms, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100 ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms, 8 ms, 7 ms, 6 ms, 5ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, or less. In some case, two or more atoms may be quantum mechanically entangled with a coherence lifetime that is within a range defined by any two of the preceding values.

[0129] In some cases, the lasers may comprise one or more continuous wave lasers. The lasers may comprise one or more pulsed lasers. The lasers may comprise one or more gas lasers, such as one or more helium-neon (HeNe) lasers, argon (Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N2) lasers, carbon dioxide (CO2) lasers, carbon monoxide (CO) lasers, transversely excited atmospheric (TEA) lasers, or excimer lasers. For instance, the lasers may comprise one or more argon dimer (Arc) excimer lasers, krypton dimer (Krc) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe2) excimer lasers, argon fluoride (ArF) excimer lasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF) excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride (XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The laser may comprise one or more dye lasers.

[0130] In some cases, the lasers may comprise 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 laser, or manganese chloride (MnCh) metal-vapor lasers.

[0131] In some cases, the lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal-doped crystal lasers, or metal-doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd:YAG) lasers, neodymium / chromium doped yttrium aluminum garnet (Nd / Cr:YAG) lasers, erbium-doped yttrium aluminum garnet (Er: YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd:YLF) lasers, neodymium-doped yttrium orthovanadate (NIXYVCh) lasers, neodymium-doped yttrium calcium oxoborate (Nd:YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped yttrium aluminum garnet (Tm:YAG) lasers, ytterbium-doped ytrrium aluminum garnet (Yb:YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho:YAG) lasers, chromium-doped zinc selenide (CrZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) lasers, erbium-doped glass (Erglass) lasers, erbium- ytterbium-codoped glass (Er / Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers.

[0132] In some cases, the lasers may comprise one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGalnP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.

[0133] In some cases, the lasers may emit continuous wave laser light. The lasers may emit pulsed laser light. The lasers may have a pulse length of at least about 1 femtoseconds (fs), 2 fs, 3 fs, 4 fs, 5 fs, 6 fs, 7 fs, 8 fs, 9 fs, 10 fs, 20 fs, 30 fs, 40 fs, 50 fs, 60 fs, 70 fs, 80 fs, 90 fs, 100 fs, 200 fs, 300 fs, 400 fs, 500 fs, 600 fs, 700 fs, 800 fs, 900 fs, 1 picosecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 nanosecond (ns), 2 ns, 3 ns, 4 ns, 5 ns, 6 ns, 7 ns, 8 ns, 9 ns, 10 ns, 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1,000 ns, or more. The lasers may have a pulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3 ns, 2 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800 fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80 fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6 fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or less. The lasers may have a pulse length that is within a range defined by any two of the preceding values.

[0134] In some cases, the lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz,80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values.

[0135] In some cases, the lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (pj), 2 pj, 3 pj, 4 pj, 5 pj, 6 pj, 7 pj, 8 pj, 9 pj, 10 pj, 20 pj, 30 pj, 40 pj, 50 pj, 60 pj, 70 pj, 80 pj, 90 pj, 100 pj, 200 pj, 300 pj, 400 pj, 500 pj, 600 pj, 700 pj, 800 pj, 900 pj, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 pj, 800 pj, 700 pj, 600 pj, 500 pj, 400 pj, 300 pj, 200 pj, 100 pj, 90 pj, 80 pj, 70 pj, 60 pj, 50 pj, 40 pj, 30 pj, 20 pj, 10 pj, 9 pj, 8 pj, 7 pj, 6 pj, 5 pj, 4 pj, 3 pj, 2 pj, 1 pj, 900 nJ, 800 nJ, 700 nJ, 600 nJ, 500 nJ, 400 nJ, 300 nJ, 200 nJ, 100 nJ, 90 nJ, 80 nJ, 70 nJ, 60 nJ, 50 nJ, 40 nJ, 30 nJ, 20 nJ, 10 nJ, 9 nJ, 8 nJ, 7 nJ, 6 nJ, 5 nJ, 4 nJ, 3 nJ, 2 nJ, 1 nJ, or less. The lasers may emit light having a pulse energy that is within a range defined by any two of the preceding values.

[0136] In some cases, the lasers may emit light having an average power of at least about 1 microwatt (pW), 2 pW, 3 pW, 4 pW, 5 pW, 6 pW, 7 pW, 8 pW, 9 pW, 10 pW, 20 pW, 30 pW, 40 pW, 50 pW, 60 pW, 70 pW, 80 pW, 90 pW, 100 pW, 200 pW, 300 pW, 400 pW, 500 pW, 600 pW, 700 pW, 800 pW, 900 pW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 pW, 800 pW, 700 pW, 600 pW, 500 pW, 400 pW, 300 pW, 200 pW, 100 pW, 90 pW, 80 pW, 70 pW, 60 pW, 50 pW, 40 pW, 30 pW, 20 pW, 10 pW, 9 pW, 8 pW, 7 pW, 6 pW, 5 pW, 4 pW, 3 pW, 2 pW, 1 pW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values.

[0137] In some cases, the lasers may emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The lasers may emit light comprising one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm, 1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm, 1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm, 1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm, 1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm, 1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370 nm, 1,380 nm, 1,390 nm, 1,400 nm, or more. The lasers may emit light comprising one or more wavelengths of at most about 1,400 nm, 1,390 nm, 1,380 nm, 1,370 n, 1,360 nm, 1,350 nm, 1,340 nm, 1,330 nm, 1,320 nm, 1,310 nm, 1,300 nm, 1,290 nm, 1,280 nm, 1,270 n, 1,260 nm, 1,250 nm, 1,240 nm, 1,230 nm, 1,220 nm, 1,210 nm, 1,200 nm, 1,190 nm, 1,180 nm, 1,170 n, 1,160 nm, 1,150 nm, 1,140 nm, 1,130 nm, 1,120 nm, 1,110 nm, 1,100 nm, 1,090 nm, 1,080 nm, 1,070 n, 1,060 nm, 1,050 nm, 1,040 nm, 1,030 nm, 1,020 nm, 1,010 nm, 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values.

[0138] In some cases, the lasers may emit light having a bandwidth of at least about 1 x 10'15nm, 2 x 10'15nm, 3 x 10'15nm, 4 x 10'15nm, 5 x 10'15nm, 6 x 10'15nm, 7 x 10'15nm, 8 x 10'15nm, 9 x 10’15nm, 1 x 10'14nm, 2 x 10'14nm, 3 x 10'14nm, 4 x 10'14nm, 5 x 10'14nm, 6 x 10'14nm, 7 x 10'14nm, 8 x 10'14nm, 9 x 10'14nm, 1 x 10'13nm, 2 x 10'13nm, 3 x 10'13nm, 4 x 10'13nm, 5 x 10’13nm, 6 x 10'13nm, 7 x 10'13nm, 8 x 10'13nm, 9 x 10'13nm, 1 x 10'12nm, 2 x 10'12nm, 3 x 10'12nm, 4 x IO’12nm, 5 x 10'12nm, 6 x 10'12nm, 7 x 10'12nm, 8 x 10'12nm, 9 x 10'12nm, 1 x 10'11nm, 2 x IO'11nm, 3 x IO'11nm, 4 x IO'11nm, 5 x IO'11nm, 6 x IO'11nm, 7 x IO'11nm, 8 x IO'11nm, 9 x IO'11nm, 1 x IO'10nm, 2 x IO'10nm, 3 x IO'10nm, 4 x IO'10nm, 5 x IO'10nm, 6 x IO'10nm, 7 x IO'10nm, 8 x IO'10nm, 9 x IO'10nm, 1 x 10'9nm, 2 x 10'9nm, 3 x 10'9nm, 4 x 10'9nm, 5 x 10'9nm, 6 x 10'9nm, 7 x 10'9nm, 8 x 10'9nm, 9 x 10'9nm, 1 x 10'8nm, 2 x 10'8nm, 3 x 10'8nm, 4 x 10'8nm, 5 x 10'8nm, 6 x 10'8nm, 7 x 10'8nm, 8 x 10'8nm, 9 x 10'8nm, 1 x 10'7nm, 2 x 10'7nm, 3 x 10'7nm, 4 x 10'7nm, 5 x 10'7nm, 6 x 10'7nm, 7 x 10'7nm, 8 x 10'7nm, 9 x 10'7nm,I x 10'6nm, 2 x 10'6nm, 3 x 10'6nm, 4 x 10'6nm, 5 x 10'6nm, 6 x 10'6nm, 7 x 10'6nm, 8 x 10'6nm, 9 x 10'6nm, 1 x IO'5nm, 2 x IO'5nm, 3 x IO'5nm, 4 x IO'5nm, 5 x IO'5nm, 6 x IO'5nm, 7 xIO'5nm, 8 x IO'5nm, 9 x IO'5nm, 1 x 10'4nm, 2 x 10'4nm, 3 x 10'4nm, 4 x 10'4nm, 5 x 10'4nm,6 x 10'4nm, 7 x 10'4nm, 8 x 10'4nm, 9 x 10'4nm, 1 x 10'3nm, or more. The lasers may emit light having a bandwidth of at most aboutl x 10'3nm, 9 x 10'4nm, 8 x 10'4nm, 7 x 10'4nm, 6 x 10'4nm, 5 x 10'4nm, 4 x 10'4nm, 3 x 10'4nm, 2 x 10'4nm, 1 x 10'4nm, 9 x IO'5nm, 8 x IO'5nm, 7 x IO'5nm, 6 x IO'5nm, 5 x IO'5nm, 4 x IO'5nm, 3 x IO'5nm, 2 x IO'5nm, 1 x IO'5nm, 9 x 10'6nm, 8 x 10'6nm, 7 x 10'6nm, 6 x 10'6nm, 5 x 10'6nm, 4 x 10'6nm, 3 x 10'6nm, 2 x 10'6nm, 1 x 10'6nm, 9 x 10'7nm, 8 x 10'7nm, 7 x 10'7nm, 6 x 10'7nm, 5 x 10'7nm, 4 x 10'7nm, 3 x 10'7nm, 2 x 10'7nm, 1 x 10'7nm, 9 x 10'8nm, 8 x 10'8nm, 7 x 10'8nm, 6 x 10'8nm, 5 x 10'8nm, 4 x 10'8nm, 3 x 10'8nm, 2 x 10'8nm, 1 x 10'8nm, 9 x 10'9nm, 8 x 10'9nm, 7 x 10'9nm, 6 x 10'9nm, 5 x 10'9nm, 4 x 10'9nm, 3 x 10'9nm, 2 x 10'9nm, 1 x 10'9nm, 9 x IO'10nm, 8 x IO'10nm, 7 x IO'10nm, 6 x IO'10nm, 5 x IO'10nm, 4 x IO'10nm, 3 x IO'10nm, 2 x IO'10nm, 1 x IO'10nm, 9 x IO'11nm, 8 x IO'11nm, 7 x IO'11nm, 6 x IO'11nm, 5 x IO'11nm, 4 x IO'11nm, 3 x IO'11nm, 2 x IO'11nm, 1 x 10’I Inm, 9 x IO’12nm, 8 x 10'12nm, 7 x 10'12nm, 6 x 10'12nm, 5 x 10'12nm, 4 x 10'12nm, 3 x 10'12nm, 2 x IO’12nm, 1 x 10'12nm, 9 x 10'13nm, 8 x 10'13nm, 7 x 10'13nm, 6 x 10'13nm, 5 x 10'13nm, 4 x 10'13nm, 3 x 10'13nm, 2 x 10'13nm, 1 x 10'13nm, 9 x 10'14nm, 8 x 10'14nm, 7 x 10'14nm, 6 x 10'14nm, 5 x 10'14nm, 4 x 10'14nm, 3 x 10'14nm, 2 x 10'14nm, 1 x 10'14nm, 9 x IO'15nm, 8 x IO’15nm, 7 x IO'15nm, 6 x IO'15nm, 5 x IO'15nm, 4 x IO'15nm, 3 x IO'15nm, 2 x IO'15nm, 1 x IO’15nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values.Certain definitions and considerations

[0139] 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 in this 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” herein is intended to encompass “and / or” unless otherwise stated.

[0140] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “at least,” “greater than” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3.

[0141] Whenever the term “no more than,” “less than,” or “less than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “no more than,” “less than,” or “less than or equal to” applies to each of the numerical values in that series of numerical values. For example, less than or equal to 3, 2, or 1 is equivalent to less than or equal to 3, less than or equal to 2, or less than or equal to 1.

[0142] Where values are described as ranges, it will be understood that such disclosure includes the disclosure of all possible sub-ranges within such ranges, as well as specific numerical values that fall within such ranges irrespective of whether a specific numerical value or specific subrange is expressly stated.

[0143] As used herein, like characters refer to like elements.

[0144] The term “about” or “approximately” may mean within an acceptable error range for the particular value, which will depend in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, “about” may mean within 1 or more than 1 standard deviation, per the practice in the art. Alternatively, “about” may mean a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value. Where particular values are described in the application and claims, unless otherwise stated the term “about” meaning within an acceptable error range for the particular value may be assumed.

[0145] As used herein, the terms “non-classical computation,” “non-classical procedure,” “non- classical operation,” any “non-classical computer” generally refer to any method, system, or computer-readable media for performing computational procedures outside of the paradigm of classical computing. A non-classical computation, non-classical procedure, non-classical operation, or non-classical computer may comprise a quantum computation, quantum procedure, quantum operation, or quantum computer.

[0146] As used herein, the terms “quantum computation,” “quantum procedure,” “quantum operation,” and “quantum computer” generally refer to any method, system, or computer-readable media for performing computations using quantum mechanical operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) on a Hilbert space represented by a quantum device. As such, quantum and classical (or digital) computation may be similar in the following aspect: both computations may comprise sequences of instructions performed on input information to then provide an output. Various paradigms ofquantum computation may break the quantum operations down into sequences of basic quantum operations that affect a subset of qubits of the quantum device simultaneously. The quantum operations may be selected based on, for instance, their locality or their ease of physical implementation. A quantum procedure or computation may then consist of a sequence of such instructions that in various applications may represent different quantum evolutions on the quantum device. For example, procedures to compute or simulate quantum chemistry may represent the quantum states and the annihilation and creation operators of electron spin-orbitals by using qubits (such as two-level quantum systems) and a universal quantum gate set (such as the Hadamard, controlled-not (CNOT), and TT / 8 rotations) through the so-called Jordan-Wigner transformation or Bravyi-Kitaev transformation.

[0147] Additional examples of quantum procedures or computations may include procedures for optimization such as quantum approximate optimization algorithm (QAOA) or quantum minimum finding. QAOA may comprise performing rotations of single qubits and entangling gates of multiple qubits. In quantum adiabatic computation, the instructions may carry stochastic or non-stochastic paths of evolution of an initial quantum system to a final one.

[0148] Quantum-inspired procedures may include simulated annealing, parallel tempering, master equation solver, Monte Carlo procedures and the like. Quantum-classical or hybrid algorithms or procedures may comprise such procedures as variational quantum eigensolver (VQE) and the variational and adiabatically navigated quantum eigensolver (VanQver).

[0149] A quantum computer may comprise one or more adiabatic quantum computers, quantum gate arrays, one-way quantum computers, topological quantum computers, quantum Turing machines, quantum annealers, Ising solvers, or gate models of quantum computing.

[0150] As used herein, the term “adiabatic” refers to any process performed on a quantum mechanical system in which the parameters of the Hamiltonian are changed slowly in comparison to the natural timescale of evolution of the system.

[0151] As used herein, the term “non-adiabatic” refers to any process performed quantum mechanical system in which the parameters of the Hamiltonian are changed quickly in comparison to the natural timescale of evolution of the system or on a similar timescale as the natural timescale of evolution of the system.Computer systems

[0152] In another aspect, provided is a system for performing state detection for non-classical computing, comprising: one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits; one or more non-classical computation units configured to perform a non-classical computation usingat least a portion of said first plurality of qubits; one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and one or more measurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0153] FIG. 6 shows a computer system 601 that is programmed or otherwise configured to operate any method, system, computer-readable media, process, or technique described herein (such as the systems, the methods, the computer-readable media, or the techniques for nondestructive atomic qubit state-resolved imaging, described herein). The computer system 601 can regulate various aspects of the present disclosure. The computer system 601 can be an electronic device of a user or a computer system that is remotely located with respect to the electronic device. The electronic device can be a mobile electronic device.

[0154] The computer system 601 includes a central processing unit (CPU, also “processor” and “computer processor” herein) 605, which can be a single core or multi core processor, or a plurality of processors for parallel processing. The computer system 601 also includes memory or memory location 610 (e.g., random-access memory, read-only memory, flash memory), electronic storage unit 615 (e.g., hard disk), communication interface 620 (e.g., network adapter) for communicating with one or more other systems, and peripheral devices 625, such as cache, other memory, data storage and / or electronic display adapters. The memory 610, storage unit 615, interface 620 and peripheral devices 625 are in communication with the CPU 605 through a communication bus (solid lines), such as a motherboard. The storage unit 615 can be a data storage unit (or data repository) for storing data. The computer system 601 can be operatively coupled to a computer network (“network”) 630 with the aid of the communication interface 620. The network 630 can be the Internet, an internet and / or extranet, or an intranet and / or extranet that is in communication with the Internet. The network 630 in some cases is a telecommunication and / or data network. The network 630 can include one or more computer servers, which can enable distributed computing, such as cloud computing. The network 630, in some cases with the aid of the computer system 601, can implement a peer-to-peer network, which may enable devices coupled to the computer system 601 to behave as a client or a server.

[0155] The CPU 605 can execute a sequence of machine-readable instructions, which can be embodied in a program or software. The instructions may be stored in a memory location, such as the memory 610. The instructions can be directed to the CPU 605, which can subsequently program or otherwise configure the CPU 605 to implement methods of the present disclosure.Examples of operations performed by the CPU 605 can include fetch, decode, execute, and writeback.

[0156] The CPU 605 can be part of a circuit, such as an integrated circuit. One or more other components of the system 601 can be included in the circuit. In some cases, the circuit is an application specific integrated circuit (ASIC).

[0157] The storage unit 615 can store files, such as drivers, libraries and saved programs. The storage unit 615 can store user data, e.g., user preferences and user programs. The computer system 601 in some cases can include one or more additional data storage units that are external to the computer system 601, such as located on a remote server that is in communication with the computer system 601 through an intranet or the Internet.

[0158] The computer system 601 can communicate with one or more remote computer systems through the network 630. For instance, the computer system 601 can communicate with a remote computer system of a user. Examples of remote computer systems include personal computers (e.g., portable PC), slate or tablet PC’s (e.g., Apple® iPad, Samsung® Galaxy Tab), telephones, Smart phones (e.g., Apple® iPhone, Android-enabled device, Blackberry®), or personal digital assistants. The user can access the computer system 601 via the network 630.

[0159] Methods as described herein can be implemented by way of machine (e.g., computer processor) executable code stored on an electronic storage location of the computer system 601, such as, for example, on the memory 610 or electronic storage unit 615. The machine executable or machine readable code can be provided in the form of software. During use, the code can be executed by the processor 605. In some cases, the code can be retrieved from the storage unit 615 and stored on the memory 610 for ready access by the processor 605. In some situations, the electronic storage unit 615 can be precluded, and machine-executable instructions are stored on memory 610.

[0160] The code can be pre-compiled and configured for use with a machine having a processer adapted to execute the code or can be compiled during runtime. The code can be supplied in a programming language that can be selected to enable the code to execute in a pre-compiled or as- compiled fashion.

[0161] In another aspect, provided is a non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of performing state detection for non-classical computing on a non-classical computer, wherein said non-classical computer is configured to execute said one or more instructions, the method comprising: (a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites; (b) performing a non-classical computation using at least a portion of said first plurality of qubits; (c) exposing a second plurality of qubits to radiation, wherein said first pluralityof qubits comprises said second plurality of qubits; and (d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

[0162] Aspects of the systems and methods provided herein, such as the computer system 601, can be embodied in programming. Various aspects of the technology may be thought of as “products” or “articles of manufacture” typically in the form of machine (or processor) executable code and / or associated data that is carried on or embodied in a type of machine readable medium. Machine-executable code can be stored on an electronic storage unit, such as memory (e.g., readonly memory, random-access memory, flash memory) or a hard disk. “Storage” type media can include any or all of the tangible memory of the computers, processors or the like, or associated modules thereof, such as various semiconductor memories, tape drives, disk drives and the like, which may provide non-transitory storage at any time for the software programming. All or portions of the software may at times be communicated through the Internet or various other telecommunication networks. Such communications, for example, may enable loading of the software from one computer or processor into another, for example, from a management server or host computer into the computer platform of an application server. Thus, another type of media that may bear the software elements includes optical, electrical and electromagnetic waves, such as 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 or the like, also may be considered as media bearing the software. As used herein, unless restricted 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.

[0163] Hence, a machine readable medium, such as computer-executable code (e.g., computer- readable media), may take many forms, including but not limited to, a tangible storage medium, a carrier wave medium or physical transmission medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in any computer(s) or the like, such as may be used to implement the databases, etc. shown in the drawings. Volatile storage media include dynamic memory, such as 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 lightwaves such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media therefore include for example: a floppy disk, a flexible disk, hard disk, magnetic tape, anyother magnetic medium, a CD-ROM, DVD or DVD-ROM, any other optical medium, punch cards paper tape, any other physical storage medium with patterns of holes, a RAM, a ROM, a PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave transporting data or instructions, cables or links transporting such a carrier wave, or any other medium 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.

[0164] The computer system 601 can include or be in communication with an electronic display 635 that comprises a user interface (UI) 640. Examples of UI’s include, without limitation, a graphical user interface (GUI) and web-based user interface.

[0165] Methods and systems of the present disclosure can be implemented by way of one or more algorithms. An algorithm can be implemented by way of software upon execution by the central processing unit 605.Certain references

[0166] This disclosure hereby incorporates by reference for all purposes, (A) M. Martinez- Dorantes, et al, “Fast Nondestructive Parallel Readout of Neutral Atom Registers in Optical Potentials”. Phys. Rev. Lett. Vol. 119, Iss. 18 (2017),' (B) Minho Kwon, et al, “Parallel low-loss measurement of multiple atomic qubits ”. Department of Physics, University of Wisconsin- Madison (2018),' (C) Margaret E. Shea, et al. “Sub-ms, nondestructive, time-resolved quantumstate readout of a single, trapped neutral atom. ’’Phys. Rev. A 102, 053101 (2020),' (D) Christopher Monroe, et al, “Adaptive and optimal imaging of quantum optical systems for quantum computing”. University of Maryland at College Park, U.S. Patent Number US 11,262,785; (E) U.S. Patent Number US 11,875,227; and (F) M.A. Norcia, et al, “Mid-circuit qubit measurement and rearrangement in a171Yb atomic array.” arXiv e-prints (2023),' (G) Alec Jenkins, et al, “Ytterbium Nuclear-Spin Qubits in an Optical Tweezer Array”. University of Colorado and National Institute of Standards and Technology, and Department of Physics, University of Colorado (2022).

[0167] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. It is not intended that the invention be limited by the specific examples provided within the specification. While the invention has been described with reference to the aforementioned specification, the descriptions and illustrations of the embodiments herein are not meant to be construed in a limiting sense. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. Furthermore, it shall beunderstood that all aspects of the invention are not limited to the specific depictions, configurations or relative proportions set forth herein which depend upon a variety of conditions and variables. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is therefore contemplated that the invention shall also cover any such alternatives, modifications, variations, or equivalents. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.EXAMPLESExample 1: Simulations of the state-resolved imaging infidelity

[0168] FIG. 7 illustrates a simulation of state-resolved imaging infidelity as a function of A for the two-level system illustrated in FIG. 2. In the simulations, the excited states are171Yb3Pi and the states of the ' So manifold are used as qubit states. In some cases, / ' is the spontaneous decay rate, d is the detuning between the energy level and the excitation laser light, and Q is the Rabi frequency.

[0169] FIG. 8 illustrates another example of state-resolved imaging infidelity as a function of magnetic field for the two-level system illustrated in FIG. 2. In the simulations, the excited states are171Yb3Pi and the states of thexSo manifold are used as qubit states.

[0170] In some cases, the photon emission rate from the excited state is given by Equation 1, r sRsc= - 2 l+452 / r2+s , ’ ( VEq 'Iuation 1) / where T is the spontaneous decay rate, 5 is the detuning between the energy level and the excitation laser light, s is parameter 2f22 / E2, defined by Rabi frequency . As such, the state- resolved imaging infidelity may then roughly become the ratio in between the number of emitted photons at the target cycling transition and the nearest off-resonant non-target transition multiplied by a constant according to Equation 2,where the constant c includes a collection and detection efficiency of the emitted photons through the imaging system and some prefactors regarding the imaging beam polarization. In some cases, if the distribution is separated well enough (e.g., the detected photon numbers are large enough), the infidelity due to the overlap between |0) and 11) atoms can have a small distribution (e.g., below 1 x 10'5) for the imaging cameras. Accordingly, tuning the constant, £, may accommodate the photon count effects.

[0171] In some cases, provided that £=0.01 and s=l, the state-resolved imaging infidelity may be calculated or determined as a function of A as shown in FIG. 7. In FIG. 8, the state-resolvedimaging infidelity is illustrated as a function of B field in the case of171Yb3Pi as excited states andxSo as qubit states. In some cases, c can be chosen to be 0.01 to account for the imaging system photon collection efficiency. In some cases, if c is chosen to be 0.001, it may be roughly 10 photons collected by the camera, which can push the photon distribution infidelity down, e.g., to below 1 x 10'5.Example 2: Experimental Results

[0172] The experimental system comprised two main vacuum regions - the “MOT chamber” and the “science chamber” - connected by a differential pumping tube. Atoms were loaded from a pre-cooled atomic beam into a two-stage magneto-optical trap (formed using the 399 nmxPi transition followed by the 556 nm3Pi narrow-line transition) in the MOT chamber. Atoms were then loaded into an optical lattice formed using 532 nm light and transported vertically by 30 cm into the science chamber.

[0173] In order to achieve a deep lattice in a power-efficient manner, the waists of the transport beams were translated synchronously with the optical lattice by moving the position of two focusing lenses, one for each of the two counterpropagating beams that form the lattice. Alignment between the two beams were actively maintained using closed-loop piezo-electric steering mirrors. Atoms were transferred into the optical tweezer array by overlapping the atoms with the array, ramping up the power in the tweezers, and then ramping down the transport lattice. This led to a typical occupancy of several atoms per tweezer. No dissipation was applied to transfer atoms from the transport lattice into the tweezers.

[0174] The two-chamber design may allow for temporally static magnetic fields. No magnetic fields were varied during any of the experimental (e.g., non-classical computation) sequences, which both avoids time-delays associated with switching, and allows for simultaneously maintaining a magnetic field gradient for MOT formation and a large and uniform bias field in the science region. The two-stage MOT operates at a constant field gradient of approximately 18 Gauss / cm in the strong direction.

[0175] After atoms were loaded into tweezers, light was applied with the same parameters as used for imaging of the rm = 1 / 2 qubit state to induce light-assisted collisions and project to a single atom per tweezer. During this time, a second tone that addresses thexSo, rm = -1 / 2 to3Pi, rm = 1 / 2 transition is applied to transfer all atoms to the rm = 1 / 2 state.

[0176] FIG. 5B (bottom) illustrates experimental data showing two subsequent single-shot images of a fully filled 10 by 3 array of atoms generated by an example of the methods and systems disclosed herein.

[0177] Table 1 illustrates measured imaging errors for base condition, e.g., without hiding light applied, and for data and ancilla qubits, with hiding light applied to data qubits at a predetermined wavelength. The hiding procedure may be similar to that disclosed in commonly owned international application no. PCT / US2023 / 026730, which is incorporated herein by reference for all purposes. Uncertainties can represent a Wilson score interval where Table 1 reports only the larger direction for visual clarity. Results can be averaged over a 10 by 7 site array. Methods herein, as illustrated in Table 1, can include characterizing imaging in terms of the accuracy with which methods can distinguish the state of the atom, the probability of leakage into the other qubit state during imaging, and the probability of atom loss. To measure loss from the imaged state, methods herein can include performing repeating images and fitting an exponential decay to the measured occupancy. In some case, apparent loss from the imaged state can include atom loss from the trap, as well as transitions to other unmeasured states, including the other qubit state. Methods herein can include estimating distinguishability based on the overlap of a double-Gaussian fit to the count histograms obtained from stochastically occupied sites. In some cases, to characterize rates of population leakage during imaging, which can be much smaller than loss rates, methods herein can include: (i) preparing atoms in either qubit state, (ii) performing a pre-image of that state, and (iii) performing a “dummy image” of each state. In (ii), for these measurements, the array may be stochastically loaded, so this pre-image can allow post-selecting on occupied sites. In (iii), performing a “dummy image” can include using default imaging intensities and detunings but a longer duration chosen to enhance the signal relative to statistical or systematic readout errors. In some cases, methods herein can include imaging the population of either qubit state to infer population transfer. Values can be inferred by dividing the measured population transfer by the ratio of the dummy image length to a default value, e.g., 5 ms. Further details on determining errors or sources of errors are provided herein elsewhere.Table 1

[0178] As illustrated in Table 1, the largest error channel can be loss of the imaged state. In some cases, for a range of sufficiently low scattering rates, methods herein can include determining a regime where the probability of loss is proportional to the number of collected photons, e.g., near about 104per photon collected, which can correspond to approximately 4 / 106per photon scattered. In some cases, about half of this loss may be explained by the predicted Raman scattering out of3Pi due to the trapping light in the approximately 350 microkelvin («I<) deep traps. In some cases, this loss mechanism can be eliminated by: 1) operating in shallower traps and imaging more slowly, or 2) re-pumping from the metastable3Po and3P2 states at the expense of state- selectivity. In some cases, additional loss in this regime may be caused by photoionization. In some cases, for higher scattering rates, which can correspond to several scattering events per trap period, the loss probability may increase sharply, which may be attributed to heating of the atoms. In some cases, methods herein can include using an operating condition with a scattering rate just below the onset of this additional loss.

Claims

CLAIMSWHAT IS CLAIMED IS:

1. A method of performing state detection for non-classical computing, comprising:(a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites;(b) performing one or more qubit gate operations on at least a portion of said first plurality of qubits;(c) performing a measurement operation, wherein said measurement operation comprises exposing a second plurality of qubits to electromagnetic energy, wherein said first plurality of qubits comprises said second plurality of qubits, wherein said electromagnetic energy is configured to selectively drive a qubit of said second plurality of qubits from an initial state to an excited state in a presence of an applied magnetic field, wherein a selectivity of a transition to said excited state is based at least in part on a strength of said applied magnetic field to said first plurality of qubits, said second plurality of qubits, or both; and(d) determining that said qubit was in said initial state, wherein said determining is based at least in part on said qubit returning to said initial state by emission of a photon in response to said electromagnetic energy in (c).

2. The method of claim 1, further comprising repeating (c) to (d) a plurality of times.

3. The method of claim 1, wherein said returning to said initial state by emission of said photon is a fluorescence transition.

4. The method of claim 1, wherein said returning to said initial state is from a manifold of excited states, wherein said strength of said magnetic field determines a separation between states in said manifold of excited states.

5. The method of claim 4, wherein said separation between states determines which states are in resonance with said radiation.

6. The method of claim 4, wherein a transition to a single state with said manifold of excited states is allowed by a selection rule for each qubit state.

7. The method of claim 1, wherein (c) comprises exposing a first subset of said second plurality of qubits to a first electromagnetic energy and then exposing a second subset of said second plurality of qubits to a second electromagnetic energy.

8. The method of claim 7, wherein said first electromagnetic energy comprises a first polarization and said second electromagnetic energy comprises a second polarization.

9. The method of any one of claims 7 or 8, further comprising:(e) determining that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation.

10. The method of claim 5, further comprising:(f) determining, based at least in part on determining at (e) that a number (n) of spatially distinct optical trapping sites of said array of spatially distinct optical trapping sites is missing a qubit.

11. The method of claim 10, wherein said first plurality of qubits has at most n more qubits than said second plurality of qubits.

12. The method of claim 1, wherein said electromagnetic energy is polarized.

13. The method of any one of claims 8-10 or 12, wherein one or more of said electromagnetic energy, said first electromagnetic energy, or said second electromagnetic energy is circularly polarized.

14. The method of any one of the preceding claims, wherein said qubit comprises a first state |0) and said second state | 1).

15. The method of claim 14, wherein said initial state is |0).

16. The method of claim 14, wherein said initial state is 11).

17. The method of any one of the preceding claims, wherein said first plurality of qubits comprises neutral atoms.

18. The method of claim 17, wherein said neutral atoms comprise a Group II element.

19. The method of claim 18, wherein said Group II element is strontium.

20. The method of claim 17, wherein said neutral atoms comprise rubidium or cesium.

21. The method of claim 17, wherein said neutral atoms comprise ytterbium.

22. The method of any one of the preceding claims, wherein said qubits comprise a temperature of at most 10 microkelvin (pK).

23. The method of any one of the preceding claims, wherein said array is two- dimensional.

24. The method of any one of claims 1-23, wherein said array is three-dimensional.

25. The method of claim 1, wherein said transition is a cycling transition.

26. The method of claim 1, wherein said qubit state comprises a first state |0) and said second state |1), wherein said excited state comprises a manifold of excited states, wherein a transition between said first state and said manifold and said second state in said manifold comprise two closed two-level systems.

27. The method of claim 26, wherein a state of said qubit is identified as one or said first state or said state by emission of said photon or by an absence of said emission of said photon.

28. The method of claim 1, wherein said returning to said initial state comprises spontaneous emission, wherein said returning to said initial state comprises stimulated emission.

29. The method of claim 1, wherein, prior to (c), the method comprises tuning said strength of said applied magnetic field to a selected measurement state.

30. The method of claim 1, wherein (d) comprises imaging said photon at a measurement unit.

31. The method of claim 1, wherein (c) to (d) is configured to perform an atom cooling operation.

32. A system for performing state detection for non-classical computing, comprising: one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits; one or more non-classical computation units configured to perform a non-classical computation using at least a portion of said first plurality of qubits; one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and one or more measurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

33. A non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of performing state detection for non-classical computing on a non-classical computer, wherein said non- classical computer is configured to execute said one or more instructions, the method comprising:(a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites;(b) performing a non-classical computation using at least a portion of said first plurality of qubits;(c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and(d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

34. A method of performing state detection for non-classical computing, comprising:(a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites;(b) performing a non-classical computation using at least a portion of said first plurality of qubits;(c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and(d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

35. The method of claim 34, wherein exposing said second plurality of qubits to said radiation comprises exposing said second plurality of qubits to a first radiation and then exposing said second plurality of qubits to a second radiation.

36. The method of claim 35, wherein said first radiation is a first light with a first polarization and said second radiation is a second light with a second polarization.

37. The method of any one of claims 35 or 36, further comprising:(e) determining that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation.

38. The method of claim 37, further comprising:(f) determining, based at least in part on determining at (e) that said first subset of said second plurality of qubits fluoresces in response to being exposed to said first radiation and that said second subset of said second plurality of qubits fluoresces in response to being exposed to said second radiation, that a number (n) of spatially distinct optical trapping sites of said array of spatially distinct optical trapping sites is missing a qubit.

39. The method of claim 38, wherein said first plurality of qubits has at most n more qubits than said second plurality of qubits.

40. The method of claim 34, wherein said radiation is light.

41. The method of claim 40, wherein said light is polarized light.

42. The method of claim any one of claims 36-38 or 41, wherein one or more of said polarized light, said first light, or said second light is circularly polarized.

43. The method of any one of the preceding claims, wherein said second plurality of qubits are exposed to said radiation while said second plurality of qubits are in a magnetic field.

44. The method of any one of the preceding claims, wherein said at least one of said second plurality of qubits that fluoresces in response to being exposed to said radiation is said first subset of said second plurality of qubits.

45. The method of claim 44, wherein said first state is |0) and said second state is I D.

46. The method of claim 44, wherein said first state is 11) and said second state is 10).

47. The method of any one of the preceding claims, wherein said first plurality of qubits comprise neutral atoms.

48. The method of claim 47, wherein said neutral atoms comprise a Group II element.

49. The method of claim 48, wherein said Group II element is strontium.

50. The method of claim 47, wherein said neutral atoms comprise ytterbium.

51. The method of any one of the preceding claims, wherein said qubits comprise a temperature of at most 10 microkelvin ( l<).

52. The method of any one of the preceding claims, wherein said array is two- dimensional.

53. The method of any one of claims 34-52, wherein said array is three-dimensional.

54. A system for performing state detection for non-classical computing, comprising: one or more optical trapping units configured to obtain an array of spatially distinct optical trapping sites, wherein said array comprises a first plurality of qubits; one or more non-classical computation units configured to perform a non-classical computation using at least a portion of said first plurality of qubits; one or more electromagnetic delivery units configured to expose a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and one or more measurement units configured to determine or predict, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

55. A non-transitory computer-readable media comprising machine-executable code comprising one or more instructions that, upon execution, implement a method of performing state detection for non-classical computing on a non-classical computer, wherein said non- classical computer is configured to execute said one or more instructions, the method comprising:(a) obtaining a first plurality of qubits in an array of spatially distinct optical trapping sites;(b) performing a non-classical computation using at least a portion of said first plurality of qubits;(c) exposing a second plurality of qubits to radiation, wherein said first plurality of qubits comprises said second plurality of qubits; and(d) determining or predicting, based at least in part on at least one of said second plurality of qubits fluorescing in response to being exposed to said radiation, that each of a first subset of said second plurality of qubits is in a first state and each of a second subset of said second plurality of qubits is in a second state.

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