Method and system for error correction in a neutral atom quantum computer

Abstract

A method for error-correcting quantum computing may include identifying that a qubit has been lost, replacing the qubit, re-implementing the qubit in a circuit, and flagging measurements taken while the qubit was missing as unreliable.

Description

[Technical Field]

[0001] cross reference This application claims the benefit of U.S. Provisional Application No. 63 / 488,407, filed March 03, 2023, which is incorporated herein by reference.

[0002] Statement of Federally Funded Research This invention was made with U.S. government support under Grant Nos. 2040527 (Phase 1) and 2134345 (Phase 2) awarded through the Convergence Accelerator Research Program of the National Science Foundation. The U.S. government has certain rights in this invention. [Background technology]

[0003] Quantum error correction can be used in quantum computing to account for errors from decoherence and noise. Quantum error correction can be important for developing larger quantum computers. Errors can occur during gate preparation, gate execution, gate measurement, etc. Summary of the Invention

[0004] The systems and methods disclosed herein may improve methods and systems for the correction of errors in quantum computers by better accounting for errors due to atom loss.

[0005] In one aspect, the present disclosure provides a method for error corrected quantum computation that includes the steps of: (a) identifying that a qubit has been lost, (b) replacing the qubit, (c) re-implementing the qubit in a circuit, and (d) flagging measurements taken while the qubit is missing as unreliable.

[0006] In some embodiments, the identifying in (a) includes using a plurality of swap gates. In some embodiments, the swap gates in the plurality of swap gates are implemented as a plurality of CNOT gates. In some embodiments, the method further includes measuring alternating atoms in the lattice, performing the plurality of swap gates to transfer data stored in the data qubits to the ancilla qubits, and measuring the swapped data qubits to identify the one or more missing atoms. In some embodiments, the identifying step in (a) comprises using a modified knock-knock protocol, the modified knock-knock protocol comprising providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified control-Z gate between the first atom and the second atom based on the Rydberg interaction; and rotating the second qubit back to a computational basis and performing the measurement.

[0007] In some embodiments, the re-implementing in (c) includes using (i) a decoder algorithm, where the decoder algorithm takes the graph and determines a set of edges. In some embodiments, the method includes, prior to (i), updating a matching graph passed to the decoder algorithm based on a predicted probability distribution of the lost qubit replaced in (b). In some embodiments, the re-implementing in (c) includes using a minimum-weight perfect matching decoder algorithm. In some embodiments, the method further includes updating the matching graph passed to the minimum-weight perfect matching decoder algorithm based on the predicted probability distribution of the lost qubit replaced in (b). In some embodiments, the method further includes, if an ancilla qubit is lost, updating the matching graph such that nodes including ancilla qubits are connected by edges corresponding to the predicted probability distributions, and, if a data qubit is lost, updating the matching graph by assigning a predicted probability distribution to each node including a data qubit. In some embodiments, each node that contains an ancillary qubit is updated.

[0008] In some embodiments, (a)-(d) are performed during a quantum computation circuit. In some embodiments, (a)-(d) are performed without measurement of each or more of the data qubits. In some embodiments, (a)-(d) are performed substantially without loss of coherence of each or more of the data qubits. In some embodiments, (d) includes flagging measurements taken during a window of time that includes a time when the qubit is missing as unreliable.

[0009] In some embodiments, the qubit is a trapped atomic qubit. In some embodiments, the trapped atomic qubit is a neutral atomic qubit. In some embodiments, the neutral atomic qubit is a Group 2 element or a Group 2-like element. In some embodiments, the Group 2 element or a Group 2-like element comprises ytterbium, rubidium, cesium, or strontium. In some embodiments, the qubit is 1 It contains qubit states that include nuclear spin states on the S0 manifold.

[0010] In some embodiments, (a) includes an operation in which a two-qubit interaction between the qubit and the missing qubit has the effect of a Pauli operation or an identity operation on the qubit. In some embodiments, the two-qubit interaction includes an excitation of a nuclear spin state of the neutral atom to a Rydberg state of the neutral atom.

[0011] In some embodiments, the re-implementing in (c) includes implementing (i) a decoder, the decoder configured to receive the matching graph and determine a set of edges, and the matching graph received by the decoder is updated based on a predicted probability distribution of the missing qubit. In some embodiments, the method further includes performing a measurement operation, the measurement operation being state-selective. In some embodiments, the measurement operation includes the qubit being measured with electromagnetic energy, the electromagnetic energy being configured to selectively drive the measured qubit from an initial state to an excited state in the presence of an applied magnetic field, the selectivity of the transition to the excited state being based at least in part on the strength of the applied magnetic field. In some embodiments, the method further includes determining that the measured qubit was in an initial state, the determining being based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

[0012] In some embodiments, the qubits are atoms and (b) is implemented using optical tweezers.

[0013] In some embodiments, the decoder includes a union find, a tensor network decoder, belief propagation with an ordered statistics decoder, a maximum likelihood decoder, or a lookup table decoder. In some embodiments, the minimum weight perfect matching includes a sparse blossom or a fusion blossom. In some embodiments, (a)-(d) include a portion of an error correcting code, and the error correcting code includes a topological code. In some embodiments, the topological code is a stabilizer code. In some embodiments, the error correcting code is a surface code, a color code, a toric code, a shor style code, or a qLDPC code. In some embodiments, the color code is a Steane code. In some embodiments, the shor style code is a Bacon-shor code. In some embodiments, the qLDPC code is a hypergraph product code.

[0014] In some embodiments, (c) includes implementing an error correction code, the error correction code implementation including a decoder configured to receive the matching graph and determine a set of edges, and the matching graph received by the decoder is updated based on a predicted probability distribution of lost qubits. In some embodiments, the decoder is configured to receive the matching graph and determine a set of edges, and the matching graph received by the decoder is updated based on a predicted probability distribution of lost qubits. In some embodiments, the error correction code is a stabilizer code. In some embodiments, each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges that correspond to possible physical errors. In some embodiments, the edges are weighted based on the likelihood of a particular error occurring. In some embodiments, an atom loss is treated as a gate error with a 50% probability of occurring.

[0015] In another aspect, this disclosure provides a method for error corrected quantum computation. The method may include implementing an error correcting code, the error correcting code implementation including a decoder, the decoder configured to receive a matching graph and determine a set of edges, and the matching graph received by the decoder is updated based on a predicted probability distribution of a lost qubit.

[0016] In another aspect, this disclosure provides a method for error-correcting quantum computing. The method may include providing a plurality of qubits and implementing an error-correcting code, the error-correcting code implementation including a decoder configured to receive a matching graph and determine a set of edges, wherein the matching graph received by the decoder is updated based on a predicted probability distribution of a missing qubit.

[0017] In some embodiments, the error correcting code includes an operation in which a two-qubit interaction between the qubit and a missing qubit has the effect of a Pauli operation or an identification operation on the qubit. In some embodiments, the qubit is a non-lost qubit. In some embodiments, the qubit is a missing qubit. In some embodiments, the method further includes, prior to (b), (i) identifying that the qubit is missing, (ii) replacing the qubit, and (iii) re-implementing the qubit in the circuit. In some embodiments, the method includes, following (b), (iv) flagging measurements taken while the qubit was missing as unreliable.

[0018] In some embodiments, the identifying step in (i) includes using a plurality of swap gates. In some embodiments, the swap gates in the plurality of swap gates are implemented as a plurality of CNOT gates. In some embodiments, the method further includes measuring alternating atoms in the lattice, performing the plurality of swap gates to transfer data stored in the data qubits to the ancilla qubits, and measuring the swapped data qubits to identify one or more missing atoms. In some embodiments, the identifying step in (i) includes using a modified knock-knock protocol, the modified knock-knock protocol further includes providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on the Rydberg interaction; rotating the second qubit back to the computational basis; and performing the measurement. In some embodiments, the re-implementing step in (iii) includes using (a) a decoder algorithm, the decoder algorithm taking the graph and determining a set of edges.

[0019] In some embodiments, the method includes, before (a), updating the matching graph passed to the decoder algorithm based on the predicted probability distribution of the lost qubit replaced in (ii). In some embodiments, the reimplementing in (iii) includes using a minimum weight perfect matching decoder algorithm. In some embodiments, the method further includes updating the matching graph passed to the minimum weight perfect matching decoder algorithm based on the predicted probability distribution of the lost qubit replaced in (ii). In some embodiments, the method further includes, if an ancilla qubit is lost, updating the matching graph so that nodes including ancilla qubits are connected by edges corresponding to the predicted probability distributions, and, if a data qubit is lost, updating the matching graph by assigning a predicted probability distribution to each node including a data qubit. In some embodiments, each node including an ancilla qubit is updated.

[0020] In some embodiments, (a)-(b) are performed during quantum computing. In some embodiments, (a)-(b) are performed without measurement of each or more of the data qubits. In some embodiments, (a)-(b) are performed substantially without loss of coherence of each or more of the data qubits. In some embodiments, the method includes, subsequent to (b), flagging measurements taken during a time window that includes the time the missing qubit was missing as unreliable.

[0021] In some embodiments, the plurality of qubits comprises trapped atomic qubits. In some embodiments, the trapped atomic qubits comprise neutral atomic qubits. In some embodiments, the neutral atomic qubits comprise Group 2 elements or Group 2-like elements. In some embodiments, the Group 2 elements or Group 2-like elements comprise ytterbium, rubidium, cesium, or strontium. In some embodiments, the plurality of qubits comprises: 1The two-qubit interaction includes an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

[0022] In some embodiments, the method further includes performing a measurement operation, where the measurement operation is state-selective. In some embodiments, the measurement operation includes applying electromagnetic energy to the qubit being measured, where the electromagnetic energy is configured to selectively drive the qubit being measured from an initial state to an excited state in the presence of an applied magnetic field, where the selectivity of the transition to the excited state is based at least in part on the strength of the applied magnetic field. In some embodiments, the method further includes determining that the qubit being measured was in the initial state, where the determining is based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

[0023] In some embodiments, the plurality of qubits comprises atomic qubits, and the atom replacement operation is performed using optical tweezers. In some embodiments, the decoder comprises a union find, a tensor network decoder, a belief propagation with order statistics decoder, a maximum likelihood decoder, or a lookup table decoder. In some embodiments, the decoder comprises a minimum weight perfect matching. In some embodiments, the decoder comprises a sparse blossom or a fusion blossom. In some embodiments, the error correcting code comprises a topological code. In some embodiments, the topological code is a stabilizer code. In some embodiments, the error correcting code is a surface code, a color code, a toric code, a shor-style code, or a qLDPC code. In some embodiments, the color code is a Steane code. In some embodiments, the shor-style code is a Bacon-shor code. In some embodiments, the qLDPC code is a hypergraph product code. In some embodiments, each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges that correspond to possible physical errors. In some embodiments, edges are weighted based on the likelihood that a particular error will occur, hi some embodiments, an atom loss is treated as a gate error with a 50% probability of occurring.

[0024] In another aspect, the present disclosure provides a system for error-correcting quantum computing, which may comprise an error-correcting code, an implementation of the error-correcting code including a decoder configured to receive a matching graph and determine a set of edges, wherein the matching graph received by the decoder is updated based on a predicted probability distribution of a lost qubit.

[0025] In some embodiments, the error correcting code includes an operation in which a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli operation or an identity operation on the qubit. In some embodiments, the qubit is a non-missing qubit. In some embodiments, the qubit is a missing qubit.

[0026] In some embodiments, the system further comprises a processor configured to implement an error correction code. In some embodiments, the processor is further configured to provide instructions to a non-classical computing system, the non-classical computing system configured to implement the instructions to (i) identify that a qubit has been lost, (ii) replace the qubit, and (iii) re-implement the qubit in the circuit. In some embodiments, the processor is further configured to (iv) flag measurements taken while the qubit is missing as unreliable. In some embodiments, (i) includes using multiple swap gates. In some embodiments, the swap gates in the multiple swap gates are implemented as multiple CNOT gates. In some embodiments, the processor is further configured to provide instructions to the non-classical computing system to measure alternating atoms in the lattice, execute the multiple swap gates to transfer data stored in the data qubits to the ancilla qubits, and measure the swapped data qubits to identify the one or more missing atoms.

[0027] In some embodiments, (i) includes using a modified knock-knock protocol, the modified knock-knock protocol including providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on the Rydberg interaction; rotating the second qubit back to the computational basis; and performing the measurement. In some embodiments, (iii) includes using (A) a decoder algorithm, the decoder algorithm taking the graph and determining a set of edges. In some embodiments, before (A), the processor is further configured to update the matching graph passed to the decoder algorithm based on a predicted probability distribution of the missing qubit replaced in (ii). In some embodiments, (iii) includes using a minimum weight perfect matching decoder algorithm. In some embodiments, the processor is further configured to update the matching graph passed to the minimum weight perfect matching decoder algorithm based on a predicted probability distribution of the missing qubit replaced in (ii). In some embodiments, the processor is further configured to: update the matching graph when an ancilla qubit is lost so that nodes including ancilla qubits are connected by edges corresponding to the predicted probability distributions; and update the matching graph when a data qubit is lost by assigning a predicted probability distribution to each node including a data qubit. In some embodiments, each node including an ancilla qubit is updated.

[0028] In some embodiments, the error correction code is configured to be implemented between the quantum computing circuit. In some embodiments, the error correction code is configured to be implemented without measuring each or more of the data qubits. In some embodiments, the error correction code is configured to be implemented without substantially losing coherence of each or more of the data qubits. In some embodiments, the processor is further configured to flag measurements taken during a time frame that includes a time when the missing qubit was missing as unreliable. In some embodiments, the system further comprises a non-classical computing system, the non-classical computing system comprising a trapped atomic qubit. In some embodiments, the trapped atomic qubit comprises a neutral atomic qubit. In some embodiments, the neutral atomic qubit comprises a Group 2 element or a Group 2-like element. In some embodiments, the Group 2 element or Group 2-like element comprises ytterbium, rubidium, cesium, or strontium. In some embodiments, the plurality of qubits comprises: 1 The two-qubit interaction includes an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

[0029] In some embodiments, the processor is further configured to provide instructions to the non-classical computing system to perform a measurement operation, where the measurement operation is state-selective. In some embodiments, the measurement operation includes applying electromagnetic energy to the qubit to be measured, where the electromagnetic energy is configured to selectively drive the qubit to be measured from an initial state to an excited state in the presence of an applied magnetic field, where the selectivity of the transition to the excited state is based at least in part on the strength of the applied magnetic field. In some embodiments, the processor is further configured to determine that the qubit to be measured was in the initial state based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

[0030] In some embodiments, the system further comprises a non-classical computing system, the non-classical computing system comprising a plurality of qubits, the plurality of qubits comprising atomic qubits, and the atomic permutation operation is performed using optical tweezers. In some embodiments, the decoder comprises a union find, a tensor network decoder, a belief propagation with order statistics decoder, a maximum likelihood decoder, or a lookup table decoder. In some embodiments, the decoder comprises a minimum weight perfect matching. In some embodiments, the decoder comprises a sparse blossom or a fusion blossom. In some embodiments, the error correcting code comprises a topological code. In some embodiments, the topological code is a stabilizer code. In some embodiments, the error correcting code is a surface code, a color code, a toric code, a shor-style code, or a qLDPC code. In some embodiments, the color code is a Steane code. In some embodiments, the shor-style code is a Bacon-shor code. In some embodiments, the qLDPC code is a hypergraph product code. In some embodiments, each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges that correspond to possible physical errors. In some embodiments, edges are weighted based on the likelihood that a particular error will occur. In some embodiments, atom loss is treated as a gate error with a 50% probability of occurring.

[0031] In another aspect, the present disclosure provides a method for error-correcting quantum computing, the method including the steps of (a) identifying that a qubit has been lost, (b) replacing the qubit, (c) re-implementing the qubit in a circuit, and (d) flagging measurements taken while the qubit was missing as unreliable.

[0032] In some embodiments, the identifying in (a) includes using a plurality of swap gates. In some embodiments, the swap gates in the plurality of swap gates are implemented as a plurality of CNOT gates. In some embodiments, the method includes measuring alternating atoms in the lattice; executing the plurality of swap gates to transfer data stored in the data qubits to the ancilla qubits; and measuring and identifying the swapped data qubits to identify the one or more missing atoms. Further includes:

[0033] In some embodiments, the identifying step in (a) comprises using a modified knock-knock protocol, the modified knock-knock protocol comprising providing a first atom to be probed using a second atom, where the second atom is an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on the Rydberg interaction; and rotating the second qubit back to the computational basis and performing the measurement.

[0034] In some embodiments, the re-implementing in (c) includes using a minimum weight perfect matching decoder algorithm. In some embodiments, the method further includes updating the matching graph that is passed to the minimum weight perfect matching decoder algorithm based on a predicted probability distribution of the lost qubit replaced in (b). In some embodiments, the method further includes, if an ancilla qubit is lost, updating the matching graph so that nodes containing ancilla qubits are connected by edges that correspond to the predicted probability distribution, and, if a data qubit is lost, updating the matching graph by assigning a predicted probability distribution to each edge connecting the data qubit to its respectively connected ancilla qubit.

[0035] Another aspect of the present disclosure provides a system comprising one or more computer processors and a computer memory coupled thereto, the computer memory including machine-executable code that, when executed by the one or more computer processors, performs any of the methods described above or elsewhere herein.

[0036] While only illustrative embodiments of the present disclosure have been shown and described, further aspects and advantages of the present disclosure will become readily apparent to those skilled in the art from the following detailed description. As will be realized, the present disclosure is capable of other and different embodiments, and its several details are capable of modifications in various obvious respects, all without departing from the present disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.

[0037] Incorporation by Reference All publications, patents, and patent applications mentioned herein are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent that the publications and patents or patent applications incorporated by reference conflict with the present disclosure contained herein, the present specification is intended to supersede and / or take precedence over any such conflicting material. [Brief explanation of the drawings]

[0038] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also referred to herein as "Figure" and "FIG") [Figure 1] 1 is a flowchart of an exemplary method for error-correcting quantum computing. [Figure 2A]FIG. 1 is a schematic diagram of an energy level diagram illustrating when a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli or identity operation on the qubit. [Figure 2B] 1 is a flowchart of an exemplary method for error-correcting quantum computation based on updating a matching graph received by a decoder. [Figure 3A] 1 is a flowchart of an example of a method for identifying atomic losses in an error correcting code, the error correcting code including performing a series of SWAP gates. [Figure 3B] FIG. 10 is an operational diagram of a method for implementing an error-correcting code that involves performing a series of SWAP gates between an ancilla qubit and a data qubit. [Figure 4] 1 is a flowchart of an example of a method for implementing an error correction code that accounts for atomic losses, where the error correction code comprises a modified knock-knock protocol. [Figure 5A] FIG. 10 is a diagram of a matching graph showing a missing ancilla qubit. [Figure 5B] FIG. 10 is a diagram of a matching graph showing missing data qubits. [Figure 5C] 1 is a flowchart of a method for reimplementing a qubit. [Figure 6A] 1 illustrates a system for error-correcting quantum computing that is programmed or otherwise configured to implement the methods provided herein. [Figure 6B] 1 illustrates a computer system that is programmed or otherwise configured to implement the methods provided herein. [Figure 7A] 10 is a plot of simulation data showing a plot of physical error rate versus logical error probability for a model that accounts for atomic losses. [Figure 7B] 10 is a plot of simulation data showing a plot of physical error rate versus logical error probability for a model that does not consider atomic losses. [Figure 7C] An overlay of the data in Figures 7A and 7B. Detailed Description of the Invention

[0039] While various embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the invention described herein may be employed.

[0040] Whenever the terms "at least," "greater than," or "greater than or equal to" precede the first number in a series of two or more numbers, the terms "at least," "greater than," or "greater than or equal to" apply to each of the numbers in the series. For example, 1, 2, or 3 or more is equivalent to 1 or more, 2 or more, or 3 or more.

[0041] Whenever the terms "no more than," "less than," or "less than or equal to" precede the first number in a series of two or more numbers, the terms "only," "less than," or "less than or equal to" apply to each and every number in the series. For example, 3, 2, or 1 or less is equivalent to 3 or less, 2 or less, or 1 or less.

[0042] Certain inventive embodiments herein contemplate numerical ranges. Where ranges exist, the ranges include the endpoints of the ranges. Moreover, all subranges and values ​​within the ranges exist as if expressly stated.

[0043] The terms "about" or "approximately" can mean within an acceptable error range for a particular value, which depends in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, "about" can mean within one standard deviation or more than one standard deviation, in accordance with practice in the art. Alternatively, "about" can mean a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value. Where a particular value is described in this application and claims, unless otherwise specified, the term "about" meaning within an acceptable error range for that particular value can be assumed.

[0044] Methods for error-correcting quantum computing The systems and methods disclosed herein may generally relate to qubit loss during error correction. There may be at least two general pieces within qubit loss during error correction. The systems and methods described herein may be directed to detecting qubit loss without destroying the data stored on the qubit. Instead of manifesting as a gate measurement error, if a qubit is lost, data that may be present is absent rather than incorrect. The systems and methods disclosed herein may be directed to identifying cases where an error is caused by a missing qubit. The systems and methods described herein may also be directed to modifying a decoder to handle loss events. For example, an error correction code may be directed to updating a calculation to account for the error. In some cases, knowledge of the error may be required to implement the error correction code. However, in other cases, the error correction code may be modified to account for the missing data without explicit knowledge that a qubit is missing.

[0045] The systems and methods disclosed herein may generally not alter the topology of the underlying surface code. The systems and methods described herein may improve methods for detecting atomic losses by compressing the underlying protocol. The systems and methods disclosed herein may enable loss detection between cycles (or perhaps less frequently) rather than after every gate. The systems and methods disclosed herein may address un-induced erasure errors in addition to, or instead of, gate-induced erasure errors.

[0046] The systems and methods of the present disclosure may improve upon other procedures, at least because they may not include or require operations that alter the topology of the underlying surface code. In some cases, the underlying surface code may be unchanged. Instead, the matching graph passed to the decoder algorithm may be updated to account for the predicted probability distribution of the lost qubit. Because the matching graph passed to the decoder is updated, the underlying decoder may also remain unchanged. Because the decoder is unchanged, the error correction code may also remain unchanged. Thus, the methods and systems of the present disclosure may be used without modifying the underlying surface code. Similarly, because no modifications to the underlying decoder and surface code are required, the methods and systems of the present disclosure may be used with a wide variety of decoders and surface codes.

[0047] In some cases, tracking syndrome measurements can be used to detect loss events (e.g., defects). See, for example, Siegel, A. et al., Adaptive Surface Code for Quantum Error Correction in the Presence of Temporary or Permanent Defects, arXiv:2211.08468v1[quant-ph]15 Nov 2022, which is incorporated herein by reference in its entirety and available at https: / / arxiv.org / pdf / 2211.08468.pdf.

[0048] In some cases, gate-induced erasure errors can be addressed by error correction. See, for example, Wu, Y., et al., Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays, Nat. Comms. Vol. 13, P. 4657 (2022), which is incorporated herein by reference in its entirety. In the above, atoms may still be present. Rather than addressing the missing atoms or qubits, the above-referenced applications can convert gate errors into erasure errors.

[0049] In some cases, noise structures in the hardware can be used for error correction. See, for example, Shay, K., et al., High threshold codes for neutral atom qubits with biased erasure errors, arXiv:2302.03063 v1 [quant-ph] 6 Feb 2023, which is incorporated herein by reference in its entirety. As with Wu, the atom may still be present. Rather than addressing the missing atom or qubit, the above-referenced application can turn a gate error into an erasure error.

[0050] Error Correction - Quantum error correction is a procedure for encoding quantum information in a distributed manner across many quantum systems, whereby the stored information is protected from local errors on the constituent systems, provided these errors are sufficiently sparse. In some cases, the stored information and the constituent systems are both two-level quantum systems or qubits. The information to be protected is encoded across many physical qubits, forming one or more logical qubits.

[0051] The process of detecting and correcting errors on logical qubits involves measuring the parity of a given set of operators acting on physical qubits and using the measured parity to diagnose and correct errors. In a simple example, a parity measurement checks the equality of two qubits, returning a true or false answer, which can be used to determine whether a correction needs to be made. Additional measurements can be made on more than two qubits. Because physical qubits cannot be measured directly without corrupting the state of the logical system, their parity is measured using ancillary qubits (i.e., ancillary qubits). Thus, there can be two types of physical qubits: data qubits, where logical information is stored, and ancillary qubits, which are used to extract the desired parity check.

[0052] In practice, an error-correction cycle consists of a series of gates to transfer parity values ​​to the ancilla qubits, followed by measurements of the ancilla qubits. This process is known as syndrome extraction. Errors can occur at any time during the syndrome extraction process, including during readout of the ancilla qubits.

[0053] One method of error correction (Shor style) uses repeated rounds of syndrome extraction to overcome read errors and build confidence in the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors occurred and which corrections need to be applied. The decoding problem is typically represented as a weighted graph or hypergraph. In this setting, each node in the graph corresponds to a set of syndrome measurements. Such a set of syndrome measurements is called a detector. Edges or hyperedges in the graph correspond to errors, and the weight of this edge corresponds to the likelihood that that error occurs. The occurrence of a given error can be expected to flip the parity of all associated detectors. The decoding problem can then be stated as follows: Given a set of detectors (nodes) whose parity differs from that expected in the absence of errors, determine the set of most likely physical errors (edges) that could cause the observed detection.

[0054] Error correction with atom loss—Quantum computers based on trapped atoms can be susceptible to errors caused by the loss of qubits. In trapped atom quantum computers, qubits can include atoms in an array. The atoms can be neutral atoms or ions. Error correction codes can generally employ repeated implementations of circuits that implement quantum computation. As circuits are implemented and reimplemented, statistics can be generated regarding the errors that occur. However, error correction codes implemented on systems with qubit loss can generally differ from other systems. For example, non-qubit loss errors can be gate errors. Similarly, loss of coherence can be represented as gate errors. For gate errors or errors similar to gate errors, the set of possible error values ​​is relatively small because the consequences of gate errors resemble measurements of the system. In some cases, the atom loss rate can be similar to or greater than the gate error rate; therefore, it can be useful to provide improved methods for correcting atom loss.

[0055] 1 is a flowchart of an exemplary method 100 for error-correcting quantum computing. In some cases, an error-correcting code that accounts for qubit loss includes identifying that a qubit has been lost 110, replacing the qubit 120, re-implementing the qubit in a circuit that may be in an incorrect state if replaced 130, and flagging measurements taken while the qubit was missing as unreliable 140.

[0056] Referring to FIG. 1 , in operation 110 of the method for error correction with atomic loss 100, atomic loss may be detected. For example, atomic loss may be detected at the end of each syndrome extraction cycle. Methods for detecting atomic loss are described herein with reference to the section "Identifying Qubit Loss." In operation 120 of the method for error correction with atomic loss 100, once a qubit is identified as lost, it may be replaced with a new qubit. The new qubit may, at least initially, be in a random state.

[0057] In operation 130 of the method 100 for error correction with atomic loss, the qubits may be reimplemented in a circuit. In some cases, operation 130 includes using a decoder algorithm. The decoder algorithm may take a graph and determine a set of edges. In some cases, operation 130 includes updating a matching graph that is passed to the decoder algorithm based on a predicted probability distribution of the lost qubits replaced in operation 120 before implementing the decoder algorithm. Methods for updating the decoder algorithm are described in the "Modifying the Decoding Algorithm" section herein.

[0058] In systems that are not affected by atom loss, the error can be a discrete Pauli error on the physical qubit. However, when qubits are stored on atoms, the atoms, and therefore the qubits they contain, can be lost. The effect on syndrome extraction in the presence of loss depends on the details of the hardware. In the case of neutral atoms using a Rydberg gate, the effect of atom loss appears as a non-interaction instead of a two-qubit gate. In fact, the two-qubit interaction between the missing atom and the resident atom can be treated as affecting the discrimination gate on the resident atom.

[0059] FIG. 2A is a schematic diagram of an energy level diagram illustrating when a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli or identity operation on the qubit. As shown, atoms A and B can be adjacent qubits. The qubit can be a trapped ion qubit. The qubit can be a trapped atom qubit. As shown, one qubit acquires a phase contingent on the state-selective excitation of the other. In some cases, the state-selective excitation is from state |1> to state |r>.

[0060] In some cases, state |r> is a Rydberg state. In some cases, state |r> is a Rydberg state of a neutral atomic qubit. When atom A is excited to a Rydberg state, atom B (if present) experiences a shift due to the Rydberg interaction. In one example, optical excitation can be tuned to the frequency difference between the |1> state and the Rydberg state. When atom A is in state |1>, atom A is driven at least transiently to a Rydberg state, and atom B (if present) experiences a shift due to the Rydberg interaction. When atom B is absent and atom A is in state |1>, there is no shift to atom B. When atom A is in state |0> and atom B is present, the energy gap is too large and nothing happens to atom A or atom B. When atom A is in state |0> and atom B is absent, the energy gap is still too large and nothing happens to atom A or atom B (if absent). Thus, the two-qubit interaction between the missing atom and the present atom can be treated as affecting the identification gate on the present atom. While this example describes the case where the two-qubit interaction with the missing atom affects the identification operation, the disclosed methods and systems also work when the two-qubit interaction with the missing atom affects a Pauli operation. The Pauli operation can include a Pauli X gate, a Pauli Y gate, or a Pauli Z gate. For example, a Pauli X gate is a single qubit rotation of π radians around the X axis. For example, a Pauli Y gate is a single qubit rotation of π radians around the Y axis. For example, a Pauli Z gate is a single qubit rotation of π radians around the Z axis. A rotation about an axis of 2π radians is an identification operation.

[0061] The above works similarly if atom A is also a missing qubit. In some cases, the qubit is a non-missing qubit. In some cases, the qubit is a missing qubit. For example, if a qubit is a missing qubit, a two-qubit gate between the two missing qubits affects the identification as well. A two-qubit operation between the atom and the missing atom affects the identification. The protocol does not propagate errors forward in time (to first order). For example, if the two-qubit operation is incomplete. Operations may propagate errors of higher order in time.

[0062] The two-qubit gate properties described with respect to FIG. 2A can be used to modify the decoder.

[0063] 2B is a flowchart of an exemplary method (200) for error-correcting quantum computing based on updating a matching graph received by a decoder. In operation (210), the method may include providing a plurality of qubits. The systems and methods described herein may be directed to detecting qubit loss without destroying data stored on the qubits. For example, it may be difficult to determine that a qubit has been lost and update the decoder mid-circuit.

[0064] In operation (220), the method may include implementing an error correction code, the error correction code implementation including a decoder configured to receive the matching graph and determine a set of edges, and the matching graph received by the decoder is updated based on the predicted probability distribution of the lost quantum bit.

[0065] The systems and methods disclosed herein can be used with various decoders. The error correction schemes (e.g., error correction code implementations) of the present disclosure can include a decoder. The decoder can decode which errors occurred on which qubits. Once identified, these errors can be tracked, and the information can be used to correct subsequent measurements using classical control software. In some cases, error correction code implementations include operations in which a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli operation or an identification operation on the qubit.

[0066] In one example, a round of an error-correcting code may include performing a two-qubit operation between a data qubit and an ancillary qubit, which may comprise part of a round of syndrome extraction.

[0067] If an ancilla qubit is lost, no change can occur to the data qubit because the two qubit gates on the qubit and the lost qubit affect the identification. However, because an ancilla qubit is lost, the round of syndrome extraction associated with that ancilla qubit does not provide a reliable syndrome value for that particular round of syndrome extraction. To handle this in decoding, the missing syndrome bit may be assigned a value associated with a random state. In the matching graph, an edge may be added to each detector containing a missing syndrome value and assigned an edge weight corresponding to a 50% error probability (this is similar to telling the decoder, "Do not trust these syndrome values").

[0068] If a data qubit is lost, none of the syndromes involving the data qubit can be viewed as reliable. Thus, for each syndrome measurement that includes the missing data qubit, the above process can be repeated for each ancillary element with which the missing data qubit was intended to interact. Thus, because multiple ancillary elements may be associated with the missing data qubit, a larger number of ancillary elements measurements may provide unreliable syndrome data depending on the particular error-correcting code. For example, in a surface code, four edges may be created. In another example, in a color code, three edges may be created.

[0069] The methods for updating the decoder described herein may be independent of the type of atom, the type of qubit, the type of error correction code, or the particular decoder used in the error correction code. If a two-qubit gate operation affects an identification or Pauli operation, the matching graph passed to the decoder may be updated as described herein.

[0070] In an operation 140 of the method 100 for error correction involving atom loss, measurements taken while a qubit was missing can be flagged as unreliable. In some cases, the operation 140 includes flagging measurements taken during a time window that includes the time the qubit was missing as unreliable. For example, the time window can include a round of syndrome measurements. It is not necessary to know exactly which measurements were made during the time window that includes the missing atom, only which set of measurements were made during the time window that includes the missing atom.

[0071] The methods and systems described herein may enable identification of a missing qubit without substantially shutting down the circuit. For example, if a set of measurements is flagged as unreliable, the circuit may continue with other measurements and return to reacquire the unreliable measurements. For example, if a data qubit is flagged as missing, the circuit may continue with other qubits while the atom is replaced, and the portion of the circuit containing that qubit may be reimplemented.

[0072] The methods and systems described herein may enable identification of missing qubits and replacement of missing qubits without measurements of each or multiple data qubits. As described above, missing qubits may be identified in a round of syndrome measurements without measurements of data qubits. Thus, atoms identified as missing may be replaced and the circuit may continue without measurements of data qubits.

[0073] The methods and systems described herein may enable identification of lost qubits without loss of coherence of each or multiple data qubits. As described above, error correction codes use measurements of syndrome qubits to identify atomic losses, and data qubits may not need to be measured during peripheral error correction. The systems and methods of the present disclosure may enable continuous correction of atomic losses. The systems and methods of the present disclosure may enable mid-circuit correction of atomic losses.

[0074] Identifying qubit losses The present disclosure provides at least two methods for identifying qubit loss; however, various methods for identifying qubit loss can be integrated into the methods and systems of the present disclosure. An exemplary method for implementing an error correction code that accounts for atomic loss can include implementing multiple swap gates. Another exemplary method for implementing an error correction code that accounts for atomic loss can include a modified knock-knock protocol.

[0075] SWAP Gate Method—In some cases, identifying missing qubits in operation (110) of method (100) includes using multiple swap gates. In some cases, the swap gates in the multiple swap gates are implemented as multiple CNOT gates. In some cases, the method further includes measuring alternating atoms in the lattice, performing multiple SWAP gates to transfer data stored in the data qubits to the ancilla qubits, and measuring the swapped data qubits to identify one or more missing atoms. For example, measuring alternating atoms in the lattice may include measuring each or more ancilla qubits to arrive at syndrome information. For example, performing multiple SWAP gates to transfer data stored in the data qubits to the ancilla qubits may include performing a series of SWAP gates between the ancilla qubits and the data qubits. For example, measuring the swapped data qubits to identify one or more missing atoms may include measuring each or more data qubits and checking for unexpected errors.

[0076] In an exemplary method for detecting a missing qubit, the missing qubit may be identified using a series of swap gates. Figure 3A is a flowchart of an example of a method (300) for implementing an error correction code that accounts for atomic losses. Figure 3B is a diagram of an operation (340) of the method (300) for implementing an error correction code, which includes performing a series of SWAP gates between an ancillary qubit and a data qubit. The SWAP gates may transfer information between paired qubits. For example, the SWAP gates may transfer information from an ancillary qubit to a data qubit.

[0077] In some cases, alternating atoms in the lattice may be measured (ancilla qubits, e.g., the circular gray atoms). A SWAP gate is then performed to transfer the data stored in the data qubits to the ancilla qubits. As a result of this operation, the ancilla qubits from the previous round of syndrome extraction become the data qubits for the next round. Following the swap gate, the former data qubits may be measured to identify the missing atom. The missing atom may then be replaced.

[0078] The operation (310) of the method (300) may include resetting an ancilla qubit. In the case of a trapped atom quantum computing process, the resetting may include optically pumping one or more atoms to a particular state. The optical pumping may be an incoherent process. The particular state may be a qubit state, e.g., a |1> state or a |0> state. In some cases, the optical pumping includes periodically pumping the qubit from a first state to a second state. For example, the qubit may be coherently pumped from the first state to the second state. When the qubit is periodically pumped, the qubit may undergo repeated excitation and decay between an upper state and a lower state. For example, the optical pumping process may include a transition from a |1> state or a |0> state to an excited state. In one example, the optical pumping process includes: 1 From the nuclear spin states on the S0 manifold, 87 Rb, 87 Sr, 171 Yb etc. 3 P1 or 3 The optical pumping process may involve a transition to a state in the P0 manifold. For example, the optical pumping process may involve a Raman transition from a |1> state or a |0> state to a virtual state. For example, the optical pumping process may involve a Raman transition from a |1> state or a |0> state to a virtual state. 1 From the nuclear spin states in the S0 manifold, 3 P1 or 3The optical pumping may include a transition to a virtual state below the P0 manifold. The optical pumping may function to reset the qubit to either the |1> or |0> state.

[0079] The operations (320) of the method (300) may include sequentially implementing two-qubit gates between each ancilla qubit and its four neighbors. As described above, the error correction sequence may include two-qubit interactions. For example, a round of an error correction code may include performing a two-qubit operation between a data qubit and an ancilla qubit. The two-qubit operation may include part of a round of syndrome extraction. As described with respect to FIG. 2A, the two-qubit interaction between a qubit and a missing qubit may have the effect of a Pauli or identity operation on the qubit. In some cases, the two-qubit gates between a qubit and its four neighbors are each applied in series.

[0080] Operation 330 of method 300 may include measuring each or multiple ancilla qubits to arrive at syndrome information. When each ancilla qubit is measured, some syndrome data may have been lost due to a missing atom. In some cases, error correction methods that do not account for atom loss may stop here. When an ancilla is measured, some syndrome data may have been lost. For example, syndrome data may have been lost due to the loss of a syndrome qubit, e.g., the loss of a syndrome atom. The syndrome data qubits identified at this stage may be replaced in the atom replacement process described herein.

[0081] In some cases, a measurement operation herein is a state-selective measurement operation. Exemplary measurement operations are described herein with respect to the "Measurement Operation" section.

[0082] Operation 340 of method 300 may include performing a series of swap gates between an ancilla qubit and a data qubit. The SWAP gates may generally be implemented as a series of CNOT gates rather than as a standard SWAP. The series of CNOT gates may include three CNOT gates. In some cases, the swap may be a single site move; however, in some cases, a larger distance SWAP may be performed. The SWAP gate may differ from physically moving qubits. For example, the SWAP gate may change the information carried by the swapped qubit without moving the atoms themselves.

[0083] FIG. 3B is a diagram of an operation (340) of a method (300) for implementing an error-correcting code, which includes performing a series of SWAP gates between an ancilla qubit and a data qubit. The SWAP gate can transfer information between paired qubits. For example, a SWAP gate may transfer information from an ancilla qubit to a data qubit. As shown, multiple SWAP gates may be implemented across the array. For example, the entire array may undergo a series of SWAP operations between adjacent qubits. For example, a portion of the array may undergo a series of SWAP operations between adjacent qubits. As a result of this operation, the ancilla qubit from a previous round of syndrome extraction becomes the data qubit for a subsequent round of syndrome extraction.

[0084] The operations 350 of the method 300 may include measuring each or a plurality of data qubits and checking for unexpected errors. In some cases, the measurement operations herein are state-selective measurement operations. Exemplary measurement operations are described herein with respect to the "Measurement Operation" section.

[0085] Operations 310-340 may be repeated. After the sequence, the error correction code may have measured a significant portion or all of the qubits in the array without measuring the data itself. During operations 310-340, missing qubits may be identified. The procedure may track which sites have missing atoms so that the algorithm may be updated. The qubit flagging operation may include an embodiment, variation, or example of operation 140 of method 100.

[0086] Operation 350 of method 300 may include qubit replacement. For example, one or more atoms may be replaced once one or more unexpected errors are identified. The qubit replacement operation may include an embodiment, variation, or example of operation 120 of method 100. The qubit replacement operation may include an embodiment, variation, or example of operation 120 of method 100. Methods for replacing qubits are described herein with respect to the section "Qubit Replacement."

[0087] In some cases, the qubits may be reimplemented in a circuit. The qubit reimplementation operation may include an embodiment, variation, or example of operation (130) of method (100). In some cases, operation (330) includes using a decoder algorithm. The decoder algorithm may take a graph and determine a set of edges. In some cases, operation (130) includes updating a matching graph passed to the decoder algorithm based on a predicted probability distribution of the missing qubit replaced in operation (120) before implementing the decoder algorithm. Methods for updating the decoder algorithm are described herein with respect to the "Modifying the Decoding Algorithm" section.

[0088] The operations (310-350) may be repeated. After the sequence, the error correcting code may have measured a significant portion or all of the qubits in the array without measuring the data itself.

[0089] Operation 350 of method 300 may include updating the algorithm to flag measurements that are suspect due to identified atom losses. Updating the decoding algorithm may include an embodiment, variation, or example of operation 220 of method 200.

[0090] Modified Knock-Knoc—In some cases, identifying in operation 110 of method 100 includes using a modified Knock-Knoc protocol. The Knock-Knoc protocol may generally include a two-qubit operation and some rotation operation. For example, the two-qubit operation in the Knock-Knoc protocol may include a CNOT operation, such as the CNOT-X-CNOT described herein. For example, the two-qubit operation in the Knock-Knoc protocol may include a controlled-Z (CZ) operation, as described herein.

[0091] For example, a modified knock-knock protocol may include providing a first atom to be probed using a second atom, where the second atom is an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on the Rydberg interaction; and rotating the second qubit back to the computational basis and performing a measurement.

[0092] In another exemplary method for detecting a missing qubit, a variation of the knock-knock protocol is used to identify the missing qubit. The knock-knock protocol may include two consecutive CNOT gates performed on a paired qubit with an X gate in the middle of the probed qubit. For example, the sequence may include CNOT-X-CNOT. A CNOT gate is a two-qubit operation, where the first qubit is typically called the control qubit and the second qubit is called the target qubit, which is the probed qubit. The CNOT gate flips the target state when the control is in state |1>. When the control is in state |0>, the target state remains unchanged. In the case of probe qubit loss, the CNOT does nothing if the target qubit is not present, regardless of whether it starts in |0> or |1>. When the target qubit is present, the CNOT gate flips the target state when the control is in state |1>. The X gate is a bit-flip gate applied to the probed qubit. If the probed qubit is missing, the operation does nothing. If the probed qubit is present, the state is flipped. Thus, the CNOT-X-CNOT sequence functions like a two-qubit gate, controlled by the presence of the qubits rather than their states. In a Rydberg variant on the procedure, the CNOT gate may include a Rydberg interaction. In some cases, the Rydberg interaction may make the arrangement more compact.

[0093] In some cases, knock-knock protocols can be used to detect loss without destroying data. See, for example, Stricker, R., et al., Deterministic Correction of Atom Loss, arXiv:2002.09532v1[quant-ph]21 Feb 2020, which is incorporated herein by reference in its entirety and available at https: / / arxiv.org / pdf / 2002.09532.pdf.

[0094] 4 is a flowchart of an example method 400 for implementing an error correction code that accounts for atomic losses. The method 400 may include a modified knock-knock protocol. The systems and methods disclosed herein propose a condensed version of the knock-knock protocol. As disclosed below, the presence of one atom (atom A) can be probed using an ancilla (atom B). In some cases, one or both of atoms A and B are qubits.

[0095] The operation (410) of the method (400) may include preparing a qubit B in a |+> state. The operation step of preparing the qubit in |+> may include optical pumping. The preparing may include optically pumping one or more atoms to a particular state. The optical pumping may be an incoherent process. The particular state may be a qubit state, e.g., a |+> state or a |-> state. In some cases, the optical pumping includes cyclically pumping the qubit from a first state to a second state. In one example, the qubit may be coherently pumped from the first state to the second state. When the qubit is cyclically pumped, the qubit may undergo repeated excitation and decay between an upper state and a lower state. For example, the optical pumping process may include a transition from a |1> state or a |0> state (similarly, from a |+> state to a |-> state) to an excited state. For example, the optical pumping process may include: 1 From the nuclear spin states on the S0 manifold, 87 Rb, 87 Sr, 171 Yb etc. 3 P1 or 3 The optical pumping process may involve a transition to a state in the P0 manifold. For example, the optical pumping process may involve a Raman transition from the |1> or |0> state (similarly from the |+> state to the |-> state) to a virtual state. For example, the optical pumping process may involve a Raman transition from the |1> or |0> state to a virtual state. 1 From the nuclear spin states in the S0 manifold, 3 P1 or 3This may include transitions to virtual states below the P0 manifold. Optical pumping may serve to reset the qubit to the |1> or |0> state (similarly from the |+> state to the |-> state).

[0096] Operation 420 of method 400 may include applying a modified controlled-Z gate between atoms A and B based on the Rydberg interaction. In some cases, in 410, rather than exciting only the |1> state of atom A to the Rydberg level, both qubit states of atom A may be excited to the Rydberg level. Thus, qubit B may acquire a phase that is conditional on the presence of atom A, rather than a particular state of qubit A.

[0097] Operation 430 of method 400 may include rotating qubit B back into a computational basis and performing a measurement, where one result indicates atom A is present and the other result indicates atom A is missing.

[0098] In some cases, a measurement operation herein is a state-selective measurement operation. Exemplary measurement operations are described herein under the heading "Measurement Operation."

[0099] The operation (440) of the method (400) may include determining whether atom A is present based at least in part on the measurement.

[0100] Optical Pumping—The systems and methods of the present disclosure may use state preparation or reset techniques to set or reset the initial state of a qubit. The qubit state of the present disclosure may be set by optical pumping. Optical pumping may be affected by an optical pumping unit. The optical pumping unit may be configured to emit light to optically pump an atomic state from an equilibrium distribution of atomic states to a non-equilibrium atomic state. For example, the optical pumping unit may be configured to emit light to optically pump an atomic state from an equilibrium distribution of atomic states to a single pure atomic state. The optical pumping unit may be configured to emit light to optically pump an atom to the ground atomic state or any other atomic state. The optical pumping unit may be configured to optically pump an atom between any two atomic states.

[0101] The optical pumping unit may include one or more light sources (such as any of the light sources described herein) configured to emit light at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1000 nm, 1010 nm, 1020 nm, 1030 nm, 1040 nm, 1050 nm, 1060 nm, 1070 nm, 1080 nm, 1090 nm, 1100 nm, 1110 nm, 1120 nm, 1130 nm, 1140 nm, 1150 nm, 1160 nm, 1 The wavelength may include one or more wavelengths of 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light has a maximum wavelength of approximately 1,000nm, 990nm, 980nm, 970nm, 960nm, 950nm, 940nm, 930nm, 920nm, 910nm, 900nm, 890nm, 880nm, 870nm, 860nm, 850nm, 840nm, 830nm, 820nm, 810nm, 800nm, 790nm, 780nm, 770nm, 760nm, 750nm, 740nm, 730nm, 720nm, 710nm, 700nm, 690nm The light may include one or more wavelengths below 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or lower. The light may include one or more wavelengths within a range defined by any two of the foregoing values.For example, the light may include one or more wavelengths in the ranges of 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm.

[0102] For example, the optical pumping process 1 From the nuclear spin states in the S0 manifold, 87 Rb, 87 Sr, 171 Yb etc. 3 P1 or 3 The light may include a transition to a state in the P0 manifold, 1 From the nuclear spin states in the S0 manifold, 87 Rb, 87 Sr, 171 Yb etc. 3 P1 or 3 The optical pumping process can be tuned to the transition frequency to a state in the P manifold. For example, 1 From the nuclear spin states on the S0 manifold, 3 P1 or 3 The light may include a transition to a virtual state below P0, 1 Nuclear spin states on the S0 manifold, 87 Rb, 87 Sr, 171 Yb etc. 3 P1 or 3 It can be tuned to the transition frequency between virtual states below the P0 manifold.

[0103] Qubit Replacement The disclosed methods and systems may replace qubits into quantum circuits after vacancies have been identified. In some cases, the qubits are atomic qubits. In some cases, the qubits are atoms trapped in spatially distinct optical trapping sites. While the examples presented herein may enumerate qubits that include neutral atoms, however, the disclosed methods and systems may be combined with various types of qubits.

[0104] In some cases, the disclosed methods and systems may replace atoms into the array after vacancies are identified. In some cases, individual atoms may be moved from a filled site to an empty site using optical tweezers. The optical tweezers may be formed from crossed acousto-optical deflectors. In some cases, the array of optical trapping sites may include a first array and a second array different from the first array. The first array may include a science array. The second array may include a reservoir array. In some cases, the first array and the second array are physically separated by a distance. In some cases, the first array and the second array are subsets of a single continuous array. To fill vacancies in the array, the atom replacement operation may include moving atoms from a second array, such as from a reservoir array to a science array, into the first array. In some cases, atoms are transferred from a second array to a first array and vacancies are created in the second array, which can be replenished from a reservoir optical trap.

[0105] The atom displacement operation can be affected by one or more atom transfer units. As disclosed herein, the systems and methods disclosed herein can use multiple arrays of optical traps (e.g., a science array, a reservoir array, an intermediate array, etc.). In some cases, the science array is different from the reservoir array. In some cases, the science array is spatially different from the reservoir array. For example, the science array can be physically separated from the science array.

[0106] The physical separation of the science array and the reservoir array can be useful in at least some respects. For example, if the science array and the reservoir array are physically separate, the reservoir array can be more easily spatially separated from the science region. This can allow loading of the reservoir array without disturbing the atoms in the science region while the reservoir region is being loaded. For example, disturbances can arise from unwanted scattering during transfer, unwanted optical shifts, etc. In some cases, an optical system separate from trap excitation can be used to transfer atoms from a first array to a second array disclosed herein. For example, the reservoir array can be loaded from a separate optical potential or array, which could disturb atoms in the science region if the reservoir array and the science array were too close. Using separate optical systems to generate the two arrays can be useful to separate the science array and the reservoir array. A separate (e.g., third) optical system can be used to further isolate the arrays for atom transfer.

[0107] In some examples, the techniques may be combined with methods for stochastic, deterministic, or quasi-deterministic loading of optical or other traps, such as those disclosed herein. In some examples, atoms in the science region may or may not rearrange when the science array is replenished. In some examples, atoms may be moved between sites by optical tweezers. In some examples, atoms may be moved between sites by optical lattices. In some examples, atoms may be moved between sites by tunneling / hopping between sites. In some examples, atoms may be moved between sites by autonomous stabilization techniques.

[0108] In some cases, atom replacement is performed using one or both of a moving optical trap or optical tweezers. In some cases, optical tweezers can be used to move single atoms (e.g., pick and place) or subsets of atoms between or within arrays. In some cases, a moving optical trap can be used to translate or compress an array. A moving optical trap can implement a tone to sweep atoms from one location to another. An atom transfer unit can be configured to transfer one or more replacement atoms from one or more atom reservoirs to one or more optical trapping sites. For example, one or more atom transfer units can comprise one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs).

[0109] The optical trapping systems disclosed herein may include one or more atom rearrangement units configured to impart an altered spatial arrangement of a plurality of atoms trapped by the optical trapping moiety based on one or more images acquired by an imaging unit. The optical trapping unit may include any number of atom rearrangement units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atom rearrangement units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units. In some cases, the science array is associated with a first spatial light modulator and the reservoir array is associated with a second spatial light modulator.

[0110] The atom rearrangement unit may be configured to change the spatial arrangement of the plurality of optical trapping sites to obtain an increased filling factor. The filling factor may be defined as the ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total number of computationally active optical trapping sites available in the optical trapping unit or a portion of the optical trapping unit. For example, the initial loading of atoms in the computationally active optical trapping sites may result in a filling factor of 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information acquired by the imaging unit, the atomic rearrangement units may achieve a filling rate of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atomic rearrangement units may achieve a packing fraction 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 atomic rearrangement units may achieve a packing fraction within a range defined by any two of the foregoing values.

[0111] Modification of the decoding algorithm The systems and methods of the present disclosure may be used in connection with error correction methodologies for quantum computing systems. The error correction schemes (e.g., error correction code implementations) of the present disclosure may include a decoder and an error correction code. The decoder can decode which errors occurred on which qubits. Once identified, these errors can be tracked, and the information can be used to correct any subsequent measurements using classical control software. The methods for updating the decoder described herein may be independent of the type of atom, the type of qubit, the type of error correction code, or the specific decoder used in the error correction code. In some cases, the error correction code may be from the class of stabilizer codes. If a two-qubit gate operation affects the identification or Pauli operation, the matching graph passed to the decoder may be updated as described herein.

[0112] The systems and methods of the present disclosure can be used with various error correction codes. The error correction code can be a Shor-style code. For example, in a Shor-style code, repeated rounds of syndrome extraction can be implemented to overcome read errors and build confidence in the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors occurred and what corrections need to be applied.

[0113] The disclosed systems and methods can be used with various stabilizer codes. The stabilizer code can be an error-correcting code using a stabilizer. The stabilizer code can be a class of error-correcting codes. The class of stabilizer codes can include toric codes, surface codes, etc. By repeatedly measuring a quantum system using a complete set of commuting stabilizers, the system can be forced into a simultaneous and unique eigenstate of all stabilizers. The stabilizers can be measured without perturbing the system, and if the measurement result changes, this corresponds to one or more qubit errors, and the quantum state is projected into a different stabilizer eigenstate by the measurement.

[0114] Error-correcting codes may include topological codes. The class of topological codes may overlap with the class of stabilizer codes. Topological codes may include surface codes, color codes, toric codes, etc. Topological codes may be referred to as homological codes. Topological codes may include arrays or lattices of qubits arranged on a surface (or higher-dimensional structure). The systems and methods of the present disclosure may generally not change the underlying topology of the topological code.

[0115] The systems and methods disclosed herein can be used with various surface codes. Surface codes can be implemented as stabilizer codes. For example, in surface code literature, surface codes can include two types of qubits: data qubits and measurement qubits (e.g., ancilla qubits). Data qubits can contain information carried by a quantum circuit, and errors in these qubits can be corrected. Measurement qubits can be used to stabilize and manipulate the quantum states of data qubits. In surface codes, measurement qubits can include two types: measurement Z qubits and measurement X qubits. These two types of qubits can be referred to as Z syndrome qubits and X syndrome qubits, respectively. Measurement Z qubits can measure Z stabilizers. Measurement X qubits can measure X stabilizers. In some cases, surface codes can be implemented with decoders. In some cases, surface codes can address errors that occur during a surface code cycle, as long as they can identify errors that occur during each surface code cycle.

[0116] The systems and methods disclosed herein may use surface codes. The surface codes disclosed herein may include, for example, variations on the minimum weight perfect matching algorithm for decoding surface codes. However, many surface codes may be applicable to the systems and methods disclosed herein. A general description of surface codes is provided, for example, in Fowler, A. G., et al., "Surface codes: Towards Practical Large-scale Quantum Computation," arXiv:1208.0928[quant-ph], 4 Aug 2012, available at https: / / arxiv.org / pdf / 1208.0928.pdf, which is incorporated herein by reference in its entirety.

[0117] The systems and methods disclosed herein may be used with various color codes. The color codes may be implemented as stabilizer codes. For example, the color codes may include Steane codes, etc. The systems and methods disclosed herein may be used with various Shor-style codes, such as Bacon-Shor codes. Shor-style codes may be implemented as stabilizer codes. The systems and methods disclosed herein may be used with various qLDPC codes, such as hypergraph product codes. qLDPC codes may be implemented as stabilizer codes.

[0118] The systems and methods disclosed herein can be used with various decoders. The disclosed error correction schemes (e.g., error correction code implementations) can include a decoder and an error correction code. The decoder can decode which errors occurred on which qubits. Once identified, these errors can be tracked, and the information can be used to correct subsequent measurements using classical control software. Decoder algorithms can include, for example, minimum weight perfect matching, union find, tensor network decoder, belief propagation with order statistics decoder, maximum likelihood decoder, and lookup table decoder. The disclosed methods and systems can be integrated with variants of minimum weight perfect matching, such as sparse bloom and fusion blossom. The decoder can incorporate a matching graph. The disclosed systems and methods can update the matching graph passed to the decoder to account for lost qubits.

[0119] In some cases, qubit loss may involve modifications to surface code techniques that do not experience qubit loss errors. For example, an error-correcting code that does not account for qubit loss errors may unexpectedly continue to track a particular qubit changing from 1 to 0 or from 0 to 1. When a qubit is lost, there is no change in state; instead, there is no value to measure.

[0120] The modification of the decoding algorithm may be a sub-operation of the operation to re-implement the qubits in the circuit. The qubit re-implementation operation may include an embodiment, variation, or example of operation 130 of method 100. In some cases, the modification of the decoding algorithm may be performed after or during a re-implementation operation, such as operation 130 of method 100.

[0121] To enhance decoding algorithms, the systems and methods disclosed herein may update existing decoders to incorporate changes in error types. In some cases, the systems and methods disclosed herein may update the matching graph passed to the decoder. In some cases, the systems and methods disclosed herein may update the matching graph passed to a minimum-weight perfect matching decoder algorithm or any other decoder algorithm that incorporates a matching graph.

[0122] In some examples, to update the matching graph, each node in the graph corresponds to a change in the value of a particular stabilizer. Specific nodes are connected by edges that correspond to possible physical errors. These edges are weighted based on the likelihood that that particular error will occur. If an atom is lost and then replaced, the loss can be treated like a gate error, which occurs with a 50% probability. If data for an ancilla qubit is lost, this procedure can change slightly.

[0123] FIG. 5A is a diagram of a matching graph showing a missing ancillary qubit. The detector involved in the missing qubit can be updated. In the illustrated example graph, nodes are detectors, and edges connect these nodes. If a missing appendage is detected, the edge between its corresponding detectors is assigned a 50% probability. In the illustrated example, the number of detectors measuring the missing ancillary qubit is two. Hook errors are introduced depending on the induced noise. If an ancillary qubit is lost, the value of the syndrome data held by the missing ancillary qubit can be selected and inserted. The matching graph can then be updated so that the node with that particular syndrome bit is connected by an edge corresponding to a 50% probability of error. In the code, there is a feed forward of the probability of possible bit flips (not errors). The value of the missing atom is random. Therefore, ancillary qubit loss errors from this point can exist as gate errors that can be addressed by various types of surface codes.

[0124] FIG. 5B is a diagram of a matching graph showing a lost data qubit. If the lost data qubit is detected, all edges between appropriate detectors are assigned a 50% probability. If a data qubit is lost, all measured syndrome values ​​involving that data qubit are unreliable. In some cases, each data qubit is included in two, three, or four syndrome measurements. The number of detectors for a particular qubit may vary depending on the specific type of decoder. The methods and systems described herein can be adapted to any number of detectors specified in the decoder, as long as each edge between detectors involved in the lost qubit is updated. If a data qubit is lost, all associated syndrome values ​​are unreliable. In this case, inserting values ​​for measured syndrome bits may not be required. Instead, the matching graph may be expanded so that all nodes affected by each unreliable ancillary element are connected by a 50% error edge. Thus, data qubit loss errors from this point may exist as gate errors that can be addressed by various types of surface codes. In the illustrated example, four edges are shown in bold.

[0125] 5C is a flowchart of a method 500 for repackaging a qubit. In some cases, modifications to the decoding algorithm may be performed after or during a repackaging operation, such as operation 130 of method 100.

[0126] In operation 510, the method 500 may include using a decoder algorithm that takes the graph and determines a set of edges. In some cases, the decoder is a minimum weight exact matching decoder algorithm.

[0127] In operation 520, method 500 may include, prior to 510, updating the matching graph that is passed to the decoder algorithm based on a predicted probability distribution of the replaced lost qubit. The permutation operation may be, for example, operation 120 of method 100. In some cases, the matching graph is passed to a minimum-weight perfect matching decoder algorithm. The matching graph may be updated based on a predicted probability distribution of the replaced lost qubit. In some cases, the predicted probability distribution includes, if an ancilla qubit is lost, updating the matching graph so that nodes containing the ancilla qubit are connected by edges corresponding to the predicted probability distribution, and, if a data qubit is lost, updating the matching graph by assigning a predicted probability distribution to each node containing a data qubit. In some cases, each node containing an ancilla qubit is updated.

[0128] The operation (520) can be generalized to various decoders and error correction codes. If a data qubit is lost, it will introduce as many pairs of vertices as there are stabilizers involved. Each data qubit may participate in several parity checks, depending on the error correction code. In the example shown in FIG. 5B, the particular lost data qubit participated in four parity checks, which correspond to four ancillaries measured twice, resulting in four random edges.

[0129] Measurement Calculation In an operation (140) of the method (100) for error correction involving atom loss, measurements made while a qubit is missing can be flagged as unreliable. In some cases, the operation (140) includes flagging measurements taken during a time window that includes the time the qubit was missing as unreliable. For example, the time window can include a round of syndrome measurements. It is not necessary to know exactly which measurements were made while the atom was missing, only which set of measurements were made during the time window that includes the missing atom. The flagging operation can include performing measurements of one or more qubits before the measurement is flagged. The measurement can result in the emission of a photon. In some cases, the measurement can be state-selective. For example, the measurement can selectively probe either the |0> state or the |1> state. After a round of measurements, it can be known whether the measured atom is missing. In one example, measurements of an ancilla atom during an error correction protocol can indicate whether the ancilla atom is missing. An identification operation in (110) (e.g., a swap gate, a knock-knock protocol, etc.) can be performed to determine whether a qubit is missing in a set of qubits that includes a data qubit, without measuring the data qubit.

[0130] In the systems and methods of the present disclosure, the fact of a lost qubit may not be heralded immediately. For example, it may become clear that a qubit is lost after completing a round of syndrome extraction, rather than immediately after making a measurement indicating the lost qubit. Once the round of syndrome extraction is complete, it may become clear that there has been a qubit loss, and in order to proceed with the computation, it may be beneficial to flag a series of measurements taken during the time frame that includes the time the qubit was missing. Each of these measurements may be flagged as unreliable. In some cases, the series of times that include a flagged qubit may not be limited to the time that the qubit was deterministically lost. The series of times that include a flagged qubit may include at least the time that the qubit was lost.

[0131] Advantageously, qubit loss may be identified before the measurement of the data qubit (and after a round of syndrome extraction). Because the data qubit has not yet been measured, it may be possible to continue the quantum circuit after replacing the lost qubit. The circuit may be adapted to reacquire or restart the portion of the computation that implicates the lost qubit. As a result, the systems and methods disclosed herein may enable loss detection between cycles (or perhaps less frequently), rather than after every gate.

[0132] For example, an error correction code may be directed to updating the calculation to account for the error. In some cases, knowledge of the error may be required to implement the error correction code. However, in other cases, the error correction code may be modified to account for the missing data without explicit knowledge that a qubit is missing.

[0133] Measurement Operation—In some cases, performing a measurement can include imaging one or more atoms, e.g., atoms in an array. Imaging can be affected by exciting the atoms into an emitting state, e.g., a fluorescing state, a spontaneously emitting state, a state undergoing stimulated emission, or a phosphorescing state. Scattered photons after imaging can be collected in a camera or detector.

[0134] In some cases, it may be useful to selectively measure qubit states. For example, a measurement operation that selectively determines whether the state of a qubit is in state |0> or state |1>. Selectively measuring qubit state sites may be useful. For example, a measurement operation that selectively determines the state of a particular qubit in an array may be useful in quantum computing. It may be useful to measure qubits without causing atomic loss of the qubit from the array. Additionally, because error correction codes may involve measurements on qubits during error correction cycles, measurements that preserve atoms in the array may be useful to improve the efficiency of the error correction codes.

[0135] The measurement operation 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 qubit states (e.g., |0>, |1>) 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 the selectivity of the imaging transition for either or both qubit states. For example, the strength of the applied magnetic field may selectively shift the upper state of the imaging transition, the lower state of the imaging transition, or both. In some cases, the applied electric field may be a low magnetic field, e.g., a magnetic field of about 500 Gauss.

[0136] In some cases, performing the measurement operation includes exposing the qubit to electromagnetic energy configured to selectively drive the qubit from an initial state to an excited state in the presence of an applied magnetic field. In some cases, the selectivity of the transition to the excited state is based at least in part on the strength of the applied magnetic field. In some cases, determining the state of the qubit is based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

[0137] Mid Circuit Measurement—The methods and systems described herein may enable identification of a missing qubit without substantially shutting down the circuit. For example, if a set of measurements is flagged as unreliable, the circuit may continue with other measurements and return to retake the unreliable measurements. For example, if a data qubit is flagged as missing, the circuit may continue with other qubits while the atom is replaced, and the portion of the circuit containing that qubit may be reimplemented.

[0138] The methods and systems described herein may enable identification of missing qubits and replacement of missing qubits without measuring each or multiple data qubits. As described above, missing qubits may be identified in a round of syndrome measurements without measuring data qubits. Thus, atoms identified as missing may be replaced and the circuit may continue without measuring data qubits.

[0139] The methods and systems described herein may enable identification of lost qubits without loss of coherence of each or multiple data qubits. As described above, because the error correction code uses measurements of syndrome qubits to identify atomic losses, data qubits may not need to be measured during around error correction. The systems and methods of the present disclosure may enable continuous correction of atomic losses. The systems and methods of the present disclosure may enable correction of atomic loss intermediate circuits.

[0140] Decoder updates in neutral atomic mass calculations Qubits—In some cases, the qubits described herein are neutral atom qubits. In some cases, the neutral atoms include a Group 2 element. In some cases, the Group 2 element is strontium. In some cases, the neutral atoms include rubidium or cesium. In some cases, the neutral atoms include ytterbium. For example, the first plurality of qubits or the second plurality of qubits can include neutral atoms. 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 can 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 include rare earth atoms (e.g., lanthanide series atoms (e.g., ytterbium, neodymium, lanthanum, erbium, etc.), alkali atoms (e.g., sodium, potassium, rubidium, cesium, etc.), alkaline earth atoms (e.g., calcium, strontium (e.g., strontium-87 atom), etc.), or any combination thereof.

[0141] In some cases, the qubits described herein may include nuclear spin qubits. The qubit states (e.g., |0>, |1>) may include nuclear spin states within a manifold of electronic states. For example, the qubit state may be a nuclear spin state on the ground state manifold. In some cases, the qubit state may include a nuclear spin state on the ground state manifold of a Group 2 element or an analog of a Group 2 element. The analog of a Group 2 element may include two valence electrons. In some cases, the ground state manifold may be 1 In some cases, the ground state manifold is 87 Rb, 87 Sr, 171 Yb etc. 1 It is in S0 state.

[0142] Qubit States—In some cases, the qubits described herein can include a first atomic state and a second atomic state. The first atomic state can include the state of a first single qubit. The second atomic state can include the state of a second single qubit. The first atomic state or the second atomic state can be elevated in energy relative to the ground atomic state of the atom, for example, within a manifold of excited states. The first atomic state or the second atomic state can be within the manifold of ground states.

[0143] The first atomic state may include a first hyperfine electronic state, and the second atomic state may include a second hyperfine electronic state different from the first hyperfine electronic state. For example, the first atomic state and the second atomic state may include a first hyperfine state and a second hyperfine state on a multiplet manifold, such as a triplet manifold, a singlet manifold, etc. The first atomic state and the second atomic state may each be: 3 P1, 3 P2, 1 The first and second atomic states may include a first hyperfine state and a second hyperfine state, such as on the S0 manifold. The first and second atomic states may be strontium-87, strontium-87, and strontium-87, respectively. 3 P1 manifold, strontium-87 3 P2, strontium-87 1 S0, ytterbium-171 3 P1, ytterbium-171 3 P2, ytterbium-171 1 of any atom described herein, such as S0 3 P1, 3 P2, 1 It may include a first hyperfine state and a second hyperfine state on the S0 manifold.

[0144] In some cases, the first atomic state and the second atomic state are first and second hyperfine states of the first electronic state. Optical excitation can be applied between the first electronic state and the second electronic state. The optical excitation can excite the first hyperfine state and / or the second hyperfine state to the second electronic state. The single qubit transition can include a two-photon transition between two hyperfine states within the first electronic state, using the second electronic state as an intermediate state. To drive the single qubit transition, a pair of frequencies detuned from the single-photon transition to the intermediate state can be applied to drive the two-photon transition. In some cases, the first and second hyperfine states are hyperfine states of the ground electronic state. The ground electronic state may not decay to a lower electronic state by spontaneous or stimulated emission. The hyperfine state can include a nuclear spin state.

[0145] In some cases, the hyperfine state is strontium-87 1 S0 or Ytterbium-171 1 The qubit transition involves the nuclear spin states of the S0 manifold, and the strontium-87 1 S0 or Ytterbium-171 1 One or both of the two nuclear spin states of S0 can be 3 P2 or 3 Detuned from the P1 manifold, or 3 P2 or 3 In some cases, the single-qubit transition is driven by strontium-87 1 S0 or Ytterbium-171 1 is a two-photon Raman transition between the nuclear spin states of S0, 3 P2 or 3 Detuned from the P1 manifold, or 3 P2 or 3via detuned states within the P1 manifold. In some cases, the nuclear spin state may be a Stark shifted nuclear spin state. The Stark shift may be optically driven. The optical Stark shift may be driven off-resonance by any, all, or a combination of single-qubit transitions, two-qubit transitions, shelving transitions, imaging transitions, etc.

[0146] In some cases, the hyperfine state includes a nuclear spin state of ytterbium.

[0147] The first atomic state may include a first nuclear spin state, and the second atomic state may include a second nuclear spin state different from the first nuclear spin state. The first and second atomic states may include, respectively, the first and second nuclear spin states of a quadrupolar nucleus. The first and second atomic states may include, respectively, the first and second nuclear spin states 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 include, respectively, the first and second nuclear spin states of any atom described herein, such as the first and second spin states of strontium-87.

[0148] For first and second nuclear spin states associated with nuclei containing spins greater than 1 / 2 (such as spin-1, spin-3 / 2, spin-2, spin-5 / 2, spin-3, spin-7 / 2, spin-4, or spin-9 / 2 nuclei), transitions between the first and second nuclear spin states may involve transitions between other spin states on the nuclear spin manifold. For example, in a spin-9 / 2 nucleus in the presence of a uniform magnetic field, all nuclear spin levels may be spaced apart by equal energy. Thus, for example, m N =9 / 2 spin state to m NTransitions designed to move atoms to the spin state of m = 7 / 2 (such as Raman transitions) also N =7 / 2 to m N =5 / 2, m N =5 / 2 to m N =3 / 2, m N =3 / 2 to m N =1 / 2, m N =1 / 2 to m N =-1 / 2, m N =-1 / 2 to m N =-3 / 2, m N =-3 / 2 to m N =-5 / 2, m N =-5 / 2 to m N =-7 / 2 to m N =-7 / 2 to m N =-9 / 2, where mN is the nuclear spin state. N =9 / 2 spin state to m N Transitions designed to move atoms to the =5 / 2 spin state (such as Raman transitions) also N =7 / 2 to m N =3 / 2, m N =5 / 2 to m N =1 / 2, m N =3 / 2 to m N =-1 / 2, m N =1 / 2 to m N =-3 / 2, m N =-1 / 2 to m N =-5 / 2, m N =-3 / 2 to m N =-7 / 2 to m N =-5 / 2 to m N =-9 / 2. Therefore, such transitions may not be selective in inducing transitions between specific spin states on the nuclear spin manifold.

[0149] Alternatively, it may be desirable to selectively effect transitions between specific first and second spin states on the nuclear spin manifold. This can be achieved by providing light from a light source that produces an AC Stark shift, which accompanies the transition between the desired first and second nuclear spin states and moves adjacent nuclear spin states off resonance. For example, m N =-9 / 2 and m N If transitions from a first nuclear spin state and a second nuclear spin state with .mu.m and .mu.m are desired, the light N =-5 / 2 spin state, which leads to an AC Stark shift of m N =-7 / 2 state and m N =-5 / 2 state can be significantly reduced. N =-9 / 2 and m N If transitions from a first nuclear spin state and a second nuclear spin state with .mu.m and .mu.m are desired, the light N =-1 / 2 spin state, which leads to an AC Stark shift of m N =-5 / 2 state and m N This effectively creates a two-level subsystem within the nuclear spin manifold that is decoupled from the rest of the nuclear spin manifold, which can greatly simplify the dynamics of the qubit system. Nuclear spin states near the edge of the nuclear spin manifold (e.g., for spin-9 / 2 nuclei, m N =-9 / 2 and m N =-7 / 2, m N =7 / 2 and m N =9 / 2, m N =-9 / 2 and m N =-5 / 2, or m N =5 / 2 and m N = 9 / 2) can be advantageous, since only one AC Stark shift is required. Alternatively, nuclear spin states further from the edge of the nuclear spin manifold (e.g., m N =-5 / 2 and m N =-3 / 2, or m N =-5 / 2 and m N=-1 / 2) can be used and two AC Stark shifts can be performed (e.g., m N =-7 / 2 and m N =-1 / 2, or m N =-9 / 2 and m N =3 / 2).

[0150] Stark shifting of the nuclear spin manifold can shift adjacent nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and the second electronic state or a state detuned therefrom. Stark shifting can reduce leakage from the first and second nuclear spin states to other states within the nuclear spin manifold. Stark shifting can be achievable up to 100 kHz with beam powers less than 10 mW. Frequency selectivity of the upper states can reduce scattering from imperfect polarization control. 3 The separation of different angular momentum states in the P1 manifold can be many gigahertz from single-qubit and two-qubit gate lights. Leakage to other states in the nuclear spin manifold can lead to decoherence. The Rabi frequency of the two-qubit transition (e.g., how fast the transition can be driven) can be faster than the decoherence rate. Scattering from intermediate states in the two-qubit transition can cause decoherence. Detuning from intermediate states can improve the fidelity of the two-qubit transition.

[0151] Qubits based on the nuclear spin state of the electronic ground state are based on long-lived metastable excited electronic states (such as strontium-87 or ytterbium-171). 3 This may allow atoms to be selectively transitioned into such states to reduce crosstalk or improve gating or detection fidelity. Such a storage or shelving process may be atom-selective using the SLM or AOD described herein. Shelving transitions may be achieved using the Strontium-87 or Ytterbium-171 states. 1 From the S0 state, strontium-87 or ytterbium-1713 P0 state or 3 This may include a transition to the P2 state.

[0152] Clock transitions (also referred to herein as "shelving transitions" or "storage transitions") can be qubit-state selective. The upper states of clock transitions can have very long natural lifetimes, e.g., greater than 1 second. The linewidth of the clock transition can be much narrower than the energy spacing of the qubits. This can enable direct spectral resolution. An ensemble can be moved from one of the qubit states to the clock state. This allows for separate readout of individual qubit states by first moving the ensemble from one qubit state to the clock state and imaging onto the qubit, then moving the ensemble from the clock state to the ground state and imaging again. In some cases, magic wavelength transitions are used to drive the clock transitions.

[0153] The shelving clock light can be atom-selective or non-atom-selective. In some cases, the clock transition is globally applied (e.g., non-atom-selective). Globally applied clock transitions may not include directing or structuring the light through a microscope objective. In some cases, the clock transition is atom-selective. Atom-selective clock transitions can improve gate fidelity by minimizing crosstalk. For example, to reduce crosstalk within atoms, atoms can be shelved to a clock state that is not affected by light. This can reduce crosstalk between adjacent qubits during the transition. To achieve atom-selective clock transitions, the light can pass through one or more microscope objectives and be structured with one or more spatial light modulators, digital micromirror devices, crossed acousto-optic deflectors, etc.

[0154] Multi-qubit gates—The entanglement units herein may enable two-qubit gates and multi-qubit gates. For example, an entanglement excitation may be configured to perform an entanglement operation between a first qubit and another qubit different from the first qubit. The entanglement unit may be configured to quantum mechanically entangle at least a first atom of the plurality of atoms with at least a second atom of the plurality of atoms. The first atom or the second atom may be in a superposition state upon quantum mechanical entanglement. Alternatively or additionally, the first atom or the second atom may not be in a superposition state upon 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 unit may be configured to quantum mechanically entangle any number of atoms as described herein.

[0155] The entanglement unit 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. A multi-qubit unit may include a two-qubit unit, a three-qubit unit, a four-qubit unit, or an n-qubit unit, where n may be 5, 6, 7, 8, 9, 10, or more. For example, a two-qubit unit may include a first atom quantum mechanically entangled with a second atom, a three-qubit unit may include a first atom quantum mechanically entangled with a second atom and a third atom, a four-qubit unit may include a first atom quantum mechanically entangled with a second atom, a third atom, and a fourth atom, etc. The first atom, the second atom, the third atom, or the fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or additionally, the first atom, the second atom, the third atom, or the fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first atom, the second atom, the third atom, and the 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.

[0156] The entanglement unit may include one or more Rydberg units. The Rydberg units may be configured to electronically excite at least a first atom into a Rydberg state or a superposition of a Rydberg state and a lower-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg units may be configured to induce one or more quantum mechanical entanglements between the Rydberg atom or dressed Rydberg atom and at least a second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (μm), 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9 μm, 10 μm, or more from the Rydberg atom or dressed Rydberg atom. The second atom can be located at a distance of up to about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atom or coated Rydberg atom. The second atom can be located at a distance from the Rydberg atom or coated Rydberg atom within a range defined by any two of the above values. The Rydberg unit can be configured to allow the Rydberg atom or coated Rydberg atom to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg unit can be configured to induce the Rydberg atom or coated Rydberg atom to relax to a lower-energy atomic state. The Rydberg unit can be configured to drive the Rydberg atom or coated Rydberg atom to a lower-energy atomic state. For example, the Rydberg unit may be configured to apply electromagnetic radiation (such as RF radiation or light radiation) to drive the Rydberg atoms or coated Rydberg atoms into a lower energy atomic state.The Rydberg unit may be configured to induce quantum mechanical entanglement between any number of the plurality of atoms.

[0157] The Rydberg unit can include one or more light sources (such as any light source described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths can be selected to correspond to wavelengths that form Rydberg atoms or coated Rydberg atoms. For example, the light can include one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. The light may include one or more wavelengths up to about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. The light may include one or more wavelengths within a range defined by any two of the above values. For example, the light may include one or more wavelengths within the range of 300 nm to 400 nm.

[0158] The Rydberg unit can be configured to induce two-photon transitions to generate entanglement. The Rydberg unit can be configured to induce two-photon transitions to generate entanglement between two atoms. The Rydberg unit can be configured to selectively induce two-photon transitions to generate entanglement between two atoms. For example, the Rydberg unit can be configured to direct electromagnetic energy (such as light energy) to specific optical trapping sites and selectively induce two-photon transitions to generate entanglement between two atoms. Two atoms can be trapped in nearby optical trapping sites. For example, two atoms can be trapped in adjacent optical trapping sites. The two-photon transitions can be induced using a first light from a first light source and a second light from a second light source, respectively. The first light source and the second light source can each include any light source described herein (such as any laser described herein). The first light source can be the same as or similar to the light source used to perform the single qubit operations described herein. Alternatively, different light sources may be used to perform single qubit operations and induce two-photon transitions to generate entanglement. The first light source may emit light containing one or more wavelengths in the visible region of the optical spectrum (e.g., within a range of 400 nm to 800 nm, or within a range of 650 nm to 700 nm). The second light source may emit light containing one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., within a range of 200 nm to 400 nm, or within a range of 300 nm to 350 nm). The first light source and the second light source may emit light having substantially equal and opposite spatially dependent frequency shifts.

[0159] The Rydberg atom or coated Rydberg atom may include a Rydberg state in which atomic interactions with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) may be strong enough to enable multi-qubit operations. The Rydberg state may include a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or more. The Rydberg state may include a principal quantum number of up to about 100, 90, 80, 70, 60, 50, or less. The Rydberg state may include a principal quantum number within a range defined by any two of the aforementioned values. The Rydberg state may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift the energy levels of the atoms.

[0160] State-selective excitation of atoms to Rydberg levels can enable the performance of multi-qubit operations. Multi-qubit operations can include two-qubit operations, three-qubit operations, or n-qubit operations (where n is 4, 5, 6, 7, 8, 9, 10, or more). Two-photon transitions can also drive atoms to the ground state ( 1 S0 ground state) to the Rydberg state (n 3 The two-photon transition can be performed using a first laser source and a second laser source, as described herein. The first laser source can emit π-polarized light, which may not change the projection of the atomic angular momentum along the magnetic field. The second laser can emit circularly polarized light, which may not change the projection of the atomic angular momentum along the magnetic field by more than one unit. This polarization can be used to excite the first and second qubit levels to the Rydberg level. However, because the Rydberg level is more sensitive to the magnetic field than the ground state, large splitting (e.g., on the order of 100 MHz) can be easily obtained. This spectral selectivity can enable state-selective excitation to the Rydberg level.

[0161] Multi-qubit operations (such as two-qubit, three-qubit, and four-qubit operations) can rely on energy shifts of levels due to van der Waals interactions as described herein. Such shifts can either prevent the excitation of one atom from being conditional on the state of the other atom, or alter the coherent dynamics of the excitations of a two-atom system to perform two-qubit operations. In some cases, "dressed states" can be generated under continuous drive to achieve two-qubit operations without requiring full excitation to the Rydberg levels (e.g., as described at www.arxiv.org / abs / 1605.05207, which is incorporated herein by reference in its entirety for all purposes).

[0162] One-qubit gates, two-qubit gates, and multi-qubit gates can be implemented by one or more nonclassical computation units. The nonclassical computation units can be configured to perform one-qubit gate operations, two-qubit gate operations, multi-qubit gate operations, and sequences and combinations thereof to perform nonclassical computations. The nonclassical computation units can include electromagnetic delivery units, such as any of the electromagnetic delivery units disclosed herein. The electromagnetic delivery units disclosed herein for cooling and trapping can be the same electromagnetic delivery units used for qubit gate operations or different electromagnetic delivery units. The electromagnetic energy can include one or more pulses, pulse sequences, or optical waveforms. The nonclassical computation units can include one or more entanglement units disclosed herein, one or more Rydberg units disclosed herein, or both. In some cases, the Rydberg units disclosed herein are an example of an entanglement unit disclosed herein that uses Rydberg excitations to generate entanglement and perform two-qubit or multi-qubit gate operations.

[0163] The non-classical computation can be configured to provide a pulse, pulse sequence, or optical waveform to perform the non-classical computation. The pulse, pulse sequence, or optical waveform can include any number of pulses, pulse sequences, or optical waveforms. For example, the pulse, pulse sequence, or optical waveform can include at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, or more pulses or subwaveforms. A pulse, pulse sequence, or optical waveform may include up to about 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 pulses or subwaveforms. A pulse, pulse sequence, or optical waveform may include a number of pulses or subwaveforms within a range defined by any two of the foregoing values. Each pulse of a pulse sequence may include any pulse shape, such as any pulse shape described herein.

[0164] Pulses, pulse sequences, or optical waveforms may be configured to reduce the time required to perform multi-qubit operations as described herein (e.g., with respect to Example 3). For example, a pulse sequence may include a time of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (μs), 2 μs, 3 μs, 4 μs, 5 μs, 6 μs, 7 μs, 8 μs, 9 μs, 10 μs, 20 μs, 30 μs, 40 μs, 50 μs, 60 μs, 70 μs, 80 μs, 90 μs, 100 μs, or more. A pulse sequence can include a time period of at most about 100 μs, 90 μs, 80 μs, 70 μs, 60 μs, 50 μs, 40 μs, 30 μs, 20 μs, 10 μs, 9 μs, 8 μs, 7 μs, 6 μs, 5 μs, 4 μs, 3 μs, 2 μs, 1 μs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. A pulse sequence can include a time period within a range defined by any two of the foregoing values.

[0165] The pulses, pulse sequences, or optical waveforms, as described herein, may be configured to increase the fidelity of multi-qubit operations. For example, the pulses, pulse sequences, or optical waveforms may be 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, 0.9 ...92, 0.999993, 0.999994, 0.999995, 0.999996, 0.999997, 0.999998, 0.999999, or higher fidelity. The pulse sequence is approximately 0.999999, 0.999998, 0.999997, 0.999996, 0.999995, 0.999994, 0.999993, 0.999992, 0.999991, 0.99999, 0.99998, 0.99997, 0.99996, 0.99995, 0.99994, 0.99993, 0.99992, 0.99991, 0.9999, 0.9998, 0.9997, 0 0.99, 0.996, 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 sequence may enable multi-qubit operations with a fidelity that is within a range defined by any two of the above values.

[0166] The pulses, pulse sequences, or optical waveforms may enable multi-qubit operations to be performed on non-adiabatic time scales while maintaining effectively adiabatic dynamics. For example, the pulse sequences may include one or more of: shortcut-to-adiabatic (STA) pulse sequences, transition-free quantum drive (TQD) pulse sequences, superadiabatic pulse sequences, anti-adiabatic drive pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamilitonian (Wah Wah) pulse sequences.For example, pulse sequences have been 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 A) 91, 012337 (2015); A. Mitra et al., "Robust Molmer-Sorenson Gate for Neutral Atoms Using Rapid Adiabatic Rydberg Dressing," www.arxiv.org / abs / 1911.04045 (2019); or L.S. Theis et al., "Counteracting Systems of Diabaticities Using DRAG Controls: The Status after 10 Years," Europhysics Letters 123(6), 60001 (2018), each of which is incorporated herein by reference in its entirety for all purposes.

[0167] The pulse, pulse sequence, or optical waveform may further include one or more optimal control pulse sequences derived from one or more procedures including gradient ascent pulse engineering (GRAPE), Krotov's, chopped basis, chopped random basis (CRAB), Nelder-Mead, gradient optimization using parametrization (GROUP), genetic algorithm, and gradient optimization of analytical control (GOAT). For example, the pulse sequence may be similar to those described in N. Khaneja et al., "Optimal Control of Coupled Spin Dynamics: Design of NMR Pulse Sequences by Gradient Ascent Algorithms," Journal of Magnetic Resonance 172(2), 296-305 (2005); or J.T. Merrill et al., "Progress in Compensating Pulse Sequences for Quantum Computation," Advances in Chemical Physics 154, 241-294 (2014), each of which is incorporated by reference in its entirety for all purposes.

[0168] In the example of a spatially distinct optical trap-trap atom quantum computer, any of the methods disclosed herein, such as methods (100, 200, 300, 400, 500), may include an initial act of providing a plurality of spatially distinct optical traps, each trap configured to trap an atom, the atom being a qubit.

[0169] In some examples, optical traps can be formed by tightly focused light (tweezers), by standing-wave lattices, or by imaged masks or gratings. Optical traps can also include various methods in which atoms are cooled with optical illumination, e.g., a laser, and a spatially varying magnetic field to create the trap. Such optical traps can be called magneto-optical traps (MOTs).

[0170] 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 1D, 2D, or 3D optical traps. In some examples, the array can be linear, two-dimensional, three-dimensional, or can include synthetic dimensions. Synthetic dimensions can include, for example, dimensions consisting of internal atomic states or motional states. The plurality of spatially distinct optical traps can include single or multiple reservoir regions. In some examples, the array can be a regular, irregular, or quasi-regular geometric shape.

[0171] 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 includes determining, based at least in part on the determination, that some spatially distinct optical trapping sites of the array of spatially distinct optical trapping sites lack a quantum bit. For example, each optical trapping site of the plurality of optical trapping sites can be spatially separated from other optical trapping sites by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 μm, 2 μm, 3 μm, 4 μm, 5 μm, 6 μm, 7 μm, 8 μm, 9 μm, 10 μm, or more. Each optical trapping site can be spatially separated from other optical trapping sites by a distance of at most about 10 μm, 9 μm, 8 μm, 7 μm, 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, 1 μm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site can be spatially separated from each other optical trapping site by a distance within a range defined by any two of the aforementioned values. In some cases, the array is three-dimensional. For example, an array of three-dimensional optical traps can be formed.

[0172] The array of cooling-spatially distinct optical trapping sites may comprise part of an atom cooling and trapping system. The atom cooling and trapping system may include one or more optical lattices. For example, the atom cooling and trapping system may include a first optical lattice followed by a second optical lattice. In some cases, the optical lattice may be filled by a reservoir trap. In some cases, the reservoir may include an unstructured or semi-structured optical trap. In some cases, atoms are trapped in a two-stage magneto-optical trap (at 399 nm) from a pre-cooled atomic beam. 1 P1 transition, followed by 556 nm 3 In some cases, atoms can then be loaded into an optical lattice formed using 532 nm light from an optical trapping system.

[0173] The qubits in the array of spatially distinct traps can be cooled to a temperature. In some cases, the qubits include a temperature of up to 10 microkelvins (μK). In some cases, the qubits include a temperature of up to 10 microkelvins (μK). In some cases, one or more atoms disposed in an optical trap 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 microkelvins. In some cases, one or more atoms disposed within the optical trap may comprise a temperature of up to 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, one or more atoms disposed within the optical trap may comprise a temperature within a range as defined by any two of the foregoing values.

[0174] Optical Tweezers—In some cases, the multiple spatially distinct optical traps include optical tweezers. The optical trapping site may include one or more optical tweezers. The optical tweezers include one or more focused laser beams to provide attractive or repulsive forces for holding or moving one or more atoms. The beam waist of the focused laser beam may include a strong electric field gradient. Atoms may be attracted or repelled along the electric field gradient toward the center of the laser beam, which may include the strongest electric field. The optical trapping site may include one or more optical tweezers sites of one or more optical arrays of tweezers. The optical trapping site may include one or more optical tweezers sites of one or more one-dimensional (1D) 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 similarly applied to optical lattices. Optical tweezers can be useful in moving atoms or arrays of atoms.

[0175] Sites - The optical trapping system can be configured to generate multiple optical trapping sites. The optical trapping system can be configured to generate multiple spatially distinct optical trapping sites. Each optical trapping system can include any number of sites disclosed herein. Each optical trapping system can include any number of trapped atoms disclosed herein.

[0176] For example, each optical trapping system may have 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, 10,000, 11,000, 12,000, 13,000, 14,000, 15,000, 16,000, 17,000, 18,000, 19,000, 21,000, 22,000, 23,000, 24,000, 25,000, 26,000, 27,000, 28,000, 29,000, 30,000, 31,000, 32,000, 33,000, 34,000, 35,000, 36,000, 37,000, 38,000, 39,000, 40,000, 41,000, 42,000, 43,000, 44,000, 45,000, 4 The optical trapping device may be configured to generate 00, 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 can accommodate up to approximately 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, and 20,000 traps. , 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 may be configured to trap a number of optical trapping sites within a range defined by any two of the foregoing values.

[0177] Each optical trapping system can be configured to trap a plurality of atoms, for example, 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, 11,000, 12,000, 13,000, 14,000, 15,000, 16,000, 17,000, 18,000, 19,000, 20,000, 21,000, 22,000, 23,000, 24,000, 25,000, 26,000, 27,000, 28,000, 29,000, 30,000, 31,000, 32,000, 33,000, 34,000, 35,000, 36,000, 37,000, 38,000, 39,000, 40,000, 41,000, 42,000, 43,000, 44,000, 45,000, 46,000, 47,000, 48,000, 49,000, 50,000, 51,000, 52,000, 5 The trap may be configured to trap 00, 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 may be configured to trap up to 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, 50,000, 60,000, 70,000, 80,000, 90,000, 10 ... The optical trapping system may be configured to trap 0, 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 may be configured to trap a number of atoms within a range defined by any two of the foregoing values.

[0178] Trap Electromagnetic Energy—In some cases, the methods and systems disclosed herein may be configured to form multiple optical trapping sites using trap electromagnetic energy (e.g., "trap excitation" herein). The trap excitation may be generated by a trap light source.

[0179] The trap excitation may include optical excitation, e.g., in a magneto-optical trap, optical tweezers, 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 includes 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 to form multiple arrays of traps with similar properties.

[0180] Atoms—In some cases, the qubits disclosed herein include neutral atom qubits. In some cases, the plurality of atoms include neutral atoms. In some cases, the plurality of atoms include Group 2 elements. In some cases, the plurality of atoms include strontium. In some cases, the plurality of atoms include Group 2-like elements. In some cases, the plurality of atoms are atoms with two valence electrons. In some cases, the plurality of atoms include ytterbium. In some cases, the plurality of atoms are qubits.

[0181] The optical trapping system can be configured to trap neutral atoms. In some cases, the optical trapping system can trap alkaline earth or alkaline earth-like atoms. In some cases, the alkaline earth or alkaline earth-like atoms include two valence electrons. In some cases, the alkaline earth or alkaline earth-like atoms include strontium or ytterbium.

[0182] In some cases, one or more atoms may include an alkali atom. One or more atoms may include a lithium (Li) atom, a sodium (Na) atom, a potassium (K) atom, a rubidium (Rb) atom, or a cesium (Cs) atom. One or more atoms may include a lithium-6 atom, a lithium-7 atom, a sodium-23 atom, a potassium-39 atom, a potassium-40 atom, a potassium-41 atom, a rubidium-85 atom, a rubidium-87 atom, or a cesium-133 atom. One or more atoms may include an alkaline earth atom. One or more atoms may include a beryllium (Be) atom, a magnesium (Mg) atom, a calcium (Ca) atom, a strontium (Sr) atom, or a barium (Ba) atom. The one or more atoms may include a beryllium-9 atom, a magnesium-24 atom, a magnesium-25 atom, a magnesium-26 atom, a calcium-40 atom, a calcium-42 atom, a calcium-43 atom, a calcium-44 atom, a calcium-46 atom, a calcium-48 atom, a strontium-84 atom, a strontium-86 atom, a strontium-87 atom, a strontium-88 atom, a barium-130 atom, a barium-132 atom, a barium-133 atom, a barium-134 atom, a barium-135 atom, a barium-136 atom, a barium-137 atom, or a barium-138 atom. The one or more atoms may include a rare earth atom. The one or more atoms may include a scandium (Sc) atom, a yttrium (Y) atom, a lanthanum (La) atom, a cerium (Ce) atom, a praseodymium (Pr) atom, a neodymium (Nd) atom, a samarium (Sm) atom, a europium (Eu) atom, a gadolinium (Gd) atom, a terbium (Tb) atom, a dysprosium (DY) atom, a holmium (Ho) atom, an erbium (Er) atom, a thulium (Tm) atom, a ytterbium (Yb) atom, or a lutetium (Lu) atom.One or more atoms may be scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-1 59 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-169 atoms, erbium-170 atoms, erbium-171 atoms, erbium-172 atoms, erbium-173 atoms, erbium-174 atoms, erbium-175 atoms, erbium-176 atoms, erbium-177 atoms, erbium-178 atoms, erbium-179 atoms, erbium-178 atoms, erbium-179 atoms, erbium-179 atoms, erbium-170 atoms, erbium-171 atoms, erbium-172 atoms, erbium-173 atoms, erbium-174 atoms, erbium-175 atoms, erbium-176 atoms, erbium-177 atoms, erbium-178 atoms, erbium-179 atoms, erbium-179 atoms, erbium-179 atoms, erbium-179 atoms, erbium-170 atoms, erbium-171 atoms, erbium-172 atoms, erbium-173 atoms, erbium-174 atoms, erbium-175 atoms, erbium-176 atoms, erbium-177 atoms, erbium-178 atoms, erbium-179 atoms, er It may include rubium-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.

[0183] In some cases, the plurality of atoms may include a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may include a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may include an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The atoms may include rare earth atoms. For example, the plurality of atoms may include lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, cesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, strontium-89 atoms, strontium-90 atoms, strontium-91 atoms, strontium-92 atoms, strontium-93 atoms, strontium-94 atoms, strontium-95 atoms, strontium-96 atoms, strontium-97 atoms, strontium-98 atoms, strontium-99 atoms, strontium-90 atoms, strontium-91 ...2 atoms, strontium-93 atoms, strontium-94 atoms, strontium-95 atoms, strontium-96 atoms, strontium-97 atoms, strontium-98 atoms, strontium-99 atoms, strontium-100 -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 atom, Samarium-152 atom, Samarium-154 atom, Europium-151 atom, Europium-153 atom, Gadolinium-154 atom, Gadolinium-155 atom, Gadolinium-156 atom, Gadolinium-157 atom, Gadolinium-158 atom, Gadolinium-160 atom, Terbium-159 atom, Dysprosium-156 atom, Dys Prosium-158 atom, Dysprosium-160 atom, Dysprosium-161 atom, Dysprosium-162 atom, Dysprosium-163 atom, Dysprosium-164 atom, Erbium-162 atom, Erbium-164 atom, Erbium-166 atom, Erbium-167 atom, Erbium-168 atom, Erbium-170 atom, Holmium-165 atom, Thulium-169 atom , ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms, which may comprise at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, Enriched to 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 isotopic abundance. The multiple atoms are lithium-6 atom, lithium-7 atom, sodium-23 atom, potassium-39 atom, potassium-40 atom, potassium-41 atom, rubidium-85 atom, rubidium-87 atom, cesium-133 atom, beryllium-9 atom, magnesium-24 atom, magnesium-25 atom, magnesium-26 atom, calcium-40 atom, calcium-42 atom, calcium-43 atom, calcium-44 atom, calcium-46 atom, calcium-48 atom, strontium-84 atom, strontium-86 atom, strontium-87 atom, strontium-88 atom, barium-130 atom, barium-132 atom, barium-133 atom, barium-134 atom, barium-135 atom,Barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium- 152 atom, samarium-154 atom, europium-151 atom, europium-153 atom, gadolinium-154 atom, gadolinium-155 atom, gadolinium-156 atom, gadolinium-157 atom, gadolinium-158 atom, gadolinium-160 atom, terbium-159 atom, dysprosium-156 atom, dysprosium-158 atom, dysprosium-160 atom, dysprosium-161 atom, dysprosium-162 atom, dysprosium-163 atom, dysprosium rhodium-164 atom, erbium-162 atom, erbium-164 atom, erbium-166 atom, erbium-167 atom, erbium-168 atom, erbium-170 atom, holmium-165 atom, thulium-169 atom, ytterbium-168 atom, ytterbium-170 atom, ytterbium-171 atom, ytterbium-172 atom, ytterbium-173 atom, ytterbium-174 atom, ytterbium-176 atom, lutetium-175 atom, or ruthenium It may include Tetium-176 atoms, which are enriched to an isotopic abundance of up to about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50% or less. The multiple atoms are lithium-6 atom, lithium-7 atom, sodium-23 atom, potassium-39 atom, potassium-40 atom, potassium-41 atom, rubidium-85 atom, rubidium-87 atom, cesium-133 atom, beryllium-9 atom,Magnesium-24 atom, Magnesium-25 atom, Magnesium-26 atom, Calcium-40 atom, Calcium-42 atom, Calcium-43 atom, Calcium-44 atom, Calcium-46 atom, Calcium-48 atom, Strontium-84 atom, Strontium-86 atom, Strontium-87 atom, Strontium-88 atom, Barium-130 atom, Barium-132 atom, Barium-133 atom, Barium-134 atom, Barium-135 atom, Barium-136 atom, Barium-137 atom, Barium Lithium-138 atom, Scandium-45 atom, Yttrium-89 atom, Lanthanum-139 atom, Cerium-136 atom, Cerium-138 atom, Cerium-140 atom, Cerium-142 atom, Praseodymium-141 atom, Neodymium-142 atom, Neodymium-143 atom, Neodymium-145 atom, Neodymium-146 atom, Neodymium-148 atom, Samarium-144 atom, Samarium-149 atom, Samarium-150 atom, Samarium-152 atom, Samarium-154 atom, Europium-151 atom, Euro Pium-153 atom, Gadolinium-154 atom, Gadolinium-155 atom, Gadolinium-156 atom, Gadolinium-157 atom, Gadolinium-158 atom, Gadolinium-160 atom, Terbium-159 atom, Dysprosium-156 atom, Dysprosium-158 atom, Dysprosium-160 atom, Dysprosium-161 atom, Dysprosium-162 atom, Dysprosium-163 atom, Dysprosium-164 atom, Erbium-162 atom, Erbium-164 atom, Erbium-166 atom The isotopic abundance may include 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, which are enriched to an isotopic abundance within a range defined by any two of the foregoing values.

[0184] In some cases, the first plurality of qubits includes neutral atoms. In some cases, the second plurality of qubits may include neutral atoms. For example, one or more atoms of the array may include one or more qubits. One or more atoms may be configured to be usable as one or more qubits. One or more qubits may be configured to perform non-classical computations. For example, one or more qubits may be configured to perform gate-based quantum computations. In another example, one or more qubits may be configured to perform quantum computations. The one or more atoms 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 atoms. The one or more atoms can include up to 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 can include a number of atoms defined by any two of the foregoing values. For example, the one or more atoms can include about 75 to about 150 atoms.

[0185] System for calculating error correction amount FIG. 6A shows a system for error-correcting quantum computing, programmed or otherwise configured to implement the methods provided herein. The system for error-correcting quantum computing may include an error-correcting code. The present disclosure provides a system for error-correcting quantum computing. The system may include an error-correcting code. An implementation of the error-correcting code may include a decoder. The decoder may be configured to receive a matching graph and determine a set of edges, and the matching graph received by the decoder may be updated based on a predicted probability distribution of the lost qubit. In some cases, the error-correcting code includes an operation in which a two-qubit interaction between the qubit and the lost qubit has the effect of a Pauli operation or an identity operation on the qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit.

[0186] In some cases, the system for error-correcting quantum computing may include a non-classical computing system (650). The non-classical computing system may be a quantum computing system. The non-classical computing system may be a trapped atom quantum computing system. The trapped atom quantum computing system may include an atom transfer unit, an atom rearrangement unit, an optical trapping unit, an imaging unit, an optical pumping unit, an entanglement unit, a Rydberg unit, a non-classical computing unit, an electromagnetic delivery unit, or any combination thereof.

[0187] In some cases, a non-classical computing system may include multiple qubits.

[0188] In some cases, the nonclassical computing system may include one or more electromagnetic delivery units. The electromagnetic delivery units may be configured to generate electromagnetic excitations to perform various operations, such as, for example, atom transfer, atom rearrangement, optical trapping, imaging, and various operations on atoms, which may include portions of nonclassical computation. Portions of the nonclassical computation on trapped atoms may include optical pumping, entanglement operations, Rydberg operations, gate operations (e.g., one-qubit operations, two-qubit operations, etc.). In some cases, the atom transfer unit may include an atom rearrangement unit. The components of the nonclassical computing system are discussed herein above with respect to the operations they implement.

[0189] In some cases, the system further includes a non-classical computing system, the non-classical computing system including a trapped atomic qubit. In some cases, the trapped atomic qubit includes a neutral atomic qubit. In some cases, the neutral atomic qubit includes a Group 2 element or a Group 2-like element. In some cases, the Group 2 element or a Group 2-like element includes ytterbium, rubidium, cesium, or strontium. In some cases, the plurality of qubits includes: 1 The two-qubit interaction involves excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

[0190] In some cases, the system further comprises a non-classical computing system, the non-classical computing system comprising a plurality of qubits, the plurality of qubits comprising atomic qubits, and the atomic permutation operation is performed using optical tweezers.

[0191] In some cases, a non-classical computing system may be configured to interact with the processor 601. The processor may be a classical processing system. The processor may be a digital processing system.

[0192] In some cases, the system further includes a processor configured to implement an error correction code. In some cases, the processor is further configured to provide instructions to a non-classical computing system, the non-classical computing system configured to implement the instructions to (i) identify that a qubit has been lost, (ii) replace the qubit, and (iii) re-implement the qubit in the circuit. In some cases, the processor is further configured to (iv) flag measurements taken while the qubit is missing as unreliable. In some cases, (i) includes using multiple swap gates. In some cases, the swap gates in the multiple swap gates are implemented as multiple CNOT gates. In some cases, the processor is further configured to provide instructions to the non-classical computing system to measure alternating atoms in the lattice, execute the multiple swap gates to transfer data stored in the data qubits to the ancilla qubits, and measure the swapped data qubits to identify the one or more missing atoms.

[0193] In some cases, (i) includes using a modified knock-knock protocol, the modified knock-knock protocol including providing a first atom to be probed using a second atom, where the second atom is an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on the Rydberg interaction; rotating the second qubit back to the computational basis; and performing a measurement. In some cases, (iii) includes using (A) a decoder algorithm, where the decoder algorithm takes the graph and determines a set of edges. In some cases, before (A), the processor is further configured to update the matching graph passed to the decoder algorithm based on a predicted probability distribution of the missing qubit replaced in (ii). In some cases, (iii) includes using a minimum-weight perfect matching decoder algorithm. In some cases, the processor is further configured to update the matching graph that is passed to the minimum-weight perfect matching decoder algorithm based on the predicted probability distribution of the lost qubit replaced in (ii). In some cases, the processor is configured to update the matching graph when an ancilla qubit is lost so that nodes including the ancilla qubit are connected by edges corresponding to the predicted probability distribution, and when a data qubit is lost, update the matching graph by assigning a predicted probability distribution to each node including a data qubit. In some cases, each node including an ancilla qubit is updated.

[0194] In some cases, the error correction code is configured to be implemented between the quantum computing circuitry. In some cases, the error correction code is configured to be implemented without measuring each or more of the data qubits. In some cases, the error correction code is configured to be implemented without substantially losing coherence of each or more of the data qubits. In some cases, the processor is further configured to flag measurements taken during a time window that includes the time when the missing qubit was missing as unreliable.

[0195] In some cases, the decoder includes a union finder, a tensor network decoder, a belief propagation with order statistics decoder, a maximum likelihood decoder, or a lookup table decoder. In some cases, the decoder includes a minimum-weight perfect matching. In some cases, the decoder includes a sparse blossom or fusion blossom. In some cases, the error-correcting code includes a topological code. In some cases, the topological code is a stabilizer code. In some cases, the error-correcting code is a surface code, a color code, a toric code, a shor-style code, or a qLDPC code. In some cases, the color code is a Steane code. In some cases, the shor-style code is a Bacon-shor code. In some cases, the qLDPC code is a hypergraph product code. In some cases, each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges that correspond to possible physical errors. In some cases, the edges are weighted based on the likelihood of a particular error occurring. In some cases, atom loss is treated as a gate error with a 50% probability of occurring.

[0196] In some cases, the processor is further configured to provide instructions to the non-classical computing system to perform a measurement operation, where the measurement operation is state-selective. In some cases, the measurement operation includes applying electromagnetic energy to the qubit to be measured, where the electromagnetic energy is configured to selectively drive the qubit to be measured from an initial state to an excited state in the presence of an applied magnetic field, where the selectivity of the transition to the excited state is based at least in part on the strength of the applied magnetic field. In some cases, the processor is further configured to determine that the qubit to be measured was in the initial state based at least in part on the qubit returning to the initial state by emission of a photon in response to the electromagnetic energy.

[0197] Digital Computer System FIG. 6B illustrates a computer system programmed or otherwise configured to implement the methods provided herein. The present disclosure provides computer systems programmed to perform the methods of the present disclosure. FIG. 6B illustrates a computer system (601) programmed or otherwise configured to perform a method of the present disclosure, such as method (100, 200, 300, 400, or 500). The computer system (601) can coordinate various aspects of the error-correcting quantum computing system of the present disclosure, such as providing control signals to perform one or more operations on a plurality of qubits. The computer system (601) can be a user's electronic device or a computer system located remotely relative to the quantum computing system. The electronic device can be a mobile electronic device.

[0198] The computer system (601) includes a central processing unit (CPU; further referred to herein as "processor" and "computer processor") (605), which may be a single-core or multi-core processor, or multiple processors for parallel processing. The computer system (601) also includes memory or memory locations (610) (e.g., random access memory, read-only memory, flash memory), an electronic storage unit (615) (e.g., hard disk), a 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 an electronic display adapter. The memory (610), storage unit (615), interface (620), and peripheral devices (625) communicate with the CPU (605) via a communication bus (solid lines), such as a motherboard. The storage unit (615) may be a data storage unit (or data repository) for storing data. The computer system (601) may be operably coupled to a computer network ("network") (630) using the communication interface (620). The network (630) may be the Internet, an internet and / or an extranet, or an intranet and / or an extranet in communication with the Internet. The network (630) may, in some cases, be a telecommunications and / or data network. The network (630) may include one or more computer servers, which may enable distributed computing, such as cloud computing.The network 630 may, in some cases, implement a peer-to-peer network using the computer system 601, which may allow devices coupled to the computer system 601 to function as clients or servers.

[0199] The CPU (605) can execute sequences of machine-readable instructions, which may be embodied in a program or software. The instructions may be stored in a memory location, such as the memory (610). The instructions are directed to the CPU (605), which can then program or otherwise configure the CPU (605) to implement the methods of the present disclosure. Examples of operations performed by the CPU (605) may include fetch, decode, execute, and writeback.

[0200] The CPU 605 may be part of a circuit, such as an integrated circuit. One or more other components of the system 601 may be included in the circuit. In some cases, the circuit is an application specific integrated circuit (ASIC).

[0201] The storage unit (615) can store files such as drivers, libraries, and saved programs. The storage unit (615) can store user data, such as user preferences and user programs. The computer system (601) may, in some cases, include one or more additional data storage units external to the computer system (601), such as located on a remote server in communication with the computer system (601) through an intranet or the Internet.

[0202] The computer system (601) can communicate with one or more remote computer systems via the network (630). For example, the computer system (601) can communicate with a user's remote computer system. Examples of remote computer systems include personal computers (e.g., portable PCs), slate or tablet PCs (e.g., Apple® iPad, Samsung® Galaxy Tab), telephones, smartphones (e.g., Apple® iPhone, Android-enabled devices, Blackberry®), or personal digital assistants. A user can access the computer system (601) via the network (630).

[0203] For example, the computer system 601 can communicate with a non-classical computing system 650. In some cases, the non-classical computing system is local to the processor 601. In some cases, the non-classical computing system is remote from the processor 601. The processor may access the processor 601 via a network 630.

[0204] The methods described herein can be implemented by machine (e.g., computer processor) executable code stored on an electronic storage location of the computer system (601), such as, for example, 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 in memory (610) for easy access by the processor (605). In some situations, the electronic storage unit (615) can be omitted, and machine-executable instructions can be stored in memory (610).

[0205] The code may be pre-compiled and configured for use with a machine having a processor adapted to execute the code, or may be compiled during run-time. The code may be supplied in a programming language that may be selected to allow the code to be executed in a pre-compiled or compiled manner.

[0206] Aspects of the systems and methods provided herein, such as the computer system (601), can be embodied in programming. Various aspects of the present technology may be considered “products” or “articles of manufacture,” typically in the form of machine (or processor) executable code and / or associated data carried on or embodied in some type of machine-readable medium. The machine-executable code may be stored in an electronic storage unit, such as memory (e.g., read-only memory, random-access memory, flash memory) or a hard disk. “Storage” type media may include any or all of the tangible memory of a computer, processor, or its associated modules, such as various semiconductor memories, tape drives, disk drives, etc., which may provide non-transitory storage for software programming at any time. All or portions of the software may, from time to time, be communicated via the Internet or various other telecommunications networks. Such communication may, for example, enable loading of the software from one computer or processor to another, e.g., from an administrative server or host computer to the computer platform of an application server. Thus, another type of medium that may carry software elements includes light waves, radio waves, and electromagnetic waves, such as those used across physical interfaces between local devices, through wired and optical landline networks, and via various air links. The physical elements that carry such waves, such as wired or wireless links, optical links, etc., may be considered software-bearing media. As used herein, unless limited to non-transitory, tangible "storage" media, terms such as computer or machine "readable medium" refer to any medium that participates in providing instructions to a processor for execution.

[0207] Thus, a machine-readable medium such as a computer-executable code may take many forms, including, but not limited to, a tangible storage medium, a carrier wave medium, or a physical transmission medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in any computer, such as may be used to implement the databases, etc., shown in the figures. Volatile storage media include dynamic memory, such as the main memory of such a computer platform. Tangible transmission media include coaxial cables, copper wire, and optical fiber, including the wires that comprise a bus within a computer system. Carrier-wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Thus, common forms of computer-readable media include, for example, a floppy disk, a flexible disk, a hard disk, magnetic tape, any other magnetic medium, a CD-ROM, a DVD or DVD-ROM, any other optical medium, punched cards, paper tape, any other physical storage medium with a pattern of holes, RAM, ROM, PROM and EPROM, FLASH-EPROM, any other memory chip or cartridge, a carrier wave transmitting data or instructions, a cable or link carrying 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.

[0208] The computer system 601 may include or communicate with an electronic display 635 that includes a user interface (UI) 640. Examples of UIs include, but are not limited to, graphical user interfaces (GUIs) and web-based user interfaces.

[0209] The methods and systems of the present disclosure may be implemented by one or more algorithms. The algorithms may be implemented by software when executed by the central processing unit (605). The algorithms may be, for example, variations, examples, or embodiments of error correction algorithms, decoders, quantum circuits, etc. [Example]

[0210] The following example illustrates the simulation results of a surface coding procedure using a minimum weight perfect matching decoder.

[0211] Circuit simulations were performed using Methods-Stim. We simulated a surface code memory experiment consisting of d rounds of syndrome extraction for a surface code with distance d. Typical distances belonged to the set {3, 5, 7, 9}. The simulations used standard depolarizing noise on single- and two-qubit gates, as well as readout errors, state-preparation errors, and per-round idle errors. We also included losses. To simulate losses, we adopted a model in which there was a uniform probability of loss for each atom for each circuit step (parallel gate or round of operation). All noise channels, including losses, were set to have equal probability throughout the simulation.

[0212] To simulate atom loss, loss events were randomly pre-generated on a per-shot basis. In Stim, any two-qubit gate containing a lost atom was replaced with an identification gate. To simulate atom replacement, we performed a qubit reset followed by a fully depolarizing noise channel for the replaced atom. The decoding graph of the circuit was pre-computed assuming no lost atoms. For each shot, the decoding graph was updated based on the atom loss event, following the procedure in the previous description. In the surface code memory experiments described herein, all edges are graph-like (i.e., not hyperedges), simplifying the graph modification procedure. However, the method can be extended to hyperedges. Thus, each lost ancillary element corresponded to a single 50% error edge between the two detectors. Since each data qubit participates in either two (boundary qubits) or four (bulk qubits) syndrome checks per round, for each missing data qubit we added two or four 50% error edges between the corresponding detectors. The simulated data and modified decoding graph were decoded using minimum-weight perfect matching implemented by Pymatching.

[0213] Results—Figures 7A, 7B, and 7C show simulation results from an experimental implementation of the systems and methods disclosed herein. Simulations were performed with and without loss. Figure 7A is a plot of simulation data showing a plot of physical error rate versus logical error probability for a model that accounts for atomic loss. The error sources considered include state preparation, state readout, single-qubit gate errors, two-qubit gate errors, qubit loss rate, and loss-correlated induced noise. Figure 7B is a plot of simulation data showing a plot of physical error rate versus logical error probability for a model that does not account for atomic loss. The error sources considered include state preparation, state readout, single-qubit gate errors, and two-qubit gate errors.

[0214] Figure 7C is an overlay of the data in Figures 7A and 7B. Results comparing lossy and lossless simulations are shown in Figure 7C. Although loss impairs thresholds, the results show that the effect is small. Furthermore, for each particular code size, the results with loss asymptotically scale with the same exponent as the lossless results, indicating that atomic loss does not impair effective code distance.

[0215] 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. The present invention is not intended to be limited by the specific examples provided herein. While described with reference to the foregoing specification, the descriptions and illustrations of the embodiments herein are not intended to be construed in a limiting sense. Numerous variations, changes, and substitutions will occur to those skilled in the art without departing from the invention. Furthermore, it is to be understood that all aspects of the invention are not limited to the specific depictions, configurations, or relative proportions set forth herein, which vary depending upon a variety of conditions and variables. It is to be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is therefore contemplated that the present invention will further encompass 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.

Claims

1. 1. A method for error-correcting quantum computing, the method comprising: (a) identifying that a qubit has been lost; (b) permuting the qubit; (c) re-implementing the qubit into a circuit; (d) flagging measurements taken while the qubit is missing as unreliable; A method comprising:

2. 2. The method of claim 1, wherein the identifying step in (a) includes using a plurality of swap gates.

3. 3. The method of claim 2, wherein the swap gates in the plurality of swap gates are implemented as a plurality of CNOT gates.

4. 3. The method of claim 2, further comprising measuring alternating atoms in a lattice; executing the plurality of swap gates to transfer data stored in data qubits to ancilla qubits; and measuring the swapped data qubits to identify one or more missing atoms.

5. 2. The method of claim 1 , wherein the identifying step in (a) comprises using a modified knock-knock protocol, the modified knock-knock protocol comprising: providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on a Rydberg interaction; and rotating the second qubit back to a computational basis and performing a measurement.

6. The method of any one of claims 1 to 5, wherein the re-implementing in (c) comprises (i) using a decoder algorithm, the decoder algorithm taking in the graph and determining a set of edges.

7. 7. The method of claim 6, comprising, prior to (i), updating the matching graph passed to the decoder algorithm based on predicted probability distributions of missing qubits replaced in (b).

8. The method of any one of claims 1 to 7, wherein the re-implementing step in (c) comprises using a minimum weight exact matching decoder algorithm.

9. 9. The method of claim 8, further comprising updating the matching graph passed to the minimum weight perfect matching decoder algorithm based on a predicted probability distribution of the missing qubits replaced in (b).

10. if an ancilla qubit is lost, updating the matching graph so that nodes containing the ancilla qubit are connected by edges corresponding to the predicted probability distributions; if a data qubit is lost, updating the matching graph by assigning the predicted probability distribution to each node that contains the data qubit; 10. The method of claim 9, further comprising:

11. The method of claim 10 , wherein each node that includes the ancillary qubit is updated.

12. 12. The method of any one of claims 1 to 11, wherein (a) to (d) are performed during quantum computing circuitry.

13. 13. The method of any one of claims 1 to 12, wherein (a)-(d) are performed without measurement of the or multiple data qubits.

14. 14. The method of any one of claims 1 to 13, wherein (a)-(d) are performed substantially without loss of coherence of the or multiple data qubits.

15. 15. The method of any one of claims 1 to 14, wherein (d) comprises flagging measurements taken during a time window that includes a time when the qubit was missing as unreliable.

16. The method of any one of claims 1 to 15, wherein the qubit is a trapped atomic qubit.

17. 17. The method of claim 16, wherein the trapped atomic qubit is a neutral atomic qubit.

18. 18. The method of claim 17, wherein the neutral atomic qubit is a Group 2 element or a Group 2-like element.

19. 19. The method of claim 18, wherein the Group 2 or Group 2-like element comprises ytterbium, rubidium, cesium, or strontium.

20. The quantum bit is １ S ０ 20. The method of claim 19, comprising qubit states comprising nuclear spin states on a manifold.

21. 21. The method of any one of claims 1 to 20, wherein (a) comprises an operation in which a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli operation or an identity operation on the qubit.

22. 22. The method of claim 21 , wherein the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

23. 23. The method of any one of claims 1 to 22, wherein the re-implementing in (c) comprises: (i) implementing a decoder configured to receive a matching graph and determine a set of edges, wherein the matching graph received by the decoder is updated based on predicted probability distributions of lost qubits.

24. The method of any preceding claim, wherein the method further comprises performing a measurement operation, the measurement operation being state-selective.

25. 25. The method of claim 24, wherein the measurement operation includes a qubit being measured in response to electromagnetic energy configured to selectively drive the qubit being measured from an initial state to an excited state in the presence of an applied magnetic field, the selectivity of the transition to the excited state being based at least in part on the strength of the applied magnetic field.

26. 26. The method of claim 25, further comprising determining that the qubit being measured was in the initial state, said determining being based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

27. 27. The method of any one of claims 1 to 26, wherein the qubits are atoms and (b) is implemented using optical tweezers.

28. 8. The method of claim 7, wherein the decoder comprises a union find, a tensor network decoder, a belief propagation with ordered statistics decoder, a maximum likelihood decoder, or a lookup table decoder.

29. 9. The method of claim 8, wherein the minimum weight perfect matching comprises sparse blossom or fusion blossom.

30. 30. The method of any preceding claim, wherein (a) to (d) comprise part of an error correcting code, said error correcting code comprising a topological code.

31. 31. The method of claim 30, wherein the topological code is a stabilizer code.

32. 32. The method of claim 30 or 31, wherein the error correction code is a surface code, a color code, a toric code, a short-style code, or a qLDPC code.

33. 33. The method of claim 32, wherein the color code is a Steane code.

34. 33. The method of claim 32, wherein the shor-style code is a Bacon-shor code.

35. 33. The method of claim 32, wherein the qLDPC code is a hypergraph product code.

36. 36. The method of any one of claims 1 to 35, wherein (c) comprises implementing an error correcting code, the error correcting code implementation comprising a decoder, the decoder configured to receive the matching graph and determine a set of edges, and wherein the matching graph received by the decoder is updated based on the predicted probability distributions of lost qubits.

37. 37. The method of claim 36, wherein the decoder is configured to receive a matching graph and determine a set of edges, and wherein the matching graph received by the decoder is updated based on predicted probability distributions of missing qubits.

38. 38. The method of claim 36 or 37, wherein the error correction code is a stabilizer code.

39. 39. The method of any one of claims 36 to 38, wherein each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges corresponding to possible physical errors.

40. 40. The method of claim 39, wherein the edges are weighted based on the likelihood that a particular error will occur.

41. 41. The method of claim 39 or 40, wherein an atom loss is treated as a gating error that occurs with a 50% probability.

42. 1. A method for error-correcting quantum computing, the method comprising: (a) providing a plurality of qubits; (b) implementing an error correction code; wherein the error correcting code implementation includes a decoder configured to receive a matching graph and determine a set of edges, and wherein the matching graph received by the decoder is updated based on predicted probability distributions of lost qubits.

43. 43. The method of claim 42, wherein the error-correcting code comprises an operation in which a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli operation or an identity operation on the qubit.

44. 43. The method of claim 42, wherein the qubit is a non-missing qubit.

45. 43. The method of claim 42, wherein the qubit is a missing qubit.

46. (b) before (i) identifying that a qubit has been lost; (ii) permuting the qubit; (iii) re-implementing the qubit into a circuit; The method of any one of claims 42 to 45, further comprising:

47. 47. The method of any one of claims 42 to 46, further comprising, following (b), the step of: (iv) flagging measurements taken while the qubit is missing as unreliable.

48. 48. The method of claim 46 or 47, wherein the identifying step in (i) comprises using a plurality of swap gates.

49. 49. The method of claim 48, wherein swap gates in the plurality of swap gates are implemented as a plurality of CNOT gates.

50. 49. The method of claim 48, further comprising measuring alternating atoms in a lattice; executing the plurality of swap gates to transfer data stored in data qubits to ancilla qubits; and measuring the swapped data qubits to identify one or more missing atoms.

51. 48. The method of claim 46 or 47, wherein the identifying step in (i) comprises using a modified knock-knock protocol, the modified knock-knock protocol further comprising: providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on a Rydberg interaction; and rotating the second qubit back to a computational basis and performing a measurement.

52. 52. A method according to any one of claims 46 to 51, wherein the re-implementing in (iii) comprises (a) using a decoder algorithm, said decoder algorithm taking in the graph and determining the set of edges.

53. 53. The method of claim 52, comprising, prior to (a), updating the matching graph passed to the decoder algorithm based on predicted probability distributions of missing qubits replaced in (ii).

54. A method according to any one of claims 46 to 53, wherein the re-implementing step in (iii) comprises using a minimum weight exact matching decoder algorithm.

55. 55. The method of claim 54, further comprising updating the matching graph passed to the minimum weight perfect matching decoder algorithm based on predicted probability distributions of missing qubits replaced in (ii).

56. if an ancilla qubit is lost, updating the matching graph so that nodes containing the ancilla qubit are connected by edges corresponding to the predicted probability distributions; if a data qubit is lost, updating the matching graph by assigning the predicted probability distribution to each node that contains the data qubit; 56. The method of claim 55, further comprising:

57. 57. The method of claim 56, wherein each node that includes the ancillary qubit is updated.

58. 58. The method of any one of claims 42 to 57, wherein (a) to (b) are performed within a quantum computing circuit.

59. 58. The method of any one of claims 42 to 57, wherein (a)-(b) are performed without measuring the or more data qubits.

60. 58. The method of any one of claims 42 to 57, wherein (a)-(b) are performed substantially without loss of coherence of the or more data qubits.

61. 59. The method of any one of claims 42 to 58, comprising, subsequent to (b), flagging measurements taken during a time window that includes a time that the missing qubit was missing as unreliable.

62. 60. The method of any one of claims 42 to 59, wherein the plurality of qubits comprises trapped atomic qubits.

63. 63. The method of claim 62, wherein the trapped atomic qubit comprises a neutral atomic qubit.

64. 64. The method of claim 63, wherein the neutral atomic qubit comprises a Group 2 element or a Group 2-like element.

65. 65. The method of claim 64, wherein the Group 2 or Group 2-like element comprises ytterbium, rubidium, cesium, or strontium.

66. the plurality of quantum bits: １ S ０ 66. The method of claim 65, comprising qubit states comprising nuclear spin states on a manifold.

67. 67. The method of any one of claims 42 to 66, wherein the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom.

68. 68. The method of any one of claims 42 to 67, further comprising the step of performing a measurement operation, wherein the measurement operation is state-selective.

69. 69. The method of claim 68, wherein the measurement operation comprises applying electromagnetic energy to a qubit to be measured, the electromagnetic energy being configured to selectively drive the qubit to be measured from an initial state to an excited state in the presence of an applied magnetic field, the selectivity of the transition to the excited state being based at least in part on the strength of the applied magnetic field.

70. 70. The method of claim 69, further comprising determining that the qubit being measured was in the initial state, said determining being based at least in part on the qubit returning to the initial state by emitting a photon in response to the electromagnetic energy.

71. 71. The method of any one of claims 42 to 70, wherein the plurality of qubits comprises atomic qubits, and the atomic replacement operation is performed using optical tweezers.

72. A method according to any one of claims 42 to 71, wherein the decoder comprises a union find, a tensor network decoder, a belief propagation with order statistics decoder, a maximum likelihood decoder, or a lookup table decoder.

73. A method according to any one of claims 42 to 71, wherein the decoder comprises a minimum weight perfect matching.

74. 74. The method of claim 73, wherein the decoder comprises Sparse Blossom or Fusion Blossom.

75. A method according to any one of claims 42 to 74, wherein the error correcting code comprises a topological code.

76. 76. The method of claim 75, wherein the topological code is a stabilizer code.

77. 77. The method of claim 75 or 76, wherein the error correction code is a surface code, a color code, a toric code, a short-style code, or a qLDPC code.

78. 78. The method of claim 77, wherein the color code is a Steane code.

79. 78. The method of claim 77, wherein the shor-style code is a Bacon-shor code.

80. 78. The method of claim 77, wherein the qLDPC code is a hypergraph product code.

81. 81. A method according to any one of claims 77 to 80, wherein each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges corresponding to possible physical errors.

82. 82. The method of claim 81, wherein the edges are weighted based on the likelihood that a particular error will occur.

83. 83. The method of claim 81 or 82, wherein an atom loss is treated as a gating error that occurs with a 50% probability.

84. 1. A system for error-correcting quantum computing, the system comprising:

1. A system comprising: an error correcting code, an implementation of the error correcting code including a decoder, the decoder configured to receive a matching graph and determine a set of edges, the matching graph received by the decoder being updated based on predicted probability distributions of lost qubits.

85. 85. The system of claim 84, wherein the error correction code comprises an operation in which a two-qubit interaction between a qubit and a missing qubit has the effect of a Pauli operation or an identity operation on the qubit.

86. 86. The system of claim 85, wherein the qubit is a non-missing qubit.

87. 86. The system of claim 85, wherein the qubit is a missing qubit.

88. 86. The system of any one of claims 84 to 85, further comprising a processor configured to implement an error correction code.

89. The processor is further configured to provide instructions to a non-classical computing system, the non-classical computing system comprising: (i) identifying that a qubit has been lost; (ii) permuting the qubits; and (iii) Re-implementing the qubit into a circuit.

90. The system of claim 88, configured to implement instructions for:

90. 90. The system of claim 89, wherein the processor is further configured to: (iv) flag measurements taken while the qubit is missing as unreliable.

91. 90. The system of claim 88 or 89, wherein (i) includes using multiple swap gates.

92. 92. The system of claim 91 , wherein a swap gate in the plurality of swap gates is implemented as a plurality of CNOT gates.

93. 93. The system of claim 92, wherein the processor is further configured to provide instructions to the non-classical computing system to measure alternating atoms in a lattice; execute the plurality of swap gates to transfer data stored in data qubits to ancilla qubits; and measure the swapped data qubits to identify one or more missing atoms.

94. 93. The system of claim 91 or 92, wherein (i) comprises using a modified knock-knock protocol, the modified knock-knock protocol comprising providing a first atom to be probed using a second atom, the second atom being an ancilla qubit; preparing the second atom in a |+> state; applying a modified controlled-Z gate between the first atom and the second atom based on a Rydberg interaction; and rotating the second qubit back to a computational basis and performing a measurement.

95. 95. The system of any one of claims 89-94, wherein (iii) comprises (A) use of a decoder algorithm, the decoder algorithm taking in the graph and determining a set of edges.

96. 96. The system of claim 95, wherein prior to (A), the processor is further configured to update the matching graph that is passed to the decoder algorithm based on predicted probability distributions of missing qubits replaced in (ii).

97. The system of any one of claims 89 to 96, wherein (iii) comprises the use of a minimum weight exact matching decoder algorithm.

98. 98. The system of claim 97, wherein the processor is further configured to update the matching graph passed to the minimum weight perfect matching decoder algorithm based on a predicted probability distribution of the missing qubits replaced in (ii).

99. the processor: if an ancilla qubit is lost, updating the matching graph so that nodes containing the ancilla qubit are connected by edges corresponding to the predicted probability distributions; 99. The system of claim 98, further configured to, if a data qubit is lost, update the matching graph by assigning the predicted probability distribution to each node that includes the data qubit.

100. 100. The system of claim 99, wherein each node that includes the ancilla qubit is updated.

101. 101. The system of any one of claims 84 to 100, wherein the error correction code is configured to be implemented between quantum computing circuits.

102. 102. A system according to any one of claims 84 to 101, wherein the error correcting code is configured to be implemented without measuring each or more data qubits.

103. 103. A system according to any one of claims 84 to 102, wherein the error correction code is configured to be implemented without substantially losing coherence of the or multiple data qubits.

104. 104. The system of any one of claims 84 to 103, wherein the processor is further configured to flag measurements taken during a time window that includes a time when the missing qubit was missing as unreliable.

105. 105. The system of any one of claims 84 to 104, wherein the system further comprises a non-classical computing system, the non-classical computing system comprising a trapped atomic qubit.

106. 106. The system of claim 105, wherein the trapped atomic qubit comprises a neutral atomic qubit.

107. 107. The system of claim 106, wherein the neutral atomic qubit comprises a Group 2 element or a Group 2-like element.

108. 108. The system of claim 107, wherein the Group 2 or Group 2-like element comprises ytterbium, rubidium, cesium, or strontium.

109. the plurality of quantum bits: １ S ０ 109. The system of claim 108, comprising a qubit state comprising nuclear spin states on a manifold.

110. 110. A system according to any one of claims 84 to 109, wherein the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of said neutral atom.

111. 111. The system of any one of claims 84 to 110, wherein the processor is further configured to provide instructions to the non-classical computing system for performing a measurement operation, the measurement operation being state-selective.

112. 112. The system of claim 111, wherein the measurement operation comprises applying electromagnetic energy to a quantum bit to be measured, the electromagnetic energy being configured to selectively drive the quantum bit to be measured from an initial state to an excited state in the presence of an applied magnetic field, the selectivity of the transition to the excited state being based at least in part on the strength of the applied magnetic field.

113. 113. The system of claim 112, wherein the processor is further configured to determine that the measured quantum bit was in the initial state based at least in part on the quantum bit returning to the initial state by emitting a photon in response to the electromagnetic energy.

114. 114. The system of any one of claims 84 to 113, further comprising a non-classical computing system, the non-classical computing system comprising a plurality of qubits, the plurality of qubits comprising atomic qubits, and wherein the atomic permutation operation is performed using optical tweezers.

115. 115. The system of any one of claims 84 to 114, wherein the decoder comprises a union find, a tensor network decoder, a belief propagation with order statistics decoder, a maximum likelihood decoder, or a lookup table decoder.

116. A system according to any one of claims 84 to 115, wherein the decoder comprises a minimum weight perfect matching.

117. 117. The system of claim 116, wherein the decoder comprises Sparse Blossom or Fusion Blossom.

118. A system according to any one of claims 84 to 117, wherein the error correcting code comprises a topological code.

119. 119. The system of claim 118, wherein the topological code is a stabilizer code.

120. 120. The system of claim 118 or 119, wherein the error correction code is a surface code, a color code, a toric code, a short-style code, or a qLDPC code.

121. 121. The system of claim 120, wherein the color code is a Steane code.

122. 121. The system of claim 120, wherein the shor-style code is a Bacon-shor code.

123. 121. The system of claim 120, wherein the qLDPC code is a hypergraph product code.

124. 124. The system of any one of claims 84 to 123, wherein each node in the matching graph corresponds to a change in the value of a particular stabilizer, and pairs of nodes are connected by edges corresponding to possible physical errors.

125. 125. The system of claim 124, wherein the edges are weighted based on the likelihood that a particular error will occur.

126. 126. The system of claim 124 or 125, wherein an atom loss is treated as a gating error with a 50% probability of occurring.

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