WO2024150142A1

WO2024150142A1 - Methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware

Info

Publication number: WO2024150142A1
Application number: PCT/IB2024/050222
Authority: WO
Inventors: Shunji Matsuura; Pooya Ronagh
Original assignee: 1QB Information Technologies Inc
Current assignee: 1QB Information Technologies Inc
Priority date: 2023-01-10
Filing date: 2024-01-09
Publication date: 2024-07-18
Anticipated expiration: 2025-07-10

Abstract

The present disclosure provides methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware. The methods may comprise: preparing one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each having a plurality of subsystems of qudits; executing said one or more quantum circuits on said quantum hardware; using a result of said quantum measurement to perform an analysis of crosstalk of said plurality of subsystems of qudits within each said one or more quantum circuits; outputting a result of said analysis, wherein said result comprises the amount of crosstalk. The systems may comprise a quantum computer having (A) a quantum chip comprising qudits and (B) a control system, and a digital computer operatively coupled to said quantum computer.

Description

METHODS AND SYSTEMS FOR CHARACTERIZING THE AMOUNT OF CROSSTALK IN A GROUP OF QUDITS IN QUANTUM HARDWARE CROSS-REFERENCE [0001] This application claims the benefit of U.S. Provisional Application Serial No. 63/479,298, filed January 10, 2023, which is incorporated herein by reference for all purposes. BACKGROUND [0002] Quantum computing holds the promise of having a significant impact on a wide range of fields, such as drug discovery, finance, security, and artificial intelligence. A critical feature that distinguishes quantum computing from classical computing is that it uses quantum superposition and entanglement during computations. These properties enable quantum computers to perform massively parallel computations as well as to have an effect on quantum states to significantly increase the success probability of finding the desired outcome. One of the main bottlenecks toward demonstrating a quantum advantage as well as building fault-tolerant quantum computers is the presence of noise. SUMMARY [0003] Quantum computer may exhibit various types of errors including, for example, incoherent errors, coherent errors, and correlated errors. Quantum states on quantum computers, that is, qubit states, are very easily able to enter into states of entanglement with their surrounding environment. The influence of this entanglement with the environment appears as stochastic noise on qubit states, and it may result in the loss of quantum information. This type of error is called incoherent error. Coherent errors come from inaccurate quantum control, such as over- or under-rotations of qubits as well as crosstalk between qubits. Coherent errors are reversible and map a pure state into another pure state. However, the infidelity of coherent errors increases quadratically in the gate number whereas that of incoherent errors increases linearly. Therefore, identifying coherent errors is of great importance. Another obstacle to building fault-tolerant quantum computers is correlated errors. Many quantum error-correcting codes are posited to work if the error rate is below a certain value, which threshold depends on the quantum error-correcting code that is used. However, thresholds may be calculated based on various assumptions. One assumption is that errors are localized. If errors are correlated, i.e., if errors on multiple physically separated qubits happen at the same time, then the quantum error correction may no longer be valid. Therefore, there is an unmet need to develop methods that allow for testing whether or not each error is localized. [0004] An example method for characterizing quantum noise is randomized benchmarking (see, e.g., Emerson et al., “Scalable noise estimation with random unitary operators,” Journal of Optics B: Quantum and Semiclassical Optics 7, no. 10, p. S347 (2005), which is incorporated by reference herein in its entirety). Randomized benchmarking uses twirling circuits which change the noise channel into a depolarization channel. Then, by measuring the decay rate of successful measurement outcome as a function of the circuit depth, one can estimate the average gate fidelity. Another example method is gate set tomography (GST) (see, e.g., Nielsen et al., “Gate Set Tomography,” Quantum 5, p.557 (2021, which is incorporated by reference herein in its entirety). GST measures the detailed information pertaining to errors while considering it impossible to separate errors in state preparation, gate operations, and measurements. [0005] Recognized herein is the need for improved methods and systems that overcome at least one of the identified drawbacks. The present disclosure provides methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware using density matrix construction and uses of the disclosed methods and systems for fabrication of quantum hardware and for controlling qudits. [0006] Disclosed herein are methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware. Disclose herein are methods and systems for noise channel characterization from the viewpoint of quantum entanglement and quantum correlations. [0007] Systems and methods disclosed herein may detect errors due to unintended quantum entanglement, unintended quantum correlation other than quantum entanglement, and unintended classical correlation. Systems and methods disclosed herein may enable the measurement of the strength of the noise as well as how the noise spreads between different qudits. [0008] In an aspect, the present disclosure provides a method for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The method may comprise: (a) preparing one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) executing said one or more quantum circuits on said quantum hardware; (c) performing a quantum measurement on at least a portion of said group of qudits; (d) performing an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits based at least in part on a result of said quantum measurement; and (e) outputting a result of said analysis, wherein said result comprises an amount of crosstalk. [0009] In some embodiments, (b) and (c) are repeated at least one time. In some embodiments, (b) comprises executing at least one copy of at least one quantum circuit of said one or more quantum circuits. In some embodiments, said crosstalk of said plurality of subsystems of qudits within said one or more quantum circuits comprises at least one member of the group consisting of: unintended entanglement between said subsystems, unintended quantum correlations between said subsystems, and unintended classical correlations between said subsystems. In some embodiments, said crosstalk comprises one of coherent noise and incoherent noise. In some embodiments, (c) comprises performing said quantum measurement in one of a Z-basis, an X-basis, and a Y-basis. In some embodiments, (c) comprises applying a unitary operation and performing said quantum measurement in a Z-basis. [0010] In some embodiments, said plurality of subsystems of qudits comprises one of an error correction code and a plaquette in an error correction code. In some embodiments, the method further comprises using said result for a parity check on said plaquette in said error correction code. In some embodiments, the method further comprises using said result for one round of error correction in said quantum hardware. In some embodiments, said error correction code comprises CSS code, surface code, colour code, triangular colour code, rotated surface code, or toric code. [0011] In some embodiments, (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one subsystem of said plurality of subsystems of qudits, computing a corresponding partially transposed matrix of said density matrix; (iii) computing eigenvalues of said partially transposed matrix; (iv) computing a negativity of said partially transposed matrix using said eigenvalues; and (v) reporting entanglement presence in case of positive negativity and entanglement absence in case of zero negativity. [0012] In some embodiments, (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one of said plurality of subsystems of qudits computing a corresponding partially transposed matrix of said density matrix; (iii) computing at least a second and a third moment of said partially transposed matrix; (iv) comparing functions of two successive moments of said at least said second and said third moments to identify said unintended entanglement presence between said at least one subsystem of said plurality of subsystems and other subsystems; and (v) reporting said entanglement presence if identified. [0013] In some embodiments, (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) based at least in part on said density matrix, computing a joint entropy of a plurality of subsystems of qudits belonging to a same quantum circuit; (iii) based at least in part on said density matrix, computing an individual entropy of each subsystem of said plurality of subsystems of qudits belong to said same quantum circuit; (iv) comparing said joint entropy with a sum of said individual entropies to identify presence of unintended entanglement of said plurality of subsystems or multi-qudit error within said plurality of subsystems; and (v) reporting comparison results. In some embodiments, said joint entropy and said individual entropy comprise at least one member of the group consisting of: entanglement entropy, Rényi entropy, and moments of density matrices of the corresponding subsystems; further wherein the density matrices of the corresponding subsystems are computed using said density matrix of a quantum circuit. [0014] In some embodiments, said one or more quantum circuits comprise a parity check quantum circuit. In some embodiments, (b) comprises preparing a quantum state in a computational basis and implementing said parity check quantum circuit at least one time. In some embodiments, the method further comprises using said result comprising said amount of crosstalk to adjust an architecture of said quantum hardware for further use in fabricating said quantum hardware. In some embodiments, the method further comprises: (i) setting control signal characteristics of said qudits based at least in part on said result comprising said amount of crosstalk; and (ii) controlling said qudits in accordance with said control signal characteristics. [0015] In another aspect, the present disclosure provides a method for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The method may comprise: (a) preparing one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) executing said one or more quantum circuits on said quantum hardware; (c) performing a quantum measurement on at least a portion of said group of qudits; and (d) directing a result of said quantum measurement to a computing device configured to perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, wherein said remote device is configured to output a result of said analysis, wherein said result comprises an amount of crosstalk. [0016] In another aspect, the present disclosure provides a method for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The method may comprise: (a) at a computing device in communication with said quantum hardware, providing an indication to prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) providing instructions to execute said one or more quantum circuits on said quantum hardware; (c) providing instructions to perform a quantum measurement on at least a portion of said group of qudits; (d) based at least in part on a result of said quantum measurement, performing an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits; and (e) outputting a result of said analysis, wherein said result comprises an amount of crosstalk. [0017] In some embodiments, said computing device is a digital computer operatively coupled to said quantum hardware. In some embodiments, said digital computer is communicatively coupled over a network. In some embodiments, said digital computer is local to said quantum hardware. [0018] In another aspect, the present disclosure provides a system for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The system may comprise: (a) a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system; and (b) a digital computer in communication with said quantum computer, said digital computer comprising a memory having instructions to at least (A) provide an indication to said control system prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (B) provide instruction to said control system to execute said one or more quantum circuits on said quantum hardware, (C) provide instructions to said control system to perform a quantum measurement on said qudits, (D) receive a result from said quantum computer and, based at least in part on said result of said quantum measurement, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (E) output a result of said analysis, wherein said result comprises an amount of crosstalk. [0019] In some embodiments, said digital computer is configured to repeat (B) and (C) at least one time. In some embodiments, at (B) said quantum hardware is directed to execute at least one copy of at least one quantum circuit of said one or more quantum circuits. In some embodiments, said crosstalk of said plurality of subsystems of qudits within said one or more quantum circuits comprises at least one member of the group consisting of: unintended entanglement between said subsystems, unintended quantum correlations between said subsystems, and unintended classical correlations between said subsystems. In some embodiments, said crosstalk comprises one of coherent noise and incoherent noise. In some embodiments, at (C), said quantum hardware is configured to perform said quantum measurement in one of a Z-basis, an X-basis, and a Y-basis. In some embodiments, at (C), said quantum hardware is configured to apply a unitary operation and performing said quantum measurement in a Z-basis. [0020] In some embodiments, said plurality of subsystems of qudits comprises one of an error correction code and a plaquette in an error correction code. In some embodiments, said instructions further comprise using said result for a parity check on said plaquette in said error correction code. In some embodiments, said instructions further comprise using said result for one round of error correction in said quantum hardware. In some embodiments, said error correction code comprises CSS code, surface code, colour code, triangular colour code, rotated surface code, or toric code. [0021] In some embodiments, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one subsystem of said plurality of subsystems of qudits, compute a corresponding partially transposed matrix of said density matrix; (iii) compute eigenvalues of said partially transposed matrix; (iv) compute a negativity of said partially transposed matrix using said eigenvalues; and (v) report entanglement presence in case of positive negativity and entanglement absence in case of zero negativity. [0022] In some embodiments, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one of said plurality of subsystems of qudits compute a corresponding partially transposed matrix of said density matrix; (iii) compute at least a second and a third moment of said partially transposed matrix; (iv) compare functions of two successive moments of said at least said second and said third moments to identify said unintended entanglement presence between said at least one subsystem of said plurality of subsystems and other subsystems; and (v) report said entanglement presence if identified. [0023] In some embodiments, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result; (ii) based at least in part on said density matrix, compute a joint entropy of a plurality of subsystems of qudits belonging to a same quantum circuit; (iii) based at least in part on said density matrix, compute an individual entropy of each subsystem of said plurality of subsystems of qudits belong to said same quantum circuit; (iv) compare said joint entropy with a sum of said individual entropies to identify presence of unintended entanglement of said plurality of subsystems or multi-qudit error within said plurality of subsystems; and (v) report comparison results. In some embodiments, said joint entropy and said individual entropy comprise at least one member of the group consisting of: entanglement entropy, Rényi entropy, and moments of density matrices of the corresponding subsystems; further wherein the density matrices of the corresponding subsystems are computed using said density matrix of a quantum circuit. [0024] In some embodiments, said one or more quantum circuits comprise a parity check quantum circuit. In some embodiments, at (B), said digital computer is configured to provide an indication to prepare a quantum state in a computational basis and implement said parity check quantum circuit at least one time. In some embodiments, said digital computer is configured to use said result comprising said amount of crosstalk to adjust an architecture of said quantum computer and, based at least in part on said architecture for further use in fabricating said quantum hardware. In some embodiments, said control system is configured to: (i) set control signal characteristics of said qudits based at least in part on said result comprising said amount of crosstalk; and (ii) control said qudits in accordance with said control signal characteristics. [0025] In another aspect, the present disclosure provides a system for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The system may comprise: a digital computer in communication with a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system, said digital computer comprising a memory having instructions to at least (C) provide an indication to said control system to prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (D) provide instruction said control system to execute said one or more quantum circuits on said quantum hardware, (E) provide instructions to said control system to perform a quantum measurement on said qudits, (F) receive a result of said quantum measurement and, based at least in part on said result, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (G) output a result of said analysis, wherein said result comprises an amount of crosstalk. [0026] In another aspect, the present disclosure provides a system for characterizing an amount of crosstalk in a group of qudits in quantum hardware. The system may comprise: a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system, wherein said quantum computer is in communication with a digital computer, wherein said digital computer comprising a memory having instructions to at least (A) provide an indication to said control system prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (B) provide instruction to said control system to execute said one or more quantum circuits on said quantum hardware, (C) provide instructions to said control system to perform a quantum measurement on said qudits, (D) receive a result from said quantum computer and, based at least in part on said result of said quantum measurement, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (E) output a result of said analysis, wherein said result comprises an amount of crosstalk. [0027] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in this art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its several details are capable of modifications in various obvious respects, all without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive. INCORPORATION BY REFERENCE [0028] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and/or take precedence over any such contradictory material. BRIEF DESCRIPTION OF THE DRAWINGS [0029] 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 may be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also “Figure” and “FIG.” herein), of which: [0030] FIG. 1 is a schematic of an example system for characterizing the amount of crosstalk in a group of qudits in quantum hardware, in accordance with some embodiments disclosed herein. [0031] FIG. 2 is a flowchart of an example method for characterizing the amount of crosstalk in a group of qudits in quantum hardware, in accordance with some embodiments disclosed herein. [0032] FIG.3 is a flowchart of an example crosstalk analysis procedure using a quantum measurement result, in accordance with some embodiments disclosed herein. [0033] FIG.4 is a flowchart of an example crosstalk analysis procedure using a quantum measurement result, in accordance with some embodiments disclosed herein. [0034] FIG.5 is a flowchart of an example crosstalk analysis procedure using a quantum measurement result, in accordance with some embodiments disclosed herein. [0035] FIG. 6 is a schematic of a circuit used for computing the negativity and the moments of a partially transposed density matrix, in accordance with some embodiments disclosed herein. [0036] FIG.7 is a graph of the negativity of a partially transposed matrix as a function of the circuit depth for various rotation angles, in accordance with some embodiments disclosed herein. [0037] FIG.8 is a graph of the difference between the second and the third moments of a partially transposed density matrix as a function of the circuit depth for various rotation angles, in accordance with some embodiments disclosed herein. [0038] FIG. 9 is a schematic of a circuit for computing the mutual information, in accordance with some embodiments disclosed herein. [0039] FIG.10 is a graph of the Rényi entropy of the union of the subsystem A and the subsystem B as a function of the circuit depth and the sum of the Rényi entropies of subsystem A and subsystem B as a function of the circuit depth in the absence of crosstalk, in accordance with some embodiments disclosed herein. [0040] FIG. 11 is a graph of the Rényi entropy of the union of subsystem A and subsystem B as a function of the circuit depth and the sum of the Rényi entropies of subsystem A and subsystem B as a function of the circuit depth in the presence of crosstalk, in accordance with some embodiments disclosed herein. DETAILED DESCRIPTION [0041] While various embodiments of the invention are shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed. [0042] Neither the Title nor the Abstract is to be taken as limiting in any way the scope of the disclosed invention(s). The title of the present application and headings of sections provided in the present application are for convenience only and are not to be taken as limiting the disclosure in any way. [0043] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. Any reference to “or” herein is intended to encompass “and/or” unless otherwise stated. [0044] The term “plurality” generally refers to “two or more,” unless expressly specified otherwise. [0045] The term “e.g.” and like terms mean “for example,” and thus do not limit the terms or phrases they explain. For example, in a sentence “the computer sends data (e.g., instructions, a data structure) over the Internet,” the term “e.g.” explains that “instructions” are an example of “data” that the computer may send over the Internet, and also explains that “a data structure” is an example of “data” that the computer may send over the Internet. However, both “instructions” and “a data structure” are merely examples of “data,” and other things besides “instructions” and “a data structure” can be “data.” [0046] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “at least,” “greater than,” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3. [0047] Whenever the term “no more than,” “less than,” or “less than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “no more than,” “less than,” or “less than or equal to” applies to each of the numerical values in that series of numerical values. For example, less than or equal to 3, 2, or 1 is equivalent to less than or equal to 3, less than or equal to 2, or less than or equal to 1. [0048] Where values are described as ranges, the disclosure includes the disclosure of all possible sub-ranges within such ranges, as well as specific numerical values that fall within such ranges irrespective of whether a specific numerical value or specific sub- range is expressly stated. [0049] Certain inventive embodiments herein contemplate numerical ranges. When ranges are present, the ranges include the range endpoints. Additionally, every sub-range and value within the range is present as if explicitly written out. [0050] The term “about” or “approximately” may mean within an acceptable error range for the particular value, which will depend in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, “about” may mean within 1 or more than 1 standard deviation, per the practice in the art. Alternatively, “about” may mean a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value. Where particular values are described in the application and claims, unless otherwise stated the term “about” meaning within an acceptable error range for the particular value may be assumed. [0051] In the following detailed description, reference is made to the accompanying figures, which form a part hereof. In the figures, similar symbols may identify similar components, unless context dictates otherwise. The illustrative embodiments described in the detailed description, figures, and claims are not meant to be limiting. Other embodiments may be used, and other changes may be made, without departing from the scope of the subject matter presented herein. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations, all of which are explicitly contemplated herein. Quantum Computing [0052] Quantum computing may be method of computing that utilizes the concepts of quantum superposition and entanglement to manipulate information, instead of the binary bits 0 and 1 in classical computers. Quantum computation may refer a method for performing computations using quantum computing. A quantum computation may comprise performing computations using quantum operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) on a Hilbert space represented by a quantum device. [0053] A quantum state may be a state of a system described by quantum mechanics. A quantum state may be described as a wavefunction or a density matrix in quantum mechanics. A qubit, a qubit, etc. may be described by a quantum state. Quantum entanglement may be the phenomenon in which, when multiple qubits interact with each other, their quantum states are entangled and may no longer be represented individually. When quantum states are entangled, only unitary transformations involving multi-partite qudits can be generated. Quantum superposition may be the principle that states that the quantum state of a qubit can be represented by adding together two or more different quantum states, each associated with a probability. In some cases, the probabilities of all states add to 1. [0054] Quantum circuits, comprising one or more quantum gates, may be designed to perform quantum computation, such as factoring large prime numbers, which may be infeasible or highly inefficient for classical computers. A circuit may be a representation of a computational model in which the computation comprises a sequence of gates. In some cases, a circuit may be used in gate model quantum computation. In some cases, a circuit may be a quantum circuit, such as a sequence of qubit gates used in a gate model quantum computation. A quantum circuit may be circuit may be a circuit used in gate model quantum computation. Quantum gates may be logical operators comprising one or multiple qubits, which can be used to perform logical operations. A quantum circuit may comprise an initial state preparation for a set of qudits, followed by performing a gate operation and measurements on it. A quantum gate operation may comprise a quantum gate, a sequence of quantum gates, or a combination of quantum gates and quantum measurements that perform an isometry on the quantum state of qubits. A quantum measurement may be a process for extracting classical information from quantum states generated on quantum devices. [0055] Classical, used in the context of computing or computation, may generally refer to binary computing. Classical computing may be computation performed using binary values using discrete bits without the use of quantum mechanical superposition and quantum mechanical entanglement. A classical computer may be a digital computer, such as a computer employing discrete bits (e.g., 0s and 1s) without the use of quantum mechanical superposition and quantum mechanical entanglement. Non-classical, used in the context of computing or computation, may refer to any method or system for performing computational procedures outside of the paradigm of classical computing. [0056] A quantum device maybe any device or system for performing computations using quantum mechanical phenomenon such as quantum mechanical superposition and quantum mechanical entanglement. A quantum device may comprise quantum hardware. Quantum hardware may comprise devices on which controllable quantum states may be realized. A quantum device may comprise a quantum computer. A quantum hardware may comprise a quantum computer. A quantum chip may be a physical device that can utilize quantum phenomena that allows the execution of quantum gates for the purpose of computing. [0057] A qubit may be a unit of quantum information processing whose quantum state is a complex unit vector of dimension 2. These two dimensions may be referred to as “0” and “1.” [0058] A data qubit may be to one of the qubits used to encode quantum information for a quantum computation. It may contain a part of an input or a part of an output state. If quantum error correction is used, it refers to a logical qubit, and if not, it refers to a physical qubit. [0059] A qudit may be the basic unit of quantum information. A qudit may be a multi- level quantum system. In some cases, a multi-level quantum system may be a qubit. There are various methods of realizing qubits and qudits using physical implementations. One method is via superconducting qubits and qudits, and another involves the use of ion trap qubits and qudits; however, other methods may be used, such as other quantum computing methods disclosed herein. In some cases, a qubit may also refer to a qudit and vice versa. The information stored in physical qubits may, in some cases, be referred to simply as qubits, and the physical qubits and physical qudits of a quantum device as vertices. [0060] A physical qubit may be a physical implementation of a qubit. A logical qubit may be a physical or abstract qubit which perform as specified in a quantum circuit. A logical qubit may be realized by one or more physical qubits. A physical qudit may be a physical implementation of a qudit. A logical qudit may be a physical or abstract qubit which perform as specified in a quantum circuit. A logical qudit may be realized by one or more physical qudits. [0061] Gates, two-qubit gates, and one-qubit gates may refer to quantum logic gates that consist of two qubits (two-qubit gate) or one qubit (one-qubit gate), and which may be used to perform logical operations. Multi-level Quantum System [0062] A multi-level quantum system may be structured in a way which operates based on quantum mechanical processes such as superposition and entanglement of quantum states. A multi-level system can include a system with two or more energy states of an artificial or natural atom, for example, the ground state (|0>) and first excited state (|1>) of a superconducting artificial atom. Such a multi-level system can have 0, 1, …, n energy states. [0063] A multi-level quantum system may be referred to as a “qudit.” Multiple qudits may be used to implement a quantum computing system. A qudit may be thought of as one of n quantum states 0, 1, ..., n – 1 or a superposition of any of the n states. Specific subcategories of qudits exist, including a system comprising two energy states, the ground state (|0>) and first excited state (|1>). These two-state systems are referred to as “qubits.” Each qubit can be placed in one of these two states. However, due to the nature of multi-level quantum systems, they can also be placed in a superposition of these two states. Entangled qubits or qudits can perform computational tasks. Entropy [0064] One of the entanglement measures may be entanglement entropy. A quantum system may be divided into two subsystems, which may be denoted as subsystem A and subsystem B. The density matrix of the total quantum system may be denoted as ^^^^ and the reduced density matrices of the subsystem A(B) may then be denoted as ^^^^ _{^^^^( ^^^^)}. The reduced density matrix of A(B) may be defined by tracing out the Hilbert space of its complement subsystems, ^^^^ _{^^^^( ^^^^)} = tr _{^^^^( ^^^^)} ^^^^ . [0065] By using the reduced density matrices, the quantum entanglement entropy S of a subsystem A(B) may be defined as ^^^^ _{^^^^( ^^^^)} = −tr ^^^^ _{^^^^( ^^^^)} log ^^^^ _{^^^^( ^^^^)}. [0066] Similarly, the Rényi entropy may be defined as

[0067] The partial transpose of a density matrix ^^^^ may be defined as follows. The basis of the subsystems may be denoted as | ^^^^ _^ ^{^} ^^{^^} ^^{^ ^^^^} ^ 〉, | ^^^^ _^^^^〉 and the components of a density matrix

[0069] The moments of the partially transposed density matrix are given by ^^^^ _^^^^ = tr[( ^^^^ ^{^^^^ ^^^^}) ^{^^^^}]. NISQ Technology – Noisy Intermediate-Scale Quantum Technology [0070] Methods and systems disclosed herein may be suitable for a noisy, intermediate scale quantum (NISQ) device. In some cases, a NISQ device with a limitation of a two- dimensional structure of a quantum chip or a limitation on how many neighbouring qubits (or qudits) each qubit (or qudit) is connected to may benefit from methods and systems disclosed herein. Quantum Device/Quantum Hardware [0071] Any type of non-classical computer, for example, a quantum computer, may be suitable for the technologies disclosed herein. In some cases, a quantum device with a limitation of a two-dimensional structure of a quantum chip or a limitation on how many neighbouring qubits (or qudits) each qubit (or qudit) is connected to may benefit from methods and systems disclosed herein. In accordance with the description herein, suitable quantum computers may include, by way of non-limiting examples: superconducting quantum computers (qubits implemented as small superconducting circuits—Josephson junctions) (Clarke et al., “Superconducting quantum bits,” Nature 453, no. 7198, pp. 1031–1042, 2008); trapped-ion quantum computers (qubits implemented as states of trapped ions) (Kielpinski et al., “Architecture for a large-scale ion-trap quantum computer,” Nature 417, no.6890, pp.709–711, 2002); optical lattice quantum computers (qubits implemented as states of neutral atoms trapped in an optical lattice) (Deutsch et al., “Quantum computing with neutral atoms in an optical lattice,” Fortschritte der Physik: Progress of Physics 48, no.9–11, pp.925–943, 2000); spin-based quantum dot computers (qubits implemented as the spin states of trapped electrons) (Imamoḡlu et al., “Quantum information processing using quantum dot spins and cavity QED,” Physical Review Letters 83, no.20, p.4204, 1999); spatial-based quantum dot computers (qubits implemented as electron positions in a double quantum dot) (Fedichkin et al., “Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot,” arXiv:quant-ph/0006097, 2000); coupled quantum wires (qubits implemented as pairs of quantum wires coupled by quantum point contact) (Bertoni et al., “Quantum logic gates based on coherent electron transport in quantum wires,” Physical Review Letters 84, no. 25, p.5912, 2000); nuclear magnetic resonance quantum computers (qubits implemented as nuclear spins and probed by radio waves) (Cory et al., “Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing,” arXiv:quant-ph/9709001, 1997); solid-state NMR Kane quantum computers (qubits implemented as the nuclear spin states of phosphorus donors in silicon) (Kane, “A silicon-based nuclear spin quantum computer,” Nature 393, no. 6681, pp. 133–137, 1998); electrons-on-helium quantum computers (qubits implemented as electron spins) (Lyon, “Spin-based quantum computing using electrons on liquid helium,” arXiv:cond- mat/0301581, 2006); molecular magnet-based quantum computers (qubits implemented as spin states) (Leuenberger et al., “Quantum Computing in Molecular Magnets,” arXiv:cond-mat/0011415, 2001); fullerene-based ESR quantum computers (qubits implemented as electronic spins of atoms or molecules encased in fullerenes) (Harneit, “Spin Quantum Computing with Endohedral Fullerenes,” arXiv:1708.09298, 2017); diamond-based quantum computers (qubits implemented as electronic or nuclear spins of nitrogen-vacancy centres in diamond) (Nizovtsev et al., “A quantum computer based on NV centers in diamond: optically detected nutations of single electron and nuclear spins,” Optics and spectroscopy 99, no. 2, pp. 233–244, 2005); Bose–Einstein condensate-based quantum computers (qubits implemented as two-component BECs) (Byrnes et al., “Macroscopic quantum computation using Bose–Einstein condensates,” arXiv:quantum-ph/1103.5512, 2011); transistor-based quantum computers (qubits implemented as semiconductors coupled to nanophotonic cavities) (Sun et al., “A single- photon switch and transistor enabled by a solid-state quantum memory,” arXiv:quant- ph/1805.01964, 2018); metal-like carbon nanospheres based quantum computers (qubits implemented as electron spins in conducting carbon nanospheres) (Náfrádi et al., “Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres,” arXiv:cond-mat/1611.07690, 2016); topological quantum computers (qubits implemented as non-Abelian anyons) (Nayak et al., “Non-Abelian Anyons and Topological Quantum Computation,” arXiv:0707.1889, 2007); photonic continuous- variable quantum computing hardware (quantum variables represented by the quadrature operators of the quantum harmonic oscillators in a quantum optical mode) (Arrazola et al., “Quantum circuits with many photons on a programmable nanophotonic chip,” Nature 591, pp. 54–60, 2021); photonic qubit-based quantum hardware (qubits implemented on pairs of optical paths) (O’Brien et al., “Photonic quantum technologies”, Nature Photonics 3, pp.687–695, 2009); quantum computing hardware based on bosonic codes (error-protected qubits or qudits are formed by embedding a finite-dimensional code space within the infinite-dimensional Fock space associated with a bosonic quantum field mode; examples include the Gottesman–Kitaev–Preskill (GKP) code, cat codes, and binomial codes, respectively) (Gottesman et al., “Encoding a qubit in an oscillator,” Physical Review A 64, p. 012310, 2001; Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, p. 010329, 2022; Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, p. 031006, 2016); quantum hardware based on coherent network computing (operating by sampling low-energy eigenstates of an Ising Hamiltonian by encoding the spins in a network of optical parametric oscillators with all- to-all connectivity; future architectures may exploit quantum entanglement for computation) (Inui et al., “Entanglement and quantum discord in optically coupled coherent Ising machines,” Physical Review A 102, p.062419, 2020; and Yanagimoto et al., “Embedding entanglement generation within a measurement-feedback coherent Ising machine”, arXiv:1906.04902, 2019); each of which is incorporated by reference herein in its entirety. [0072] Methods and systems disclosed herein may be suitable for a device which simulates a quantum computer. For example, methods and systems disclosed herein may be suitable for a device which exploits quantum mechanical properties, but which comprises a limited number of gate operations or which does not implement a series of qubit gate operations. In some cases, a device which simulates a quantum computer with a limitation of a two-dimensional structure of a quantum chip or a limitation on how many neighbouring qubits (or qudits) each qubit (or qudit) is connected to may benefit from methods and systems disclosed herein. Digital Computer [0073] In some cases, the digital computer comprises one or more hardware central processing units (CPU) that carry out the digital computer’s functions. In some cases, the digital computer further comprises an operating system (OS) configured to perform executable instructions. In some cases, the digital computer is connected to a computer network. In some cases, the digital computer is connected to the Internet such that it accesses the World Wide Web. In some cases, the digital computer is connected to a cloud computing infrastructure. In some cases, the digital computer is connected to an intranet. In some cases, the digital computer is connected to a data storage device. [0074] In accordance with the description herein, suitable digital computers may include, by way of non-limiting examples, server computers, desktop computers, laptop computers, notebook computers, sub-notebook computers, netbook computers, netpad computers, set-top computers, media streaming devices, handheld computers, Internet appliances, mobile smartphones, tablet computers, personal digital assistants, video game consoles, and vehicles. Smartphones may be suitable for use in some cases of the method and the system described herein. Select televisions, video players, and digital music players, in some cases with computer network connectivity, may be suitable for use with one or more variations, examples, or embodiments of the systems and the methods described herein. Suitable tablet computers may include those with booklet, slate, and convertible configurations. [0075] In some cases, the digital computer comprises an operating system configured to perform executable instructions. The operating system may be, for example, software, comprising programs and data, which manages the device’s hardware and provides services for execution of applications. Suitable server operating systems include, by way of non-limiting examples, FreeBSD, OpenBSD, NetBSD®, Linux, Apple® Mac OS X Server®, Oracle® Solaris®, Windows Server®, and Novell® NetWare®. Suitable personal computer operating systems may include, by way of non-limiting examples, Microsoft® Windows®, Apple® Mac OS X®, Apple® macOS®, UNIX®, and UNIX- like operating systems such as GNU/Linux®. In some cases, the operating system is provided by cloud computing. Suitable mobile smart phone operating systems may include, by way of non-limiting examples, Nokia® Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google® Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS, Linux®, and Palm® WebOS®. Suitable media streaming device operating systems may include, by way of non-limiting examples, Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®. Suitable video game console operating systems may include, by way of non-limiting examples, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®. [0076] In some cases, the digital computer comprises a storage and/or memory device. In some cases, the storage and/or memory device is one or more physical apparatuses used to store data or programs on a temporary or permanent basis. In some cases, the device comprises a volatile memory and requires power to maintain stored information. In some cases, the device comprises non-volatile memory and retains stored information when the digital computer is not powered. In some cases, the non-volatile memory comprises flash memory. In some cases, the non-volatile memory comprises dynamic random-access memory (DRAM). In some cases, the non-volatile memory comprises ferroelectric random-access memory (FRAM). In some cases, the non-volatile memory comprises phase-change random-access memory (PRAM). In some cases, the non- volatile memory comprises resistive random-access memory (RRAM). In some cases, the device comprises a storage device including, by way of non-limiting examples, CD- ROMs, DVDs, flash memory devices, magnetic disk drives, magnetic tapes drives, optical disk drives, and cloud computing based storage. In some cases, the storage and/or memory device comprises a combination of devices, such as those disclosed herein. [0077] In some cases, the digital computer comprises a display used for providing visual information to a user. In some cases, the display comprises a cathode ray tube (CRT). In some cases, the display comprises a liquid crystal display (LCD). In some cases, the display comprises a thin film transistor liquid crystal display (TFT-LCD). In some cases, the display comprises an organic light-emitting diode (OLED) display. In some cases, an OLED display comprises a passive-matrix OLED (PMOLED) or active-matrix OLED (AMOLED) display. In some cases, the display comprises a plasma display. In some cases, the display comprises a video projector. In some cases, the display comprises a combination of devices, such as those disclosed herein. [0078] In some cases, the digital computer comprises an input device to receive information from a user. In some cases, the input device comprises a keyboard. In some cases, the input device comprises a pointing device including, by way of non-limiting examples, a mouse, trackball, trackpad, joystick, game controller, or stylus. In some cases, the input device comprises a touch screen or a multi-touch screen. In some cases, the input device comprises a microphone to capture voice or other sound input. In some cases, the input device comprises a video camera or other sensor to capture motion or visual input. In some cases, the input device comprises a Kinect®, Leap Motion®, or the like. In some cases, the input device comprises a combination of devices, such as those disclosed herein. Systems and Methods for Characterizing an Amount of Crosstalk [0079] The present disclosure provides methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware using density matrix construction and uses of the disclosed methods and systems for fabrication of quantum hardware and for controlling qudits. [0080] Disclosed herein are methods and systems for characterizing the amount of crosstalk in a group of qudits in quantum hardware. Disclose herein are methods and systems for noise channel characterization from the viewpoint of quantum entanglement and quantum correlations. [0081] Systems and methods disclosed herein may detect errors due to unintended quantum entanglement, unintended quantum correlation other than quantum entanglement, and unintended classical correlation. Systems and methods disclosed herein may enable the measurement of the strength of the noise as well as how the noise spreads between different qudits. [0082] An advantage of the methods and systems disclosed herein is that they can identify coherent errors. [0083] Another advantage of the methods and systems disclosed herein is that they can identify unintended entanglement between the subsystems, unintended quantum correlation between the subsystems, and unintended classical correlation between the subsystems. [0084] Another advantage of the methods and systems disclosed herein is that they may be used in an error correction code. [0085] Another advantage of the methods and systems disclosed herein is that they may be used for parity checks on a plaquette in an error correction code. [0086] Another advantage of the methods and systems disclosed herein is that the analysis results comprising the amount of crosstalk may be used to adjust the quantum hardware’s architecture. [0087] Another advantage of the methods and systems disclosed herein is that they may be used for controlling the qudits. [0088] Another advantage of the methods and the systems disclosed herein is that they can characterize the amount of crosstalk in a group of qudits in quantum hardware. [0089] Now referring to FIG. 1, there is shown a schematic of an example system for characterizing the amount of crosstalk in a group of qudits in quantum hardware. The system comprises i) a classical computer which in this embodiment is a digital computer 8, and ii) a quantum computer 10. The digital computer 8 may be any digital computer disclosed elsewhere herein. [0090] Crosstalk may comprise unintended unitary and classical transformations which involves multiple qubits. In some cases, crosstalk may comprise unintended entanglement between subsystems of qudits, unintended quantum correlation between subsystems of qudits, or unintended classical correlations between subsystems of qudits. An amount of crosstalk may comprise a value or values which quantify the strength of crosstalk. [0091] In some cases, the quantum computer 10 comprises a quantum chip 12. In some cases, the quantum computer 10 comprises a control system 14. The quantum computer 10 may be operatively connected to the digital computer 8 by way of connection between the control system 14 and communication ports 28. The quantum computer 10 may comprise any quantum computer such as any quantum device or quantum hardware disclosed herein. [0092] In some cases, the digital computer 8 is for providing instructions to the quantum computer 10 using the communication ports 28 and the control system 14. [0093] In some cases, the digital computer 8 comprises a processing device 20, a display device 24, an input device 26, the communication ports 28, and a memory 22. The processing device 20, the display device 24, the input device 26, the communication ports 28, and the memory 22 may be of various types, such as any type disclosed elsewhere herein. The memory 22 comprises a computer program executable by the processing device 20. In some cases, the communication ports 28 communicate with the quantum computer 10 via the control system 14. [0094] Now referring to FIG. 2, there is shown a flowchart of an example method for characterizing the amount of crosstalk in a group of qudits in quantum hardware. In some cases, the group of qudits comprises an error correction code. The error correction code may be of various types. The error correction code includes but is not limited to CSS code, surface code, colour code, triangular colour code, rotated surface code, and toric code. CSS code is an error correction code invented by Calderbank, Shor, and Steane. CSS code comprises two classical linear codes, one for detecting bit-flip errors and the other for detecting phase-flip errors. Surface code, colour code, triangular colour code, rotated surface code, and toric code are topological error correction codes. CSS code allows having a set of stabilizer generators that are either Z-type or X-type Pauli strings. (Calderbank et al., “Good quantum error-correcting codes exist,” Physical Review A 54, p. 1096, 1996; and Steane, “Error Correcting Codes in Quantum Theory,” Physical Review Letters 77, p.793, 1996, each of which is incorporated by reference herein for all purposes). [0095] An error correcting code or error correction code may be a quantum code which has the capacity to correct errors in quantum processes. A plaquette may be a group of qudits that forms a closed loop. In some cases, the group of qudits comprises a plaquette in an error correction code. Topological error correcting codes may be defined on a lattice, such as a square lattice or a triangular lattice on a two-dimensional surface. Qudits may be located on the vertices of the lattice whereas the edges may represent the connectivity of qudits. A plaquette may be defined as an individual surface of this lattice. [0096] The method disclosed with respect to FIG.2 may be used for parity checks on the plaquette in an error correction code. In some cases, quantum error correction codes may be stabilizer codes. For example, CSS code, surface code, colour code, triangular colour code, rotated surface code, and toric code may be stabilizer codes. Stabilizer codes may comprise a measurement the parity of stabilizer elements. This parity can be used to determine whether a quantum state contains errors and the locations of errors. In some cases, it may be convenient to locate qubits on a lattice in a way that each plaquette of lattice corresponds to each stabilizer. [0097] The method disclosed with respect to FIG.2 may be used for one round of error correction. To perform error correction, a measurement of the parities of all plaquettes may be made. For example, in the case of CSS code, the CSS code may comprise a measurement of the parities of phase-flip errors as well as those of bit-flip errors. [0098] In some cases, the crosstalk comprises coherent or incoherent noise. Coherent noise may comprise unintended unitary transformations on qudits. Coherent noise may cause coherent errors. In some cases, coherent errors may come from inaccurate quantum control, such as over- or under-rotations of qubits as well as crosstalk between qubits. Coherent errors may be reversible and map a pure state into another pure state. Incoherent noise may comprise noise that is not coherent. Incoherent noise may cause incoherent errors. The influence of interaction with the environment appears as stochastic noise on qubit states, and it may result in the loss of quantum information. This type of error is called incoherent error. [0099] Still referring to FIG.2 and according to processing operation 202, one or more quantum circuits are prepared on quantum hardware. In some cases, the one or more quantum circuits each has a plurality of subsystems of qudits. The quantum hardware may be of various types, such as any quantum computer disclosed elsewhere herein. In some cases, the quantum hardware is the quantum computer 10 disclosed herein with respect to FIG. 1. In some cases, quantum circuits may be efficiently simulable on classical computers. In some cases, the circuit comprises alternating layers of random single qudit Clifford gates and a parity check circuit. First, a set of numbers ^^^^ determining the circuit depth may be selected. ^^^^ alternating layers of random single qudit Clifford gates and the parity check circuit may then be applied. Then, a single layer of random single qudit Pauli gates may be applied. Then, a reverse of the first ^^^^ alternating layers of the random single qudit Clifford gates and the parity check circuit may be applied. [0100] According to processing operation 204, the one or more quantum circuits is executed on the quantum hardware. In some cases, the quantum hardware is the quantum computer 10 disclosed herein with respect to FIG.1. The quantum hardware may be of various types, such as any quantum computer disclosed elsewhere herein. In some cases, at least one copy of at least one quantum circuit of the one or more quantum circuits is executed. The circuit may be executed multiple times. The same process may be repeated for different values of the circuit depth, which is a function of ^^^^. [0101] According to processing operation 206, a quantum measurement is performed on the qudits. In some cases, a control system, such as the control system 14 disclosed herein with respect to FIG.1, is used for the readouts of quantum measurements. In some cases, the quantum measurement is performed in the Z-basis, the X-basis, or the Y-basis. In some cases, a unitary operation is applied, and the quantum measurement is performed in the Z-basis. In some cases, the applied unitary operation is random. One embodiment for realizing this is to apply a layer of random single qudit Clifford gates, which change the measurement bases. For instance, the Hadamard gate ^^^^ = ^{1 1 1} 2^� 1₋₁ ^{� changes the} measurement basis from the Z-basis to the X-basis. [0102] In some cases, processing operations 204 and 206 are repeated at least one time. At each set of measurements, a bitstring of the output may be obtained. Collection of the bitstrings obtained by repeating the executions of the circuit provides information for the density matrix generated by the circuit. Using the set of bitstrings as well as the random Clifford gates used to determine the axes of the measurements, a snapshot of the density matrix may be constructed. [0103] A set of Clifford operators, denoted by ^^^^ _^^^^ , where ^^^^ is a qudit index, may be applied before measurement. The resultant bitstring outcome may be denoted by

. Then, the snapshot may be given by

[0104] The expectation value of the snapshot may agree with the density matrix: ^^^^( ^^�^^) = ^^^^ (see, e.g., Huang et al., “Predicting Many Properties of a Quantum System from Very Few Measurements,” Nature Physics 16, p. 1050, 2020, which is incorporated by reference herein for all purposes). The expectation value may be taken over random Clifford gates ^^^^ _^^^^. [0105] Still referring to FIG. 2 and according to processing operation 208, a quantum measurement result may be used to perform an analysis of crosstalk of the plurality of subsystems of qudits within each quantum circuit. The analysis of crosstalk of the plurality of subsystems of qudits within each quantum circuit may be performed according to different examples, such as any example disclosed herein. The analysis of crosstalk of the plurality of subsystems of qudits within each quantum circuit may be performed using a digital computer. The digital computer may be of various types, such as the digital computer 8 disclosed with respect to FIG.1. [0106] In some cases, the crosstalk of the plurality of subsystems of qudits within the one or more quantum circuits includes unintended entanglement between the subsystems. In some cases, in superconducting qubit based quantum computers, a microwave signal used to operate a two-qudit gate leaks outside the connector of the corresponding qudits, which may result in changing the quantum states of other qudits different from the two- qudits on which a gate operation was intended to be applied. [0107] According to processing operation 210, the analysis results comprising the amount of the crosstalk is output. In some cases, the analysis results are output using a digital computer. The digital computer may be of various types, such as the digital computer 8 disclosed herein with respect to FIG. 1. In some cases, the amount of crosstalk is measured in terms of the entanglement entropy ^^^^_experiment, the. negativity ^^^^_experiment , or the moments ^^^^_experiment. In the absence of noise and entanglement, these values take specific values, ^^^^_ideal, ^^^^_ideal, and ^^^^_ideal, which may be computed efficiently on a classical computer in the setup disclosed herein. These values may also be computed from the density matrices obtained from quantum devices through shadow tomography. The deviation of these values from the theoretical values, namely, Δ ^^^^ = ^^^^_ideal − ^^^^_experiment,Δ ^^^^ = ^^^^_ideal

represent the amount of crosstalk comprising unintended entanglement, unintended quantum, or unintended classical correlations. [0108] The methods disclosed with respect to FIG. 2 may be used for fabrication of quantum hardware. In some cases, the analysis results comprising the amount of crosstalk are used to adjust the architecture of the quantum hardware. For instance, in superconducting quantum computers, the presence of unintended quantum entanglement may indicate that microwave pulses for single or multi-qudit gate operations change the frequencies of neighbouring qudits. In trapped ion quantum computers, the presence of unintended quantum entropy may indicate that lasers are not sufficiently localized. In order to reduce the amount of crosstalk, in the case of superconducting qubits optimize the pulses and scheduling can be optimized, or in the case of trapped ion quantum computers the focus of the lasers or locations of the ions can be changed. [0109] The methods disclosed with respect to FIG.2 may be used for controlling qudits. In some cases, the analysis results comprising the amount of crosstalk are used to set control signal characteristics. Then, processing operations of the method disclosed with respect to FIG. 2 may be performed for the qudits controlled using set control signal characteristics. The control signal characteristics may be adjusted based on the updated analysis results. This procedure may be repeated one or more times. In some cases, gate operations of a superconducting qubit are controlled by a time dependent qubit Hamiltonian. For instance, a qubit Hamiltonian is given by ^^^^ = ^^^ _^^_^^^( ^^^^) ^^^^ _^^^^ + ^^^ _^^_^^^( ^^^^) ^^^^ _^^^^ + ^∑ _{^^^^∈ ^^^^ ^^^^ ^^^^( ^^^^) ^^^ ^^^^^ ^^^^ ^^^^ ^^^^ ⊗ ^^^^ ^^^^ . Here, ^^^ ^^^^^} ⁽ _^^^^ ⁾ _{, ^^^ ^^^^^( ^^^^) are control pulses of a single qubit operation} for qubit ^^^^; ^^^^ _^^^^, ^^^^ _^^^^ represent Pauli operators acting on qubit ^^^^; ^^^^ ^^^^ ^^^^( ^^^^) represents a set of qubits connected to qubit ^^^^; and the crosstalk ^^^ _^^_{^^^ ^^^^} represents a ^^^^ ^^^^ crosstalk coupling between qubit ^^^^ and its neighbour qubit ^^^^. The crosstalk ^^^ _^^_{^^^ ^^^^} may be estimated by the methods disclosed herein. In order to reduce the ^^^^ ^^^^ crosstalk while operating a single- qubit gate ^^^^ _^^^^ , the time dependent functions

are optimized so that the evolution of the quantum state with respect to qubit Hamiltonian ^^^^ represents the desired single qubit gate ^^^^ _^^^^, while suppressing the crosstalk ^^^ _^^_{^^^ ^^^^} as much as possible. In some cases, the control signal characteristics are updated in real time or near real time in response to a change in the amount of crosstalk. [0110] Now referring to FIG. 3, there is shown a flowchart of an example crosstalk analysis procedure using a quantum measurement result. The procedure may be used as part of processing operation 208 disclosed herein with respect to FIG.2. [0111] According to processing operation 302, the quantum measurement result is used to construct a density matrix of a quantum system for each of the one or more quantum circuits. ^{[0112] In some cases, quantum circuits are designed so that the initial quantum state} _{|00 … 0^ is mapped to the same state |00 … 0^ in the absence of errors.} [0113] A single qudit Clifford gate on ^^^^-th qudit right before the measurement may be denoted as ^^^^ _^^^^ and its measurement result as ^^^^ _^^^^ . For instance, in the case of Hadamard ^{gates (which are Clifford gates), ^^^^ ^^^^ =}

^{If the measurement result is 0, then} _{| ^^^^ ^^^^ > = |0 >=�1} _{0� . From these data, a snapshot of the density matrix may be} ^{constructed as} ^_{^�^^ =⊗ 3 ^^^^+} _{^^^^ | ^^^^ ^^^^〉〈 ^^^^ ^^^^| ^^^^ ^^^^ − ^^^^ .} [0114] The expectation value of this snapshot over the random Clifford gates and measurements agrees with the density matrix ^^^^( ^^�^^) = ^^^^ . For instance, if a qubit is measured in the computational basis and the measurement result is |0^ , then ^^�^^ = ^{�2 0} 0₋₁ ^{Hadamard gate is applied before the measurement and the measurement} _{result is |0^, then ^^�^^ =�} ^{1/2 3/2} 3_{/2 1/2�.} [0115] According to processing operation 304, for at least one subsystem of the plurality of subsystems of qudits, a corresponding partially transpose matrix of the density matrix is computed. For example, a set of qudits may be denoted by a subsystem A and the rest of the qudits of the system may be denoted by a subsystem B. The snapshot ^�^^^of the density matrix disclosed in processing operation 302 has indices of qudits in subsystem A as well as subsystem B. The partial transpose of ^�^^^ may be constructed by exchanging the column index and the row index of subsystem A. The partial transpose of a density matrix ^^^^ may be defined as follows. If the bases of subsystems A and B are denoted by ^^^^ ^{^^^^}〉, then the components of the density matrix ^{^^^^ ^^^^} ^_^^^ are

^^^^ _^^^^ | ^^^^| ^^^^ _^^^^ , ^^^^ _^^^^ 〉 and the partial transpose of the density matrix corresponding to subsystem A is ^^^^ ^{^^^^ ^^^^} (_{^^^^ ^^^^, ^^^^ ^^^^)} = 〈 ^^^^ _^ ^{^} ^^{^} ^^{^} ^^{^} , ^^^^ ^{^^^^} ^_^^^ | ^^^^| ^^^^ _^^ ^{^^} ^^{^^} ^ , ^^^^ _^ ^{^} ^^{^^} ^^{^} ^ 〉. [0116] According to processing operation 306, the eigenvalues of the partially transposed matrix are computed. The density matrix may be calculated as an expectation value of the snapshots ^^�^^ of the density matrix and denoted by ^^^^⁽ ^^�^^⁾ = ^^^^. The partial transpose of the density matrix may be denoted as ^^^^ ^{^^^^ ^^^^ ^^^^ ^^^^} (_{^^^^ ^^^^, ^^^^ ^^^^)} and calculated using ^^^^_{( ^^^^ ^^^^, ^^^^ ^^^^)} = 〈 ^^^^ _^ ^{^} ^^{^} ^^{^} ^^{^} , ^^^^ ^{^^^^} ^_^^^ | ^^^^| ^^^^ _^^ ^{^^} ^^{^^} ^ , ^^^^ _^ ^{^} ^^{^^} ^^{^} ^ 〉 for the obtained density matrix

^^^^ ^{^^^^ ^^^^ ^^^^} ^_^^^ | ^^^^| ^^^^ _^^^^ , ^^^^ _^^^^ 〉 . Alternatively, it may be calculated from the partial transpose of the snapshot ^^�^^ ^{^^^^ ^^^^} = ^�^^^ ^{^^^^} ^_^^^ ⊗ ^�^^^ _^^^^, by evaluating its expectation value using ^^^^ ^{^^^^ ^^^^} = ^^^^⁽ ^^�^^ ^{^^^^ ^^^^)}. Eigenvalues of the partial transpose of the density matrix are obtained by solving the characteristic equation of the partial transpose of the density matrix. [0117] According to processing operation 308, the negativity of the partially transposed matrix is computed using the eigenvalues. The negativity is the sum of the negative eigenvalues of the partial transpose of the density matrix ^^^^ ^{^^^^ ^^^^}. It may be computed from the eigenvalues obtained in processing operation 306. [0118] One example of a quantum circuit for computing the negativity and the moments of a partially transposed density matrix is shown in FIG.6. The first layer of the circuit comprises single qubit rotations, the second layer comprises unitary transformations representing the parity check circuit, the third layer comprises Hadamard, phase, and Pauli X gates, all of which are Clifford gates. The last layer represents the measurements. [0119] Shown in FIG. 6 is a three-qubit system, labelled q0, q1, and q2. A subsystem A comprises q0 and q2, whereas a subsystem B comprises q2. First, single-qubit gates are applied on all three qubits. Then, a parity check circuit is applied an odd number of times (three in FIG. 6). Then, random single-qubit gates are applied (H, S, and X in FIG. 6, where H represents the Hadamard gate, S represents the phase gate, and X represents the Pauli X gate) before the measurement. After running this circuit a number of times, the density matrix may be calculated by using shadow tomography. From the obtained density matrix, the negativity of a partially transposed matrix may be calculated. The results are shown, as a function of the number of parity checks, in FIG. 7, which is a graph of the negativity of a partially transposed matrix as a function of the circuit depth for various rotation angles. The model includes Pauli errors as well as crosstalk. The negativity has a positive value when there is entanglement between subsystem A and subsystem B. In an ideal case where there is no crosstalk, the negativity is 0 for the initial rotation angle being zero. However, when there is crosstalk between qubits, the negativity can have positive values as shown in FIG.7. [0120] According to processing operation 310, the presence of entanglement is reported in the case of positive negativity and the absence of entanglement is reported in the case of zero negativity. If the negativity is positive, then it means that the quantum state is entangled. This is an indication of the presence quantum crosstalk. The presence of entanglement may be reported using a digital computer. The digital computer may be of various types, such as the digital computer 8 disclosed herein with respect to FIG.1. [0121] Now referring to FIG. 4, there is shown a flowchart of an example crosstalk analysis procedure using a quantum measurement result. The procedure may be used as part of processing operation 208 disclosed herein with respect to FIG.2. [0122] According to processing operation 402, the quantum measurement result is used to construct a density matrix of a quantum system for each of the one or more quantum circuits. A set of Clifford operators, denoted by ^^^^ _^^^^ , where ^^^^ is a qudit index, may be applied before the measurement. The resultant bitstring outcome may be denoted by ^^^^ _^^^^. ^{Then, the snapshot is given by:}

The density matrix is calculated as an expectation value of the snapshot ^^^^ = ^^^^⁽ ^^�^^⁾. [0123] According to processing operation 404, for at least one subsystem of the plurality of subsystems of qudits, a corresponding partially transposed matrix of the density matrix is computed. [0124] For example, a set of qudits may be denoted by subsystem A and the rest of the qudits of the system may be denoted by subsystem B. The snapshot ^^�^^ of the density matrix disclosed in processing operation 302 disclosed with respect to FIG.3 has indices of qudits in subsystem A as well as subsystem B. The partial transpose of ^^�^^ may be constructed by exchanging the column index and the row index of subsystem A. The partial transpose of a density matrix ^^^^ may be defined as follows. If the bases of subsystems A and B are denoted by� ^^^^ _^^ ^{^^} ^^{^^} ^ ^〉,� ^^^^ ^{^^^^} ^_^^^ ^〉, then the components of the density matrix are

^^^^ ^{^^^^} ^_^^^ | ^^^^| ^^^^ _^^ ^{^^} ^^{^} ^^{^} , ^^^^ _^ ^{^} ^^{^^} ^^{^} ^ 〉 and the partial transpose of the density matrix corresponding to subsystem A is ^{^^^^}

^^^^ _^^^^ 〉. Alternatively, it may be calculated from the partial transpose of the snapshot ^^�^^ ^{^^^^ ^^^^} = ^�^^^ ^{^^^^} ^_^^^ ⊗ ^�^^^ _^^^^, by evaluating its expectation value using ^^^^ ^{^^^^ ^^^^} = ^^^^( ^^�^^ ^{^^^^ ^^^^}). [0125] According to processing operation 406, at least the second and the third moments of the partially transposed matrix are computed. The moments of the partial transpose of the density matrix may be obtained from the expression of the density matrix snapshot ^^�^^ using ^_{^^^ ^^^^ = ^^^^} ⁽ _tr ^[ _^^�^^ ^{^^^^])} _, where E represents the expectation value. [0126] According to processing operation 408, functions of two successive moments of at least the second and the third moments are compared to identify the unintended entanglement presence between the at least one subsystem and other subsystems. More precisely, when the n-th power of the n-th moment is greater than the (n – 1)-th power of the (n + 1)-th moment, ( ^^^^ _^^^^) ^{^^^^} > ( ^^^^ _^^^^+1) ^{^^^^−1}, then the quantum state is an entangled state. [0127] According to processing operation 410, the presence of entanglement is reported if identified. [0128] Now referring to FIG. 8, there is shown a graph of the difference between the second and the third moments of partially transposed density matrix as a function of the circuit depth for various rotation angles. The circuit and the subsystems used are the circuit and the subsystems disclosed with respect to FIG.6. When ( ^^^^₂)² > ^^^^₃, there is entanglement between subsystem A and subsystem B. If the rotation angle of the single qubit gates at the beginning of the circuit is zero, the final state should have no entanglement in the absence of errors. However, an entangled state can be created if there is crosstalk between the subsystems. The solid curve with circular markers takes positive values for a range of the circuit depth, which is an indication of the presence of crosstalk. [0129] In some cases, the inequality may fail to detect quantum entanglement if the incoherent noise becomes larger. This problem can be mitigated by using the fact that the incoherent errors are single qubit depolarization noise due to the random Clifford gates. The incoherent errors cause the expectation values of the Pauli operators to decay exponentially. By rescaling the expectation value to remove the exponentially decaying part the incoherent errors may be separated from coherent errors. [0130] In some cases, the presence of entanglement may be reported using a digital computer. The digital computer may be of various types, such as the digital computer 8 disclosed herein with respect to FIG.1. [0131] Now referring to FIG. 5, there is shown a flowchart of an example crosstalk analysis procedure using a quantum measurement result. The procedure may be used as part of processing operation 208 disclosed herein with respect to FIG.2. [0132] According to processing operation 502, the quantum measurement result is used to construct a density matrix of a quantum system for each of the one or more quantum circuits. [0133] For each circuit, random single qudit Clifford gates may be inserted before the measurement. A single qudit Clifford gate on ^^^^-th qudit right before the measurement may be denoted as ^^^^ _^^^^ and its measurement result as ^^^^ _^^^^ . For instance, in the case of ^{Hadamard gates (which are Clifford gates), ^^^^} _^^^^ ^{= 1}

_{1 −1} ^{�. If the measurement result} is 0, then | ^^^^ _^^^^ > = |0 > From these data, a snapshot of the density matrix may be

^{constructed as}

_. [0134] The expectation value of this snapshot over the random Clifford gates and measurements agrees with the density matrix ^^^^( ^^�^^) = ^^^^ . For instance, if a qubit is measured in the computational basis and the measurement result is |0^ , then ^^�^^ = b^{efore the measurement and the measurement}

[0135] According to processing operation 504, the density matrix is used to compute the joint entropy of at least two subsystems of qudits within the same quantum circuit. [0136] In some cases, one or more qudits from the qudits in quantum hardware is selected and denoted as a subsystem A, and one or more qudits from the rest of the qudits as a subsystem B. The quantum hardware may be of various types, such as any quantum computer disclosed elsewhere herein. In some examples, the quantum hardware may be the quantum computer 10 disclosed herein with respect to FIG.1. [0137] The joint density matrix of subsystem A and subsystem B may be denoted as ^^^^ _{^^^^ ^^^^}. This density matrix may be computed by taking the expectation value of the snapshot _{^^�^^ ^^^^ ^^^^, i. e. , ^^^^ ^^^^ ^^^^ = ^^^^} ⁽ _{^^�^^ ^^^^ ^^^^} ⁾ _{. The joint entanglement entropy is given by ^^^^ ^^^^ ^^^^ =} −tr( ^^^^ _{^^^^ ^^^^} log ^^^^ ¹ ^_{^^^ ^^^^}), and the Rényi entropy is given by ^^^^( ^^^^ _{^^^^ ^^^^}) = − ^{^^^^} ^_^^^−1 log tr _{^^^^ ^^^^} ^^^^ _{^^^^ ^^^^} . [0138] According to processing operation 506, the density matrix is used to compute the individual entropy of each of the at least two subsystems of qudits within the same quantum circuit. The individual entropy is computed from the snapshot of the density matrix for each subsystem ^^^^ _^^^^ = ^^^^( ^^�^^ _^^^^), ^^^^ _^^^^ = ^^^^⁽ ^^�^^ _^^^^ ⁾. The entanglement entropy is , and the Rényi entropy is ^^^^( ^^^^ _{^^^^( ^^^^)}) =

[0139] Now referring to FIG. 9, there is shown a schematic of a circuit used for computing the mutual information. More precisely, FIG. 9 shows a four-qubit circuit with alternating layers of single qubit rotations and CNOT gates. In this circuit, a subsystem A comprises the qubits q0 and q1, and a subsystem B comprises the qubits q2 and q₃. In the ideal situation where there is no error, subsystem A and subsystem B have no classical correlations, and no quantum correlations (including quantum entanglement). [0140] Still referring to FIG 5., in some cases, the snapshot of the density matrix may be obtained in the same way as in a processing operation such as the processing operation 306 disclosed herein with respect to FIG.3. A density matrix of subsystem A, a density matrix of subsystem B, and a density matrix of the joint system A and B are constructed. The Rényi entropies may be obtained by plugging the density matrix into the definition of the Rényi entropy, wherein the density matrix may be found using the equation ^^^^ _{^^^^ ^^^^} = ^^^^( ^^�^^ _{^^^^ ^^^^}). The joint entanglement entropy is ^^^^ _{^^^^ ^^^^} = −tr( ^^^^ _{^^^^ ^^^^} log ^^^^ _{^^^^ ^^^^}) , and the Rényi entropy

[0141] Now referring to FIG. 10, there is shown a graph of the Rényi entropy of the union of a subsystem A and a subsystem B as a function of the circuit depth and the sum of the Rényi entropies of subsystem A and subsystem B as a function of the circuit depth in the absence of crosstalk. The noise model used in the numerical analysis is the Pauli noise model. This noise model does not include any crosstalk between the two subsystems. The graph in FIG. 10 indicates that the two entropies coincide. When the circuit has crosstalk between subsystem A and subsystem B, the result changes. Now referring to FIG. 11, there is shown a graph of the Rényi entropy of the union of a subsystem A and a subsystem B as a function of the circuit depth and the sum of the Rényi entropies of subsystem A and subsystem B as a function of the circuit depth in the presence of crosstalk. More precisely, FIG.11 shows that the summation of the entropies of subsystem A and subsystem B is greater than the entropy of the union of subsystem A and subsystem B in the presence of crosstalk. Therefore, the difference in entropy

or mutual information, can be used to identify the presence of correlations between the subsystems. [0142] In some cases, quantum discord is used as a measure of quantum correlations. Quantum discord may be defined by ^^^^ = mi ^^{^^^} ^^^n^

_{− ^^^^( ^^^^ ^^^^ ^^^^) , where} _^^^^ ^^^^ _^^ ^{^} ^_^ ^{^^^} is the positive operator-valued measures in subsystem A, ^^^^ _^^^^ = tr( ^^^^ _^^ ^{^} ^_^ ^{^^^} ^^^^ _{^^^^ ^^^^}) is the probability of the ^^^^ -th outcome, and ^^^^ _{^^^^| ^^^^} = tr( ^^^^ _^^ ^{^} ^_^ ^{^^^} ^^^^ _{^^^^ ^^^^})/ ^^^^ _^^^^ is the conditional state of subsystem B. The benefit of using quantum discord is that it enables classical correlations and quantum correlations to be distinguished. [0143] Still referring to FIG. 5 and according to processing operation 508, the joint entropy and the sum of the individual entropies are compared to identify the presence of unintended entanglement between the at least two subsystems or multi-qudit error within the at least two subsystems. [0144] The joint and individual entropies may be of various types, such as any type disclosed elsewhere herein. In some cases, the joint and individual entropies are entanglement entropies. In some cases, the joint and individual entropies are the Rényi entropies. [0145] For example, a two-qubit system may be considered in which the crosstalk changes the qubit state into the Bell state (|00^ + |11^)/√2. A subsystem A is the first qubit and a subsystem B is the second qubit. The entanglement entropy of the two qubit system denoted by ^^^^( ^^^^ _{^^^^ ^^^^}) is 0, whereas the sum of the entanglement entropies of the _{individual systems denoted by ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{+ ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{is 2. Therefore, the mutual information} _^^^^ ⁽ _{^^^^ ^^^^} ⁾ _{+ ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{− ^^^^} ⁽ _{^^^^ ^^^^ ^^^^} ⁾ _{= 2 > 0 identify the presence of entanglement between the} two subsystems. [0146] The Rényi entropy of the two qubit system denoted by ^^^^⁽ ^^^^ _{^^^^ ^^^^} ⁾ is 0, whereas the _{sum of the Rényi entropies of the individual system denoted by ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{+ ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{is 2.} _{Therefore, ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{+ ^^^^} ⁽ _{^^^^ ^^^^} ⁾ _{− ^^^^} ⁽ _{^^^^ ^^^^ ^^^^} ⁾ _{= 2 > 0 , which identify the presence of} entanglement between the two subsystems. [0147] According to processing operation 510, the results of the comparison are reported. When the sum of the entropies of subsystem A and subsystem B is not equal to the entropy of the joint system consisting of A and B, this indicates the presence of quantum or classical crosstalk. In some cases, the results of the comparison may be reported using a digital computer. The digital computer may be of various types, such that the digital computer 8 disclosed with respect to FIG.1. [0148] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

CLAIMS: 1. A method for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said method comprising: (a) preparing one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) executing said one or more quantum circuits on said quantum hardware; (c) performing a quantum measurement on at least a portion of said group of qudits; (d) performing an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits based at least in part on a result of said quantum measurement; and (e) outputting a result of said analysis, wherein said result comprises an amount of crosstalk.

2. The method of claim 1, wherein (b) and (c) are repeated at least one time.

3. The method of claim 1, wherein (b) comprises executing at least one copy of at least one quantum circuit of said one or more quantum circuits.

4. The method of claim 1, wherein said crosstalk of said plurality of subsystems of qudits within said one or more quantum circuits comprises at least one member of the group consisting of: unintended entanglement between said subsystems, unintended quantum correlations between said subsystems, and unintended classical correlations between said subsystems.

5. The method of claim 1, wherein said crosstalk comprises one of coherent noise and incoherent noise.

6. The method of claim 1, wherein (c) comprises performing said quantum measurement in one of a Z-basis, an X-basis, and a Y-basis.

7. The method of claim 1, wherein (c) comprises applying a unitary operation and performing said quantum measurement in a Z-basis.

8. The method of claim 1, wherein said plurality of subsystems of qudits comprises one of an error correction code and a plaquette in an error correction code.

9. The method of claim 8, further comprising using said result for a parity check on said plaquette in said error correction code.

10. The method of claim 8, further comprising using said result for one round of error correction in said quantum hardware.

11. The method of claim 8, wherein said error correction code comprises CSS code, surface code, colour code, triangular colour code, rotated surface code, or toric code.

12. The method of claim 1, wherein (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one subsystem of said plurality of subsystems of qudits, computing a corresponding partially transposed matrix of said density matrix; (iii) computing eigenvalues of said partially transposed matrix; (iv) computing a negativity of said partially transposed matrix using said eigenvalues; and (v) reporting entanglement presence in case of positive negativity and entanglement absence in case of zero negativity.

13. The method of claim 1, wherein (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one of said plurality of subsystems of qudits computing a corresponding partially transposed matrix of said density matrix; (iii) computing at least a second and a third moment of said partially transposed matrix; (iv) comparing functions of two successive moments of said at least said second and said third moments to identify said unintended entanglement presence between said at least one subsystem of said plurality of subsystems and other subsystems; and (v) reporting said entanglement presence if identified.

14. The method of claim 1, wherein (d) comprises: (i) constructing a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) based at least in part on said density matrix, computing a joint entropy of a plurality of subsystems of qudits belonging to a same quantum circuit; (iii) based at least in part on said density matrix, computing an individual entropy of each subsystem of said plurality of subsystems of qudits belong to said same quantum circuit; (iv) comparing said joint entropy with a sum of said individual entropies to identify presence of unintended entanglement of said plurality of subsystems or multi-qudit error within said plurality of subsystems; and (v) reporting comparison results.

15. The method of claim 14, wherein said joint entropy and said individual entropy comprise at least one member of the group consisting of: entanglement entropy, Rényi entropy, and moments of density matrices of the corresponding subsystems; further wherein the density matrices of the corresponding subsystems are computed using said density matrix of a quantum circuit.

16. The method of claim 1, wherein said one or more quantum circuits comprise a parity check quantum circuit.

17. The method of claim 16, wherein (b) comprises preparing a quantum state in a computational basis, and implementing said parity check quantum circuit at least one time.

18. The method of claim 1, further comprising using said result comprising said amount of crosstalk to adjust an architecture of said quantum hardware for further use in fabricating said quantum hardware.

19. The method of claim 1, further comprising: (i) setting control signal characteristics of said qudits based at least in part on said result comprising said amount of crosstalk; and (ii) controlling said qudits in accordance with said control signal characteristics.

20. A method for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said method comprising: (a) preparing one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) executing said one or more quantum circuits on said quantum hardware; (c) performing a quantum measurement on at least a portion of said group of qudits; and (d) directing a result of said quantum measurement to a computing device configured to perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, wherein said remote device is configured to output a result of said analysis, wherein said result comprises an amount of crosstalk.

21. A method for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said method comprising: (a) at a computing device in communication with said quantum hardware, providing an indication to prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits; (b) providing instructions to execute said one or more quantum circuits on said quantum hardware; (c) providing instructions to perform a quantum measurement on at least a portion of said group of qudits; (d) based at least in part on a result of said quantum measurement, performing an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits; and (e) outputting a result of said analysis, wherein said result comprises an amount of crosstalk.

22. The method of claim 20 or 21, wherein said computing device is a digital computer operatively coupled to said quantum hardware.

23. The method of claim 22, wherein said digital computer is communicatively coupled over a network.

24. The method of claim 22, wherein said digital computer is local to said quantum hardware.

25. A system for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said system comprising: (a) a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system; (b) a digital computer in communication with said quantum computer, said digital computer comprising a memory having instructions to at least (A) provide an indication to said control system prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (B) provide instruction to said control system to execute said one or more quantum circuits on said quantum hardware, (C) provide instructions to said control system to perform a quantum measurement on said qudits, (D) receive a result from said quantum computer and, based at least in part on said result of said quantum measurement, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (E) output a result of said analysis, wherein said result comprises an amount of crosstalk.

26. The system of claim 25, wherein said digital computer is configured to repeat (B) and (C) at least one time.

27. The system of claim 25, wherein at (B) said quantum hardware is directed to execute at least one copy of at least one quantum circuit of said one or more quantum circuits.

28. The system of claim 25, wherein said crosstalk of said plurality of subsystems of qudits within said one or more quantum circuits comprises at least one member of the group consisting of: unintended entanglement between said subsystems, unintended quantum correlations between said subsystems, and unintended classical correlations between said subsystems.

29. The system of claim 25, wherein said crosstalk comprises one of coherent noise and incoherent noise.

30. The system of claim 25, wherein, at (C), said quantum hardware is configured to perform said quantum measurement in one of a Z-basis, an X-basis, and a Y-basis.

31. The system of claim 25, wherein, at (C), said quantum hardware is configured to apply a unitary operation and performing said quantum measurement in a Z-basis.

32. The system of claim 25, wherein said plurality of subsystems of qudits comprises one of an error correction code and a plaquette in an error correction code.

33. The system of claim 32, wherein said instructions further comprise using said result for a parity check on said plaquette in said error correction code..

34. The system of claim 32, wherein said instructions further comprise using said result for one round of error correction in said quantum hardware.

35. The system of claim 32, wherein said error correction code comprises CSS code, surface code, colour code, triangular colour code, rotated surface code, or toric code.

36. The system of claim 25, wherein, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one subsystem of said plurality of subsystems of qudits, compute a corresponding partially transposed matrix of said density matrix; (iii) compute eigenvalues of said partially transposed matrix; (iv) compute a negativity of said partially transposed matrix using said eigenvalues; and (v) report entanglement presence in case of positive negativity and entanglement absence in case of zero negativity.

37. The system of claim 25, wherein, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result of said quantum measurement; (ii) for at least one of said plurality of subsystems of qudits compute a corresponding partially transposed matrix of said density matrix; (iii) compute at least a second and a third moment of said partially transposed matrix; (iv) compare functions of two successive moments of said at least said second and said third moments to identify said unintended entanglement presence between said at least one subsystem of said plurality of subsystems and other subsystems; and (v) report said entanglement presence if identified.

38. The system of claim 25, wherein, at (D), said digital computer is configured to: (i) construct a density matrix of a quantum system for each of said one or more quantum circuits based at least in part on said result; (ii) based at least in part on said density matrix, compute a joint entropy of a plurality of subsystems of qudits belonging to a same quantum circuit; (iii) based at least in part on said density matrix, compute an individual entropy of each subsystem of said plurality of subsystems of qudits belong to said same quantum circuit; (iv) compare said joint entropy with a sum of said individual entropies to identify presence of unintended entanglement of said plurality of subsystems or multi-qudit error within said plurality of subsystems; and (v) report comparison results.

39. The system of claim 38, wherein said joint entropy and said individual entropy comprise at least one member of the group consisting of: entanglement entropy, Rényi entropy, and moments of density matrices of the corresponding subsystems; further wherein the density matrices of the corresponding subsystems are computed using said density matrix of a quantum circuit.

40. The system of claim 25, wherein said one or more quantum circuits comprise a parity check quantum circuit.

41. The system of claim 40, wherein, at (B), said digital computer is configured to provide an indication to prepare a quantum state in a computational basis and implement said parity check quantum circuit at least one time.

42. The system of claim 25, wherein said digital computer is configured to use said result comprising said amount of crosstalk to adjust an architecture of said quantum computer and, based at least in part on said architecture for further use in fabricating said quantum hardware.

43. The system of claim 25, wherein said control system is configured to: (i) set control signal characteristics of said qudits based at least in part on said result comprising said amount of crosstalk; and (ii) control said qudits in accordance with said control signal characteristics.

44. A system for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said system comprising: a digital computer in communication with a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system, said digital computer comprising a memory having instructions to at least (C) provide an indication to said control system to prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (D) provide instruction said control system to execute said one or more quantum circuits on said quantum hardware, (E) provide instructions to said control system to perform a quantum measurement on said qudits, (F) receive a result of said quantum measurement and, based at least in part on said result, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (G) output a result of said analysis, wherein said result comprises an amount of crosstalk.

45. A system for characterizing an amount of crosstalk in a group of qudits in quantum hardware, said system comprising: a quantum computer comprising (A) a quantum chip comprising qudits, and (B) a control system, wherein said quantum computer is in communication with a digital computer, wherein said digital computer comprising a memory having instructions to at least (A) provide an indication to said control system prepare one or more quantum circuits on said quantum hardware, wherein said one or more quantum circuits each comprises a plurality of subsystems of qudits, (B) provide instruction to said control system to execute said one or more quantum circuits on said quantum hardware, (C) provide instructions to said control system to perform a quantum measurement on said qudits, (D) receive a result from said quantum computer and, based at least in part on said result of said quantum measurement, perform an analysis of crosstalk of said plurality of subsystems of qudits within each of said one or more quantum circuits, and (E) output a result of said analysis, wherein said result comprises an amount of crosstalk.