Matrix Product State - based Decoder for Stabilizer Codes under Device Noise for Quantum Computing and Information Processing

Abstract

A strengthened matrix product state-based decoder is generated and adopted to almost optimally detect and correct errors in a quantum computing and information processing system. The decoder takes as input a detection level error model that describes a set of physical error channels and error detection. This error model is improved using experimental data.

Description

Technical Field

[0001] Priority Claim This application claims the benefit of priority of U.S. Provisional Patent Application No. 63 / 389,165, filed Jul. 14, 2022, entitled "MATRIX PRODUCT STATE-BASED DECODERS FOR STABILIZER CODES UNDER DEVICE NOISE FOR QUANTUM COMPUTING AND INFORMATION PROCESSING", which is incorporated herein by reference in its entirety. This application also claims the benefit of priority of U.S. Provisional Patent Application No. 63 / 436,313, filed Dec. 30, 2022, entitled "MATRIX PRODUCT STATE-BASED DECODERS FOR STABILIZER CODES UNDER DEVICE NOISE FOR QUANTUM COMPUTING AND INFORMATION PROCESSING", which is incorporated herein by reference in its entirety.

[0002] The present disclosure generally relates to quantum computing systems, and more particularly, to decoders for quantum error correction.

Background Art

[0003] Quantum computing is a computational method that utilizes quantum effects, such as superposition and entanglement of the ground state, to perform certain calculations more efficiently than classical digital computers. In contrast to digital computers that store and manipulate information in the form of bits, such as "1" or "0", a quantum computing system can manipulate information using quantum bits ("qubits"). A qubit may refer to a quantum device that enables superposition of data in multiple states, such as both the "0" and "1" states, and / or the superposition itself of data in multiple states. According to conventional terminology, the superposition of the "0" state and the "1" state in a quantum system can be expressed, for example, as

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Summary of the Invention

[0004] Aspects and advantages of embodiments of the present disclosure are shown in part in the following description, or can be learned from the description, or can be learned through the practice of the embodiments.

[0005] One exemplary aspect of the present disclosure is directed to a method of operating a quantum computing system (QCS). More specifically, embodiments may be used to detect and / or correct quantum errors that occur in a QCS. The method may include accessing an error model of the QCS. The error model may encode a set of error channels corresponding to a set of quantum error modes. A set of detected quantum errors may be accessed. The set of detected quantum errors may be encoded in a set of experimental measurements performed in the QCS. A tensor network may be generated. The tensor network may encode a correlation between quantum error modes of the set of quantum error modes and detected quantum errors of the set of detected quantum errors. A matrix product state (MPS) may be generated based on the tensor network. To generate the MPS, the tensor network may be contracted via tensor contraction. A fault tolerant system having a decoder and an error correction code (e.g., a code) may be generated for the QCS. At least, the decoder of the fault tolerant system may be based on the MPS. That is, the MPS may affect the decoder of the fault tolerant system. The error correction code may be deployed to the QCS. When deployed, the error correction code may be employed to detect errors that occur during the runtime of the QCS. The error correction code may be employed to correct the detected errors.

[0006] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0007] These and other features, aspects, and advantages of the various embodiments of the present disclosure will become better understood with reference to the following detailed description of the invention and the accompanying claims. The accompanying drawings, which are incorporated herein and constitute a part of this specification, illustrate exemplary embodiments of the present disclosure and, together with the detailed description, serve to explain the relevant underlying principles.

[0008] A detailed description of embodiments directed to those skilled in the art is described in this specification with reference to the following attached drawings.

Brief Description of the Drawings

[0009]

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Modes for Carrying Out the Invention

[0010] Exemplary aspects of the present disclosure are directed to an enhanced system and method for calibrating composite quantum gates (e.g., two-qubit quantum gates) in a quantum computing system. Quantum gates can be the building blocks of quantum circuits implemented by a quantum computing system for quantum computing and quantum information processing. As used herein, the term "composite quantum gate" may refer to an object (e.g., a quantum circuit, device, and / or structure) that performs an operation on multiple input qubits (e.g., two qubits, three qubits). Throughout, when discussing composite quantum gates, the terms "quantum gate" and simply "gate" may refer to a composite quantum gate. These two terms may also be further adopted to refer to single-qubit quantum gates (e.g., quantum gates that perform an operation on a single qubit). When a distinction is necessary in the discussion, the terms composite gate and single-qubit gate may be adopted. Various embodiments and physical implementations of composite quantum gates are discussed in "Demonstrating a Continuous Set of Two-Qubit Gates for Near-Term Quantum Algorithms, Phys. Rev. Lett. 125, 120504 (2020)" by Foxen et al., the contents of which are hereby incorporated by reference in their entirety. This incorporated paper can be viewed on the archive server at https: / / arxiv.org / abs / 2001.08343.

[0011] Quantum computing and information processing (QC&IP) may require separating the wave function of the physical manifestation of an information encoding mechanism (e.g., qubits) from other parts of the universe. For example, the quantum states associated with the qubits of a system are relatively fragile because they are exposed to decoherence. Thus, the noise threshold that a QC&IP system can tolerate is relatively low. Further, quantum error correction (QEC) can be an important feature of a QC&IP system in some embodiments. A QEC mechanism detects and corrects errors in quantum signals, where the quantum signals are encoded in the "interference" pattern of coherent qubits. That is, when a QEC-enabled system detects an error, rather than simply re-executing the computation, the system may deploy a mechanism to correct the detected error. A QEC code may provide such a mechanism. Classical error correction codes (e.g., Hamming codes) typically encode redundant bits of information of the signal being processed. This redundant information enables the reconstruction of classical information that is corrupted during storage, transmission, and / or processing of the signal.

[0012] QEC codes are somewhat similar to classical error correction codes. However, in contrast to encoding redundant information in "extra" bits, QEC encoding may "spread" or distribute information across multiple qubits through entanglement of the quantum states of the qubits. The act of "spreading" a signal across multiple qubits (e.g., encoding the information to be processed) is generally called encoding. The act of extracting the information that has been spread across multiple qubits is generally called decoding. Thus, encoding and decoding are important aspects in implementing a QEC protocol for building an error-corrected quantum computer.

[0013] A quantum decoder is generally responsive to estimating whether a logical error has occurred in a QC&IP system. More specifically, the decoder takes as input a particular set of experimental measurements and an error model. The decoder can be (or generate) a probabilistic description of the likelihood of an error occurring in the QC&IP system. The probabilistic description can be based on the set of experimental measurements and the error model. The individual elements of the set of experimental measurements may be referred to as detection events and / or detectors.

[0014] Exemplary embodiments are directed to a reinforced decoder that uses tensor contraction to generate a way to probabilistically describe, almost exactly, situations in which errors are likely to occur in a QC&IP system. Mapping a set of detection events and an error model to a description of the error can be computationally complex. For example, the computational resources required to analytically compute this mapping scale exponentially as the size of the QEC code increases linearly.

[0015] Rather than directly computing this mapping, conventional decoders often employ heuristics to generate an approximate mapping. Such heuristics often result in non-ideal QEC performance. For example, a conventional decoder may attempt to generate the mapping by searching for a valid error configuration that matches the set of experimental measurements. Such conventional decoders generate the mapping via graph matching techniques. These conventional decoders generate only an approximate mapping, since only a subset of the error configurations that match the set of experimental measurements are mapped.

[0016] In contrast to conventional decoders, exemplary embodiments of the present disclosure employ a tensor network, more specifically, a matrix product state. As will be described below, the exemplary embodiments generate and evaluate a tensor network via tensor contraction (e.g., contraction of one or more indices in a tensor and / or tensor product). The present embodiments include a matrix product state decoder that can decode almost optimally. The matrix product state decoder can also function as a framework for an approximate decoder. The enhanced decoder takes as input a detector-level error model that describes the physical error channel. This error model can be improved using experimental data.

[0017] Aspects of the present disclosure provide several technical effects and advantages. For example, the enhanced decoder can map a physical error channel and a detector-level error model that describes how they affect which measurements to a tensor network. Conventional decoders have only been successful with finely tuned error models. The resulting tensor network can be evaluated almost exactly (via tensor contraction) by an enhanced algorithm. This algorithm works by mapping irregular tensor network contractions to an efficient matrix product state (MPS) evolution. The enhanced MPS includes hypotheses (e.g., initial educated guesses) for encoding a probability distribution and logical bit information associated with detectors (e.g., experimental measurements or detection events). The hypotheses can be updated (e.g., trained) via machine learning techniques. Once trained, the hypotheses may be used to efficiently evaluate the likelihood of different logical measurement output results. The likelihoods (or probabilities) of different logical measurement output results can then be used to determine whether a logical error has occurred. As input, the decoder takes in an error model (e.g., encoding of the probabilities of different measurement output results). The error model may be updated (or trained) iteratively by leveraging additional experimental data and MPS hypotheses.

[0018] FIG. 1 shows an exemplary quantum computing system 100. System 100 is an example of a system of one or more classical computers and / or quantum computing devices at one or more locations, and may implement the systems, components, and techniques described below. Those skilled in the art will understand that other quantum computing devices or systems may be used without departing from the scope of the present disclosure using the disclosure provided herein.

[0019] System 100 includes quantum hardware 102 that communicates data with one or more classical processors 104. The classical processor 104 may be configured to execute computer-readable instructions stored in one or more memory devices to perform operations such as any of the operations described herein. The quantum hardware 102 includes components for performing quantum computing. For example, the quantum hardware 102 includes a quantum system 110, one or more control devices 112, and one or more readout devices 114 (e.g., readout resonators). The quantum system 110 may include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubit 120). In some embodiments, the multi-level quantum subsystem may include superconducting qubits such as flux qubits, charge qubits, transmon qubits, gmon qubits, and spin-based qubits.

[0020] The type of multi-level quantum subsystem utilized by system 100 can vary. For example, in some cases, it may be convenient to include one or more readout devices 114 (plural) attached to one or more superconducting qubits, such as transmon, flux, Gmon, Xmon, or other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (which can prepare states without the need for qubits) may be used. Further examples of multi-level quantum subsystems include fluxonium qubits, silicon quantum dots, or phosphorus impurity qubits.

[0021] A quantum circuit can be constructed and applied to a register of qubits included within quantum system 110 via a plurality of control lines coupled to one or more control devices 112. Exemplary control devices 112 operating on the qubit register can implement and use quantum gates or quantum circuits having a plurality of quantum gates, such as Pauli gates, Hadamard gates, controlled NOT (CNOT) gates, controlled phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. One or more control devices 112 can be configured to operate within quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some embodiments, the multi-level quantum subsystem can be a superconducting qubit, and the control device 112 can be configured to provide control pulses to the control lines to generate a magnetic field to adjust the frequency of the qubit.

[0022] The quantum hardware 102 may further include a readout device 114 (e.g., a readout resonator). Measurement results 108 obtained via a measurement device may be provided to a classical processor 104 for processing and analysis. In some embodiments, the quantum hardware 102 can include a quantum circuit, and the control device(s) 112 and readout device(s) 114 may implement one or more quantum logic gates operating on the quantum system 102 through physical control parameters (e.g., microwave pulses) that are transmitted through wires included in the quantum hardware 102. Further examples of control devices include any waveform generator where a digital-to-analog converter (DAC) creates a signal.

[0023] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and transmit the measurement results 108 to the classical processor 104. Further, the quantum hardware 102 may be configured to receive data from the classical processor 104 that specifies physical control qubit parameter values 106. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the operation of the control device(s) 112 and readout device(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing the voltage strengths of one or more DACs included within the control device 112 and, in response, update the action of the DAC on the quantum system 110. The classical processor 104 may be configured to initialize the quantum system 110 to an initial quantum state, for example, by transmitting data to the quantum hardware 102 that specifies an initial set of the parameters 106.

[0024] In some embodiments, the readout device(s) 114 are of elements of a quantum system, such as qubits

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[0025] In some embodiments, the quantum system 110 may include, for example, a plurality of qubits 120 arranged within a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in FIG. 1 includes 4x4 qubits, but in some embodiments, the system 110 may include fewer or more qubits. In some embodiments, the plurality of qubits 120 may interact through a multi-qubit coupler, such as qubit coupler 124. The qubit coupler may define the nearest-neighbor interactions between the plurality of qubits 120. In some embodiments, the strength of the multi-qubit coupler is an adjustable parameter. In some cases, the multi-qubit coupler included in the quantum computing system 100 may be a coupler with a fixed coupling strength.

[0026] In some embodiments, the plurality of qubits 120 may include data qubits such as qubit 126 and measurement qubits such as qubit 128. Data qubits are qubits that participate in the calculations being performed by system 100. Measurement qubits are qubits that can be used to determine the output results of the calculations performed by the data qubits. That is, during the calculation, the unknown state of the data qubits is transmitted to the measurement qubits using appropriate physical operations and measured via appropriate measurement operations performed on the measurement qubits.

[0027] In some embodiments, each qubit of the plurality of qubits 120 may be operated using respective operating frequencies such as an idling frequency and / or an interaction frequency and / or a readout frequency and / or a reset frequency. The operating frequencies can vary from qubit to qubit. For example, each qubit may idle at a different operating frequency. The operating frequencies of qubits 120 may be selected before the calculations are performed.

[0028] FIG. 1 shows one exemplary quantum computing system that can be used to implement the methods and operations according to the exemplary aspects of the present disclosure. Other quantum computing systems may be used without departing from the scope of the present disclosure.

[0029] FIG. 2 shows a visual depiction of an exemplary error model 200 according to an exemplary embodiment of the present disclosure. As shown in FIG. 2, the error model 200 can be encoded via a bipartite graph 210 that includes an error channel 220 and a detector 230. The error channel 220 includes graph nodes on the left side of the graph 210, and the detector 230 includes graph nodes on the right side of the graph 200. The error channel 220 can be a physical error channel. The error model 200 can be a detector-level error model that encodes a physical error channel 220 (e.g., corresponding to the nodes on the left side of the bipartite graph). The error model 200 can further encode a detector 230 (e.g., the corresponding nodes on the right side of the bipartite graph) where each of the error channels affects 220. The bipartite graph 200 may be referred to as a circuit-level Tanner graph.

[0030] As shown in FIG. 2, each error channel 220 (e.g., the left nodes or vertices) can have an associated probability. The detectors 230 (e.g., the right nodes (or vertices)) are indexed such that D0 through DN correspond to detector (or measurement) variables. Thus, the right nodes (or vertices) indexed as D0 through DN may be referred to as detector bits. The right nodes (s) indexed as L0 through LM correspond to logical bit variables. The right nodes indexed as L0 through LM may be referred to as logical bits. The logical bit variables can change their values by activating the error channels connected to them. Each detector or logical bit changes its value if an odd number of adjacent error channels are activated.

[0031] On the right side of FIG. 2, a decoding process 240 is shown. More specifically, the sum of the probabilities of all possible error configurations can match the outputs of some detectors and logical bits and is equal to the likelihood of that configuration. By comparing the likelihoods corresponding to all possible output results of all logical measurements (e.g., comparison of likelihoods 250), the likelihood of the maximum likelihood logical bits can be estimated. Thus, it is possible to infer whether a logical error has occurred with a certain reliability. The bipartite graph 210 in FIG. 2 may be representative of a multi-dimensional tensor network. Calculating this exponentially large sum of probabilities involves contracting the underlying tensor network (e.g., via tensor contraction). A protocol can be adopted to efficiently contract this tensor network. The protocol converts the multi-dimensional tensor network in FIG. 2 into a circuit corresponding to the development of a one-dimensional tensor network called a matrix product state (MPS) or tensor train.

[0032] FIG. 3 shows an example of local correlations between detectors within an error model. The locality of the correlations between detectors indicates a finite maximum bond dimension that is used to parameterize the contraction of the tensor network, as will be explained below.

[0033] Focusing on FIG. 4, it shows that the multi-dimensional tensor network in FIG. 2 (e.g., graph 210) efficiently contracts to a one-dimensional matrix product state (MPS) 460. The circuit diagram 410 in FIG. 4 encodes the tensor network (e.g., graph 210 in FIG. 2). In the circuit diagram 410, the nodes on the left side of graph 210 correspond to the error channel 420 of the tensor network in FIG. 2, while the vertical lines represent the detector bits and logical bits of the underlying tensor network, which are labeled with L and D to indicate the corresponding logical bits and detector bits. The set of error channels 420 forms a one-dimensional MPS 460 that can be developed from left to right by contracting with the detector. By setting the maximum bond dimension that allows the development of MPS 460 to involve contraction, the tensor network can be approximately evaluated when a series of detection events and logical bit values are given. In practice, the bond dimensions required for small to medium-sized error correction codes are not very large.

[0034] In related decoding techniques, the contraction of this tensor network is utilized to generate an MPS over detector and logical bit variables before the corresponding values are assigned. The MPS can be directly used as a hypothesis for their joint probability distribution over these variables. It is also possible to train this hypothesis to match experimental data.

[0035] FIG. 5 shows a constructive approach for generating matrix product states that approximate the binding probability according to various embodiments. Accordingly, the present embodiment can be adopted to implement a decoder that is closest (e.g., enhanced) to mapping the prior distribution to a tensor network. Various embodiments of the prior distribution are discussed in "Suppressing Quantum Error by Scaling a Surface Code Logical Qubit, arXiv:2207.06431v2 [quant-ph], (July 2021)" by Acharya et al., the contents of which are incorporated herein by reference in their entirety. This incorporated paper can be viewed on the archive server https: / / arxiv.org / abs / 2207.06431v2.

[0036] Given the configuration of detection events and the selection of changes in the logical frame, the contraction of this tensor network can be used to estimate the binding probability. This is done by summing the probabilities of all error configurations that are compatible with the set of detection events and logical frame changes considered. This decoder infers a more reasonable change in the logical frame by comparing the likelihoods of both output results. If there is more than one logical qubit, the decoder infers the most reasonable change in the logical frame from among all 2(# logical qubits) output results. In contrast to conventional decoders, the embodiment uses device-level noise as input correlation instead of gate-level noise.

[0037] The resulting contraction complexity of the tensor network increases exponentially with d 2 where d is the distance of the code. In practice, the contraction is approximated by matrix product state (MPS) evolution (shown in FIGS. 4-5) using a finite maximum bond dimension χ. The complexity of this approximate contraction increases with χ 3 and.

[0038] FIG. 6 shows experimental data demonstrating the convergence of the logical error probability as a function of the coupling distance. That is, plot 600 in FIG. 6 shows the convergence of the logical error probability as a function of the parameter χ. As shown in FIG. 6, the convergence 610 of plot 600 occurs at approximately χ = 30. This convergence 610 is achieved by successively contracting the tensor network from the error channel tensor towards the detector tensor. Thus, in various embodiments, a coupling parameter value of χ = 30 may be employed to decode the error.

[0039] When generating plot 600 in FIG. 6, the logical error probability of an experiment with 50,000 shots at distance 5, Z basis, and 25 rounds is used as a function of the maximum bond dimension χ used in the tensor network contraction. Error bars indicate the standard error of the mean. It is observed that the logical error probability stabilizes at χ = 30.

[0040] FIG. 7 shows a flowchart of an exemplary method 700 for detecting and correcting quantum errors in a quantum computing system according to an exemplary embodiment of the present disclosure. Method 700 begins at block 702 where a quantum error model may be accessed. The quantum error model may encode a set of quantum error channels. The quantum error model may further encode one or more relationships between the quantum error channels and the output results of the detectors. At block 704, a set of output results of the detectors is accessed. The set of detected quantum errors may include a set of experimental measurements. At block 706, a tensor network is generated. The tensor network may encode the correlation between the modes of the quantum errors and the output results of the detectors. The correlation may be based on the quantum error model and the measured output results of the detectors. At block 708, a matrix product state (MPS) protocol may be generated based on the contraction of the tensor network. At block 710, the MPS protocol may be deployed to infer the presence of one or more quantum errors in the quantum computing system.

[0041] Additional embodiments One exemplary aspect of the present disclosure is directed to a method of operating a quantum computing system (QCS). More specifically, embodiments may be used to detect and / or correct quantum errors that occur in a QCS. The method may include accessing an error model of the QCS. The error model may encode a set of error channels corresponding to a set of quantum error modes. A set of detected quantum errors may be accessed. The set of detected quantum errors may be encoded in a set of experimental measurements performed in the QCS. A tensor network may be generated. The tensor network may encode a correlation between the quantum error modes of the set of quantum error modes and the detected quantum errors of the set of detected quantum errors. A matrix product state (MPS) may be generated based on the tensor network. To generate the MPS, the tensor network may be contracted via tensor contraction. A fault-tolerant system having a decoder and an error correction code (e.g., a code) may be generated for the QCS. At least, the decoder of the fault-tolerant system may be based on an MPS protocol. That is, the MPS protocol may affect the decoder of the fault-tolerant system. The error correction code may be deployed to the QCS. When deployed, the error correction code may be employed to detect errors that occur during the runtime of the QCS. The error correction code may be employed to correct the detected errors.

[0042] In at least one embodiment, the MPS protocol can be deployed to infer errors that occur during the runtime of the QCS. The MPS protocol can be deployed to correct errors that occur during the runtime of the QCS. The MPS protocol can be generated by contracting a tensor network. Contracting the tensor work can include summing probabilities associated with a subset of error modes that are compatible with a detected set of quantum errors. Generating the MPS protocol can be parameterized by a maximum bond dimension. The maximum bond dimension can be set to a value of about 30. The MPS protocol can be generated by adopting a hypothesis that encodes a probability distribution associated with a detected set of quantum errors and logical bit information. The method can further include training the hypothesis.

[0043] In various embodiments, the method further includes encoding an error model and a detected set of quantum errors in a bipartite graph. A circuit diagram can be generated. The bipartite graph can be generated based on the generated circuit diagram. Generating the circuit diagram can be based on the bipartite graph. The MPS protocol can be generated based on the circuit diagram.

[0044] The digital, classical, and / or quantum topics described herein, as well as the embodiments of digital functional operations and quantum operations, can be implemented in digital electronic circuits, appropriate quantum circuits, or more generally, in a quantum computing system, tangibly implemented digital and / or quantum computer software or firmware, digital and / or quantum computer hardware including the structures disclosed herein and their structural equivalents, or one or more combinations thereof. The term "quantum computing system" can include, but is not limited to, a quantum computer / computing system, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0045] The digital and / or quantum subject matter embodiments described herein may be implemented as one or more digital and / or quantum computer programs, i.e., as one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits or a single qubit, or a combination of one or more of them. Alternatively or in addition, the program instructions may be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing apparatus.

[0046] The terms quantum information and quantum data refer to information or data transmitted by, stored in, or held within a quantum system, and the smallest non-trivial system is the qubit, i.e., the system that defines the unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational basis states are identified with the ground state and the first excited state, but it is understood that other setups are possible where the computational states are identified with higher levels of excitation (e.g., qubits).

[0047] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and includes, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, or multiple digital and quantum processors or computers, and combinations thereof, and all kinds of devices, apparatuses, and machines for processing digital and / or quantum data. The apparatus may also be, or further include, special purpose logic circuits, such as FPGAs (field programmable gate arrays), ASICs (application specific integrated circuits), or quantum simulators, i.e., quantum data processing apparatuses designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the ability to perform universal quantum computing. The apparatus may also optionally include, in addition to the hardware, code that creates an execution environment for digital and / or quantum computer programs, such as processor firmware, protocol stacks, database management systems, operating systems, or code that constitutes one or more combinations thereof.

[0048] A digital or classical computer program may also be referred to or described as a program, software, software application, module, software module, script, or code, and can be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, and it can be deployed in any form, either as a stand-alone program or in the form of a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program may also be called a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, can be converted into an appropriate quantum programming language, or can be written in a quantum programming language, such as QCL, Quipper, Cirq, etc.

[0049] Digital and / or quantum computer programs may correspond to files in a file system, but they do not necessarily have to. The program may be stored in a part of a file that holds other programs or data, such as one or more scripts stored in a document in a markup language, in a single file dedicated to the program of interest, or in multiple cooperating files, such as files that store one or more modules, subprograms, or portions of code. Digital and / or quantum computer programs may be deployed to be executed on one digital or one quantum computer, or on multiple digital and / or quantum computers located in one place, or on multiple computers distributed across multiple locations and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system, such as qubits. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0050] The processes and logical flows described herein may be implemented by one or more programmable digital and / or quantum computers with one or more digital and / or quantum processors, and, as necessary, executing one or more digital and / or quantum computer programs that perform functions by operating on input digital and quantum data to generate output. The processes and logical flows may also be implemented by special purpose logic circuits, such as FPGAs or ASICs, or quantum simulators, or by a combination of special purpose logic circuits or quantum simulators and one or more programmed digital and / or quantum computers, and the apparatus may be implemented as special purpose logic circuits, such as FPGAs or ASICs, or quantum simulators.

[0051] A system of one or more digital and / or quantum computers or processors being "configured to" or "operable to" perform a particular operation or action means that the system has installed thereon software, firmware, hardware, or a combination thereof that causes the system to perform the operation or action during operation. One or more digital and / or quantum computer programs being configured to perform a particular operation or action means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer may receive from a digital computer instructions that, when executed by a quantum computing device, cause the device to perform an operation or action.

[0052] Digital and / or quantum computers suitable for the execution of digital and / or quantum computer programs may be based on general-purpose or special-purpose digital and / or quantum microprocessors, or both, or any other kind of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data, such as photons, or a combination thereof.

[0053] Some exemplary elements of a digital and / or quantum computer are a central processing unit for implementing or executing instructions and one or more memory devices for storing instructions as well as digital and / or quantum data. The central processing unit and the memory can be supplemented by or incorporated into special-purpose logic circuits or quantum simulators. Generally, a digital and / or quantum computer is operatively coupled to include, receive from, transfer digital and / or quantum data to, or do both to one or more mass storage devices for storing digital and / or quantum data, such as, for example, magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0054] Digital and / or quantum computer program instructions and digital and / or quantum computer-readable media suitable for storing digital and / or quantum data include, by way of example, all forms of non-volatile digital and / or quantum memories, media, and memory devices, including semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices, magnetic disks, such as internal hard disks or removable disks, magneto-optical disks, and CD-ROM and DVD-ROM disks, as well as quantum systems, such as trapped atoms or electrons. It is understood that a quantum memory is a device capable of storing quantum data for a long time with high fidelity, such as, for example, an optical-matter interface where light is used for transmission and matter is used for storing and preserving quantum properties of quantum data such as superposition or quantum coherence.

[0055] The various systems described herein, or control of portions thereof, can be implemented as a digital and / or quantum computer program product comprising instructions stored on one or more tangible, non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems described herein, or portions thereof, can each be implemented as an apparatus, method, or electronic system comprising one or more digital and / or quantum processing devices and memory for storing executable instructions for performing the operations described herein.

[0056] Although this specification contains many detailed examples, these should not be construed as limiting the scope of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments. The specific features described in the context of individual embodiments may also be implemented in combination within a single embodiment. Conversely, the various features of the invention described in the context of a single embodiment may also be provided separately or in any suitable sub-combination in a plurality of embodiments. Furthermore, features may be described above as acting in certain combinations and may initially be claimed as such, but one or more features from a claimed combination may in some cases be excised from that combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0057] Similarly, the operations are presented in the drawings in a particular order, but this should not be construed as requiring that the operations be performed in the particular order or sequence shown, or that all of the illustrated operations be performed, to achieve the desired result. In certain circumstances, multitasking and parallel processing may be advantageous. Further, the separation of the various system modules and components in the above-described embodiments should not be construed as requiring such separation in all embodiments, and the above-described program components and systems may generally be integrated together into a single software product or packaged into multiple software products as should be understood.

[0058] Particular embodiments of the invention have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims may be performed in a different order and still result in a desirable outcome. As one example, the process shown in the accompanying figures need not be in the particular order or series of orders shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for operating a quantum computing system (QCS), comprising: accessing an error model of the QCS, the error model encoding a set of error channels and a relationship with output results of detectors; accessing a set of measured detector output results; generating a tensor network encoding a correlation between the set of quantum error modes of the set of quantum error modes and the set of measured detector output results; generating a matrix product state (MPS) protocol based on the tensor network.

2. The method of claim 1, further comprising deploying the MPS protocol to detect errors occurring during runtime of the QCS.

3. The method of claim 2, further comprising deploying the MPS protocol to correct errors occurring during runtime of the QCS.

4. The method of claim 1, wherein the MPS protocol is generated by contracting the tensor network.

5. The method of claim 4, wherein contracting the tensor work comprises summing probabilities associated with a subset of the error modes that are compatible with the detected set of quantum errors.

6. The method of claim 1, wherein generating the MPS protocol is parameterized by a maximum bond dimension.

7. The method of claim 6, wherein the maximum bond dimension is set to a value of about 30.

8. The method of claim 1, wherein the MPS protocol is generated by adopting a hypothesis encoding a probability distribution associated with the detected set of quantum errors and logical bit information.

9. The method of claim 8, further comprising training the hypothesis.

10. generating a circuit diagram based on the error model and the detected set of quantum errors; generating a bipartite graph based on the circuit diagram; The method of claim 1, further comprising generating the MPS protocol based on at least one of the circuit diagram and the bipartite graph.

11. A system comprising: one or more memory devices, the one or more memory devices, when executed by the one or more processors, Accessing an error model of a quantum computing system (QCS), wherein the error model encodes a set of error channels corresponding to a set of quantum error modes, and accessing a set of detected quantum errors encoded by a set of experimental measurements performed on the QCS, and generating a tensor network encoding a correlation between the quantum error modes of the set of quantum error modes and the detected quantum errors of the set of detected quantum errors, and generating a matrix product state (MPS) protocol based on the tensor network, and adopting the MPS protocol to generate an error-corrected QCS to calculate a likelihood of a logical error occurring, a system storing computer-readable instructions for causing an operation to be performed.

12. The operation further includes deploying the MPS protocol to detect errors occurring during runtime of the QCS, the system according to claim 11.

13. The operation further includes deploying the MPS protocol to correct errors occurring during runtime of the QCS, the system according to claim 11.

14. The MPS protocol is generated by contracting the tensor network, the system according to claim 11.

15. Contracting the tensor network includes summing probabilities associated with a subset of the error modes compatible with the set of detected quantum errors, the system according to claim 14.

16. Generating the MPS protocol is parameterized by a maximum bond dimension, the system according to claim 11.

17. The maximum bond dimension is set to a value of about 30, the system according to claim 16.

18. The operation further includes generating a circuit diagram based on the error model and the set of detected quantum errors, and generating a bipartite graph based on the circuit diagram, and further includes generating the MPS protocol based on at least one of the circuit diagram and the bipartite graph, the system according to claim 11.

19. One or more non-transitory computer-readable media storing instructions for operating a quantum computing system (QCS), wherein when the instructions are executed by one or more processors, the one or more processors are caused to access the error model of the QCS, wherein the error model encodes a set of error channels and a relationship with output results of detectors, access a set of measured output results of detectors, generate a tensor network encoding a correlation between the quantum error modes of the set of quantum error modes and the set of measured output results of detectors, generate a matrix product state (MPS) protocol based on the tensor network. One or more non-transitory computer-readable media. **Claim 20** The operation includes generating a circuit diagram based on the error model and the set of detected quantum errors, generating a bipartite graph based on the circuit diagram, The computer-readable medium according to claim 19, further comprising generating the MPS protocol based on at least one of the circuit diagram and the bipartite graph.

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