Models for determining logical error rates for quantum error correction codes operating on qubit arrays with non-uniform physical error rates

Abstract

One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The method includes accessing a set of physical error rates. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits. The set of physical error rates is non-uniform. A predicted logical error rate for a quantum error correction (QEC) code is determined. The QEC code corresponds to the set of qubits. Determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code. The QCS is operated in accordance with the predicted logical error rate for the QEC code.

Description

PRIORITY

[0001] This application claims priority to U.S. Provisional Application No. 63 / 561,631, entitled MODELS FOR DETERMINING LOGICAL ERROR RATES FOR QUANTUM ERROR CORRECTION CODES OPERATING ON QUBIT ARRAYS WITH NON-UNIFORM PHYSICAL ERROR RATES, filed on March 5, 2024, the contents of which of herein incorporated in their entirety. FIELD

[0002] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to efficient quantum memories via shadow tomography. BACKGROUND

[0003] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that enables the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0) + b |1) The “0” and “1” states of a digital computer are analogous to the |0) and |1) basis states, respectively of a qubit. SUMMARY

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

[0005] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The method includes accessing a set of physical error rates. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits. The set of physical error rates is nonuniform. A predicted logical error rate for a quantum error correction (QEC) code is determined. The QEC code corresponds to the set of qubits. Determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code. The QCS is operated in accordance with the predicted logical error rate for the QEC code.

[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 various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the related principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0009] FIG. 1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0010] FIG. 2 provides a workflow for the estimation of a logical error rate of a quantum error correction code, according to the various embodiments.

[0011] FIG. 3A shows limitations of the example logical error rate estimator model of FIG. 2.

[0012] FIG. 3B shows that multiple shortest paths exist for the purpose of estimating a logical error rate, according to various embodiments.

[0013] FIG. 4A shows a message passing scheme, according to various embodiments.

[0014] FIG. 4B shows a set of physical error rates that serve as input to a message passing algorithm, according to the various embodiments.

[0015] FIG. 5 shows examples of the three types of message passing arrows, according to various embodiments.

[0016] FIG. 6A shows an entire set of message-passing arrows after the uppermost row has been reached in a message passing algorithm, according to the various embodiments.

[0017] FIG. 6B shows the vectorization of a message passing algorithm, according to the various embodiments.

[0018] FIG. 6C provides a training workflow to employ supervised training to learn the trainable parameters of the message passing algorithm.

[0019] FIGS. 7A-7B are provided to show the supremacy of the embodiments over previous methods for estimating the LER of a QEC code.

[0020] FIG. 8 shows a method for operating a quantum computing system, according to various embodiments. DETAILED DESCRIPTION

[0021] Example aspects of the present disclosure are directed to methods, architectures, and hardware configurations for predicting logical error rates for quantum error correction codes operating on qubit arrays with non-uniform physical error rates. One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The QCS may include a set of qubits. The method includes accessing a set of physical error rates. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a separate qubit of the set of qubits and the set of physical error rates is non-uniform. A predicted logical error rate (LER) for a quantum error correction (QEC) code operating on the set of qubits may be determined based on the set of physical error rates and a logical error rate model for the QEC code. The QCS may be operated based on the predicted logical error rate for the QEC code.

[0022] The logical error rate (LER) is a primary metric for assessing the quality of quantum error correction (QEC) code. To that end, the embodiments are directed to the generation and implementation of a predictive model that predicts the (or estimates) LER for a given QEC code. In the following discussion, a surface code is assumed for the QEC code. However, the embodiments are not so limited, and some embodiments may contemplate other QEC codes (e.g., a color code). Also, the following discussion assumes that the QEC code is implemented on a 2D array of physical qubits (PQs). At least a subset of the set of PQs that are arranged in the 2D array form a logical qubit (LQ). Because the predictive model estimates the LER for a QEC code (and a given set of qubits), the predictive model may be referred to throughout as an LER estimator (or simply an estimator).

[0023] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The method includes accessing a set of physical error rates. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits. The set of physical error rates is nonuniform. A predicted logical error rate for a quantum error correction (QEC) code is determined. The QEC code corresponds to the set of qubits. Determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code. The QCS is operated in accordance with the predicted logical error rate for the QEC code.

[0024] The embodiments have many applications. For instance, the embodiments may be employed in quantum control optimization. The LER message passing embodiments can be used as a metric for optimization of gate frequencies or readout pulse parameters or some other control parameters that affect this metric. The embodiments can be employed in quantum processor optimization. When a sufficiently complete model of a processor is developed it is possible to optimize this LER model with respect to architectural design parameters to adapt the processor specifically for the quantum error correction. The embodiments may be employed to biased sampling. Below the threshold, the probability of a logical error under many decoders (for example MWPM) is exponentially small in the code distance. In other words, the cases which are "difficult" for most reasonable decoders become extremely rare. When classically sampling errors, this motivates shifting the distribution of errors used for evaluating decoders (either for benchmarking performance or training) to the rarer but more ambiguous cases. Effective use of the biased sampling requires choosing a shifted distribution which decreases the variance of decoder performance, and whose probabilities can be accurately evaluated. Ideas parallel to those used in the design of the message-passing LER estimator play an important role in the practical design of one such scheme.

[0025] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the space-complexity and time-complexity of this model are both O(d), which is scalable to large code distances, unlike the straightforward approach based on the Monte Carlo simulation of the system which has exponential complexity. Furthermore, the embodiments’ predictive accuracy is significantly higher than that of the alternative heuristic models, such as the ones based on simple averaging of the physical error rates. The training time and sample size required for training of the embodiments are significantly smaller than for neural networks or other models without strong inductive bias. The embodiments have multiple applications: in the processor design, in the optimization of gate frequencies for QEC algorithms using a snake algorithm, in the training of machine learning decoders. Quantum Computing Systems

[0026] FIG. 1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0027] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can 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 computation. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spin-based qubits, and the like. The superconducting qubits may be located in a cryostat to cool the qubits to superconducting temperatures (e.g., less than about 3 Kelvin). However, aspects of the present disclosure are not limited to superconducting qubits. In some examples, any suitable qubit structure may be used without deviating from the scope of the present disclosure, such as photonic qubits, trapped ion qubits, spin qubits, neutral atom qubits, quantum dot qubits, molecular qubits, or other qubits.

[0028] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (e.g., with which states may be prepared without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0029] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or more control devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0030] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout devices(s) 114 may implement one or more quantum logic gates that operate on the quantum system 102 through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0031] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout devices(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0032] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0) and |1) states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state |0) or the state |1), due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0033] In some embodiments, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the twodimensional grid 122 depicted in FIG. 1 includes 4x4 qubits, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0034] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit t. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit. That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0035] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed.

[0036] FIG. 1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure. Logical Error Rate Estimation

[0037] The logical error rate (LER) is a primary metric for assessing the quality of quantum error correction (QEC) code. To that end, the embodiments are directed to the generation and implementation of a predictive model that predicts the (or estimates) LER for a given QEC code. In the following discussion, a surface code is assumed for the QEC code. However, the embodiments are not so limited, and some embodiments may contemplate other QEC codes (e.g., a color code). Also, the following discussion assumes that the QEC code is implemented on a 2D array of physical qubits (PQs). At least a subset of the set of PQs that are arranged in the 2D array form a logical qubit (LQ). Because the predictive model estimates the LER for a QEC code (and a given set of qubits), the predictive model may be referred to throughout as an LER estimator (or simply an estimator).

[0038] FIG. 2 provides a workflow 200 for the estimation of a logical error rate of a quantum error correction code, according to the various embodiments. Workflow 200 employs an LER estimator 220, that implements a LER estimator model. In workflow 200, a set of physical error rates 210 is provided as input to the LER estimator 220. The set of physical error rates 210 may include one or more error rates for each PQ in a LQ. For instance, the set of physical error rates 210 may include an error rate for each channel of Pauli-noise for each PQ in the LQ. In FIG. 2, the set of physical error rates 210 is represented via a “heat map” superimposed on a 2D array of PQs. The heat map provides a visual indication of the set of physical error rates 210. Thus, the set of physical error rates 210 may be visualized as a heat map over a 2D arrays of PQs, as shown in FIG. 2. In the 2D array of PQs, PQs are represented via circles, and qubit couplers (represented by “rectangular boxes”) that couple qubits along the diagonals of the 2D array. In the heat map, the darker the shading of a circular (representing a qubit), the higher the one or more physical error rates are for the corresponding qubit. As the heat map superimposed on the set of PQs indicates, the spatial distribution of the physical error rates may be nightly non-uniform. Thus, the set of physical error rates 210 includes a 2D spatial distribution of physical error rates. In the 2D array of PQs, some of the PQs may be data qubits of the QEC code, while other PQs are measure qubits of the QEC code.

[0039] The LER estimator includes and / or implements a LER estimator model. An example LER estimator model 222 is shown in FIG. 2. In workflow 200 and via the LER estimator model, the LER estimator 220 determines (e.g., calculates and / or generates) and outputs the LER 230 for the LQ formed by the set of PQs. Thus, the embodiments model the LER (e.g., estimate the LER) via workflow 200. That is, the embodiments employ the LER estimator 220 (and a corresponding LER estimator model) to model (or predict) the LER for a set of PQs and a corresponding QEC code.

[0040] Modeling the LER for any given set of PQs (and a QEC code) is an important ingredient in designing and optimizing the quantum processors for the QEC code. As an alternative approach to determining the LER, Monte Carlo (MC) simulation can be employed. To determine (or estimate) the LER via MC methods, errors may be injected onto the qubits, where the errors are injected in accordance to the set of physical error rates 210. From the injected errors, error syndromes are generated. The error syndromes are provided to a decoder for the QEC code. The results of the decoder and knowledge of the injected errors are employed to determine the LER. Rather than MC sampling, to determine the LER, the embodiments employ a LER estimator model described below. Due to exponential reduction of the LER with code distance, this MC approach becomes infeasible for large QEC codes (especially in the deep subthreshold regime) because of the divergent sample complexity. In order to overcome the exponential scaling required for Monte Carlo sampling, the embodiments employ practical heuristic models (e.g., the LER estimator model) that can predict LER without performing the full Monte Carlo simulation of the QEC. As noted above, an example LER estimator model 222 is shown in FIG. 2.

[0041] Logical operators on a LQ are formed via “paths” of PQs from a boundary of the LQ to an opposing boundary of the LQ. Thus, any logical error that occurs on the LQ can be mapped to a path of PQs from a boundary of the LQ to an opposing boundary of the LQ. The probability of the logical error scales as approximately the product of the individual physical error rates along the path. Because physical error rates are then 1.0, longer paths tend to have lesser probabilities. Thus, when determining the LER, the “shortest” paths are considered, and the “higher order” longer paths may be neglected via the embodiments. As described in detail below, in the embodiments, the LER estimator model may generate a sum over all such possible shortest paths. Because the physical error rates are “accumulated” along a path, paths with significantly faulty (or error-probe) PQs tend to dominate the estimation of the LER. This is why the geometrical distribution of the set of physical error rates 210 may be of importance when calculating the LER.

[0042] In contrast to traversing all such possible shortest paths, the example LER estimator model 222 of FIG. 2 does not consider the geometrical distribution of the set of physical error rates 210. Rather, the example LER estimator model 222 “averages” the set of physical error rates 210 without considering the spatial distributions of the set of physical error rates 210. In example LER estimator model 222, F[p] is a mean of the set of physical error rates 210 and d is the QEC code distance. The example LER estimator model 222 has two parameters that can be calibrated based on properties of the decoder of the QEC: p(h and C. p(h is a threshold error rate. Note that when p(h > E [p], then the LER “blows-up” with increasing code distance. Example LER estimator model 222 is based on the mean of value of the set of physical error rates 210 and is independent of the spatial distributions for nonuniform physical error rates. That is, example LER estimator model 222 does not incorporate the spatial information about input physical error rates. Accordingly, example LER estimator model 222 may work well for simple textbook examples of the surface code with uniform (or nearly-uniform) physical error rates. However, it may not be accurate in realistic systems where some qubits can be significantly worse than others. In particular, the example LER estimator model 222 does not take into account the relative layout of “outlier” qubits in the grid (e.g., the darkly shaded qubits in the heat map), since it performs averaging.

[0043] Accordingly, the embodiments employ a heuristic (but physically motivated) LER estimator model that is accurate, fast, and scalable to large code distances. The LER estimator model of the embodiments consumes the geometric information embedded in the set of physical error rates 210 and generates a more accurate LER 230.

[0044] FIG. 3A shows limitations of the example logical error rate estimator model 222 of FIG. 2. For instance, FIG. 3A shows a cumulative distribution function (CDF) 300 for a set of error rates (e.g., the set of error rates 210 of FIG. 2). As discussed above, the LER is highly dependent on the geometrical arrangement of the outlier qubits 302. As shown in CDF 300, the outlier qubits 302 are those PQs with significantly higher physical error rates than most of the qubits. Unless noted otherwise, in the following discussion, the term qubit may refer to a PQ. For reasons indicated above, the geometrical arrangement of these outlier qubits 302 drive the estimate of the LER. For instance, FIG. 3A also shows four separate sets of physical error rates 310 and their corresponding LERs. Each of the four separate sets of physical error rates 310 corresponds to a common CDF (e.g., CDF 300). However, as shown via the heat maps in FIG. 3, the geometric arrangement of the physical error rates are different in each of the four separate sets of physical error rates 310. The corresponding LER is given for each of the geometrical arrangements. Via inspection of the LERs, it is noted that the LER is highly dependent on the geometrical distribution of the outlier qubits 302.

[0045] More specifically, logical operators in a QEC code, such as a surface code, correspond to the “shortest paths” linking two opposite boundaries in the QEC code. Thus, the LER is dependent on each “shortest path” linking two opposing boundaries of the QEC code. The following discussion assumes paths linking the top and lower boundaries of the QEC code. When one or more paths include multiple outlier qubits 302, the LER can be significantly greater than when multiple outlier qubits 302 are included in the same path. That is, to achieve a low LER, one hopes that the worst (e.g., most error prone) qubits do not line up on a single shortest path linking two code boundaries, since such a path corresponds to a logical operator and will become a critical contributor to the LER.

[0046] FIG. 3B shows that multiple shortest paths exist for the purpose of estimating a logical error rate, according to various embodiments. Note that at each step in each path in the multiple shortest paths 320 goes along one of two possible diagonals (e.g., qubit couplers) when traversing between the upper boundary and the lower boundary of the QEC code. This is because a logical operator of the QEC code must follow qubits that are couplable.

[0047] The LER estimator models of the embodiments perform a summation over all possible such shortest paths (or trajectories) that link the two opposing boundaries. A “score” is assigned to each shortest path. The score assigned to each path is the product of the physical error rates for all the PQs in the path. Thus, the shortest paths with multiple outlier qubits in the path will dominate the summation over all possible shortest paths (or trajectories). The score that the LER estimator models of the embodiments assigns to each shortest path roughly corresponds to the probability of a logical error happening on this path. The LER estimator model then sums the contributions of all such shortest paths.

[0048] Note that the number of possible shortest paths scales exponentially with the code distance. To avoid the exponential scaling of having to sum over a number of paths of scales exponentially with code distance, the LER estimator models of the embodiments employ a “messaging” algorithm to determine the score assigned to each shortest path. Via the messaging algorithm, the scaling of the summation is linear in both time and space, with respect to the code distance. Message Passing Algorithms

[0049] The summation over paths can be done efficiently using an algorithm that belongs to a class of message passing algorithms. A passing matrix and a local function that can be applied at each node after receiving the messages is detailed below. In many cases, such computation can be vectorized over the messages, and the vector encoding messages may be propagated from one boundary of the QEC code to the opposing boundary. Hence, the space and time complexity of this predictive model both scale linearly with the code distance.

[0050] FIG. 4A shows a message passing 400 scheme, according to various embodiments. Message passing 400 is employed by the embodiments to efficiently compute the sum of the score assigned to each possible shortest pass, where the number of possible shortest paths scales exponentially with the code distance. The arrows show the passing of a message from a lower layer of qubits to the subsequent layer of qubits. As noted above, the messages are passed along the diagonals (e.g., defined via the qubit couplers) such that each qubit in a lower layer (except qubits on the two opposing vertical boundaries) passes its message to two separate qubits in the subsequent layer. As will be discussed in the context of message passing rules, the arrows in FIG. 4A are referred to as message-passing arrows and transmit “messages”. The messages are scalars that propagate between the data qubits following the paths that are exemplified with arrows in the message passing 400 of FIG. 4. Because logical operators operate on data qubits (and the shortest paths correspond to logical operators), the messages are passed from data qubit to data qubit, the arrows in FIG. 4 skip over the measure qubits. Depending on the logical operator, messages will propagate in the left-right or the down-up direction. Reversing the direction by 180 degrees (i.e. to right-left or up-down) does not have any effect.

[0051] For the sake of simplicity, the following discussion focuses on a non-limiting example of the surface code memory with the code distance dX and dZ for X and Z logical Pauli operators, implemented with the CZ entangling gates. For such an example, the input physical errors rates include data qubit depolarization probabilities, measure qubit bit-flip probabilities, and CZ gate two-qubit depolarizing probabilities. The data qubit depolarization probabilities are error probabilities during the measurement time. The qubit array shape for data qubit probabilities is (dX, dZ). The array shape for measure qubit probability is (dX+1,dZ+1). The array shape for CZ gate probabilities is (2*dX, 2*dZ).

[0052] FIG. 4B provides a visualization of these three error channels (or error types). More specifically, FIG. 4B shows a set of physical error rates 500 that serve as input to a message passing algorithm, according to the various embodiments. As shown in the key of FIG. 5, each error channel has a separate shading. Also, each error channel has a scale (e.g., often visualized as a heat map) that indicates a value for the physical error rate for that channel. However, in FIG. 5, due to limitations of black and white drawings, the heat map that indicates the actual values for the physical error rates are not shown. If the qubit or a coupler is not present, the corresponding physical error rate value is 0. As described below, the embodiments may employ arrays for vectorizing the message passing heuristic.

[0053] The messages are scalars that propagate between the measure qubits (e.g., see the shading in FIG. 4B) following the paths that are exemplified with arrows in the figures. Depending on the logical operator, messages will propagate in the left-right or the down-up direction. Reversing the direction by 180 degrees (i.e. to right-left or up-down) does not have any effect. The various message passing algorithms employed by the various embodiments use a set of message passing rules and / or heuristics. Seven such message passing rules will now be discussed. The following discussion assumes a logical operator that extends between the top and lower boundaries and the message passing is going from bottom boundary to the top boundary. Thus, messages are passed from data qubits on the bottom layer to data qubits on a layer above. In the following discussion, the messaging rules employ three types of message passing arrows.

[0054] FIG. 5 shows examples of the three types of message passing arrows, according to various embodiments. More specifically, FIG. 5 shows a set of physical error rates 510, which have been previously described (e.g., see the set of physical error rates 410 of FIG. 4B). Three types of messaging arrows are shown superimposed on the set of physical error rates 510. The three types of messaging arrows include: message-passing arrows (e.g., message passing arrow 512), message-modification arrows (e.g., message-modification arrow 514), and message-feedback arrows (e.g., message-feedback arrow 516). Message-passing arrows are “straight” arrows in FIG. 5 and are oriented along the diagonals. As described below, a message-passing arrow passes from a data qubit on a lower row and terminates on another qubit on an upper row, while passing through a measure qubit on an intermediate row. Each data qubit (except those near a vertical boundary) emanates two message-passing arrows: one to the left-leaning diagonal and one to the right-leaning diagonal. Message modification arrows follow similar paths to those of message-passing arrows, except they are curved to avoid the intermediate measure qubit. The zoom-in view on FIG. 5 shows a zoomed-in view of the message-modification arrow 514 that starts from a first data qubit 522 and ends on a second data qubit 524. The message-modification arrow 514 is curved to avoid the intermediate measure qubit 526. As shown, the diagonals are defined via qubit couplers. The qubit couplers couple measure and data qubits. For instance, qubit coupler 528 couples the second data qubit 524 to the measure qubit 526. The message-feedback arrows (e.g., message-feedback arrow 516) begins are a data qubit and “curls around” to feedback on itself, in order to provide a “feedback” message.

[0055] Rule 0 (Message Initialization): The following discussion assumes a logical operator that extends between the top and lower boundaries and the message passing is going from bottom boundary to the top boundary. The messages are initialized at the data qubit “nodes” in the bottom layer or row (e.g., see FIG. 4B). At the beginning of the message passing algorithm, each message (emanating from a data qubit in the bottom row) is initialized to 1.0.

[0056] Rule 1 : When the message passes from one data qubit node to the next data qubit node along a message-passing arrow (e.g., message-passing arrow 512), it gets multiplied by a function of that data qubit’s depolarizing error probability p (e.g., a value included in the set of physical errors that are employed as inputs in the model, see FIG. 4B). One non-limiting embodiment of such a function of the depolarizing probability of a data qubit is: f(p) = / ?0 • expU^ • Zn(p) + P2 ' ^(1 — p)), where / ?0, pi, and P2 are trainable parameters.

[0057] Rule 2: In addition to the effect of the message-passing arrows described in Rule 1, there is a “parallel” message-modification arrow (e.g., message-modification arrow 514) that also crosses the same data qubit node as corresponding message-passing arrow. The effect of a message-modification arrow is to modify the “message” sent by the corresponding message-passing arrow (e.g., f(p)). In some embodiments, the modification to the message-passing arrow message is: / (p)+= Y^Pcz,oW • exp (Pcz,i • HPczW) + P CZ,2 • ln(1 - Pcz®)), where the index = 1 — 4 runs through the four qubit couplers surrounding the measure qubit that is “avoided” by the curved message-modification arrow (e.g., see the zoom-in view of FIG. 5). pcz® are the CZ physical error rates for the surrounding qubit couplers, the values of which are included in the input set of physical error rates 510. P^j^ for J' = 0 — 2 and i = 1 — 4 are twelve trainable parameters of the model.

[0058] Rule 3: Data qubit nodes wait until they receive messages from both diagonal neighbors from the previous layer (or row), and then they sum these messages together to form a new message.

[0059] Rule 4: This summed message is then multiplied by a “feedback message,” via a corresponding message-feedback arrow (e.g., message-feedback arrow 516) according to a feedback function g(q"), that is dependent on the data qubit’s “measurement” error rate (q), the value for which is included in the input set of physical error rates 510. The feedback message is passed to the data qubit via the corresponding message-feedback arrow. One nonlimiting example of a feedback message is gO) = exp(fi4 • ln(1 — —¢ / )), with one additional trainable parameter P4.

[0060] Rule 5: After the transformation in Rule 4, the message is transmitted to the nodes in the next layer, following the same Rules 1-4 again, until it reaches the last layer.

[0061] Rule 6: In the last layer (e.g., uppermost row), all messages are summed together to obtain an estimate of the LER. Note that the functional form of f(p) and g(q) is informed by the rigorous analytical bound on the LER for the matching decoders. Vectorization of Message Passing Algorithms

[0062] The message passing rules described above can be efficiently vectorized for faster execution. This may be useful if the LER estimator is to be embedded into the optimization loop, for instance in the Snaked algorithm which may be used for configuring the operating frequencies of the processor. FIG. 6A shows an entire set of message-passing arrows 600 after the uppermost row has been reached in a message passing algorithm, according to the various embodiments. So that the visualization of message passing is not too cluttered, in FIG. 6A, the indications of the message-modification arrows and the messagefeedback arrows have been suppressed.

[0063] FIG. 6B shows the vectorization of a message passing algorithm, according to the various embodiments. FIG. 6B shows a basic LER calculation 610. The basic LER calculation 610 includes an inner loop and an outer loop. Each iteration of the inner loop corresponds to one of the possible shortest paths of the set of all possible shortest paths. More specifically, each interaction of the inner loop calculates the product of the physical error rates for the data qubits that form the corresponding shortest path. The outer loop calculates the sum over all such possible paths. As noted above, the number of all possible shortest paths scales exponentially with the distance of the QEC code. Thus, the scaling law 15 for the basic LER calculation 610 is an exponential scaling law. As discussed throughout, the embodiments use various messenger passing rules (e.g., rules 0-6 as discussed above) to avoid such exponential scaling. The message passage algorithm of the embodiments may be vectorized as shown in vectorization of message passing algorithm 630.

[0064] To implement the vectorization of the message passing algorithm 630, a parameterization of the message passing matrix 620 may be performed. When parameterizing the message passing matrix 620, a probability matrix 622 is transformed in a parameterized passing matrix 624, via the above message passing function f(p) = / ?0 • exp(f^ • Zn(p) + / ?2 • ln(1 — p)), where / ?0,^, and / ?2 are trainable parameters. As discussed above, in some embodiments, the message passing function may be modified via the modification function: f (p)+= I^Pcz.o^ • exp (pCz,i • ^{Pcz(}} + Ccz,2 • In (1 — Pcz (i) )), that includes twelve additional trainable parameters. The vectorization of the message passing algorithm 630 shown in FIG. 6B shows the transformation of the message from a lower layer to an upper layer. The vectorization of the message passing algorithm 630 employs the parameterized passing matrix 624 and the feedback function shown above: g(q) = exp(f>4 • ln(1 — ^Q)), with one additional trainable parameter / ?4.

[0065] The message passing LER model is parametrized with ~10-20 parameters, which can be trained using standard techniques from modern supervised machine learning. FIG. 6C provides a training workflow 680 to employ supervised training to learn the trainable parameters of the message passing algorithm. Results

[0066] FIGS. 7A-7B are provided to show the supremacy of the embodiments over previous methods for estimating the LER of a QEC code. More specifically, FIG. 7A shows various cumulative distribution functions for the error probabilities that are used to estimate various logical error rates, according to the various embodiments. More specifically, FIG. 7A shows a first CDF 700, a second CDF 702, a third CDF 704, a fourth CDF 706, a fifth CDF 708. a sixth CDF 710, and a seventh CDF 712. FIG. 7B shows a comparison between results of the embodiments and previous results. The results presented in FIG. 7B demonstrates that the embodiments outperform previous methods.

[0067] More specifically, FIG. 7B shows three plots: arithmetic mean 720, geometric mean 730, and message passing 740. All three plots show a relationship between calculating the LER based on the seven CDFs shown in FIG. 7A via two different methods. In all three plots, the y-axis represents calculating the LER based on Monte Carlo (MC) simulations, which are discussed above. The x-axis represents estimating the same LER based on the estimator models discussed herein. More specifically, the x-axis of the arithmetic mean 720 plot corresponds to calculating the LER based on the example LER estimator model 222 of FIG. 2, where the arithmetic mean is used to calculate E[p]. The x-axis of the geometric mean 730 plot corresponds to calculating the LER based on the example LER estimator model 222 of FIG. 2, where the geometric mean is used to calculate E[p]. The x-axis of the message passing 740 plot corresponds to calculating the LER based on the message passing algorithms discussed herein (e.g., see FIGS. 3B-6C).

[0068] For each CDF of FIG. 7B, the spatial distributions of the “outlier” qubits have been varied, such that the LER is estimated multiple times via the MC methods (e.g., see FIG. 3A). Because the example LER estimator model 222 does not employ the spatial distributions, this results in “vertical bands” of LER values (calculated via the MC methods) on the arithmetic mean 720 and the geometric mean 730 plots (which do not account for the spatial distributions). If the MC methods and LER estimator model methods agree, the results would lie on the y=x line on the log-log plots of FIG. 7B. Deviation from this straight line indicates a discrepancy between the MC methods and the estimator model methods. At least for the arithmetic mean 720 and the geometric mean 730 plots, although much more expensive to calculate, the MC methods provide a more accurate determination of the LER (e.g., for a given spatial distribution) compared to the estimator model method.

[0069] Visual inspection of FIG. 7B confirms significant agreement between the MC methods and the LER estimator model efforts of the embodiments. Furthermore, the “vertical bands” as seen in the arithmetic mean 720 and geometric mean 730 plots, essentially disappear in the message passing 740 plot of the embodiments. Accordingly, the message passing methods of the embodiments, which employ the spatial distributions of the outlier qubits, clearly outperform the methods that employ the example estimator model 222, which do not account for the spatial distributions. Furthermore, the message passing methods of the embodiments provide an accurate estimate of the LER and are significantly cheaper to calculate than the MC methods. Methods

[0070] FIG. 8 shows a flowchart for a method that may be implemented via a quantum computing system (e.g., quantum computing system 100 of FIG. 1), a classical computing system, or a combination thereof. More particularly, FIG. 8 shows a method 800 for operating a quantum computing system, according to various embodiments. The quantum computing system (QCS) may include a set of qubits (e.g., the plurality of qubits 120 of FIG. 1). Method 800 begins at block 802, where a set of physical error rates is accessed. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits. The set of physical error rates is non-uniform. At block 804, a predicted logical error rate is determined for a quantum error correction (QEC) code that corresponds to the set of qubits, wherein determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code. At block 806, the QCS is operated in accordance with the predicted logical error rate for the QEC code.

[0071] The qubit array is a two-dimension (2D) qubit array arranged in a 2D spatial grid of qubits. The set of physical error rates has a non-unform spatial density over the 2D spatial grid of qubits.

[0072] The QEC code is a surface code that defines a logical qubit over a contiguous subset of the set of qubits that is contained within a set of boundaries over the 2D spatial grid of qubits. The predicted logical error rate is a logical error rate for the logical qubit.

[0073] When the QEC code is implemented, the logical qubit has a logical error when each qubit included in a path of qubits between two boundaries of the set of boundaries has a physical error, and wherein the two boundaries are opposing boundaries of the set of boundaries.

[0074] Each possible path between each possible set of opposing boundaries has a probability determined based on the physical error rates of the qubits connecting the possible path.

[0075] A subset of a set of possible paths between possible opposing boundaries of a set of possible opposing boundaries is a set of shortest paths. Implementing the logical error rate model includes determining the probability for each shortest path of the set of possible paths based on the physical error rates of the qubits included in the shortest path. The predicted logical error rate for the QEC is determined based on summing the probability for each shortest path of the set of possible paths.

[0076] Summing the probability for each shortest path of the set of possible paths is based on a message passage algorithm.

[0077] The message passage algorithm has a set of message passage rules.

[0078] The set of message passage rules is vectorized for faster execution.

[0079] The method further includes determining values for a set of parameters for the logical error rate model based on a supervised machine learning algorithm and a set of training data determined from a measurement of a logical error rate.

[0080] The method further includes operating the QCS based on employing the predicted logical error rate for calibration of grate frequencies of the QCS.

[0081] The method further includes operating the QCS based on employing the predicted logical error rate for calibration of readout pulse parameters of the QCS.

[0082] The method further includes operating the QCS based on employing the predicted logical error rate for calibration of architectural design parameters of a quantum processor of the QCS.

[0083] The method further includes operating the QCS based on employing the predicted logical error rate for decreasing a variance of a decoder performance of the QEC code Additional Embodiments

[0084] Another non-limiting embodiment includes a computing system. The computing system includes one or more processors and one or more memory devices. The one or more memory devices store computer-readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating a quantum computing system (QCS) that includes a set of qubits. The operations include accessing a set of physical error rates. Each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits. The set of physical error rates is non-uniform. A predicted logical error rate for a quantum error correction (QEC) code is determined. The QEC code corresponds to the set of qubits. Determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code. The QCS is operated in accordance with the predicted logical error rate for the QEC code.

[0085] The embodiments have many applications. For instance, the embodiments may be employed in quantum control optimization. The LER message passing embodiments 19 can be used as a metric for optimization of gate frequencies or readout pulse parameters or some other control parameters that affect this metric. The embodiments can be employed in quantum processor optimization. When a sufficiently complete model of a processor is developed it is possible to optimize this LER model with respect to architectural design parameters to adapt the processor specifically for the quantum error correction. The embodiments may be employed to biased sampling. Below threshold, the probability of a logical error under many decoders (for example MWPM) is exponentially small in the code distance. In other words, the cases which are "difficult" for most reasonable decoders become extremely rare. When classically sampling errors, this motivates shifting the distribution of errors used for evaluating decoders (either for benchmarking performance or training) to the rarer but more ambiguous cases. Effective use of the biased sampling requires choosing a shifted distribution which decreases the variance of decoder performance, and whose probabilities can be accurately evaluated. Ideas parallel to those used in the design of the message-passing LER estimator play an important role in the practical design of one such scheme.

[0086] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0087] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., 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, data processing apparatus. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0088] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qubits) are possible.

[0089] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0090] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0091] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0092] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0093] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0094] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0095] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0096] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magnetooptical disks; and CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0097] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0098] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0099] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0100] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for operating a quantum computing system (QCS) that comprises a set of qubits arranged a qubit array, the method comprising:accessing a set of physical error rates, wherein each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits and the set of physical error rates is non-uniform; anddetermining a predicted logical error rate for a quantum error correction (QEC) code that corresponds to the set of qubits, wherein determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code.

2. The method of claim 1, further comprising:operating the QCS in accordance with the predicted logical error rate for the QEC code.

3. The method of claim 1, wherein the qubit array is a two-dimension (2D) qubit array arranged in a 2D spatial grid of qubits such that the set of physical error rates has a nonunform spatial density over the 2D spatial grid of qubits.

4. The method of claim 3, wherein the QEC code is a surface code that defines a logical qubit over a contiguous subset of the set of qubits that in contained within a set of boundaries over the 2D spatial grid of qubits and the predicted logical error rate is a logical error rate for the logical qubit.

5. The method of claim 4, wherein when the QEC code is implemented, the logical qubit has a logical error when each qubit included in a path of qubits between two boundaries of the set of boundaries has a physical error, and wherein the two boundaries are opposing boundaries of the set of boundaries.

6. The method of claim 5, wherein each possible path between each possible set of opposing boundaries has a probability determined based on the physical error rates of the qubits connecting the possible path.

7. The method of claim 6, wherein a subset of a set of possible paths between possible opposing boundaries of a set of possible opposing boundaries is a set of shortest paths and implementing the logical error rate model comprises:determining the probability for each shortest path of the set of possible paths based on the physical error rates of the qubits included in the shortest path; anddetermining the predicted logical error rate for the QEC based on summing the probability for each shortest path of the set of possible paths.

8. The method of claim 6, wherein summing the probability for each shortest path of the set of possible paths is based on a message passage algorithm.

9. The method of claim 8, wherein the message passage algorithm has a set of message passage rules.

10. The method of claim 9, wherein the set of message passage rules is vectorized for faster execution.

11. The method of claim 1, further comprising:determining values for a set of parameters for the logical error rate model based on a supervised machine learning algorithm and a set of training data determined from a measurement of a logical error rate.

12. The method of claim 1, further comprising:operating the QCS based on employing the predicted logical error rate for calibration of grate frequencies of the QCS.

13. The method of claim 1, further comprising:operating the QCS based on employing the predicted logical error rate for calibration of readout pulse parameters of the QCS.

14. The method of claim 1, further comprising:operating the QCS based on employing the predicted logical error rate for calibration of architectural design parameters of a quantum processor of the QCS.

15. The method of claim 1, further comprising:operating the QCS based on employing the predicted logical error rate for decreasing a variance of a decoder performance of the QEC code.

16. A computing system comprising:one or more processors;one or more memory devices, the one or more memory devices storing computer-readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating a quantum computing system (QCS) that includes a set of qubits, the operations comprising:accessing a set of physical error rates, wherein each physical error rate of the set of physical error rates corresponds to a physical error rate of a qubit of the set of qubits and the set of physical error rates is non-uniform;determining a predicted logical error rate for a quantum error correction (QEC) code that corresponds to the set of qubits, wherein determining the predicted logical error rate is based on the set of physical error rates and a logical error rate model for the QEC code; andoperating the QCS in accordance with the predicted logical error rate for the QEC code.

17. The computing system of claim 16, wherein the computing system is integrated in the QCS.

18. The computing system of claim 16, wherein the set of qubits is a two-dimension (2D) qubit array arranged in a 2D spatial grid of qubits such that the set of physical error rates has a non-unform spatial density over the 2D spatial grid of qubits.

19. The computing system of claim 18, wherein the QEC code is a surface code that defines a logical qubit over a contiguous subset of the set of qubits that is contained within a set of boundaries over the 2D spatial grid of qubits and the predicted logical error rate is a logical error rate for the logical qubit.

20. The computing system of claim 19, wherein when the QEC code is implemented, the logical qubit has a logical error when each qubit included in a path of qubits between two boundaries of the set of boundaries has a physical error, and wherein the two boundaries are opposing boundaries of the set of boundaries.

Images