WO2025184591A1

WO2025184591A1 - Context aware fidelity estimation for surface code circuits implemented by quantum computing systems

Info

Publication number: WO2025184591A1
Application number: PCT/US2025/017969
Authority: WO
Inventors: Dripto Mazumdar DEBROY; Elie GENOIS; Jonathan Arthur GROSS; Zhang Jiang
Original assignee: Google LLC
Current assignee: Google LLC
Priority date: 2024-03-01
Filing date: 2025-02-28
Publication date: 2025-09-04
Anticipated expiration: 2026-09-01

Abstract

This disclosure includes a method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of circuit-slices. Each circuit-slice is a circuit-slice of a quantum circuit. The set of qubits is subdivided into a first subset of qubits and a second subset of qubits. The first subset of qubits is a set of qubits-to-probe. The second subset of qubits is a set of neighboring qubits. Each neighboring-qubit neighbors at least one qubit-to-probe in the quantum circuit. A tomography dataset is generated based on a set of qubit measurements. Each qubit measurement corresponds to measuring each qubit-to-probe subsequent to operating at least one circuit-slice on the set of qubits. A set of fidelities is estimated for the set of qubits based on the tomography dataset. The set of fidelities corresponds to a context of the quantum circuit.

Description

CONTEXT AWARE FIDELITY ESTIMATION FOR SURFACE CODE CIRCUITS

IMPLEMENTED BY QUANTUM COMPUTING SYSTEMS

PRIORITY

[0002] This application claims priority to U.S. Provisional Application No 63/560.500 entitled CONTEXT AWARE FIDELITY ESTIMATION FOR SURFACE CODE CIRCUITS IMPLEMENTED BY QUANTUM COMPUTING SYSTEMS, filed on March 1, 2024, the contents of which of herein incorporated in their entirety.

FIELD

[0003] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to determining the fidelity of quantum error correction (QEC) circuits.

BACKGROUND

[0004] 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 “I” 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 11) basis states, respectively of a qubit.

SUMMARY

[0005] 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.

[0006] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The QCS includes a set of qubits. The method includes generating a set of circuit-slices. Each circuit-slice in the set of circuit-slices is a circuit-slice of a quantum circuit that implements a quantum error correction (QEC) code. The set of qubits is subdivided into a first subset of qubits and a second subset of qubits. The first subset of qubits is a set of qubits-to-probe. The second subset of qubits is a set of neighboring qubits. Each neighboring-qubit of the set of neighboring-qubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit. A tomography dataset is generated based on a set of qubit measurements. Each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to-probe of the set of qubits-to-probe subsequent to operating at least one circuit-slice of the set of-circuit-slices on the set of qubits. A set of fidelities is estimated for the set of qubits based on the tomography dataset. The set of fidelities corresponds to a context of the quantum circuit.

[0007] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0008] 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

[0009] Detailed discussion of embodiments directed to one of ordinary⁷ skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0010] FIG. 1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0011] FIG. 2A illustrates a flowchart for a method for preparing a quantum computing system for generating a tomography dataset that is used to determine a context-aware fidelity⁷ estimation of a quantum error correction code circuit, according to various embodiments.

[0012] FIG. 2B illustrates a flowchart for a method for generating a tomography dataset, according to various embodiments.

[0013] FIG. 2C illustrates a flowchart for a method for operating a circuit-slice to generate tomography data, according to various embodiments. [0014] FIG. 3 shows a set of qubit and a quantum error correction circuit that operates on the set of qubits, according to various embodiments.

[0015] FIGS. 4A-4F demonstrate the slicing of the quantum error correction circuit of FIG. 3 into an ordered set of circuit-slices, according to various embodiments.

[0016] FIG. 5 shows examples of determining the initial-bases for neighboring-qubits, according to various embodiments.

[0017] FIG. 6 shows three compatible sets of qubits for a standard surface code circuit, according to various embodiments.

[0018] FIG. 7 illustrates a flowchart for a method for determining Pauli-error rates, according to various embodiments.

[0019] FIG. 8 shows gate error labels for an identity gate, according to various embodiments.

[0020] FIG. 9 shows an example of error propagations for a CZ gate when one the affected qubits is in an eigenstate of the X-basis and when one of the affected qubits is in an eigenstate of the Y-basis.

[0021] FIG. 10 shows gate error labels for a CZ gate, according to various embodiments. [0022] FIG. 11 shows a method for operating a quantum computing system, according to various embodiments.

DETAILED DESCRIPTION

[0023] Example aspects of the present disclosure are directed to methods, architectures, and hardware configurations for estimating a fidelity⁷ for quantum error correction (QEC) code (e.g.. surface code) circuits. The estimate of the fidelity⁷ is aware of the context of the circuit (e.g.. the circuit used for the underlying quantum computation). In a fault-tolerant quantum computer, much of the time spent by a given qubit is on executing stabilizer extraction circuits for the QEC code. As such, maximizing performance in this particular context is of significant importance as quantum computing devices are scaled. The embodiments include Surface Code CAFE (context aware fidelity estimation) measurements (or experiments). The term Surface-Code CAFE (SC- CAFE) refers to a series of measurements (or experiments) that determine (or at least estimate) component error rates for single- and two-qubit operations in the exact (or at least similar) context they find themselves in within a surface code circuit. [0024] The embodiments isolate gate error parameters and extract the parameters by measuring increasingly large subsections of a surface code circuit (e.g., circuit-layers) and analyzing the output after additional layers. By subtracting the errors which were occurring in previous iterations, the error contribution of the last (or current) layer of the experiment may be isolated and/or estimated. To simplify the analysis and prevent an explosion of experimental resources, the qubits are initialized in single-qubit product states in which controlled-Z (CZ) entangling operations act as product operators. As a result, the gate errors in the system may be approximated (or estimated) as product operators. These initial states are referred to as "constrained" and can be employed to find sets of compatible qubits which share the constraints. [0025] Some non-limiting embodiments include methods that are targeted at the surface code syndrome extraction round. The non-limiting embodiments described herein is a sort of “circuit slicing” mode of the experiment. It provides single qubit fidelities and Pauli error rates for each operation in the surface code experiment in context, where “single qubit” means that the circuit is analyzed under conditions where there is no entanglement generated, allowing a probing of all Pauli channels for the individual qubits.

[0026] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The QCS includes a set of qubits. The method includes generating a set of circuit-slices. Each circuit-slice in the set of circuit-slices is a circuit-slice of a quantum circuit that implements a quantum error correction (QEC) code. The set of qubits is subdivided into a first subset of qubits and a second subset of qubits. The first subset of qubits is a set of qubits-to-probe. The second subset of qubits is a set of neighboring qubits. Each neighboring-qubit of the set of neighboring-qubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit. A tomography dataset is generated based on a set of qubit measurements. Each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to-probe of the set of qubits-to-probe subsequent to operating at least one circuit-slice of the set of-circuit-slices on the set of qubits. A set of fidelities is estimated for the set of qubits based on the tomography dataset. The set of fidelities corresponds to a context of the quantum circuit.

[0027] Another 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 selecting a first subset of the set of qubits. A second subset of the set of qubits is disjointed from the first subset of qubits. Each qubit of the second subset of qubits has a neighbor in the qubit array that is included in the first subset of qubits. For each qubit of the first subset of qubits, a Pauli state of a set of Pauli states is selected. For each qubit of the first subset of qubits, the qubit is prepared in the selected Pauli state. For each qubit of the second subset of qubits, the qubit is prepared in a Pauli state of the set of Pauli states based on the selected Pauli state of the neighboring qubit of the first subset of qubits. A first portion of a quantum algorithm is executed. The quantum algorithm has an associated quantum circuit implemented on the QCS. The first portion of the quantum algorithm includes a first portion of the quantum circuit operating on the first subset of qubits and the second subset of qubits. In response to executing the first portion of the quantum algorithm, a first set of measurements is determined. Determining the first set of measurements includes, for each qubit of the first subset of qubits, measuring the qubit in a Pauli basis of a set of Pauli basis that corresponds to the selected Pauli state of the qubit. A set of error rates for the set of qubits is determined. Determining the error rates is based on the first set of measurements. The set of error rates for the set of qubits corresponds to the quantum circuit.

[0028] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments may be employed to determine physical error rates in the context of the circuit that implements a quantum algorithm, including a quantum error correction (QEC) code executed during the execution of the quantum algorithm. The error rates have both a spatial and temporal dimension based on the quantum circuit associated with the quantum algorithm. The employment of resources of the quantum computing system (e.g., the qubits) may be tailored based on these error rates.

Quantum Computing Systems

[0029] 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.

[0030] 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, spinbased 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.

[0031] The t pe 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. [0032] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 1 10 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.

[0033] 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.

[0034] 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.

[0035] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0) and 11) 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 11), 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.

[0036] 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 two-dimensional 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.

[0037] 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.

[0038] 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.

[0039] 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.

Determining Context-Aware Fidelity Estimations of Quantum Error Correction Code Circuit

[0040] The embodiments are directed towards methods, systems, and architectures for determining context-aware fidelity estimations (CAFE) of quantum circuits implementing quantum error correction (QEC) codes. Such QEC codes include but are not limited to 2D topological codes such as surface codes, color codes, and the like. Such QEC codes may also include but are not limited to ID codes such as repetition codes. Other QEC codes considered by the embodiments may include other codes, such as Shor’s code, and the like. The following discussion is targeted towards quantum circuits for extracting the syndrome of a surface code. However, as noted above, the embodiments are not limited to surface codes. The following discussion is also directed towards ‘'circuit-slicing” embodiments (e.g., see FIGS. 3A-3F), however other embodiments may not “slice” QEC circuits as discussed below. When providing CAFE, some embodiments may provide single qubit fidelities and Pauli error rates for each operation in the QEC code (e.g., a surface code) experiment in context, where “single qubit” may mean that the circuit is analyzed under conditions, where there is no entanglement generated between “neighboring” qubits, allowing for probing all channels for individual qubits.

[0041] In the following discussion, the three Pauli bases are referred to: the X-basis (e.g., the Hadamard basis), the Y-basis, and the Z-basis (e.g., the computational basis). Each of the three Pauli bases has two eigenstates: a positive eigenstate and a negative eigenstate. The positive eigenstate ofthe Z-basis is notated as |0 > (e.g., |0 > has an eigenvalue of +1 with respect to a Z- operator) and the negative eigenstate of the Z-basis is notated as 11 > (e.g., 11 > has an eigenvalue of -1 wi th respect to a Z-operator). The positive eigenstate of the X-basis is notated as |+> (e.g., |+> has an eigenvalue of +1 w ith respect to an X-operator) and the negative eigenstate of the X- basis is notated as |— > (e.g., |— > has an eigenvalue of -1 with respect to an X-operator). The positive eigenstate of the Y-basis is notated as | + i > (e.g., | + i > has an eigenvalue of + 1 with respect to a Y -operator) and the negative eigenstate of the Y -basis is notated as | — i > (e.g., | — i > has an eigenvalue of -1 with respect to a Y-operator). Thus, there are six Pauli states: |0 > , 11 >, |+>, |->, | + I >, | - i >.

[0042] FIGS. 2A-2C illustrate flowcharts for methods for determining a context-aware fidelity estimation (CAFE) of a quantum error correction code (QEC) circuit, according to various embodiments. More specifically, FIG. 2A illustrates a flowchart for a method 200 for preparing a quantum computing system (QCS) for generating a tomography dataset that is used to determine a context-aware fidelity estimation of a quantum error correction code circuit, according to various embodiments. FIG. 2B illustrates a flowchart for a method 220 for generating a tomography dataset, according to various embodiments. FIG. 2C illustrates a flow-chart for a method 260 for operating a circuit-slice to generate tomography data, according to various embodiments. Note that method 200 of FIG. 2A "‘flows” into method 220 of FIG. 2B. Furthermore, method 220 “calls” method 260 of FIG. 2C. Methods 200, 220, and 260 may be performed on a quantum computing system (QCS) system that includes a set of qubits (e.g., QCS 100 of FIG. 1). The set of qubits may be arranged in a physical and/or virtual layout. The physical virtual layout of the set of qubits may be a ID, 2D, 3D. or higher dimensional layout: e.g., a ID or 2D physical or virtual array. Thus, it may be said that each qubit of the set of qubits has one or more neighboring qubits in the set of qubits. Methods 200, 220, and 260 are discussed in conjunction with FIG. 3 and FIGS. 4A-4F. [0043] FIG. 3 show s a set of qubit 300 and a quantum error correction (QEC) circuit 310 that operates on the set of qubits 300, according to various embodiments. The set of qubits 300 includes six qubits. In the various embodiments, the set of qubits may include more than or less than six qubits. As shown via the shading (and non-shading) of the individual qubits in the set of qubits 300, and as will be discussed in conjunction with block 202 of method 200 of FIG. 2A, in FIG. 3 the set of qubits 300 has been subdivided into a first subset and a second subset. The first (shaded) subset is a set of qubit-to-probe, and the second (non-shaded) subset is a set of neighboring qubits. Each qubit is assigned an initial-state and an initial-basis. In FIG. 3 (and FIGS. 4A-4F) each qubit-to-probe is marked with an initial-state and each neighboring-qubit is marked with an initial-basis. The initial-basis may be inferred from the initial-basis. As will be discussed below, in generating a tomography dataset, each qubit-to-probe will be measured on an expected-basis.

[0044] The QEC circuit 310 includes an ordered set of circuit-layers. The ordered set of circuit layers includes a first circuit-layer 312, a second circuit-layer 314, a third circuit-layer 316, a fourth circuit-layer 318, and a fifth circuit-layer 320, where the first circuit-layer 312 is first in the ordered set of circuit-layers and the fifth circuit-layer 320 is the last circuit-layer in the ordered set of circuit layers. Note that QEC circuit 310 may include additional layers, as indicated by the three dots placed after the fifth circuit-layer 320. Various QEC circuits of the various embodiments may include more than or less than five layers. QEC circuit 310 includes Hadamard gates, X gates, and controlled-Z (CZ) gates, which are indicated by shading of the gates. Note that QEC circuit 310 may include additional and/or other single-qubit and multi-qubit gates.

[0045] Method 200 begins, at block 202, where a QEC circuit (e.g., QEC circuit 310 of FIG. 3) is sliced into an ordered set of circuit-slices. As discussed above in conjunction with FIG. 3, the QEC circuit includes an ordered set of circuit-layers. Each circuit-slice of the ordered set of circuit-slices includes a subset of the set of circuit-layers. Furthermore, each circuit-slice of the ordered set of circuit-slices (except for a zeroth-slice) includes a previous circuit-slice in the ordered set of circuit-slices and a next circuit-layer in the ordered set of circuit-layers. The zeroth circuit-layer in the ordered set of circuit-slices includes a null set of circuit-layers (e.g., the zeroth circuit-slice is the first circuit-slice in the ordered set of circuit-slices).

[0046] FIGS. 4A-4F demonstrate the slicing of the quantum error correction circuit 310 of FIG. 3 into an ordered set of circuit-slices, according to various embodiments. Each of FIGS. 4A- 4F shows the set of qubits 300. Taken in the aggregate and in sequence, FIGS. 4A-4F illustrate the ordered set of circuit-slices generated at block 202 of method 200, where each FIG. in FIGS. 4A- 4F shows a separate circuit-slice of the ordered set of circuit-slices. The ordering of the ordered set of circuit-slices (e.g., generated via the slicing of the QEC circuit 310 of FIG. 3) is the same as the ordering of the circuit-slices illustrated in the ordering of the FIGS, of FIGS. 4A-4F. As will be discussed in further conjunction with FIGS. 2A-2C. each circuit-slice in the ordered set of circuit slices operates on the set of qubits 300. Prior to the circuit-slice operating on the set of qubits 300, each qubit is prepared in an assigned initial-state (e.g., assigned in block 206 of method 200). The initial-states of the qubits-to-probed are indicated in FIGS. 4A-4F. After the circuit-slice operates on the set of qubits 300, each qubit-to-probe of the set of qubits-to-probe is measured in an ‘"expected-basis, where the expected-basis are determined and assigned in block 230 of method 220. This generates an ordered set of qubit-to-probe measurements. The qubit-to-probe measurements are aggregated into a tomography dataset.

[0047] FIG. 4A shows the zeroth circuit-slice 400 of the ordered set of circuit-slices and a zeroth set of qubits-to-probe measurements 420. The zeroth circuit-slice 400 includes the null set of the set of circuit-layers of the QEC circuit 310. FIG. 4B shows the first circuit-slice 402 of the ordered set of circuit-slices and a first set of qubits-to-probe measurements 422. The first circuitslice 402 includes the previous circuit-slice (the zeroth circuit-slice 400 of FIG. 4A) and the next circuit layer (e.g., the first circuit-layer 312 of FIG. 3). FIG. 4C shows the second circuit-slice 404 of the ordered set of circuit-slices and a second set of qubits-to-probe measurements 424. The second circuit-slice 404 includes the previous circuit-slice (the first circuit-slice 402 of FIG. 4B) and the next circuit layer (e.g.. the second circuit-layer 314 of FIG. 3). FIG. 4D shows the third circuit-slice 406 of the ordered set of circuit-slices and a third set of qubits-to-probe measurements 426. The third circuit-slice 406 includes the previous circuit-slice (the second circuit-slice 404 of FIG. 4C) and the next circuit layer (e.g., the third circuit-layer 316 of FIG. 3). FIG. 4E shows the fourth circuit-slice 408 of the ordered set of circuit-slices and a fourth set of qubits-to-probe measurements 428. The fourth circuit-slice 408 includes the previous circuit-slice (the third circuit-slice 406 of FIG. 4D) and the next circuit layer (e.g., the fourth circuit-layer 318 of FIG. 3). FIG. 4F shows the fifth circuit-slice 410 of the ordered set of circuit-slices and a fifth set of qubits- to-probe measurements 430. The fifth circuit-slice 410 includes the previous circuit-slice (the fourth circuit-slice 408 of FIG. 4E) and the next circuit layer (e.g., the fifth circuit-layer 320 of FIG. 3). In the ordered set of circuit-slices, the zeroth circuit-slice 400 is the first circuit-slice and the fifth circuit-slice 410 is the last circuit-slice.

[0048] Returning to method 200 of FIG. 2A, in block 204, a set of qubits-to-probe is selected from the set of qubits. Upon selecting the set of qubits-to-probe, the set of qubits is subdivided into two disjoint subsets of qubits. The first subset of qubits is the set of qubits-to- probe, and the second subset of qubits is a set of neighboring-qubits. Each neighboring-qubit of the set of neighboring-qubits is a (virtual and/or physical) neighbor to one or more qubits-to-probe of the set of qubits-to-probe. Each neighboring-qubit of the set of neighboring-qubits corresponds to the qubit-to-probe of the set of qubits-to-probe that the neighboring-qubit neighbors. Selecting the set of qubits-to-probe is discussed at least in conjunction with FIG. 6. FIGS. 3-4F shows the subdivision of the set of qubits 300 into the selected set of qubits-to-probe and the set of neighboring-qubits via shading (and non-shading) of the set of qubits 300. [0049] In block 206, one of the six Pauli-states is assigned to each of the qubits-to-probe of the set of qubits-to-probe. The six Pauli states include: |0 >, | 1 >, |+>, |— >, | + i >, | — i >. As show n in FIGS. 3-4F, one of the qubits-to-probe is assigned the Pauli state |+> and the other qubit-to-probe is assigned the Pauli state 11 >. The assignment of the initial-state to the qubits-to- probe may be constrained by the QEC circuit. For instance, if the QEC circuit includes X-basis and Z-basis stabilizers (as in common in many QEC codes), the assignment of Pauli-states may be limited to the eigenstates of the X-basis and the Z-basis. Each of the six Pauli states is either a positive-valued eigenstate or a negative-valued eigenstate of one of three Pauli-bases. The three Pauli bases include the Z-basis (e.g., which may be referred to as the computational basis), the X- basis (e.g., which may be referred to as the Hadamard basis) and the Y-basis. Thus, each Pauli basis corresponds to exactly two of the Pauli-states. For instance, the Pauli-state |0 > is the positive-valued eigenstate of the Z-basis and the Pauli-state 11 > is the negative-valued eigenstate of the Z-basis. Thus, the Z-basis corresponds to the two Pauli states: |0 > and 11 >. The Pauli- state |+> is the positive-valued eigenstate of the X-basis and the Pauli-state |— > is the negativevalued eigenstate of the X-basis. Thus, the X-basis corresponds to the two Pauli states: |+> and |— >. The Pauli-state | + i > is the positive-valued eigenstate of the Y-basis and the Pauli-state | — i > is the negative-valued eigenstate of the Y-basis. Thus, the Y-basis corresponds to the tw-o Pauli states: | + i > and | — I >. For reasons that will become apparent below, the Pauli-state assigned to each qubit-to-probe may be referred to as the qubit-to-probe’s initial state.

[0050] In block 208, the Pauli-basis that corresponds to the Pauli-state that is assigned to the qubit-to-probe is also assigned to the qubit-to-probe. In FIGS. 3-4F, the X-basis is assigned to the qubit-to-probe that is assigned the |+> Pauli-state and the Z-basis is assigned to the qubit-to- probe that is assigned the 11 > Pauli-state.

[0051] In block 210, the tomography dataset is generated. More specifically, method 200 flows to block 222 of method 220 of FIG. 2. Thus, it may be said that method 200 “calls’" method 220 at block 210.

[0052] Turning attention to method 220 of FIG. 2B, it is observed that blocks 222-236 form an “innermost” loop that loops over the set of circuit-slices. Thus, a counter index (e.g., i) may be used, where the index runs from 0 < i < n — 1, where n is the number of circuit-slices in the set of circuit-slices. Each time through the loop between blocks 222-236, n may be incremented by a value of 1. At block 222, the next circuit-slice (e.g., the circuit slice that corresponds to the value of ri) of the set of circuit-slices is selected. Thus, for instance, the first time through this loop, the zeroth circuit-slice 400 of FIG. 4A is selected. The second time through this loop, the first circuit- slice 402 of FIG. 4B is selected, and so on. This loop terminates, at decision block 236, after the last circuit-slice of the set of circuit-slices has been selected. Furthermore, block 206 of method 200 and decision block 238 of method 220 form a “middle’’ loop and block 204 of method 200 and decision block 240 form an “outermost” loop. Thus, methods 220 and 220 include three nested loops.

[0053] At block 224, an initial-basis is assigned to each neighboring-qubit of the set of neighboring qubits based on the circuit-slice that was selected at block 222. Various embodiments of determining and assigning an initial-basis is discussed at least in conjunction with FIG. 5. However, briefly here, the initial-basis is determined and assigned to a neighboring-qubit is done such that, when the selected circuit-slice operates on the set of qubits, entanglement between the neighboring-qubit and the corresponding qubit-to-probe that neighbors the neighboring-qubit is avoided. The initial-basis assigned to a neighboring-qubit may be one of the three Pauli-bases. In FIGS. 3-4F, the Pauli-basis assigned to each neighboring-qubit of the set of neighboring-qubits is marked. For instance, each neighboring-qubit that neighbors the qubit-to-probe that is assigned the |+> Pauli-state is assigned the X-basis. Each neighboring-qubit that neighbors the qubit-to-probe that is assigned the 11 > Pauli-state is assigned the Z-basis.

[0054] At block 226, each neighboring-qubit is assigned an initial-state. The initial-state assigned to a neighboring qubit is either the positive-valued or the negative-valued eigenstate of the Pauli-basis that is assigned to the neighboring qubit (e.g., see block 224). Thus, the initial-state assigned to a neighboring-qubit may be one of the six Pauli-states.

[0055] At block 228, for each qubit-to-probe, an expected-state is determined and assigned to the qubit-to-probe. Determining the expected-state for a qubit-to-probe is based on the initialstate assigned to the qubit-to-probe (e.g., see block 206 of method 200) and the selected circuitslice (e.g., see block 222 of method 220). To determine the expected-state of a qubit-to-probe, an analysis of the selected circuit-slice operating on the set of qubits may be performed. When performing the analysis of operating the circuit-slice on the set of qubits, each qubit may be assumed to be initialized in the initial-state assigned to the qubit and each qubit is an “error-free” qubit. Furthermore, the circuit-slice may be assumed to operate on each qubit (e.g., prepared in its assigned initial-state), and each gate in the circuit-slice may operate on the corresponding qubits without errors (e.g., error-free gates). When the selected circuit-slice operates on the set of qubits without error, the qubit-to-probe is transformed into the expected-state. That is, a determination may be made such that, if the selected circuit-slice operates on the set of qubits (e.g.. when the set of qubits have been initialized (or prepared) in the initial-states discussed above), the state that the qubit-to-probe is expected to be in after the circuit-slice operates on the set of qubits (e.g., in an error-free fashion) is referred to as the qubit-to-probe’ s expected state and is assigned to the qubit- to-probe. The expected-state determined for each qubit-to-probe is the state expected in the absence of errors (or noise) in the selected circuit-slice.

[0056] At block 230, each qubit-to-probe is also assigned with an expected-basis. The expected-basis is a basis that has an eigenstate that is equivalent to the expected-state. Depending on the details of the selected circuit-slice, the expected-state may be a Pauli-state and the expected- basis may be a Pauli-basis.

[0057] At block 232, the selected circuit-slice operates on the set qubits based on the initial-states assigned to the set qubits and the expected basis assigned to the set of qubits-to-probe. Various embodiments of operating the circuit-slice on the set of qubits are discussed in conjunction with at least method 260 of FIG. 2C.

[0058] At block 234, the tomography dataset is updated based on operating the selected circuit-slice on the set of qubits. Various embodiments of operating the circuit-slice on the set of qubits are discussed in conjunction with at least method 260 of FIG. 2C.

[0059] At decision block 236. it is decided whether the circuit-slice selected at block 222 is the last circuit-slice in the ordered set of circuit-slices. If the selected circuit-slice is the last circuit-slice in the ordered set of circuit-slices, then method 220 goes to decision block 238. If the selected circuit-slice is not the last circuit-slice in the ordered set of circuit-slices, then method 220 returns to block 222 to select the next circuit-slice in the ordered set of circuit-slices.

[0060] At decision block 238, it is decided whether sufficient statistics for the set of qubits- to-probe has been generated in the tomography dataset. If sufficient statistics for the set of qubits- to-probe have been generated, method 220 returns to block 206 of method 200 of FIG. 2A to select another set of qubits-to-probe. If sufficient statistics have been generated for the set of qubits-to- probe, then method 220 flows to decision block 240.

[0061] At decision block 240, it is determined whether each qubit of the set of qubits has been selected at least once as a qubit-to-probe. If there are one or more qubits that have not yet been selected as a qubit-to-probe, then method 220 goes to block 204 of method 200 to select additional qubits-to-probe. Otherwise, method 220 flows to block 242.

[0062] At block 242, context-aware fidelities are determined/estimated based on the tomography dataset. Various embodiments of estimating context-aware fidelities are discussed at least in conjunction with FIGS. 7-10. However, the tomography dataset may be analyzed to determine the Pauli error rates. The determination of the Pauli error rates can be used to extract the fidelities between the single measured qubit states and their ideal states (e.g., the expected-states), as well as extract Pauli error rates for each gate in the circuit, in context. These error rates may not be the standard fidelities and Pauli error rates described for gates, as they are single qubit channels. As such, a two qubit gate’s error channel is described by a pair of single qubit Pauli channels, or alternatively the fidelity of each qubit individually.

[0063] Turning attention to method 260 of FIG. 2C. method 260 is called from block 232 of method 220 of FIG. 2B. Method 260 starts at block 262. where each qubit-to-probe is prepared in the assigned initial-state (e g., one of the six Pauli-states). FIGS. 3-4F show that the qubits-to- probe have been prepared in the assigned initial-states.

[0064] At block 264, each neighboring-qubit is prepared in the initial-state assigned to the neighboring-qubit.

[0065] At block 266, the selected circuit-slice operates on the set of qubits, where each qubit has been prepared in the corresponding assigned initial-state.

[0066] At block 268, each qubit-to-probe is measured in the expected-basis assigned to the qubit-to-probe (e.g., the expected-basis assigned to the qubit-to-probe in block 230 of method 220). [0067] At block 270, the measurements of the qubits-to-probe are compared to the expected-states (e.g., determined and assigned at block 228 of method 220) of the qubits-to-probe. [0068] At block 272, the tomography dataset is updated based on the comparisons between the measurements of the qubits-to-probe to the expected-states of the qubits-to-probe. For instance, the measurements and/or the comparisons of the measurements may be included in the tomography dataset. Method 260 returns to decision block 236 of method 220.

Assigning Initial-Bases and Initial-States to Neighboring-Qubits.

[0069] FIG. 5 shows examples of determining the initial-bases for neighboring-qubits, according to various embodiments. That is, FIG. 5 provides examples of how, for some Clifford circuits, including standard surface code stabilizer measurement circuits, the initial-bases (and the initial-states) of neighboring-qubits are determined and then assigned (e.g., as assigned at blocks 224 and 226 of method 220 of FIG. 2B), according to some embodiments. In block 224 of method 220 of FIG. 2B, the initial-bases for the neighboring-qubits are determined and assigned such that no entanglement is generated between a qubit-to-probe and the neighboring-qubits that neighbor the qubit-to-probe. Avoiding entanglement between neighboring qubits allows for the analysis of the qubit gate errors as if they were separate single qubit channels (e.g., so that errors need not be analyzed as correlated errors in pairs of entangled qubits). Thus, the neighboring-qubits may be referred to as “constrained qubits’" as their initial Pauli-bases are constrained by the structure of the circuit.

[0070] More specifically, FIG. 5 shows how initial-bases for the neighboring-qubits can be determined and assigned (e.g., in block 224 of method 220) to prevent entanglement between different qubits. Since the qubits-to-probe (e.g., the qubits marked with the initial-state i i will have multiple initial states fed into it (via the loop between block 206 of method 200 of FIG. 2A and block 240 of method 220 of FIG. 2B), all the neighboring-qubits that are subject to a CZ gate (e.g., where the partner qubit is a qubit-to-probe) should be in a Z-basis eigenstate (at the input to a CZ gate), to prevent entanglement. This desired property⁷ constrains the initial-bases of the neighboring-qubits.

[0071] FIG. 5 shows an initial circuit 500. The initial circuit 500 may be a circuit-slice as discussed above. In FIG. 5, each of the two-qubit gates are CZ gates. However, other two-qubit gates (e.g., CNOT gates) may be included in a circuit-slice. The qubits-to-probe in FIG. 5 are marked with the initial-state: t i assigned to them (e.g., via block 206 of method 200 of FIG. 2A). Even though the two qubits-to-probe are marked with identical initial -states, the two qubits-to-be may be prepared in separate initial-states. The other qubits in FIG. 5 are neighboring-qubits. As noted above, the initial-bases for the neighboring qubits are determined (and assigned) such that the two-qubit gates do not generate entanglement between the qubits-to-probe and the neighboringqubits. That is, the initial-bases for the neighboring-qubits are selected and/or determined to avoid entanglement with the qubits-to-probe (e.g.. which are subject to being initialized in the initialcircuit 500 to any of the six Pauli-states).

[0072] The first step to determine initial-bases for the neighboring-qubits is to identify constraints 520 in the initial-circuit 500. In the case where the potentially-entangling two-qubit gates are CZ gates, to avoid entanglement, at least one of the two qubits should be in the Z-basis (e.g., not in a superposition of the two eigenstates of the Z-basis, such as but not limited to an eigenstate of the X-basis or an eigenstate of the Y-basis). Since the qubits-to-probe can be in various states as methods 200 and 220 are looped over, the neighboring-qubits, at the input of a CZ gate, should not be in an eigenstate of the Z-basis. This is shown in the identifying constraints 520 of FIG. 5. The second step in determining the initial-bases for the neighboring-qubits includes propagating these identified constraints through 540 the entire initial -circuit 500 (or the circuitslide). As shown in FIG. 5, this results in determining the initial-base for the “upper” neighboringqubit as the Z-basis. Due to the presence of the Hadamard gate (e.g.. which may transform a Z- basis eigenstate into an X-basis eigenstate and vice-versa) in the qubit-line for the “lower” neighboring-qubit (which changes the basis for the lower neighboring-qubit), the initial-basis for the lower neighboring-qubit should be the X-basis.

[0073] It can be shown that qubits sufficiently far apart from each other constrain their neighbors in ways that do not interfere, allowing them to be probed simultaneously. However, it is also possible, especially in circumstances like the surface code stabilizer measurement circuit which are regular in structure, for next-nearest-neighboring qubits to constrain their shared neighbors in identical ways, allowing them to be characterized together. This is shown in FIG. 5. [0074] In this way, all the qubits in a surface code stabilizer measurement circuit can be probed using three compatible groups of qubits, regardless of distance. FIG. 6 shows three compatible sets of qubits for a standard surface code circuit, according to various embodiments. The three compatible sets of qubits include all data qubits 600, all X-measure qubits 620, and all Z- measure qubits. All data qubits 600 can be calibrated together. Then the measure qubits (e.g., X- measure qubits 620 and Z-measure qubits 640) can be calibrated according to their corresponding basis. Qubits in black are being characterized, while the unshaded and shaded qubits are prepared in the X-basis or Z-basis, respectively.

[0075] As an example, when the measurement qubits are prepared in the |+) state, they hit the initial Hadamard operation in the initial circuit 500 of FIG. 5 and then spend the entangling rounds in the |0) state. This means that they do not entangle with their neighbors. As a result, all data qubits can be simultaneously probed by constraining all measure qubits to the X-basis initially and having them act as walls blocking entanglement from being generated.

[0076] Thus, in some embodiments, at block 204 of method 200 of FIG. 2 A, when selecting the set of qubit-to-probe, the set of all data qubits may be selected, the set of all X-basis measurement qubits may be selected, or the set of Z-basis measurement qubits may be selected. Thus, the loop between block 204 of method 200 and decision block 240 of method 220 of FIG. 2B may be iterated over at least three times.

Analysis of the Tomography Dataset

[0077] Context-dependent gate and qubit-fidelity information about each operation of a quantum circuit may be extracted from the tomography dataset generated. As a reminder, the tomography dataset includes a comparison of the expected-states and the measured states of the qubits-to-probe. This section discusses how the tomography dataset can be processed and analyzed to provide estimations of gate fidelity, as well as a form of Pauli-error rates for the qubits, in the context of the actual quantum circuit used in a quantum algorithm (including the QEC circuit used for fault tolerance). These fidelities and Pauli-error rates can be attributed to the gate errors in the system, but they do not follow the standard definitions in the literature, as the embodiments separate them into single qubit channels. For single qubit operations this does not cause issues, but for two-qubit operations, the reduction in state fidelity between a qubit and its expected-state is being measured and then attributing that to the entangling operation. This leads to each two-qubit gate having two "infidelities” and two sets of "Pauli error rates”, as opposed to the standard model. Below, how this approximation fares relative to the standard case is discussed.

Extracting State Fidelities

[0078] One fidelity quantity to determine (or estimate) is the fidelity between a qubit’s measured-state (e.g.. as measured in block 268 of method 260) and the qubit’s expected-state (e.g., as determined and assigned in blocks 228 and 230 of method 220). These determinations may be dependent on the choices of qubit initialization (e.g., the initial-state of a qubit-to-probe as assigned in block 208 of method 200), constrained qubit initialization, and circuit depth (e.g., the number of layers in the cunent circuit-slice). For a given compatible qubit set and choice of depth, the below discussion considers up to 72 experiments, corresponding to six choices of initial Pauli-states for the qubits-to-probe, six choices of measurement Pauli-states, and two choices of initial eigenvalue for the constrained qubits (e.g., the neighboring qubits). The initial-states define what the final states (e.g., the expected-states) should be for the measurements of the qubits-to-probe (e.g., the expected-states determined in block 228). The fidelity between the final state (e.g., the measured- state) and these ideal values (e.g., the expected-state) by averaging the probability of getting the correct outcomes in the two deterministic Pauli measurements (for a qubit which should end in |0), this would be |0) and 11)). By averaging these sub-fidelities for all 12 choices of initial-state and constrained qubit eigenvalue, a measurement (or estimation) of the cumulative average gate fidelity for that qubit-to-probe may be determined.

[0079] Some embodiments may then subtract the consecutive fidelity measurements to assign infidelities to each gate. This approximation may fail when coherent errors are brought into consideration, as a coherent revival will come out as a negative infidelity, but possibly by considering the purity of the qubit state at each depth this could be avoided. This approximation may also fail when considering errors which propagate over two-qubit operations. In the next section, the process of subtracting these error rates out when correctly tracking errors is discussed. Extracting Pauli Error Rates

[0080] Measuring the state fidelity for each initial-basis allows for the extraction of information about the basis of the error. As show n in FIG. 7, extracting such information (e.g., determining and/or estimating Pauli-error rates) is done via three stages. FIG. 7 illustrates a flowchart for a method 700 for determining Pauli-error rates, according to various embodiments. Method 700 begins at block 702, where the fidelities for each initial-basis are extracted separately for each qubit of the set of qubits. In some embodiments, the fidelities for each of the six Pauli- states for each qubit is determined at block 702. At block 704, the fidelities for each qubit are remapped to the fidelities for the state that the qubit is in at that moment in the circuit. This may be performed for each layer in the quantum circuits and for each qubit. Thus, remapping the fidelities of the qubits may be based on the quantum circuit. That is, if a qubit is assigned an initial-state corresponding to the X-basis (e.g., |+>) and, in the first circuit-layer, the qubit is operated on by a Hadamard gate, the qubit in now in an eigenstate of the Z-basis (e.g., 11 >). Then the subsequent initial X-basis fidelities are mapped to Z-basis fidelities, and vice versa, until the qubit hits another Hadamard gate, remapping the Z-basis fidelities to X-basis fidelities. In block 706, Pauli-error rates are determined/estimated and assigned to each gate in the quantum circuit. Determining the Pauli-error rates is based on subtracting the cumulative Pauli-error rates for a qubit from those in the previous round. The remainder of this section discusses this subtraction step at block 706 in more detail.

[0081] For simplicity, first consider a case in a quantum circuit where a gate is occurring that has the action of an overall identity operation (e.g., the identity' gate I). How ever, due to noise it ends up having a weak Pauli-error channel attached (e.g., the identity gate causes a Pauli-rotation around the Bloch sphere). FIG. 8 demonstrates an identity gate inducing Pauli -errors on a qubit. More specifically, FIG. 8 shows gate error labels 800 for an identity gate, according to various embodiments. The cumulative Pauli-error rates before the identity gate are labeled as {Px,n-i> Py,n-i’ Pz,n-iJ (where n may be an index for the current circuit-layer and/or the current circuit-slice). As shown in FIG. 8, the error rates after the identity gate are labeled as {Px,n> Py,n> Pz,n}- The 8^0a' is to solve for the gate-error rates {p_Xi3, P_y,_g> Pz_fg}- where g is an index for the gate type. The gate-error rates (e.g., indexed via ) are not cumulative, but rather (as shown below in the case where the gate-errors are relatively small) represent that "‘delta⁷’ between the cumulative rates at the (n — 1) circuit-layer and the n circuit-layer. That is, the gate-error rates (e.g., indexed via g) are the error-rates (or gate-error probabilities) introduced into the qubit lines via a gate of type g. Using these cumulative probabilities, equations for the final probabilities {Px,g> Py,g> Pz,g} may be written using the other two sets (e.g.. the before-the-gate cumulative probabilities {p_x,n-i> p_y,_n-i> Pz,n-i ^ancl the after-the-gate cumulative probabilities

{Px,n> Py,n> Pz,n})- FIG. 8 shows eq (1), which includes one form of these equations for the final probabilities. Since eq (1) is a linear system of equations, eq (1) may be written in matrix form as eq (2), also shown in FIG. 8. Using linear algebra techniques, if the matrix of previous-round probabilities is defined as A, the matrix of eq (2) can be inverted to solve for the gate-errors {Px,g> Py,g> Pz.g}- FIG. 8 also shows eq (3), which employs eq (2) to solve for the gate-errors. It can be shown that if the assumption {p_{x g}, p_y>g, p_Zig} « 1 holds, then the approximation A » 1₃ holds and the gate error rates simplify to just being the difference between the two probability matrices.

[0082] The same analysis can be used for other single-qubit Pauli operators (or single-qubit gates), as at least some do not change the Pauli-basis of a state. For gates like S or H, there is a slight complexity added due to the fact that the ideal gate operation does interchange Pauli bases, but that essentially amounts to interchanging the previous-round probabilities as appropriate.

[0083] A more complex case is considered for surface code circuits is the two-qubit CZ gate. Due to its potentially-entangling operations, CZ -gates may inject errors from the qubits that propagate across the gate and must be taken into account. The construction of surface code-CAFE (SC-CAFE) experiments guarantees that the neighboring qubit is in either |0) or |1). That is. as discussed above, the initial-bases for the neighboring-qubits may be selected and/or assigned to avoid entanglement. Depending on that state, one can look for errors relative to the frame of an I or Z gate. Once this is dealt with, one may also include the errors which propagate from the other qubit in the CZ gate. An IX or IY error which hits a CZ gate propagates to a ZX or ZY error on the other side. FIG. 9 shows an example of propagating errors through a two-qubit gate, according to various embodiments. More specifically, FIG. 9 shows an example of error propagations 900 for a CZ gate when one the affected qubits is in an eigenstate of the X-basis and when one of the affected qubits is in an eigenstate of the Y-basis. Errors may be propagated in two-qubit gates other than CZ gates in a similar manner. This means that these error-rates may be also accounted for when calculating the gate Pauli-errors. This leads to a form equation to eq (1) of FIG. 8, as shown in eq (4) in FIG. 10. FIG. 10 show-s gate error labels 1000 for an CZ gate, according to various embodiments. FIG. 10 is similar to FIG. 8, but rather than showing gate error labels 800 for an identity gate, FIG. 10 shows gate error labels 1000 for a CZ gate, according to various embodiments. FIG 10 also shows eq (4) which is analogous to eq (1) of FIG. 8, except for a two- qubit gate (e.g., a CZ gate). With two cases accounted for, the case when the other qubit does not have an X or Y error and the case where it does (refer to the error probabilities of the other qubit using shown in FIG. 10):

[0084] Eq (4) of FIG. 10, like in the identity gate example (e.g., eq (1)), can be inverted into a linear equation for the gate error rates as a function of the other parameters, however in this case one may invert two matrices, one for each case mentioned above.

Removing State Preparation and Measurement Errors

[0085] In order to extract gate errors, the embodiments also address state preparation and measurement (SPAM) errors. For instance, the zeroth circuit-slice (e.g., see zeroth circuit-slice 400 of FIG. 4A) does not include the first circuit layer (e.g., e.g., see first circuit-layer 312 of FIG. 3) of the quantum circuit (e.g., see QEC circuit 310 of FIG. 3). Furthermore, when removing the impact of a neighboring qubit during the CZ gate, the errors measured on that neighboring qubit in the previous round will include SPAM errors, but only state preparation errors and gate errors during the rest of the circuit prior to the CZ actually will propagate over. To account for this, some embodiments are enabled to separate state preparation errors from measurement errors. Other embodiments may assume that SPAM errors are dominated by the measurement side. These embodiments may analyze these contributions by themselves and subtract them out from the {Px> Py> Pz quantities that are measured before doing the calculations outlined above.

Methods

[0086] FIG. 11 shows a flowchart describing methods 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. 11 shows a method 1100 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 1100 begins at block 1102, where a set of circuit-slices is generated. Each circuit-slice in the set of circuitslices is a circuit-slice of a quantum circuit that implements a quantum error correction (QEC) code. At block 1104, the set of qubits is sub-divided into a first subset of qubits and a second subset of qubits. The first subset of qubits is a set of qubits-to-probe. The second subset of qubits is a set of neighboring qubits. Each neighboring-qubit of the set of neighboring-qubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit. At block 1106, a tomography dataset is generated. Various embodiments of generating a tomography dataset are discussed at least in conjunction with FIGS. 2A-2C. However, briefly here, generating the tomography dataset is based on a set of qubit measurements. Each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to-probe of the set of qubits-to-probe subsequent to operating at least one circuit-slice of the set of-circuit-slices on the set of qubits. At block 1108, a set of fidelities for the set of qubits is estimated. Estimating the set of fidelities is based on the tomography dataset. The set of fidelities corresponds to a context the quantum circuit. Various embodiments of estimating a set of fidelities for the set of qubits are discussed at least in conjunction with FIGS. 7-10.

[0087] In some embodiments, generating the tomography dataset includes assigning an initial-state to each qubit-to-probe of the set of qubits-to-probe. A first circuit-slice of the set of circuit-slices operates on the set of qubits. Prior to operating the first circuit-slice on the set of qubits, each qubit-to-probe of the set of qubits-to-probe has been prepared in the initial-state assigned to the qubit-to-probe. Subsequent to operating the first circuit-slice on the set of qubits, a first subset of the set of qubit-measurements is generated via measuring each qubit-to-probe of the set of qubits-to-probe. The set of qubit-measurements is updated to include the first subset of qubit-measurements. The tomography dataset is updated based on the updated set of qubitmeasurements.

[0088] The initial-state assigned to each qubit-to-probe of the set of qubits-to-probe is a

Pauli-state of a set of Pauli-states.

[0089] The set of circuit-slices is an ordered set of circuit-slices. Generating the tomography dataset further includes selecting a second circuit-slice of the set of circuit-slices. The second circuit-slice is subsequent to the first circuit-slice in the ordered set of circuit-slices. The second circuit-slice operates on the set of qubits. Prior to operating the second circuit-slice on the set of qubits, each qubit-to-probe of the set of qubits-to-probe has been prepared in the initial-state assigned to the qubit-to-probe. Subsequent to operating the second circuit-slice on the set of qubits, a second subset of the set of qubit-measurements is generated via measuring each qubit-to- probe of the set of qubits-to-probe. The set of qubit-measurements is updated to include the second subset of qubit-measurements. The tomography dataset is updated based on the updated set of qubit-measurements .

[0090] The quantum circuit includes an ordered set of circuit-layers. Each circuit slice of the set of circuit-slices includes a separate subset of the set of layers. The first circuit-slice includes a first subset of the set of layers. The second circuit slice includes a second subset of the set of circuit-layers. The second subset of circuit-layers includes the first subset of circuit-layers and at least one additional circuit layer that is not included in the first subset of circuit-layers. [0091] Generating the tomography dataset further includes assigning an initial-basis to each neighboring-qubit of the set of neighboring-qubits. The initial-basis assigned to each neighboringqubit is determined based on a set of constraints corresponding to the quantum circuit. An initialstate is assigned to each neighboring-qubit based on the initial-basis assigned to the neighboringqubit. The first circuit-slice of the set of circuit-slices operated on the set of qubits. Prior to operating the first circuit-slice on the set of qubits, each neighboring-qubit of the set of neighboring-qubits has been prepared in the initial-state assigned to the neighboring-qubit.

[0092] Determining the initial-basis for each neighboring-qubit of the set of neighboringqubits includes determining the set of constraints based on analyzing the quantum-circuit. Each constraint of the set of constraints corresponds to a separate two-qubit gate included in the quantum-circuit. Each constraint of the set of constraints is propagated through the quantum circuit. Each constraint is propagated forward and backward from the corresponding two-qubit gate and through the quantum circuit. For each neighboring-qubit of the set of neighboring-qubits, the initial-basis is determined based on propagating each constraint of the set of constraints through the quantum circuit.

[0093] Each constraint of the set of constraints includes avoiding entanglement of the corresponding neighboring qubit with a corresponding qubit-to-probe of the set of qubits-to-probe that the corresponding two qubit-gate operates on.

[0094] The initial-basis assigned to each neighboring-qubit of the set of neighboring-qubits is a Pauli-basis.

[0095] In some embodiments, the method may further include assigning an initial-state to each neighboring-qubit of the set of neighboring qubits based on the initial-basis assigned to the neighboring-qubit. Prior to operating the first circuit-slice on the set on the set of qubits, each neighboring-qubit of the set of neighboring-qubits is prepared in the initial-state assigned to the neighboring-qubit.

[0096] The initial-state assigned to each neighboring-qubit of the set of neighboring-qubits is an eigenstate of the initial-basis assigned to the neighboring-qubit.

[0097] In some embodiments, the method may further include prior to operating the first circuit-slice on the set of qubits, for each qubit-to-probe of the set of qubits-to-probe, assigning expected-state to the qubit-to-probe based on the initial-state assigned to the qubit-to-probe and the first circuit-slice. Prior to operating the first circuit-slice on the set of qubits, for each qubit-to- probe of the set of qubits-to-probe, an expected-basis is expected to the qubit-to-probe based on the expected-state assigned to the qubit-to-probe. The first subset of the set of qubit-measurements is generated via measuring each qubit-to-probe of the set of qubits-to-probe in the expected-basis that is assigned to the qubit-to-probe.

[0098] The expected-expected state assigned to each qubit-to-probe of the set of qubits-to- probe is a state that is expected the qubit-to-probe to be in, subsequent to the first circuit-slice operating on the set of qubits, when the first circuit-slice operates on the set of qubits without error or without noise.

[0099] The expected-state assigned to each qubit-to-probe of the set of qubits-to-probe is an eigenstate of the expected-basis assigned to the qubit-to-probe.

[0100] The tomography dataset includes a measurement of each qubit-to-probe, and the expect-state assigned to each qubit-to-probe.

[0101] The set of fidelities includes a fidelity for each possible initial-basis for each qubit of the set of qubits and a Pauli-error rate for each gate of a set of gates included in the quantum circuit.

[0102] Estimating the set of fidelities for the set of qubits includes extracting the fidelity of each possible-basis for each qubit of the set of qubits based on a tomography algorithm applied to the tomography dataset. The fidelities for each qubit of the set of qubits are remapped to each gate of the set of gates based on an expected-state that the qubit is expected to be in at that gate in the quantum circuit, wherein the remapping is based on the quantum circuit.

[0103] The Pauli-error rate for each gate of a set of gates included in the quantum circuit are assigned based on subtracting a cumulative Pauli-error rate for a qubit from the in a previous round.

[0104] In some iterations of method 1100, the set of qubits-to-probe includes each data qubit in the QEC code.

[0105] In other iterations of method 1100, the set of qubits-to-probe includes each measure qubit in a first stabilizer ty pe of the QEC code.

[0106] A two-qubit quantum logic gate included in the quantum circuit operates on each neighboring-qubit of the set of neighboring-qubits and the at least one qubit-to-probe of the set of qubits-to-probe.

[0107] Other embodiments are directed to a quantum computing system (QCS). The QCS includes a quantum processor that includes a set of qubits. The QCS also includes one or more memory devices. The one or more memory devices store computer-readable instructions that when executed by the one or more quantum processors cause the one or more processors to perform operations for operating the QCS. The operations include generating a set of circuit-slices. Each circuit-slice in the set of circuit-slices is a circuit-slice of a quantum circuit that implements a quantum error correction (QEC) code. The set of qubits is subdivided into a first subset of qubits and a second subset of qubits. The first subset of qubits is a set of qubits-to-probe. The second subset of qubits is a set of neighboring qubits. Each neighboring-qubit of the set of neighboringqubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit. A tomography dataset is generated based on a set of qubit measurements. Each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to-probe of the set of qubits- to-probe subsequent to operating at least one circuit-slice of the set of-circuit-slices on the set of qubits. A set of fidelities for the set of qubits is estimated based on the tomography dataset. The set of fidelities corresponds to a context the quantum circuit.

Additional Embodiments

[0108] Another non-limiting embodiment includes a method for operating a quantum computing system (QCS) (e.g., the QCS 100 of FIG. 1). The QCS includes a set of qubits. The method includes selecting a first subset of the set of qubits, wherein a second subset of the set of qubits is disjointed from the first subset of qubits and each qubit of the second subset of qubits has a neighbor in the qubit array that is included in the first subset of qubits. The method includes for each qubit of the first subset of qubits, selecting a Pauli state of a set of Pauli states. The method includes for each qubit of the first subset of qubits, preparing the qubit in the selected Pauli state. The method includes for each qubit of the second subset of quits, preparing the qubit in a Pauli state of the set of Pauli states based on the selected Pauli state of the neighboring qubit of the first subset of qubits. The method includes executing a first portion of a quantum algorithm that has an associated quantum circuit implemented by the QCS, wherein the first portion of the quantum algorithm includes a first portion of the quantum circuit operating on the first subset of qubits and the second subset of qubits. The method includes in response to executing the first portion of the quantum algorithm, determining a first set of measurements, wherein determining the first set of measurements includes, for each qubit of the first subset of qubits, measuring the qubit in a Pauli basis of a set of Pauli basis that corresponds to the selected Pauli state of the qubit. The method includes determining a set of error rates for the set of qubits based on the first set of measurements, wherein the set of error rates corresponds to the quantum circuit. [0109] The method includes executing a second portion of the quantum algorithm, wherein the second portion of the quantum algorithm includes a second portion of the quantum circuit operating on the first subset of qubits and the second subset of qubits. The method includes in response to executing the second portion of the quantum algorithm, determining a second set of measurements, wherein determining the second set of measurements includes, for each qubit of the first subset of qubits, measuring the qubit in a Pauli basis of the set of Pauli basis that corresponds to the selected Pauli state of the qubit. The method includes determining the set of error rates for the set of qubits based on the first set of measurements and the second set of measurements.

[0110] The method includes selecting a third subset of the set of qubits, wherein a fourth subset of the set of qubits is disjointed from the third subset of qubits and each qubit of the fourth subset of qubits has a neighbor in the qubit array that is included in the first subset of qubits. The method includes for each qubit of the third subset of qubits, selecting a Pauli state of the set of Pauli states. The method includes for each qubit of the third subset of qubits, preparing the qubit in the selected Pauli state. The method includes for each qubit of the fourth subset of quits, preparing the qubit in a Pauli state of the set of Pauli states based on the selected Pauli state of the neighboring qubit of the third subset of qubits. The method includes executing the first portion of the quantum algorithm. The method includes in response to executing the first portion of the quantum algorithm, determining a third set of measurements, wherein determining the third set of measurements includes, for each qubit of the third subset of qubits, measuring the qubit in the Pauli basis that corresponds to the selected Pauli state of the qubit. The method includes determining the set of error rates for the set of qubits based on the first set of measurements, the second set of measurements, and the third set of measurements.

[0111] The method includes executing the second portion of the quantum algorithm, wherein the second portion of the quantum algorithm includes the second portion of the quantum circuit operating on the third subset of qubits and the fourth subset of qubits. The method includes in response to executing the second portion of the quantum algorithm, determining a fourth set of measurements, wherein determining the fourth set of measurements includes, for each qubit of the third subset of qubits, measuring the qubit in a Pauli basis of the set of Pauli basis that corresponds to the selected Pauli state of the qubit. The method includes determining the set of error rates for the set of qubits based on the first set of measurements, the second set of measurements, the third set of measurements, and the fourth set of measurements.

[0112] The method includes for each qubit of the first subset of qubits, selecting another Paili state of a set of Pauli states. The method includes for each qubit of the first subset of qubits, preparing the qubit in the other selected Pauli state. The method includes for each qubit of the second subset of quits, preparing the qubit in a Pauli state of the set of Pauli states based on the other selected Pauli state of the neighboring qubit of the first subset of qubits. The method includes executing the first portion of the quantum algorithm. The method includes in response to executing the first portion of the quantum algorithm, determining a second set of measurements, wherein determining the second set of measurements includes, for each qubit of the first subset of qubits, measuring the qubit in a Pauli basis of a set of Pauli basis that corresponds to the other selected Pauli state of the qubit. The method includes determining the set of error rates for the set of qubits based on the first set of measurements and the second set of measurements.

[0113] It should be understood that a cardinality of the first subset of qubits is two.

[0114] It should be understood that the array of qubits is a two-dimension (2D) array of qubits.

[0115] It should be understood that the quantum algorithm includes an implementation of a quantum error correction (QEC) code.

[0116] It should be understood that the QEC code is a surface code.

[0117] It should be understood that error rates of the set of error rates are physical qubit error rates.

[0118] 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.

[0119] Implementations of the digital and/ or quantum subj ect 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.

[0120] 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.

[0121] 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.

[0122] 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..

[0123] A digital and/or quantum computer program may, but need not, correspond to a fde 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.

[0124] 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.

[0125] 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. [0126] 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 memoiv. or a random access memory⁷, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0127] 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.

[0128] 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 nonvolatile digital and/or quantum memory⁷, media and memory⁷ 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; magneto-optical 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 fidelity⁷ 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.

[0129] 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 memory⁷ to store executable instructions to perform the operations described in this specification. [0130] 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.

[0131] 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.

[0132] 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

WHAT IS CLAIMED IS:

1. A method for operating a quantum computing system (QCS) that includes a set of qubits, the method comprising: generating a set of circuit-slices, wherein each circuit-slice in the set of circuit-slices is a circuit-slice of a quantum circuit; subdividing the set of qubits into a first subset of qubits and a second subset of qubits, wherein the first subset of qubits is a set of qubits-to-probe and the second subset of qubits is a set of neighboring qubits, and each neighboring-qubit of the set of neighboring-qubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit; generating a tomography dataset based on a set of qubit measurements, wherein each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to- probe of the set of qubits-to-probe subsequent to operating at least one circuit-slice of the set of circuit-slices on the set of qubits; and estimating a set of fidelities for the set of qubits based on the tomography dataset, wherein the set of fidelities corresponds to a context of the quantum circuit.

2. The method of claim 1, wherein generating the tomography dataset comprises: assigning an initial-state to each qubit-to-probe of the set of qubits-to-probe; operating a first circuit-slice of the set of circuit-slices on the set of qubits, wherein prior to operating the first circuit-slice on the set of qubits, each qubit-to-probe of the set of qubits-to-probe has been prepared in the initial-state assigned to the qubit-to-probe; subsequent to operating the first circuit-slice on the set of qubits, generating a first subset of the set of qubit-measurements via measuring each qubit-to-probe of the set of qubits-to-probe; updating the set of qubit-measurements to include the first subset of qubit-measurements; and updating the tomography dataset based on the updated set of qubit-measurements.

3. The method of claim 2, wherein the initial-state assigned to each qubit-to-probe of the set of qubits-to-probe is a Pauli-state of a set of Pauli-states.

4. The method of claim 2, wherein the set of circuit-slices is an ordered set of circuitslices and generating the tomography dataset further comprises: selecting a second circuit-slice of the set of circuit-slices, wherein the second circuit-slice is subsequent to the first circuit-slice in the ordered set of circuit-slices; operating the second circuit-slice on the set of qubits, wherein prior to operating the second circuit-slice on the set of qubits, each qubit-to-probe of the set of qubits-to-probe has been prepared in the initial-state assigned to the qubit-to-probe; subsequent to operating the second circuit-slice on the set of qubits, generating a second subset of the set of qubit-measurements via measuring each qubit-to-probe of the set of qubits-to-probe; updating the set of qubit-measurements to include the second subset of qubitmeasurements; and updating the tomography dataset based on the updated set of qubit-measurements.

5. The method of claim 4, wherein the quantum circuit includes an ordered set of circuit-layers, each circuit slice of the set of circuit-slices includes a separate subset of the set of layers, the first circuit-slice includes a first subset of the set of layers, the second circuit slice includes a second subset of the set of circuit-layers, and the second subset of circuit-layers includes the first subset of circuit-layers and at least one additional circuit layer that is not included in the first subset of circuit-layers.

6. The method of claim 2, wherein generating the tomography dataset further comprises: assigning an initial-basis to each neighboring-qubit of the set of neighboring-qubits, wherein the initial-basis assigned to each neighboring-qubit is determined based on a set of constraints corresponding to the quantum circuit; assigning an initial-state to each neighboring-qubit based on the initial-basis assigned to the neighboring-qubit; and operating the first circuit-slice of the set of circuit-slices on the set of qubits, wherein prior to operating the first circuit-slice on the set of qubits, each neighboring-qubit of the set of neighboring-qubits has been prepared in the initial-state assigned to the neighboring-qubit.

7. The method of claim 6, wherein determining the initial-basis for each neighboringqubit of the set of neighboring-qubits comprises: determining the set of constraints based on analyzing the quantum circuit, wherein each constraint of the set of constraints corresponds to a separate two-qubit gate included in the quantum circuit; propagating each constraint of the set of constraints through the quantum circuit, wherein each constraint is propagated forw ard and backward from the corresponding two-qubit gate and through the quantum circuit; and for each neighboring-qubit of the set of neighboring-qubits, determining the initialbasis based on propagating each constraint of the set of constraints through the quantum circuit.

8. The method of claim 7, wherein each constraint of the set of constraints includes avoiding entanglement of the corresponding neighboring qubit with a corresponding qubit-to-probe of the set of qubits-to-probe that the corresponding tw o qubit-gate operates on.

9. The method of claim 7, w herein the initial-basis assigned to each neighboring-qubit of the set of neighboring-qubits is a Pauli-basis.

10. The method of claim 6, further comprising: assigning an initial-state to each neighboring-qubit of the set of neighboring qubits based on the initial-basis assigned to the neighboring-qubit; and prior to operating the first circuit-slice on the set on the set of qubits, preparing each neighboring-qubit of the set of neighboring-qubits in the initial-state assigned to the neighboringqubit.

11. The method of claim 10, wherein the initial-state assigned to each neighboring-qubit of the set of neighboring-qubits is an eigenstate of the initial-basis assigned to the neighboringqubit.

12. The method of claim 2, further comprising: prior to operating the first circuit-slice on the set of qubits, for each qubit-to-probe of the set of qubits-to-probe, assigning expected-state to the qubit-to-probe based on the initialstate assigned to the qubit-to-probe and the first circuit-slice; prior to operating the first circuit-slice on the set of qubits, for each qubit-to-probe of the set of qubits-to-probe. assigning an expected-basis to the qubit-to-probe based on the expected-state assigned to the qubit-to-probe; and generating the first subset of the set of qubit-measurements via measuring each qubit-to- probe of the set of qubits-to-probe in the expected-basis that is assigned to the qubit-to-probe.

13. The method of claim 12. wherein the expected state assigned to each qubit-to-probe of the set of qubits-to-probe is a state that is expected the qubit-to-probe to be in, subsequent to the first circuit-slice operating on the set of qubits, when the first circuit-slice operates on the set of qubits without error or without noise.

14. The method of claim 12, wherein the expected-state assigned to each qubit-to-probe of the set of qubits-to-probe is an eigenstate of the expected-basis assigned to the qubit-to-probe.

15. The method of claim 12, wherein the tomography dataset includes a measurement of each qubit-to-probe and the expected state assigned to each qubit-to-probe.

16. The method of claim 1, wherein the set of fidelities includes a fidelity' for each possible initial-basis for each qubit of the set of qubits and a Pauli-error rate for each gate of a set of gates included in the quantum circuit.

17. The method of claim 16, wherein estimating the set of fidelities for the set of qubits comprises: extracting the fidelity of each possible-basis for each qubit of the set of qubits based on a tomography algorithm applied to the tomography dataset; remapping the fidelities for each qubit of the set of qubits to each gate of the set of gates based on an expected-state that the qubit is expected to be in at that gate in the quantum circuit, wherein the remapping is based on the quantum circuit; and assigning the Pauli-error rate for each gate of a set of gates included in the quantum circuit based on subtracting a cumulative Pauli-error rate for a qubit from the in a previous round.

18. The method of claim 1, wherein the set of qubits-to-probe includes each data qubit in a quantum error correcting (QEC) code.

19. The method of claim 1, wherein the set of qubits-to-probe includes each measure qubit in a first stabilizer type of a quantum error correcting (QEC) code.

20. The method of claim 1, wherein a two-qubit quantum logic gate included in the quantum circuit operates on each neighboring-qubit of the set of neighboring-qubits and the at least one qubit-to-probe of the set of qubits-to-probe.

21. The method of claim 1, wherein the quantum circuit implements a quantum error correction (QEC) code

22. A quantum computing system (QCS) comprising: a quantum processor that includes a set of qubits; one or more memory devices, the one or more memory devices storing computer- readable instructions that when executed by the one or more quantum processors cause the one or more processors to perform operations for operating the QCS, the operations comprising: generating a set of circuit-slices, wherein each circuit-slice in the set of circuit-slices is a circuit-slice of a quantum circuit; subdividing the set of qubits into a first subset of qubits and a second subset of qubits, wherein the first subset of qubits is a set of qubits-to-probe and the second subset of qubits is a set of neighboring qubits, and each neighboring-qubit of the set of neighboring-qubits neighbors at least one qubit-to-probe of the set of qubits-to-probe in the quantum circuit; generating a tomography dataset based on a set of qubit measurements, wherein each qubit measurement of the set of qubits measurements corresponds to measuring each qubit-to-probe of the set of qubits-to-probe subsequent to operating at least one circuit-slice of the set of circuit-slices on the set of qubits; and estimating a set of fidelities for the set of qubits based on the tomography dataset, wherein the set of fidelities corresponds to a context of the quantum circuit.