Using Quantum Subspace Expansion to Decode Errors

Abstract

Methods, systems, and apparatuses for correcting the results of quantum computations. In one aspect, the method includes selecting a quantum error correction code for a quantum computation, where the quantum error correction code is defined by a plurality of stabilizer generators; determining a set of symmetry operators, the determination including: selecting a subset of the stabilizer generators, determining the sum between the identity operator and each selected stabilizer generator, and multiplying the determined sums to form a sum of terms, where each term in the sum is equal to a corresponding symmetry operator; using the determined set of symmetry operators to measure a projective correction of a physical observable for an output quantum state of the quantum computation, where the physical observable corresponds to the result of the quantum computation; and using the measured projective correction of the physical observable to determine a corrected result of the quantum computation.

Description

Technical Field

[0001] This specification generally relates to quantum computing, and in particular, to methods and apparatuses for correcting the results of quantum computing. Background Art

[0002] Quantum error correction codes are used in quantum computing to protect quantum information from decoherence and other quantum noise. Some quantum error correction codes use syndrome measurements to diagnose errors that corrupt quantum states. A syndrome measurement is a multi-qubit measurement of a corresponding parity operator that does not disturb the quantum information in the encoded state. The result of the syndrome measurement - the error syndrome - can be used as part of a fast feedback mechanism to decode and recover from errors. For example, a unitary recovery operator can be applied to a quantum state conditioned on the error syndrome to prevent errors from propagating through the computation. Summary of the Invention

[0003] This specification describes methods and systems for using post-processing techniques based on quantum subspace expansion to mitigate and decode errors on logical qubits.

[0004] In general, one innovative aspect of the subject matter described in this specification can be implemented as a method for correcting the results of quantum computing, the method including: selecting a quantum error correction code to perform quantum computing, where the quantum error correction code is defined by a corresponding set of stabilizer generators; determining a set of symmetry operators, the determination including: selecting a subset of the set of stabilizer generators; for each stabilizer generator in the selected subset, determining the sum between the identity operator and the stabilizer generator; and multiplying the determined sums to form a summation of terms, where each term in the summation is equal to a corresponding symmetry operator; using the determined set of symmetry operators to measure a projective correction of a physical observable of the output quantum state of the quantum computing, where the physical observable corresponds to the result of the quantum computing; and using the measured projective correction of the physical observable to determine a corrected result of the quantum computing.

[0005] Other implementations of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers can be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, the software, firmware, hardware, or a combination thereof causing the system to perform these actions in operation. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing apparatus, cause the apparatus to perform these actions.

[0006] The foregoing and other embodiments may each optionally include, individually or in combination, one or more of the following features. In some embodiments, the projective correction of a physical observable for an output quantum state of the quantum computation using the determined set of symmetry operators includes measuring wherein represents the sum of symmetry operators with uniform coefficients in the determined set of symmetry operators, and Γ represents the physical observable.

[0007] In some embodiments, the projective correction of a physical observable for an output quantum state of the quantum computation using the determined set of symmetry operators includes: selecting one or more operator pairs, where each operator pair includes i) a corresponding component of the physical observable, and ii) a corresponding symmetry operator from the determined set of symmetry operators; for each selected operator pair: performing a quantum computation on an initial quantum state to obtain the output quantum state, and measuring the selected operator pair for the output quantum state to obtain a corresponding measurement result.

[0008] In some embodiments, determining a correction result for an output of the quantum computation includes using the obtained measurement results to determine a correction result for the output of the quantum computation.

[0009] In some embodiments, using the obtained measurement results to determine a correction result for an output of the quantum computation includes calculating a linear combination of the obtained measurement results.

[0010] In some embodiments, selecting one or more operator pairs includes randomly sampling one or more operator pairs according to a random sampling scheme.

[0011] In some embodiments, the result of the quantum computation includes an expected value of the physical observable.

[0012] In some embodiments, the physical observable includes a weighted sum of Pauli operators, and wherein the component of the physical observable includes a Pauli operator in the sum of Pauli operators.

[0013] In some embodiments, measuring the selected operator pair for the output quantum state to obtain a corresponding measurement result includes measuring Γ for the output quantum state j M k where Γ j represents the component of the physical observable in the selected pair and M k represents the symmetry operator in the selected pair.

[0014] In some embodiments, the size of a selected subset of the set of stabilizer generators depends on one or more of i) a target computational accuracy or ii) a target computational cost.

[0015] In some embodiments, the quantum code includes a stabilizer code, a surface code, a Shor code, a Bacon-Shor code, or a toric code.

[0016] In some embodiments, the quantum computation is performed using a noisy intermediate-scale quantum computer.

[0017] Generally, another innovative aspect of the subject matter described in this specification can be implemented as a method for correcting the result of a quantum computation, the method including: selecting a quantum error-correcting code to perform the quantum computation, where the quantum error-correcting code is defined by a corresponding set of stabilizer generators; selecting a linear combination of stabilizer operators generated by the stabilizer generators; determining values of coefficients of the stabilizer operators in the linear combination, the determination including solving a generalized eigenvalue problem of a corresponding quantum error-correcting code Hamiltonian; using the stabilizer operators in the linear combination of stabilizer operators to measure an expansion correction of a physical observable for an output quantum state of the quantum computation, where the physical observable corresponds to the result of the quantum computation; and using the measured expansion correction of the physical observable and the determined values of the coefficients of the stabilizer operators in the linear combination to determine a corrected result of the quantum computation.

[0018] Other embodiments of these aspects include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers can be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, the software, firmware, hardware, or a combination thereof causing the system to perform these actions in operation. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing apparatus, cause the apparatus to perform these actions.

[0019] The foregoing and other embodiments can each optionally individually or in combination include one or more of the following features. In some embodiments, solving the generalized eigenvalue problem includes: preparing multiple copies of the output quantum state of the quantum computation; measuring components of a quantum error-correcting code Hamiltonian for corresponding copies of the output quantum state; measuring components of an overlap matrix of stabilizer operators for corresponding copies of the output quantum state; and using the measured components of the quantum error-correcting code Hamiltonian and the measured components of the overlap matrix of stabilizer operators to determine an eigenvalue matrix and an eigenvector matrix.

[0020] In some embodiments, using a stabilizer operator in a linear combination of stabilizer operators to measure an expansion correction of a physical observable for an output quantum state of the quantum computation includes measuring a corresponding value for the output quantum state to obtain a corresponding measurement result, where Γ represents the physical observable and M k represents a stabilizer operator.

[0021] In some embodiments, using the expansion correction of the measured physical observable and the value of the coefficient of the stabilizer operator in the determined linear combination to determine a correction result of the quantum computation includes: summing the measurement results, where each measurement result in the sum is multiplied by a corresponding determined coefficient.

[0022] In some embodiments, the result of the quantum computation includes an expected value of a physical observable.

[0023] In some embodiments, the physical observable includes a weighted sum of Pauli operators, and where a component of the physical observable includes a Pauli operator in the sum of Pauli operators.

[0024] In some embodiments, the quantum code includes a stabilizer code, a surface code, a Shor code, a Bacon-Shor code, or a toric code.

[0025] In some embodiments, the quantum computation is performed using a noisy intermediate-scale quantum computer.

[0026] Generally, another innovative aspect of the subject matter described in this specification can be implemented as a method for correcting the result of a quantum computation represented by a problem Hamiltonian, the method including: identifying a set of known symmetry operators of the problem Hamiltonian; generating a set of expansion operators, the generating including: for each symmetry operator, determining a sum between the identity operator and the symmetry operator; and multiplying the determined sums to form a summation of terms, where each term in the summation is equal to a corresponding expansion operator; measuring matrix elements of some or all of the expansion operators in the generated set; and using the measured matrix elements to determine a correction value of the output of the quantum computation.

[0027] Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers can be configured to perform particular operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which in operation causes the system to perform these actions. One or more computer programs can be configured to perform particular operations or actions by including instructions that, when executed by a data processing apparatus, cause the apparatus to perform these actions.

[0028] The foregoing and other implementations can each optionally individually or in combination include one or more of the following features. In some implementations, the symmetry operators in the identified set of approximate symmetry operators include operators that do not commute with the problem Hamiltonian.

[0029] In some implementations, the set of known symmetry operators includes operators with two different eigenvalues.

[0030] In some implementations, the method further includes identifying a set of approximate symmetry operators for the problem, and wherein generating the set of expansion operators includes: for each symmetry operator and for each approximate symmetry operator, determining the sum between the identity operator and the symmetry operator or approximate symmetry operator; and multiplying the determined sums to form a summation of terms, where each term in the summation is equal to the corresponding expansion operator.

[0031] The subject matter described in this specification can be implemented in a particular way so as to achieve one or more of the following advantages.

[0032] Systems implementing the decoding techniques described in this specification can reduce errors in quantum computing to improve the precision of quantum computing. For example, the presently described techniques can be used to mitigate errors and improve the precision of quantum computing in near-term applications of quantum computing such as quantum chemistry or general quantum circuits. Additionally, the techniques described in this specification do not require strict stabilizer measurements, alleviating the resource and control requirements of non-local measurements, and do not require a feed-forward mechanism. As a result, the complexity of the operations required to perform quantum computing with reduced errors and improved precision is also reduced.

[0033] In addition, the techniques described herein can be used to optimize quantum codes on quantum devices to improve the performance of quantum devices. It is speculated that one of the best uses of early quantum devices may be to tune quantum error-correcting codes under actual device conditions. As the system size grows, it becomes extremely difficult to model the real noise within the device, and studying which codes perform well under natural conditions and how to optimize them may contribute to achieving fully fault-tolerant computing. In fact, understanding the biased noise sources can greatly increase the threshold of a given code. The techniques described in this specification provide ways to experimentally study the encoding through post-processing while removing the complexities of fault-tolerant syndrome measurements or fast feedback. This allows for the experimental exploration of a greater variety of codes before worrying about these final details. Simple gate sequences can be run in the logical space with known results, and the post-processing decoder described herein can be used to study the decay of errors when adding stabilizers. This limitation will inform the propagation of logical errors in the system and allow for code optimization before a fully fault-tolerant protocol is available.

[0034] The decoding techniques described in this specification greatly benefit from the fact that the parity measurements do not need to be geometrically local to be implemented in a realistic environment. This allows for the exploration and utilization of codes that are not geometrically local on the architecture used, which may have improved properties in terms of distance and rate compared to geometrically local codes. In addition, they naturally allow for the implementation of recently proposed fermion-based codes (such as the Majorana loop stabilizer code or the Bravayi-Kitaev ultrafast variant, which are considered good candidates for near-term simulation) without the need for complex decoding circuits or ancillas for syndrome measurements.

[0035] In addition, since the method proposed in this specification is a post-processing method, it is fully compatible with the extrapolation techniques introduced to reduce errors. In these techniques, additional noise is artificially introduced to extrapolate to the noise floor.

[0036] The method for mitigating errors and studying error-correcting codes described in this specification can be implemented to achieve a pseudo-threshold of p≈0.60 under the [[5;1;3]] code in a depolarizing channel. This method has the potential to play a role in developing and optimizing quantum codes under realistic noise conditions and can eliminate errors in early application plans.

[0037] The techniques described in this specification can be applied to computational problems encoded using quantum codes, but can also be generalized to non-encoded problems by using physical symmetries, approximate symmetries, non-commuting operators, or asymmetries. In some cases, for example, when physical or approximate symmetries are available, the physical or approximate symmetries can also be used to improve the efficiency and / or accuracy of the decoding techniques described herein for encoded problems.

[0038] Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 An example system for performing quantum computing with reduced errors is depicted.

[0040] Figure 2A is a flowchart of a first example process for performing quantum computing with reduced errors using random subspace expansion.

[0041] Figure 2B shows a schematic diagram of an algorithm for performing quantum computing with reduced errors using random subspace expansion.

[0042] Figure 3A is a flowchart of an example process for performing quantum computing with reduced errors using deterministic subspace expansion.

[0043] Figure 3B shows a schematic diagram of an algorithm for performing quantum computing with reduced errors using deterministic subspace expansion.

[0044] Figure 4 is a flowchart of an example process for performing quantum computing in the space of a problem Hamiltonian.

[0045] Like reference numerals and names in the different figures indicate like elements. DETAILED DESCRIPTION

[0046] Overview

[0047] This specification describes techniques for mitigating errors on quantum codes that are capable of mitigating errors on encoded logical qubits using efficient classical post-processing, without complex syndrome measurements or additional qubits outside of the logical qubits. This greatly simplifies the experimental exploration of quantum codes on near-term devices, eliminating the need for locality of syndrome or fast feed-forward and allowing the study of the performance aspects of the codes on actual devices.

[0048] These techniques include a general structure that uses projections in quantum error-correcting codes in post-processing to correct observables. This structure can also be extended to approximate projections within subspaces. The structure then allows for the correction of some logical errors in the code space, the correction of physical unencoded Hamiltonians without engineered symmetries, the use of symmetries when the desired quantum numbers are unknown, and the generalization of corrections derived from approximate symmetries.

[0049] Example Hardware

[0050] Figure 1 Depicts an example quantum computing system 100. The example system 100 is an example of a system that can be implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more of the locations where the systems, components, and techniques described below can be implemented.

[0051] The system 100 includes quantum hardware 102 that communicates data with a classical processor 104. For convenience, the classical processor 104 and the quantum computing hardware 102 are shown as separate entities; however, in some embodiments, the classical processor 104 may be included within the quantum computing hardware 102. For example, the quantum computing hardware 102 may include one or more components for performing classical computing operations.

[0052] The system 100 is configured to perform classical computing using a combination of the quantum hardware 102 and the classical processor 104 for quantum computing. For example, the system may be configured to perform operations in accordance with example processes 200, 300, and 400 described below with reference to Figure 2A , Figure 3A and Figure 4 and described in.

[0053] The quantum hardware 102 includes a plurality of qubits 110 and a control device 112 for controlling the qubits 110 and enabling algorithmic operations or quantum computing to be performed.

[0054] The qubits 110 are physical qubits, e.g., physical devices that behave as two-state quantum systems, for performing algorithmic operations or quantum computing. Since physical qubits are vulnerable to decoherence and other sources of error, multiple physical qubits can be used to create a single logical qubit, where the logical qubit is executed as specified in a quantum algorithm or circuit. For example, in stabilizer quantum error correction codes, a logical qubit is a group of physical qubits that protects information using redundancy through symmetry. For example, a five-qubit [[5,1,3]] stabilizer error correction code is a code with a distance of 3, where the code with a distance of 3 encodes 1 logical qubit into 5 physical qubits and protects against any single qubit error.

[0055] The specific physical implementation of qubits 110 included in quantum computing hardware 102 and how they interact depend on various factors, including the type of quantum computing being performed by quantum computing hardware 102. For example, in some embodiments, qubits may include qubits physically implemented via atomic, molecular, or solid-state quantum systems. In other embodiments, qubits may include superconducting qubits, such as Gmon qubits or semiconductor qubits. In other embodiments, ion traps, photonic devices, or superconducting cavities may be used, with which states can be prepared without the need for qubits. Additional examples of physical implementations of qubits include fluxgate qubits, silicon quantum dots, or phosphorus-doped qubits.

[0056] The type of control device 112 included in quantum hardware 102 depends on the type of qubits 110 included in quantum hardware 102. For example, in some cases, qubits 110 may be frequency tunable. In these cases, each qubit may have an associated operating frequency, where the operating frequency can be adjusted using one or more control devices 112 (e.g., an excitation pulse generator and control lines coupling the qubit to the excitation pulse generator). Example operating frequencies include qubit idle frequency, qubit interaction frequency, and qubit readout frequency. Different frequencies correspond to different operations that a qubit can perform. For example, setting the operating frequency to the corresponding idle frequency can put the qubit into a state where it does not strongly interact with other qubits and can be used to perform single-qubit gates. As another example, in the case where qubits interact via a coupler with fixed coupling, the qubits can be configured to interact by setting their respective operating frequencies to some gate-dependent frequencies that are detuned from their common interaction frequency. In other cases, for example, when qubits interact via a tunable coupler, the qubits can be configured to interact by setting the parameters of their respective couplers to achieve interaction between the qubits, and then by setting the respective operating frequencies of the qubits to some gate-dependent frequencies that are detuned from their common interaction frequency. Such interactions can be performed in order to execute multi-qubit gates.

[0057] Control device 112 may also include a device configured to perform measurements on qubits 110 and provide the measurement results to classical processor 104 for processing and analysis, such as a readout resonator. For example, control device 112 may include a device configured to perform measurements of a physical observable Γ on one or more of physical qubits 110. Additionally, control device 112 may include a device configured to perform physical observables on one or more of physical qubits 110 based on a determined set of projection operators A device for projective correction measurement, as described below in conjunction with Figures 2A - 4 as described.

[0058] The classical processor 104 includes components for performing classical computations (e.g., the classical post-processing procedures described in this specification).

[0059] Programming the Hardware: Correcting Logical Observables in Post - Processing

[0060] Figure 2A is a flowchart of an example process 200 that uses random subspace expansion to perform quantum computations with reduced errors. For convenience, process 200 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 the system 100, can perform process 200.

[0061] The system selects a quantum error correction code to perform the quantum computation (step 202). For a system of n physical qubits, the quantum error correction code uses entanglement to encode a set of k < n logical qubits to improve robustness against possible errors. A code that requires at least Pauli operators of weight d to trigger a logical error is said to have distance d. These three numbers n, k, d can be used to define the quantum error correction code, denoted by the symbol [[n,k,d]]. Formally written as A set of 2k logical operators of

[0062]

[0063] where represents the set of check operators extracted from the stabilizer group, where the check operators can be used to deduce error syndromes. More specifically, represents the set of stabilizer generators, and S represents the complete stabilizer group that implies such that the minimum set of check operators is the stabilizer generators, but additional operators from the stabilizer group can be added, e.g., as in single-shot error correction techniques. Thus, the quantum error correction code selected by the system can be defined by the set of stabilizer generators . In some embodiments, the selection of the quantum error correction code can be performed by the classical processor of the system.

[0064] Example quantum codes include general stabilizer codes, surface codes, Shor codes, Bacon-Shor codes, or toric codes. Selecting a quantum code can also include selecting attributes of the quantum code, e.g., how many qubits are used in the encoding or how logical errors occur. For example, if the system selects a surface code, then the number of qubits used to encode the logical qubit (which determines the code distance) can also be selected. As another example, if the system selects a surface code, then the number of errors that occur as logical errors in the physical system can also be selected. As another example, the system can select a quantum code and a concatenation level (i.e., how many times the codes are chained together to increase robustness). If the quantum code can also determine how the physical qubits will be laid out on the quantum device, then the quantum code and attributes are selected. Thus, the particular quantum code and code attributes selected by the system can depend on several factors, including the type, size, and structure of the quantum device performing the quantum computation and the type of quantum computation being performed.

[0065] Traditional error correction is performed by measuring the stabilizer operators (e.g., using ancillary qubits and using the resulting syndrome information to decode the error and recover from the error). However, in some embodiments, it may not be possible to perform such dynamic measurement, decoding, and recovery, e.g., when performing quantum computations using NISQ devices. In such embodiments, projection operators are used to eliminate errors from the quantum state.

[0066] The system determines a set of symmetry operators (step 204). The symmetry operators are determined using the stabilizer form and projection operators. The members of the stabilizer group have eigenvalues ±1, and each can be used to construct a corresponding projection where the projection removes the components of the state outside the +1 eigenspace or the code space of the stabilizer. This can be used to construct projections that are linear combinations of symmetries (which in turn are based on the product of projection operators / projections) to eliminate target errors outside the code space. However, this cannot eliminate logical errors that occur within the code space. Quantum error detection typically discards results based on syndrome measurements without using error correction. However, as described in more detail below, the techniques described in this specification avoid the need for direct syndrome measurements - direct syndrome measurements can be cumbersome on geometrically local qubit layouts and challenging to perform in a fault-tolerant manner for complex codes. In some embodiments, the set of symmetry operators can be determined by the classical processor of the system.

[0067] For a stabilizer group with generator S i a complete projection can be formed from When taking over all generators, the expression ∑ i M i is the sum of all elements of the stabilizer group with constant coefficients, and the constant coefficients can be fixed to 1 / 2m to obtain

[0068]

[0069] where m represents the number of stabilizer generators used. For the case of full projection, this would be the complete stabilizer group containing 2 m terms. As described in more detail below, while this is typically an exponential number of terms, when using a random sampling scheme to apply corrections, the number of terms is not an explicit factor in the cost. Instead, the correction cost will depend on the amount of state outside the code space. This group structure allows for projective correction of the density matrix on noisy medium-scale quantum computing devices

[0070] In other words, to generate the set of symmetric operators, the system selects a subset of m stabilizer generators and, for each stabilizer generator in the selected subset, determines the sum (or difference) between the identity operator and the stabilizer generator. The determined sums (or differences) are then multiplied together to form a sum of terms, where each term in the sum is equal to the corresponding symmetric operator. The expansion of equation (2) generates the set of symmetric operators. For example, if the selected subset of the set of stabilizer generators includes two stabilizer generators S1, S2, then the sum (I + S1) and the sum (I + S2) are multiplied together to form the sum (I + S1)(I + S2) = I + S1 + S2 + S1S2, where each term I, S1, S2, S1S2 is equal to the corresponding symmetric operator M i .

[0071] In some embodiments, the system may determine the sum between the identity operator and the stabilizer generator instead of the difference, as this eliminates things in the -1 eigenspace of the stabilizer. However, in other embodiments, the system may determine the difference.

[0072] The size m of the selected subset of the stabilizer generator group depends on various factors. For example, the size of the selected subset may depend on the target accuracy or the target measurement and circuit repetition cost. For example, in some cases, the number of terms that can be effectively measured in real time may be limited. Larger subsets generally produce more accurate solutions but have higher measurement and circuit repetition costs.

[0073] Although the above symmetric operators are exponential in number, they provide several technical advantages, e.g., as opposed to traditional measurements of stabilizer generators which may not be exponential in number. Example process 200 uses transverse, destructive measurements to avoid the need for measurement syndrome qubits and associated fault-tolerant gadgets. Thus, while stabilizer generators commute, transverse measurements of their components may not. For example, while X1X2 and Z1Z2 commute, transverse measurements of Z1 and Z2 to determine Z1Z2 destroy the ability to recover X1X2. The techniques described herein allow bypassing this difficulty via random operator sampling and work with codes of arbitrary structure. This avoidance of using ancillary transverse measurement schemes provides a relatively high pseudo-threshold for a given code.

[0074] The system uses the determined set of symmetric operators to measure the projective correction of a physical observable for the output quantum state of the quantum computation (step 206). The measurement can be performed on one or more qubits of the quantum hardware by a control device of the quantum hardware. The system uses the projective correction of the measured physical observable to determine the correction result of the quantum computation (step 208). In some embodiments, the correction result can be determined by a classical processor of the system. The correction result can be output by the system and / or used as an input for further classical and / or quantum computations.

[0075] For example, the output of a quantum computation can be obtained by determining the expectation value <Γ> of a physical observable Γ. Thus, to determine the corrected value of the output of a quantum computation, the system can perform repeated state preparation (performing a quantum computation on an initial encoded quantum state to obtain an output quantum state ρ) and measurement of the prepared state according to equations (3) and (4) below, where represents the projective correction of the physical observable.

[0076]

[0077] In equation (3), ρ represents the quantum state of the quantum system after the quantum computation has been completed, represents the expansion in the subspace around the quantum state ρ, Γ represents a logical Hermitian operator, expressed as a sum of Pauli operators Γ i and Γ = ∑ i γ i Γ i The operator Γ commutes with the stabilizer group element M i and if is in the selected set of operators, it can be rewritten as a single sum over these operators, which will be repeated.

[0078] The repeated state preparation and measurement of the prepared state according to equations (3) and (4) can include: selecting one or more pairs of operators, where each pair includes i) the corresponding component Γ of the physical observablei and ii) a corresponding symmetry operator M from a set of defined symmetry operators i ; for each selected pair of operators: perform a quantum computation on an initial (encoded logical) quantum state to obtain an output (encoded logical) quantum state, and measure the selected pair of operators on the output quantum state ρ to obtain a corresponding measurement result. This process can be repeated multiple times to obtain an average over all repetitions representing a measured value with a target accuracy. The corresponding measurement results can be summed to obtain a correction result for the output <Γ> of the quantum computation.

[0079] Since the expansion of the sum in equation (3) may contain a large number of terms, a random scheme for sampling corrections that maximizes efficiency can be implemented. In particular, reducing the list of terms in the expansion of the sum and measuring each term to a fixed accuracy may lead to poor scaling. Instead, the scheme reflects the fact that if the state ρ is fully contained in the code space, the measurement of c should be 1 and the variance should be 0. This means that a reasonable random sampling of the terms should converge quickly, depend less on the actual number of terms, and depend on the amount of the state ρ.

[0080] The random scheme for sampling these corrections and the associated cost are as follows. Project the logical operator Γ onto the code space of the selected error-correcting code, giving where the symbol is absorbed into Γ i , where ∑ i γ i = 1 and γ i ≥ 0. The terms on the right-hand side of this equation can be enumerated with a bit string χ = (χ1,…,χ m ) to give where Γ χ,j = Tr[ρΓ j S χ , where As described above, it is assumed here that each M j is an encoded symmetry and commutes with the computational Hamiltonian and physical observables. To randomly sample Tr[ρΓ j S χ , the coefficients γ j of the terms are used as a normalized probability distribution. In addition, M i can be sampled according to a uniform distribution, and M i can also be sampled according to an importance sampling scheme, where, based on the device error model, some symmetries M iIntelligently weight the probabilities, and at the same time, correct the measurement results for this selection. Implementing the importance sampling scheme can reduce the variance (and the number of required quantum experiments) without biasing the results.

[0081] An example difference between the measurement scheme described above with reference to equations (3)-(4) and traditional error correction / detection is that syndrome measurements are not performed, so no additional ancillary qubits are required. While the traditional method advantageously uses non-destructive measurements, it also has several drawbacks - particularly for near-term implementations. For example, the traditional method typically requires more qubits and more non-locality. As another example, the traditional method is less effective in detecting / correcting errors and has a poor pseudo-threshold. By eliminating the need for these syndrome measurements, rather than adding additional operators and measuring all qubits independently, the applicability is improved. For example, less precise gates are required for the technique to still be effective. In other words, in the measurement scheme described above with reference to equations (3)-(4), there is no need to "seriously" measure the stabilizer because the technique includes a post-processing procedure - the information in the state can be destroyed by measuring the qubits across Pauli operators. For example, if the Pauli operator X1Z2Z3X4 is used as the stabilizer, a true stabilizer measurement would require using an ancilla to extract only the ±1 measurement. However, in the currently described scheme, repeated preparations of the state can be freely used, and <x1z2z3x4>Any unbiased estimator, including those that may disrupt the encoded state. This greatly simplifies the use of codes with non-local stabilizer measurements.

[0082] Figure 2B An algorithmic schematic 250 of an example process 200 is shown, where the example process 200 uses random subspace expansion to perform quantum computing with reduced errors. As described above, an expansion in the subspace around the prepared quantum state ρ (e.g., an initial quantum code state or an initial quantum code state that has had a sequence of quantum gates applied to it) is used to increase the expected value <Γ> of a logical observable, without the need for ancilla-based syndrome measurements or feed-forward. The observable Γ in the logical space is represented as a sum of Pauli operators Γ k and a symmetry M k naturally indicated by the system or from the stabilizer group S is chosen. When using random subspace expansion, terms from equation (3) are randomly selected according to a random sampling scheme, i.e., the operators {Γ j , M k} are randomly selected according to a random sampling scheme. For each selected term, the state ρ is prepared and the corresponding operator is measured using that state. The measurement results for each selected term are used to combine and correct the expected value <Γ>. The random sampling, measurement, and summation of terms are repeated until the corrected expected value obtained satisfies a convergence criterion.

[0083] Programming the Hardware: Relaxing Projections to Subspace Expansions

[0084] Figure 3A is a flowchart of an example process 300 that uses deterministic subspace expansion to perform quantum computing with reduced errors. For convenience, process 300 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 system 100, can perform process 300.

[0085] The system selects a quantum error correction code to perform quantum computing (step 302). The selection of a quantum error correction code to perform quantum computing was described in detail above with reference to step 202 of example process 200.

[0086] The system determines a set of stabilizer operators (step 304). Determining the set of stabilizer operators was described above with reference to step 204 of example process 200. In step 204 of example process 200, the system constructs symmetric operators from a quantum error-correcting code. Step 304 of example process 300 is based on step 204 of example process 200. However, in step 304, the construction of the projection operators is relaxed based on an approximation of the projection operators within the subspace (the operators used to construct the stabilizer operators do not need to be strictly projections, or if they are projections, they do not need to commute with the corresponding code Hamiltonian). Thus, the projection operators are also referred to as spreading operators. This relaxation provides greater flexibility and power because the relaxed construction variably minimizes errors. This means that the spreading operators can break known symmetries, thus obtaining better solutions in the corresponding metrics (e.g., energy). For example, it may be possible to replace unreliable qubits with fixed values to automatically minimize errors, but this is never an exact symmetry of the problem (unless the problem is trivial).

[0087] More specifically, in step 304, the above equation (2) is generalized. In equation (2), the coefficient c i is chosen to be uniform, i.e., c i = 1 / 2 m . However, if the sequence given by the RHS of equation (2) is truncated according to a linear ansatz, this is no longer the case, and the complete projection is given by the general expression:

[0088]

[0089] where L represents the number of terms in the linear ansatz and the stabilizer operators, and M i is an element of the stabilizer group as described above with reference to example process 200 - however, in equation (5), unlike equation (2), no longer needs to be true. Choosing a linear ansatz constructed from elements of the stabilizer group helps to incorporate symmetries and ensures the existence of spreading operators for correcting observables during post-processing.

[0090] Thus, the system chooses a linear ansatz by selecting a linear combination of stabilizer operators and determines the values of the corresponding coefficients c i . For example, the system can proceed as in step 204 of example process 200 and select a subset of m stabilizer generators for a quantum error-correcting code. Then, the system can truncate the selected subset of stabilizer generators. For example, the selected subset can be truncated in order to reduce the cost associated with performing measurements. This can be achieved by identifying elements in the selected subset that do not contribute to the computation as much as other terms and removing these identified terms from the selected subset. For example, the system can select a set of operators M i , and then iteratively: solve the generalized eigenvalue equation described by the following reference equations (7)-(9) to determine the corresponding coefficients c of the selected group i , remove the operators with the determined coefficients less than a predetermined threshold (e.g., less than 10 -3 ), and add a new operator M to the group i to generate a new fixed-size group. This iterative process can be repeated until a group of operators with large enough coefficients is found.

[0091] Determining the coefficient c i can be formulated as the task of minimizing the distance to the code space according to the normalization constraint. Using the Hamiltonian formula of the code space, this is equivalent to approximating the ground state of the code space by the following formula

[0092]

[0093] This optimization usually depends on both the state ρ and the choice of H c Both. From the linear ansatz and the normalization constraint, this problem is equivalent to the minimization of a quadratic form on the sphere with a non-orthogonal metric. The solution to this problem is given by the solution of the generalized eigenvalue problem given in the following equations (7)-(9).

[0094] HC = SCE (7)

[0095]

[0096] In equations (7)-(9), H represents the action of the code Hamiltonian in the stabilizer projection basis, the matrix S is the overlap or metric matrix defining the subspace geometry, C represents the eigenvector matrix, and E represents the diagonal matrix of eigenvalues. The columns of C corresponding to the lowest values of the diagonal matrix E contain the coefficients c i . In other words, the columns of C represent the eigenvectors, and the eigenvector associated with the lowest eigenvalue (e.g., the ground state) has entries representing c i . For example, if equations (7)-(9) are solved using a solver that produces ordered eigenvectors, the coefficients are contained in the first column of C. For the case where the stabilizer operator M i is constructed from projections of the generators, the solution is consistent with the solution obtained using the example procedure 200. In other cases, an optimal solution is obtained that interpolates between different numbers of projections in the given subspace. This expansion with respect to the state can be called quantum subspace expansion (QSE). The N M ×N M The ground state eigenvector of the eigenvalue problem forms the optimal solution to the above problem within this subspace. In some cases, the optimal solution to this eigenvalue problem may not be a strict projection. However, this is not necessarily undesirable. For example, in some cases, lower energy states can be found for a particular problem Hamiltonian corresponding to a rotation in the logical space.

[0097] The system uses the expansion correction of the physical observable Γ of the output quantum state ρ of the quantum computation by measuring the stabilizer operator in the linear combination of stabilizer operators. Where the physical observable corresponds to the result of the quantum computation (step 306). The expansion correction of the physical observable of the output quantum state of the quantum computation by measuring the stabilizer operator in the linear combination of stabilizer operators includes measuring the corresponding To obtain the corresponding measurement result, where Γ represents the physical observable and M k Represents the stabilizer operator.

[0098] The system uses the expansion correction of the measured physical observable and the value of the coefficient of the stabilizer operator in the determined linear combination to determine the correction result of the quantum computation (step 308). The determination of the correction result of the output of the quantum computation with a fixed c was described above with reference to step 206 of example process 200. For example, the system will sum each i Of the obtained measurement results, where each measurement result is multiplied by the corresponding coefficient determined by solving the generalized eigenvalue problem. Alternatively, the system can use the determined coefficients to construct a representation of the operator in the same basis defined by the expansion operator and use it to perform further symmetric projections or improve the estimation of the logical observable.

[0099]

[0100] Figure 3B In traditional quantum error correction, the degeneracy of the ground state of the full code Hamiltonian prevents the reference to a single state within the code space. This makes it impossible to eliminate logical errors with the above process. However, when considered in combination with a problem Hamiltonian from a quantum physical system (such as an electronic system), if the goal is to prepare the eigenstate of this Hamiltonian or minimize its energy, then it becomes possible to correct logical errors. As an example, given a single encoded spin with a problem Hamiltonian Where the state is wrongly found in

[0100] Figure 3B Shows an algorithmic schematic 350 of an example process 300 for performing quantum computing with reduced error using deterministic subspace expansion. As described above, expansion in the subspace around the prepared quantum state ρ is used to improve the expected value <Γ> of the logical observable, without the need for ancilla-based syndrome measurements or feedforward. The observable Γ in the logical space is represented as a sum of Pauli operators Γ k and a symmetry M k natural to the system or from the stabilizer group S k is selected. When using deterministic subspace expansion, the set of M is expanded to include non-symmetries (i.e., is no longer necessarily proportional to i ), and the corresponding expected value for ρ can be evaluated using multiple repetitions to form a representation of the operator in the subspace around ρ. These matrices define a generalized eigenvalue problem, the solution C of which defines the optimal projection in the basis of the operator M

[0101] Programming the Hardware: Leveraging Corrections of Uncoded Systems

[0102] Figure 2A and the corrected expected value <Γ> of the target observable. Figure 3A Example processes 200 and 300 for p and

[0103] Figure 4 respectively describe decoding within an error correction code that redundantly encodes quantum information via engineered symmetries. However, due to the encoding in the execution of the gates, this strategy involves some overhead. In some recent experiments, it may be more practical to work directly in the space of the physical problem Hamiltonian H Figure 1 .

[0104] is a flowchart of an example process 400 for performing quantum computing in the space of the problem Hamiltonian. For convenience, process 400 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 system 100, can perform process 400.

[0104] The system identifies a set of known symmetry operators for the problem Hamiltonian (step 402a). These known symmetry operators can be pre-known or can be determined using a dedicated algorithm. For example, in the case of an interacting fermion system, the identified set of symmetry operators can include the total fermion number, the total spin, and S z One or more of the components, or symmetries associated with the spatial degrees of freedom in the system. In some embodiments, the symmetry operator can be determined by a classical processor of the system. In some embodiments, the symmetry operator can be received as an input to the system and / or determined from an input to the system.

[0105] In some embodiments, the symmetry operators in the identified set can include operators that have only two distinct eigenvalues, also referred to as the symmetry of the problem Hamiltonian. Selecting such symmetry operators can reduce the cost associated with implementing these operators. For example, the number symmetry problem in a fermionic Hamiltonian can result in eigenvalues that vary from 0 to the number of spin-orbitals in the system. Thus, to select the correct particle number, the projection removes all components except the desired particle number N p other than, or where represents the number operator over all fermionic modes of the system. This can lead to a large number of terms that are expensive to implement, e.g., requiring more qubits, additional gate depth, or increased computational effort. Examples of symmetries include the number symmetry operator (where, in the Jordan-Wigner representation, it takes the form of ∏ i Z i ), or the up-spin (α) and down-spin (β) number parities ∏ i∈a Z i and ∏ i∈β Z i . These operators generate the full number parity and provide additional capabilities in their projections.

[0106] The system uses the symmetry operator to generate one or more projection / expansion operators (such as step 204 of example process 200 or step 304 of example process 300 (step 404)). For example, the identified symmetry can be used in place of the stabilizer generator S i which can generate the operator M i .

[0107] In the case where the system identifies a symmetry operator at step 402a but the symmetry subspace to which the target state belongs is unknown, it will be automatically selected. For example, in the case where the system selects the symmetry F of the Hamiltonian H p defined by [H p ,F]=0, but does not know which of the two eigenspaces the exact ground state belongs to, applying or can lead to different results. Connectivity with other symmetries can complicate this problem. However, the QSE process described herein can automatically select between the two to find the optimal choice.

[0108] Due to this property, the system can also identify a set of approximate symmetry operators of the problem Hamiltonian, or a set of more general operators that do not commute with the Hamiltonian but have a known structure with respect to the problem at step 402a (step 402b). For example, for the local fermion occupation operator given by Z under the Jordan-Wigner transformation i E[H p ,Z i ≠0 and the exact state has some components on orbital i. However, the sites in the fermionic simulation problem are typically approximately ordered in terms of energy and possible occupations in the so-called natural orbital basis. This means that some sites are less likely to be occupied than others, and they incur a disproportionately large energy error compared to similar errors on other sites. In this example, the QSE process can automatically decide whether to apply the projection by balancing the contribution of site i to the exact wave function with the energy penalty caused by its additional occupation under noise Although this particular projection is simple enough that when applied exactly, it amounts to removing qubits, one can imagine a paired occupation projection on the highest energy orbitals, such as that can effectively remove errors caused by the incorrect occupation of the highest energy orbitals, which cannot be mitigated by simple truncation of qubits.

[0109] The system determines the matrix elements of some or all of the projection / unfolding operators in the projection / unfolding operator (step 406). The matrix elements can be determined by performing measurements on one or more qubits after performing the quantum computation. The measurements can be performed by a control device of the quantum hardware on one or more qubits of the quantum hardware. As referenced above Figure 2A and Figure 3A described, the measured matrix elements are used to determine a correction value for the output of the quantum computation.

[0110] The digital and / or quantum subject matter and the implementation of digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuits, suitable quantum circuits, or more generally, a quantum computing system, tangibly implemented digital and / or quantum computer software or firmware, digital and / or quantum computer hardware including the structures disclosed in this specification and their structural equivalents, or a combination of one or more of them. The term "quantum computing system" can include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0111] ​The implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium, the digital and / or quantum computer program instructions for execution by, or to control the operation of, a 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, or a combination of one or more of them. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, the artificially generated propagated signal being generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing apparatus.

[0112] The terms quantum information and quantum data refer to information or data carried, held, or stored by a quantum system, where the smallest non-trivial system is a qubit, i.e., the system that defines the unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems can include multi-level systems, e.g., having two or more energy 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 state and the first excited state, however, it should be understood that other settings where the computational states are identified with higher excited states are also possible.

[0113] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and includes all kinds of devices, equipment, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be or further include dedicated logic circuits, such as a field programmable gate array (FPGA), an application-specific integrated circuit (ASIC), or a quantum simulator, i.e., a quantum data processing apparatus designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform general quantum computing. The apparatus can optionally include, in addition to the 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.

[0114] A digital computer program, which may also be referred to as or described as a program, software, software application, module, software module, script, or code, can be written in any form of programming language (including compiled or interpreted languages, 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 as or described as a program, software, software application, module, software module, script, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and be translated into a suitable quantum programming language, or can be written in a quantum programming language (e.g., QCL or Quipper).

[0115] A digital and / or quantum computer program may or may not correspond to a file in a file system. The program can be stored in a part of a file that holds other programs or data, such as one or more scripts stored in a markup language document, stored in a single file dedicated to the program in question, or stored in multiple cooperating files (e.g., files that store one or more modules, subroutines, or code portions). A digital and / or quantum computer program can be deployed to execute on one digital or one quantum computer, or on multiple digital and / or quantum computers 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 can transmit quantum data using quantum systems (e.g., qubits). Generally, a digital data communication network cannot transmit quantum data, whereas a quantum data communication network can transmit both quantum data and digital data.

[0116] The processes and logical flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating in conjunction 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 logical flows can also be performed by special-purpose logic circuitry (e.g., FPGA or ASIC) or a quantum simulator, and the apparatus can also be implemented as special-purpose logic circuitry, or by a combination of special-purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers.

[0117] For a system of one or more digital and / or quantum computers, being "configured" to perform a particular operation or action means that the system has installed on it software, firmware, hardware, or a combination thereof, which in operation causes the system to perform those operations or actions. For one or more digital and / or quantum computer programs configured to perform a particular operation or action, it means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by the quantum computing device, cause the device to perform the operation or action.

[0118] A digital and / or quantum computer suitable for executing digital and / or quantum computer programs can be based on a general-purpose or special-purpose digital and / or quantum processor or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (such as photons), or a combination thereof.

[0119] The components of a digital and / or quantum computer are a central processing unit for executing or running instructions and one or more storage devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented or incorporated with dedicated logic circuits or quantum simulators. Typically, a digital and / or quantum computer will also include one or more mass storage devices or be operatively coupled to receive digital and / or quantum data from, or transfer digital and / or quantum data to, one or more mass storage devices for storing digital and / or quantum data, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer does not necessarily require such devices.

[0120] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media, and memory devices, including, for example: semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. It can be understood that a quantum memory is a device capable of storing quantum data with high fidelity and high efficiency for a long time, such as an optical-matter interface where light is used for transmission and a substance for storing and preserving quantum characteristics (such as superposition or quantum coherence) of quantum data.

[0121] The control of the various systems or portions thereof described in this specification can be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification can each be implemented as an apparatus, method, or system that can include one or more digital and / or quantum processing devices and a memory storing executable instructions to perform the operations described in this specification.

[0122] Although this specification contains many specific implementation details, these should not be construed as limitations on the scope of what is claimed, but rather as descriptions of features that are specific to particular implementations. Some features described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, the various features described in the context of a single implementation can also be implemented separately in multiple implementations or in any suitable sub-combination. Additionally, although features may be described above as acting in some combinations and even initially claimed as such, in some cases, one or more features in a claimed combination can be deleted from that combination, and the claimed combination can be directed to a sub-combination or a variant of a sub-combination.

[0123] Similarly, although operations are depicted in the figures in a particular order, this should not be understood as requiring that the operations be performed in the particular order shown or in sequential order, or that all of the illustrated operations be performed, to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Additionally, the separation of the various system modules and components in the above-described embodiments should not be understood as required in all embodiments, and it should be understood that the described program components and systems can generally be integrated in a single software product or packaged into multiple software products.

[0124] Particular implementations of the subject matter have been described. Other implementations are within the scope of the described claims. For example, the acts recited in the claims can be performed in a different order and still achieve the desired result. As one example, the processes described in the figures do not necessarily need the particular order or sequence shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for correcting the result of a quantum computation, the method comprising: selecting, by one or more classical processors, a quantum error correction code to perform the quantum computation, wherein the quantum error correction code is defined by a corresponding set of stabilizer generators; determining, by the one or more classical processors, a set of symmetric operators, the determining comprising: selecting a subset of the set of stabilizer generators; for each stabilizer generator in the selected subset, determining the sum between the identity operator and the stabilizer generator; and multiplying the determined sums to form a sum of terms, wherein each term in the sum is equal to a corresponding symmetric operator; using quantum computing hardware, measuring a projective correction of a physical observable for the output quantum state of the quantum computation using the determined set of symmetric operators, wherein the physical observable corresponds to the result of the quantum computation; and determining, by the one or more classical processors, a corrected result of the quantum computation using the measured projective correction of the physical observable, wherein the result of the quantum computation includes an expected value of the physical observable.

2. The method according to claim 1, wherein measuring the projection correction of a physical observable for the output quantum state of the quantum computation using the determined set of symmetry operators comprises measuring wherein represents the sum of the symmetry operators with uniform coefficients in the determined set of symmetry operators, and Γ represents the physical observable.

3. The method according to claim 1 or claim 2, wherein measuring a projective correction of a physical observable for the output quantum state of the quantum computation using the determined set of symmetric operators comprises: selecting one or more operator pairs, wherein each pair includes i) a corresponding component of the physical observable, and ii) a corresponding symmetric operator from the determined set of symmetric operators; for each selected operator pair: performing a quantum computation on an initial quantum state to obtain the output quantum state, and measuring the selected operator pair for the output quantum state to obtain a corresponding measurement result.

4. The method according to claim 3, wherein determining a corrected result of the output of the quantum computation comprises using the obtained measurement results to determine a corrected result of the output of the quantum computation.

5. The method according to claim 4, wherein using the obtained measurement results to determine a corrected result of the output of the quantum computation comprises calculating a linear combination of the obtained measurement results.

6. The method according to claim 3, wherein selecting one or more operator pairs comprises randomly sampling one or more operator pairs according to a random sampling scheme.

7. The method according to claim 3, wherein the physical observable comprises a weighted sum of Pauli operators, and wherein a component of the physical observable comprises a Pauli operator in the sum of Pauli operators.

8. The method according to claim 3, wherein measuring the selected pair of operators for the output quantum state to obtain corresponding measurement results includes measuring Γ for the output quantum state j M k , where Γ j represents a component of the physical observable in the selected pair and M k represents the symmetric operator in the selected pair.

9. The method according to claim 1, wherein the size of the subset of the set of selected stabilizer generators depends on one or more of i) a target computational accuracy or ii) a target computational cost.

10. The method according to claim 1, wherein the quantum error correction code comprises a stabilizer code, a surface code, a Shor code, a Bacon - Shor code, or a toric code.

11. A method for correcting the result of a quantum computation, the method comprising: selecting, by one or more classical processors, a quantum error correction code to perform the quantum computation, wherein the quantum error correction code is defined by a corresponding set of stabilizer generators; The one or more classical processors select a linear combination of stabilizer operators generated by the stabilizer generator; The one or more classical processors determine values of coefficients of the stabilizer operators in the linear combination, the determining including solving a generalized eigenvalue problem of a corresponding quantum error correction code Hamiltonian; Using quantum computing hardware, use the stabilizer operators in the linear combination of stabilizer operators to measure an expansion correction of a physical observable for an output quantum state of the quantum computation, where the physical observable corresponds to a result of the quantum computation; And The one or more classical processors use the measured expansion correction of the physical observable and the values of the coefficients of the stabilizer operators in the determined linear combination to determine a correction result of the quantum computation, where the result of the quantum computation includes an expected value of the physical observable.

12. The method according to claim 11, wherein solving the generalized eigenvalue problem includes: Preparing multiple copies of the output quantum state of the quantum computation; Measuring components of the quantum error correction code Hamiltonian for corresponding copies of the output quantum state; Measuring components of an overlap matrix of the stabilizer operators for corresponding copies of the output quantum state; Using the measured components of the quantum error correction code Hamiltonian and the measured components of the overlap matrix of the stabilizer operators to determine an eigenvalue matrix and an eigenvector matrix.

13. The method according to claim 11 or claim 12, wherein the expansion correction of the physical observable for the output quantum state of the quantum computation using the stabilizer operator in the linear combination of the stabilizer operators comprises measuring the corresponding value for the output quantum state to obtain a corresponding measurement result, where Γ represents the physical observable and M k represents the stabilizer operator.

14. The method according to claim 13, wherein using the measured expansion correction of the physical observable and the values of the coefficients of the stabilizer operators in the determined linear combination to determine the correction result of the quantum computation includes: Summing the measurement results, where each measurement result in the sum is multiplied by a corresponding determined coefficient.

15. The method according to claim 11, wherein the physical observable includes a weighted sum of Pauli operators, and wherein components of the physical observable include the Pauli operators in the sum of Pauli operators.

16. The method according to claim 11, wherein the quantum error correction code includes a stabilizer code, a surface code, a Shor code, a Bacon-Shor code, or a toric code.

17. The method according to claim 11, wherein the quantum computation is performed using a noisy intermediate-scale quantum computer.

18. An apparatus for correcting a result of a quantum computation, comprising: Quantum hardware; And One or more classical processors; where the apparatus is configured to perform operations of a method including any one of claims 1 to 17.

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