Virtual distillation for quantum error mitigation

Abstract

Methods, systems, and apparatus, among other things, for determining an expectation value of error mitigation for a target observable of a noisy quantum state. In one aspect, a method includes obtaining a plurality of copies of a noisy quantum state, measuring a tensor product of the M copies of the noisy quantum state to compute an expectation value of a target observable of an entangled quantum state, where M ≥ 1 and eigenvalues corresponding to non-dominant eigenstates of the noisy quantum state are suppressed by a factor of M in a spectral decomposition of the entangled quantum state, and determining an expectation value of error mitigation for the target observable of the noisy quantum state using the computed expectation value of the target observable of the entangled quantum state.

Description

[0001] Cross-reference to related applications

[0002] This application claims the benefit of priority to U.S. Provisional Patent Application No. 63 / 112,593, filed November 11, 2020, the entire contents of which are incorporated herein by reference. Technical Field

[0003] This manual relates to quantum computing. Background Technology

[0004] Quantum error-correcting codes are used in quantum computing to protect quantum information from errors caused by decoherence and other quantum noise. Quantum error-correcting codes diffuse information encoded in a logical qubit to a highly entangled state across several physical qubits. For example, Shor codes are a type of quantum error-correcting code that can correct phase-flip and bit-flip errors using a 9-qubit circuit that requires 8 auxiliary qubits to correct 1 qubit. Summary of the Invention

[0005] This specification describes a technique for mitigating errors in noisy quantum computing using virtual quantum state distillation.

[0006] In general, an innovative aspect of the subject matter described in this specification can be implemented in a method for determining the expected value of error mitigation for a target observable with respect to a noisy quantum state, the method comprising: obtaining multiple copies of the noisy quantum state; measuring the tensor product of the M copies of the noisy quantum state to calculate the expected value of the target observable with respect to the entangled quantum state, wherein M ≥ 1, and the eigenvalues ​​corresponding to the non-dominant eigenvectors of the noisy quantum state are suppressed exponentially by M in the spectral decomposition of the entangled quantum state; and using the calculated expected value of the target observable with respect to the entangled quantum state to determine the expected value of error mitigation for the target observable with respect to the noisy quantum state.

[0007] Other implementations of this aspect include corresponding classical and quantum computer systems and 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 quantum and classical computers can be configured to perform a specific operation or action by software, firmware, hardware, or a combination thereof installed on the system that causes the system to perform actions during operation. One or more computer programs can be configured to perform a specific operation or action by including instructions that, when executed by a data processing device, cause that device to perform the action.

[0008] The foregoing and other implementations may optionally, individually or in combination include one or more of the following features. In some implementations, the noise experienced by each copy of the noisy quantum state comprises the same form and intensity.

[0009] In some implementations, entangled quantum states are formed by ρ M / Tr(ρ M The expression is given, where ρ represents the noise quantum state.

[0010] In some implementations, the method further includes: measuring the tensor product of M copies of the noisy quantum state to calculate the expectation of the identity operator with respect to the entangled quantum state, wherein determining the expectation of error mitigation for the target observable with respect to the noisy quantum state using the calculated expectation with respect to the target observable of the entangled quantum state comprises: i) dividing the expectation of the target observable with respect to the entangled quantum state by ii) the expectation of the identity operator with respect to the entangled quantum state.

[0011] In some implementations, measuring the tensor product of M copies of the noisy quantum state to calculate the expected value of the target observable with respect to the entangled quantum state includes: for each measurement repetition in a first number of measurement repetitions, measuring i) the target observable and ii) the product of a cyclic shift operator of the tensor product of M copies of the noisy quantum state to obtain a first measurement result; using the first measurement result to calculate the expected value of the product of i) the target observable and ii) the product of a cyclic shift operator of the tensor product of M copies of the noisy quantum state; and normalizing the expected value of the product of i) the target observable and ii) the product of a cyclic shift operator of the tensor product of M copies of the noisy quantum state to obtain an expected value of error mitigation for the target observable with respect to the noisy quantum state.

[0012] In some implementations, normalizing the expected value of the target observable includes: for each measurement repetition in a second number of measurement repetitions, measuring the cyclic shift operator of the tensor product with respect to M copies of the noisy quantum state to obtain a second measurement result; using the second measurement result to calculate the expected value of the cyclic shift operator of the tensor product with respect to M copies of the noisy quantum state; and dividing the expected value of the product of i) the target observable and ii) the cyclic shift operator of the tensor product with respect to M copies of the noisy quantum state by the expected value of the cyclic shift operator of the tensor product with respect to M copies of the noisy quantum state.

[0013] In some implementations, performing a measurement on the tensor product of M copies of a noisy quantum state includes performing parallel measurements.

[0014] In some implementations, M ≥ 2, and the target observable acts on a single qubit.

[0015] In some implementations, performing a measurement on the tensor product of M copies of a noisy quantum state includes, for each of K measurement repetitions: applying a diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain an evolving quantum state, wherein the diagonalization operator diagonalizes the product of i) a cyclic shift operator and ii) a symmetric version of the target observable and the cyclic shift operator; and measuring the product of the cyclic shift operator and the target observable with respect to the evolving quantum state to obtain a corresponding measurement for each qubit in the evolving quantum state.

[0016] In some implementations, the method further includes: using the measurement results to calculate the expectation of the product of the cyclic shift operator with respect to the evolving quantum state and the target observable; using the measurement results to calculate the expectation of the cyclic shift operator with respect to the evolving quantum state; and dividing the expectation of the product of the cyclic shift operator with respect to the evolving quantum state and the target observable by the expectation of the cyclic shift operator with respect to the evolving quantum state.

[0017] In some implementations, the application of a cyclic shift operator couples the qubits in the first copy of the noisy quantum state to the corresponding qubits in each subsequent copy of the noisy quantum state.

[0018] In some implementations, performing a measurement on the tensor product of M copies of a noisy quantum state includes performing a serial measurement.

[0019] In some implementations, M ≥ 2, and the target observable acts on two or more qubits, the target observable comprising multiple tensor products of a qubit operator.

[0020] In some implementations, performing a measurement on the tensor product of M copies of a noisy quantum state includes: for each tensor product of a one-qubit operator: for each measurement repetition of a first number of measurement repetitions: applying a first diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain an evolving quantum state, wherein the first diagonalization operator diagonalizes the product of i) the tensor product of the one-qubit operator and ii) the product of the cyclic shift operator, and measuring the product of the tensor product of the cyclic shift operator and the one-qubit operator with respect to the evolving quantum state to obtain a corresponding first measurement result for each qubit in the evolving quantum state; for each measurement repetition of a second number of measurement repetitions: applying a second diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain a second evolving quantum state, wherein the second diagonalization operator diagonalizes the cyclic shift operator, and measuring the cyclic shift operator with respect to the second evolving quantum state to obtain a corresponding second measurement result for each qubit in the second evolving quantum state.

[0021] In some implementations, the method further includes: calculating the expected value of the product of the target observable and the cyclic shift operator using the first measurement result; calculating the expected value of the cyclic shift operator using the second measurement result; and dividing the expected value of the product of the target observable and the cyclic shift operator by the expected value of the cyclic shift operator.

[0022] In some implementations, performing a measurement on the tensor product of M copies of a noisy quantum state to calculate the expected value of a target observable about the entangled quantum state involves performing ancilla-assisted measurements.

[0023] In some implementations, the target observable includes arbitrary observables.

[0024] In some implementations, M ≥ 2, and performing auxiliary bit-assisted measurements includes, for each measurement repetition in multiple measurement repetitions: preparing two quantum registers, wherein each quantum register includes a noisy quantum state and an auxiliary qubit prepared in the 0 state; performing a lossless measurement using a cyclic shift operator, including: applying a Hadamard gate to the auxiliary qubit, applying a cyclic shift operator to the two quantum registers with the auxiliary qubit in the 1 state, and measuring the auxiliary qubit in the X basis to obtain a corresponding first measurement result; and measuring the qubits in the two quantum registers with respect to a symmetric version of the target observable to obtain a corresponding second measurement result.

[0025] In some implementations, the method further includes: calculating the expected value of the target observable using the second measurement result; calculating the expected value of the cyclic shift operator using the first measurement result; and dividing the expected value of the target observable by the expected value of the cyclic shift operator.

[0026] The subject matter described in this specification can be implemented in a particular manner to achieve one or more of the following advantages.

[0027] Meaningful computation using recent quantum computers can be challenging due to the relatively high error rates of these devices. While quantum error correction promises to enable quantum computing with arbitrarily small noise levels, the resulting computational overhead is too large to be practical at present. The error mitigation techniques described here improve the quality of noisy computations without the large overhead of traditional quantum error correction. Error mitigation strategies reduce the effects of random errors caused by noise, such as those inherent to recent devices and random errors inherent in random quantum algorithms implemented on error-free devices. Furthermore, because error mitigation strategies are particularly well-suited for implementation on recent devices (e.g., NISQ devices), they can be applied to a wide range of technological applications, including molecular and material simulations, optimization processes, and quantum neural networks.

[0028] Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of this subject matter will become apparent from the specification, drawings, and claims. Attached Figure Description

[0029] Figure 1 A block diagram of an example system is shown, which performs virtual distillation to mitigate errors when calculating the expected value of an observable with respect to a noisy quantum state.

[0030] Figure 2 An example quantum computing system is shown.

[0031] Figure 3 This is a flowchart of an example process for determining the expected value of error mitigation for a target observable with respect to a noisy quantum state.

[0032] Figure 4 This is a flowchart of an example process for performing virtual distillation through diagonalization measurement.

[0033] Figure 5 This is a flowchart of an example process for selecting a virtual distillation protocol.

[0034] Figure 6 This is a flowchart of an example process of virtual distillation for serial measurement of two or more copies of a noisy quantum state, where the target observable acts on two or more qubits.

[0035] Figure 7 An example simulation (ansatz) is shown for numerical optimization of the diagonalization gate.

[0036] Figure 8 This is a graph showing the reduction in qDRIFT coherence costs through virtual distillation in the Heisenberg model.

[0037] The same labels and names in different drawings indicate the same components. Detailed Implementation

[0038] This specification describes a technique for mitigating errors in noisy quantum computing. Specifically, the technique described here enables efficient approximation of computational outputs generated without noise. The expected value of an observable about an approximate purified version of a noisy quantum state is reconstructed without explicitly preparing a purified quantum state. This method is referred to herein as “virtual distillation,” where the term “virtual” is used to emphasize that the purified form of the quantum state is not actually prepared (as in a conventional distillation scheme). More specifically, the expected value of an observable about an entangled quantum state, denoted as: , is measured using a collective measurement of M copies of the noisy quantum state ρ.

[0039]

[0040] where ρ = ∑ i p i |i><i| is the spectral decomposition (e.g., density matrix) of the noisy quantum state ρ, and Tr denotes the trace operation. By this method, the relative weights of the non-dominant eigenvectors are exponentially suppressed in terms of M. This represents an improvement over other methods that require explicit preparation of an approximately purified state, which in general achieve only linear suppression in terms of M. Thus, virtual distillation approximates the corrected expectation value of the observable O (e.g., the expectation value with incoherent errors suppressed) as the trace of the product of M copies of the spectral decomposition of the observable and the noisy quantum state divided by the trace of M copies of the spectral decomposition of the noisy quantum state, e.g.,

[0041]

[0042] As M increases, the resulting estimator converges exponentially rapidly to ρ towards the closest pure state. This is in contrast to other methods that achieve only linear convergence to the corrected state. Thus, the techniques described herein provide a technical improvement over these other methods in that the corrected state can be obtained in a shorter amount of time.

[0043] In this specification, the operations described are operations on multiple copies of the same noisy state. It is assumed that the noise experienced by different copies has the same form and strength. If this assumption is relaxed, as long as the copies are not entangled before virtual distillation, the effective state corresponding to the product of the density matrices of the individual copies is still measured. The letter N indicates the number of qubits in a single system, and the letter M indicates the number of copies (sometimes called subsystems). Superscripts in parentheses indicate operators acting on multiple systems. For example, the cyclic shift operator between M copies is denoted as S (M) . A bold superscript without parentheses indicates on which copy the operator / observable acts, e.g., O 1 indicates that the observable acts on subsystem 1. Superscripts without bold or parentheses generally indicate exponentiation. Subscripts on operators generally indicate on which qubit in the system the operator acts. An exception is when the subscript is more generally used as an index in a sum, which will be clear to those skilled in the art from the context and the presence of the symbol Σ.

[0044] Example operating environment

[0045] Figure 1A block diagram 100 illustrates an example system for performing virtual distillation to mitigate errors when computing the expected value of an observable with respect to a noisy quantum state. The quantum computing system 102 obtains multiple copies of the noisy quantum state 104. For example, the quantum computing system 102 may include a quantum computer performing quantum computation. A portion of the quantum computation may include measuring the output quantum state to determine the properties of the quantum state (e.g., the expected value of the target observable) and obtaining the result of the quantum computation. In the absence of noise, the output quantum state would be error-free. Therefore, by repeatedly preparing and measuring the output quantum state, an error-free expected value can be obtained. However, in practice, quantum computation performed by a quantum computer is likely to be noisy quantum computation, for example, in the case of a NISQ device. Therefore, the output quantum state is typically a noisy quantum state, and the expected value obtained by repeatedly preparing and measuring the output quantum state will be noisy. Therefore, the quantum computer must correct the expected value to obtain a more accurate and meaningful quantum computation result.

[0046] To calculate the expected value of the correction, the quantum computer performs a collective measurement on M copies of the noisy quantum state ρ to determine the expected value of the target observable of the entangled quantum state given by equation (1) above. That is, the quantum computer determines the expected value given by equation (2) above.

[0047] To calculate the numerator and denominator of equation (2), use the following equation.

[0048]

[0049] In equation (3), O i The observable O acting on (arbitrary) subsystem i, while S (M) Indicates the cyclic shift operators on M systems, i.e.

[0050]

[0051]

[0052] This identity can be proven by expanding the right-hand side and following the index. Without loss of generality, we can choose index i to be equal to 1, thus obtaining...

[0053]

[0054] Setting O to equal the identity operator II gives an equivalent expression for the denominator of equation (2), namely

[0055] Therefore, in order to calculate the numerator and denominator of equation (2), a quantum computer is configured to perform computational tasks. and These operations include preparing noisy quantum states (i.e., states). M copies of the tensor product 106, and the application of quantum circuitry 108 to the tensor product 106, for example, designed to implement a cyclic shift operator OS with respect to the tensor product 106. (M) A quantum circuit for measuring the product of the target observable and the tensor product, or a quantum circuit designed to measure the cyclic shift operator with respect to the tensor product 106. After multiple measurements are repeated 110, the quantum computer can post-process the obtained measurement results 112 to determine the expected value of the product of the cyclic shift operator with respect to the tensor product of M copies of the noisy quantum state and the target observable. And the corresponding normalization factor, i.e. The normalization factor can correspond to the expectation of the identity operator with respect to the entangled quantum state. The error mitigation estimate of the expectation of the observable 114 can be obtained by dividing the expectation o of the product of the cyclic shift operator of the tensor product of the M copies of the noisy quantum state and the target observable by the normalization factor, as given in equation (2).

[0056] As described above, the quantum computer applied to the quantum circuit 108 of the tensor product 106 includes the cyclic shift operator S (M) Diagonalization or representation of the cyclic shift operator S (M) In some implementations, quantum circuit 108 may include additional quantum gates to optimize the evaluation of the numerator and denominator of equation (2). For example, see the following reference. Figure 5 As described in more detail, in example quantum circuit 116, the quantum circuit also includes a single-layer two-qubit diagonalization gate 118 capable of being applied in parallel. Typically, quantum circuit 108 can vary depending on various factors, such as the number of copies of the noisy states included in the tensor product 106, the type of the target observable (e.g., whether the target observable is a single-qubit operator or a multi-qubit operator), and whether additional quantum computing resources are available for the quantum computer (e.g., whether auxiliary bit-assisted measurement is feasible in the experimental setup). See below for reference. Figure 5 A more detailed description of the variant of virtual distillation.

[0057] Figure 2 An example quantum computing system 200 is depicted for performing virtual distillation to mitigate errors when calculating the expected value of an observable with respect to a noisy quantum state. The example quantum computing system 200 is an example of a system implemented as a classical and quantum computer program on one or more classical computers and quantum computing devices at one or more locations, wherein the systems, components, and techniques described herein can be implemented.

[0058] Example quantum computing system 200 includes quantum computing device 202. According to some implementations, quantum computing device 202 can be used to perform the quantum computing operations described herein. Quantum computing device 202 is intended to represent various forms of quantum computing devices. In some implementations, quantum computing device 202 is a noisy quantum computing device, such as an NISQ device. The components shown herein, their connections and relationships, and their functionality are merely exemplary and do not limit the ways in which the invention described and / or claimed herein can be implemented.

[0059] Example quantum computing device 202 includes a qubit assembly 252 and a control and measurement system 204. The qubit assembly includes multiple qubits, such as qubit 206, for performing algorithmic operations or quantum computing. Although Figure 2 The qubits shown are arranged in a rectangular array, but this is a schematic illustration and not a limitation. The qubit assembly 252 also includes adjustable coupling elements, such as coupler 208, which allows interaction between the coupled qubits. Figure 2 In the schematic diagram, each qubit is tunably coupled to each of its four neighboring qubits via a corresponding coupling element. However, this is an example arrangement of qubits and couplers; other arrangements are also possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include tunable coupling between more than two qubits.

[0060] Each qubit can be a two-level quantum system or device with levels representing logical values ​​0 and 1. The specific physical implementation of multiple qubits and how they interact with each other depends on a variety of factors, including the type of quantum computing device included in example system 200 or the type of quantum computation being performed by the quantum computing device. For example, in an atomic quantum computer, qubits can be implemented using atoms, molecules, or solid-state quantum systems (e.g., hyperfine atomic states). As another example, in a superconducting quantum computer, qubits can be implemented using superconducting qubits or semiconductor qubits (e.g., superconducting transmon states). As yet another example, in an NMR quantum computer, qubits can be implemented using nuclear spin states.

[0061] In some implementations, quantum computing can be performed by preparing qubits in an initial state, for example, by resetting each qubit or performing quantum operations on the qubits to obtain a target quantum state, and applying a unitary operator sequence to the initial state. Applying unitary operators to the quantum state can include applying a corresponding sequence of quantum logic gates to the qubits. Example quantum logic gates include: single-qubit gates, such as Pauli-X, Pauli-Y, Pauli-Z (also known as X, Y, Z), Hadamard, and S gates; two-qubit gates, such as SWAP gates, controlled X, controlled Y, controlled Z (also known as CX, CY, CZ); and gates involving three or more qubits, such as Toffoli gates. The quantum logic gates can be implemented by applying control signals 210 generated by the control and measurement system 204 to the qubits and couplers.

[0062] For example, in some implementations, the qubits in the qubit assembly 252 can be frequency-tunable. In these examples, each qubit can have an associated operating frequency, which can be adjusted by applying voltage pulses via one or more drive lines coupled to the qubit. Example operating frequencies include qubit idle frequency, qubit interaction frequency, and qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency can put the qubit into a state in which it does not interact strongly with other qubits and can be used to perform single-qubit gates. As another example, in the case where the qubits interact via couplers with fixed coupling, the qubits can be configured to interact by setting their respective operating frequencies to a gate-related frequency that is detuned to their common interaction frequency. In other cases, for example, when the qubits interact via tunable couplers, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to allow interaction between the qubits, and then by setting the respective operating frequencies of the qubits to a gate-related frequency that is detuned to their common interaction frequency. Such interactions can be performed to perform multi-qubit gates.

[0063] The type of control signal 210 used depends on the physical implementation of the qubit. For example, the control signal may include RF or microwave pulses in an NMR or superconducting quantum computer system or light pulses in an atomic quantum computer system.

[0064] Quantum computing can be accomplished by measuring the state of qubits, for example, using quantum observables such as X or Z, with corresponding control signals 210. The measurement causes a readout signal 212, representing the measurement result, to be transmitted back to the measurement and control system 204. Depending on the physical scheme of the quantum computing device and / or qubits, the readout signal 212 may include RF, microwave, or optical signals. For convenience, Figure 2 The control signal 210 and readout signal 212 shown are illustrated to address only selected elements (i.e., top and bottom rows) of the qubit assembly, but during operation, the control signal 210 and readout signal 212 can address each element in the qubit assembly 252.

[0065] The control and measurement system 204 is an example of a classical computer system that can be used to perform various operations (as described above) and other classical subroutines or computations on the qubit component 252. The control and measurement system 204 includes one or more classical processors (e.g., conventional processor 214), one or more memories (e.g., memory 216), and one or more I / O units (e.g., I / O unit 218) connected by one or more data buses. The control and measurement system 204 can be programmed to send a sequence of control signals 210 to the qubit component, for example, to perform a selected series of quantum gate operations, and to receive a sequence of readout signals 212 from the qubit component, for example, as part of performing a measurement operation.

[0066] Processor 214 is configured to process instructions executed within control and measurement system 204. In some implementations, processor 214 is a single-threaded processor. In other implementations, processor 214 is a multi-threaded processor. Processor 214 is capable of processing instructions stored in memory 216.

[0067] Memory 216 stores information within control and measurement system 204. In some implementations, memory 216 includes computer-readable media, volatile memory cells, and / or non-volatile memory cells. In some cases, memory 216 may include storage devices capable of providing mass storage to system 204, such as hard disk drives, optical disk drives, storage devices shared by multiple computing devices over a network (e.g., cloud storage devices), and / or some other mass storage device.

[0068] Input / output device 218 provides input / output operations to control and measurement system 204. Input / output device 218 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers to send control signals 210 to the qubit components and receive readout signals 212 from the qubit components, which is suitable for the physical scheme of a quantum computer. In some implementations, input / output device 218 may also include one or more network interface devices, such as Ethernet cards, serial communication devices (e.g., RS-232 ports), and / or wireless interface devices (e.g., 802.11 cards). In some implementations, input / output device 218 may include a driver configured to receive input data and send output data to other external devices (e.g., keyboards, printers, and display devices).

[0069] Although Figure 2 An example control and measurement system 204 has been illustrated, but the subject matter and functional operations described in this specification can be implemented in other types of digital electronic circuits, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in a combination of one or more of them.

[0070] Example system 200 also includes example classic processor 250. Depending on some implementations, classic processor 250 can be used to perform the classic computational operations described in this specification. Figure 2 In this embodiment, classical processor 250 is shown as separate from quantum computing device 202; however, in some implementations, classical processor 250 may be included in quantum computing device 202, for example, in processor 214.

[0071] Programming the hardware: Virtual distillation

[0072] Figure 3 It is used to determine the expected value of error mitigation for the target observable O with respect to the noisy quantum state ρ. <o> corrected A flowchart of example process 300 is provided. For convenience, process 300 will be described as being executed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a system appropriately programmed according to this specification... Figure 2 The quantum computing device 200 can execute process 300.

[0073] The system acquires multiple noisy quantum states ρ (step 302). For example, the system can perform quantum computation to prepare for the replication of the noisy quantum states or retrieve a replication of the noisy quantum states from a quantum memory. In some implementations, each replication of the noisy quantum states is subjected to noise of the same form and intensity. The number of replications can be selected based on various factors, including the capabilities of the quantum device performing process 300, the type of the target observable, and / or the target precision of the desired value for output error mitigation.

[0074] This system is effective against noisy quantum states (e.g.) The tensor product of M ≥ 1 copies of the entangled quantum state is measured to calculate the expected value of the target observable (step 304). For example, the system calculates as given in equation (3) above. An entangled quantum state is a quantum state in which the eigenvalues ​​corresponding to the non-dominant eigenvectors of a noise quantum state are suppressed exponentially by M in the spectral decomposition of the entangled quantum state. For example, an entangled quantum state can be the quantum state ρ given by the above equation (1). M .

[0075] In order to perform the measurement, the system can measure i) the target observable and ii) the quantum state with respect to noise for each measurement repetition in a first number of measurement repetitions. A cyclic shift operator for the tensor product of M copies (e.g., measuring O) i S (M) The system then uses the product of the first measurement result and the product of the second measurement result to obtain the first measurement result. The system can then use the first measurement result to calculate the product of the first measurement result and the product of the product of the second measurement result and the product of .... The expected value of the product of the cyclic shift operator and the target observable, for example, calculating...

[0076] As described in reference equation (3) above, regarding The expectation of the product of the cyclic shift operator and the target observable is equal to the expectation of the target observable with respect to the entangled quantum state. Therefore, in step 304, the system calculates the expectation of the target observable with respect to the entangled quantum state without explicitly preparing the entangled quantum state.

[0077] System use about The expected value of error mitigation for the target observable with respect to the noisy quantum state is determined by the calculated expected value of the product of the cyclic shift operator and the target observable (step 306). For example, the system can determine the expected value by dividing the expected value by the product of the cyclic shift operator and the target observable. The expected value of the cyclic shift operator, for about The expectation of the product of the cyclic shift operator and the target observable is normalized. The normalized expectation provides an approximation of the expectation of the target observable with respect to the error mitigation of the noisy quantum state, as described above with reference to equations (1)-(3).

[0078] In order to calculate about To determine the expected value of the cyclic shift operator, the system performs a process similar to that described in step 304, where the observable is set equal to the identity operator. That is, for each measurement repetition in the second number of measurement repetitions, the system measures with respect to the noisy quantum state. A cyclic shift operator of M replicated tensor products is used to obtain a second measurement result. The system can then use the second measurement result to calculate information about the quantum state. The expected value of the cyclic shift operator, for example

[0079]

[0080] Example process 300 describes a general application of virtual distillation. However, in some implementations, the quantities in the numerator and denominator of Equation 2 can be evaluated in many different ways. For example, in some implementations, the steps of the process can be combined and / or implemented in parallel. See below for reference. Figure 5 A detailed description of an example variant of example procedure 300.

[0081] Programming the hardware: Virtual distillation using diagonalization measurements

[0082] Figure 4 This is a flowchart of an example process 400 for performing virtual distillation via diagonalization measurements. For convenience, process 400 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, appropriately programmed according to this specification... Figure 2 The quantum computing device 200 can execute process 400.

[0083] Example procedure 400 is particularly well-suited for implementations where M=2 and the target observable O acts on a single qubit. For convenience, example procedure 400 is described as the case where the target observable O is the Pauli Z operator acting on a single qubit. However, example procedure 400 can also be applied to other single-qubit observables, for example, by applying appropriate single-qubit rotations prior to the virtual distillation process. For example, the Pauli X operator can be measured using a procedure for measuring the Pauli Z operator after applying a Hadamard gate to the appropriate qubit in each replication.

[0084] The system obtains multiple copies of the noisy quantum state ρ (step 402). Step 402 is similar to step 302 of example process 300, and for the sake of brevity, the details will not be repeated.

[0085] This system is effective against noisy quantum states (e.g.) The tensor product of two copies of the entangled quantum state is measured to calculate the expected value of the target observable. In step 304 of example procedure 300, the relation in equation (3) is used to calculate the desired expected value. In example procedure 400, instead of directly using the relation in equation (3), the system determines a symmetric version of the target observable, defined as follows:

[0086]

[0087] For the specific case where the target observable O is the Pauli Z operator acting on a single qubit, this means

[0088]

[0089] Then, equation (3) can be used to illustrate...

[0090]

[0091] Using symmetric observables is advantageous because

[0092] [O (M) ,S (M) ]=0 (9)

[0093] Or in that case, |Z k (2) ,S (2) |=0.

[0094] S (2) and Z k (2) Both are factored into tensor products of operators acting individually on each pair of qubits, where the i-th pair contains the i-th qubit from each system. Therefore, operators that are factored using the same structure can be used simultaneously on S. (2) and Perform diagonalization. The unitary representation of the two qubits performing this diagonalization on the i-th pair is as follows: The matrix representation of this gate is given below:

[0095]

[0096] The phase selection of the matrix elements can be varied. Then, operator B... (2) It can be defined as

[0097]

[0098] As expected, the unit diagonalizes the factors constituting the observable.

[0099]

[0100]

[0101] This diagonalization is particularly easy to achieve when each qubit from the first copy of ρ is adjacent to a corresponding qubit from the second copy.

[0102] Then, the process for measuring the observables required to estimate the numerator and denominator of equation (8) simplifies to applying N two-qubit gates of a single layer in parallel and measuring them in the computational basis (step 404). In fact, because B (2) For all N values ​​of k, for Z k (2) S (2) Diagonalization was performed, so estimates of all N operators Z were collected simultaneously. k The data required to mitigate the expected value of the error. By applying appropriate single-qubit rotation before performing virtual distillation, arbitrary single-qubit observables on each qubit can be accessed.

[0103] To develop some intuition, we use the spectral decomposition of ρ to express... It is helpful to consider the two independent components of the resulting sum.

[0104]

[0105] Calculating the measurement probability and expectation is a linear operation on the density matrix, therefore these two components can be considered separately. The component with state i=j in S (2) In the +1 eigenspace, and leading to S (2) The measurement, which is based on probability This generates an eigenvalue of +1. In the case that i ≠ j, It is a uniform superposition of symmetric and antisymmetric states.

[0106]

[0107] For this component of the state, for S (2) The measurements produce +1 and -1 with equal probability, and (2) >=0. Combining these two cases, give the expected equation. For S (2) O (2) The measurements follow a similar pattern.

[0108] It is interesting to compare this behavior with the stabilizer theory of quantum error correction. In the stabilizer form, errors are detected by projecting the measurement onto one or more symmetric -1 eigenspaces. In the method described here, the error is equally supported in the eigenspace of the symmetry of the measured quantity.

[0109] Back Figure 4 To perform a measurement on the tensor product of two copies of a noisy quantum state to compute the expected value of a target observable with respect to the entangled quantum state, for each measurement repetition in multiple measurement repetitions, the system applies a diagonalization operator to the tensor product of M copies of the noisy quantum state to obtain an evolving quantum state, wherein the diagonalization operator diagonalizes the product of i) the cyclic shift operator and ii) the symmetric version of the cyclic shift operator with respect to the target operator. The system then measures the product of the cyclic shift operator and the target observable with respect to the evolving quantum state to obtain the corresponding measurement result for each qubit in the two copies of the noisy quantum state.

[0110] The system then uses the measurement results to calculate the expected value given by equation (8), and further calculates the expected value of error mitigation for the target observable of the noisy quantum state (step 406). For example, the system can use the measurement results to calculate the expected value of error mitigation for the target observable of the noisy quantum state. The expected value of the product of the cyclic shift operator and the target observable is calculated using the measurement results. The expected value of the cyclic shift operator will be about The expected value of the product of the cyclic shift operator and the target observable divided by the value of the cyclic shift operator and the target observable. The expected value of the cyclic shift operator.

[0111] The following is given by using M=2 and O=Z i Example algorithm for performing virtual distillation using diagonalized measurements in the following case:

[0112] Input: Number of repeated measurements K, 2K copies of the N-qubit noise quantum state ρ (two copies at a time).

[0113] Output: An estimate of the error mitigation for each qubit in ρ;

[0114] ​For each qubit i∈1,…,N, let E i =0.

[0115] Set D = 0.

[0116] fork∈Kdo

[0117] Perform any SWAP operation required to couple each qubit in the first copy of ρ to the corresponding qubit in the second copy.

[0118] A two-qubit gate as defined in equation (10) is applied between each qubit i in the first replication and the corresponding qubit in the second replication.

[0119] Two states are measured on a computational basis.

[0120] set up and Let represent the measurement results of the i-th qubit in the first and second replications of ρ, respectively.

[0121] for i∈1,…,Ndo

[0122]

[0123] end for

[0124]

[0125] end for

[0126] return{ <Z i > corrected :=E i / D}

[0127] Programming the hardware: Selecting different virtual distillation protocols

[0128] Figure 5 This is a flowchart of an example process 500 for selecting a virtual distillation protocol. For convenience, process 500 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a system appropriately programmed according to this specification... Figure 2 The quantum computing device 200 can execute process 500.

[0129] The system determines whether an auxiliary bit-assisted measurement is feasible in the available experimental setup (step 502). In response to determining that an auxiliary bit-assisted measurement is not feasible, the system determines whether the target observable is a single-qubit observable (step 504). In response to determining that the target observable is not a single-qubit observable, the system executes a virtual distillation protocol for serial measurement of two or more copies of the noisy quantum state (step 506).

[0130] Measuring observables with more than one qubit can be challenging. This challenge arises due to the use of Equation 8, and particularly due to the choice to use a symmetric version of the observable (defined in Equation 6). Using a symmetric version of a multi-qubit observable means it is impossible to perform the required diagonalization using a separate unitary tensor product across each pair of qubits. As an example, consider the observable O = Z. i Z j (This analysis is equally valid for any other operator consisting of tensor products of (more than one) single-qubit Pauli operators). The product of the symmetric observable and the commutative operator for M=2 yields...

[0131]

[0132] This operator cannot be factored into a tensor product of operators supporting single qubit pairs, nor can it be diagonalized by an operator factored in this way.

[0133] However, instead of using equation (8) to determine the expected value of the correction for O, the asymmetric form introduced in equation (2) can be used. Returning to O = Z i Z j For example, the task is to calculate the numerator and denominator of the following equation:

[0134]

[0135] Unlike the symmetric observables of equation (16), the operator Factorize into tensor products on N pairs of qubits (one pair is a qubit from the first system and the corresponding qubit from the second system). It is not Hermitian, but because it is unitary, it has a quantity. It can be estimated by applying circuitry to diagonalize it and measuring it in a computational basis. Because Factorization is the product of two qubit operators, and so is the diagonalization of circuits. The details of diagonalization depend on the operator being measured.

[0136] because With S (2) Since there is no commutation, the numerator and denominator of equation (17) cannot be calculated simultaneously. Nor can the expected value of the correction corresponding to different choices of i and j be measured simultaneously. More generally, regardless of the number of qubits involved, it is impossible to measure any single tensor product of a qubit operator at a time.

[0137] Figure 6 This is a flowchart of an example process 600 for serial measurement of two or more copies of a noisy quantum state, wherein the target observable acts on two or more qubits. For convenience, process 600 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, appropriately programmed according to this specification... Figure 2 The quantum computing device 200 can execute process 600. Some of the steps described below are similar to those described in the example process 300, and for the sake of brevity, details are not repeated.

[0138] The system obtains multiple copies of the noisy quantum state (step 602). The system decomposes the target observable into multiple tensor products, wherein each tensor product includes a single-qubit operator (step 604). The system performs serial measurements on the tensor products of the M copies of the noisy quantum state (step 606). For each tensor product of the single-qubit operator, the system performs a first number of measurement repetitions. Each measurement repetition includes: applying a first diagonalization operator to the tensor products of the M copies of the noisy quantum state to obtain a first evolved quantum state, wherein the first diagonalization operator diagonalizes the product of i) the tensor product of the single-qubit operator and ii) the product of the cyclic shift operator, and measuring the product of the tensor products of the cyclic shift operator and the single-qubit operator with respect to the first evolved quantum state to obtain a corresponding first measurement result for each qubit in the first evolved quantum state (step 606a).

[0139] Furthermore, for each measurement repetition in the second number of measurement repetitions, the system applies a second diagonalization operator to the tensor product of M copies of the noisy quantum state to obtain a second evolved quantum state, wherein the second diagonalization operator diagonalizes the cyclic shift operator. The system measures the cyclic shift operator with respect to the second evolved quantum state to obtain a corresponding second measurement result for each qubit in the second evolved quantum state (step 606b).

[0140] The system uses the obtained measurement results to calculate the expected value of the error mitigation for the target observable with respect to the noisy quantum state (step 608). For example, the system can use the first measurement result to calculate the error mitigation for the target observable with respect to the noisy quantum state. The expected value of the product of the cyclic shift operator and the target observable is calculated using the second measurement result regarding... The expected value of the cyclic shift operator, and will be about The expected value of the product of the cyclic shift operator and the target observable divided by the value of the cyclic shift operator and the target observable. The expected value of the cyclic shift operator.

[0141] Back Figure 5 In response to determining that the target observable is a single-qubit observable, the system executes a virtual distillation protocol to measure two or more copies of the noisy quantum state in parallel (step 508).

[0142] For example, as described in example process 400 above, the system can perform virtual distillation by diagonalizing the measurement when M=2. As another example, the system can perform virtual distillation by diagonalizing the measurement when M>2 by generalizing example process 400 to higher powers of ρ.

[0143] Similar to the commutation operator, the cyclic shift operator S between M N-qubit systems (M) Factorize into the tensor product of N M-qubit gates. Specifically, factorize into the tensor product of N single-qubit cyclic shift operators. Symmetricized operator. With operator S (M) Easier. Therefore, Z k (M) S (M) and S (M) It can be diagonalized simultaneously, even if S (M) For M>2, it is unitary, not Hermitian. Because Z k (M) and S (M) Both can be factored into tensor products over N M-tuples of qubits, so diagonalizing these operators unitarily and then factoring them into tensor products of M-qubit operators in the same way is also possible. Therefore, the above reference can be applied. Figure 4 The process described is 400.

[0144] Similar concerns regarding the expected value of the correction for determining the multi-qubit observable, as described in step 506 above for the two-copy (M=2) case, apply to this generalization protocol. The currently developed tool allows for simultaneous estimation of Tr(Z) for all values ​​of m. k ρ M ), Tr(ρ M The same applies. If we want to reconstruct Tr(Pρ) from a certain multi-qubit Pauli operator P... M If we use a generalization of equation (17), then we can use the operator required to measure a specific P at a time.

[0145] For the specific case where M=3, the corresponding circuit can be optimized to simultaneously... and Diagonalize (so that example procedure 400 can be applied when M=3). Figure 7 Showing the use of pairs and Diagonalization gates for diagonalization (For example, an example hypothesis (ansatz) for numerical optimization of a gate that satisfies properties similar to those satisfied by the gate defined in equation (10). Figure 7 middle, Let represent the two-qubit gate parameters of the i-th arbitrary two-qubit gate. The parameters of the four two-qubit gates allow an approximate error of 5E⁻⁵ (measured by the Frobenius norm of the difference between the exact and approximate matrices) when using the following equation:

[0146]

[0147]

[0148] Back Figure 5 In response to determining in step 502 that auxiliary bit-assisted measurement is feasible, the system performs virtual distillation on any observable, wherein any number M copies of the noisy quantum state are measured in parallel (step 510).

[0149] When virtual distillation is performed using auxiliary bit-assisted measurement, the form of the observable being measured is unrestricted, and it is possible to simultaneously measure operators acting on overlapping subsets of qubits. Therefore, virtual distillation using auxiliary bit-assisted measurement is compatible with known techniques for efficiently measuring large numbers of commutative operators.

[0150] To perform virtual distillation using auxiliary bit-assisted measurement on M=2 copies of a noisy quantum state, the system can use the commutation operator S. (2) The system approximates the numerator and denominator of Equation 8 using non-destructive measurements. For each measurement repetition in multiple measurement repetitions, the system prepares (or otherwise obtains) two quantum registers, each prepared in a noisy quantum state ρ and including an auxiliary qubit prepared in that 0 state. The system performs S... (2) The system performs non-destructive measurements, such as using commutation or Hadamard testing. Specifically, it applies a Hadamard gate to the auxiliary qubit, applies a cyclic shift operator to both registers while the auxiliary qubit is in a 1 state, and measures the auxiliary qubit in the X basis to obtain a first measurement result. Because S (2) The factorization is a tensor product of two qubit swapping gates, and its controlled version is similarly factored into a series of N Fredkin (controlled swapping) gates. In some implementations, compiling this circuit may require additional steps (e.g., using a series of CNOT gates to extend a single auxiliary qubit to a GHz state) to handle the constrained connectivity of recent devices.

[0151] Under the assumption that the target observable can be symmetric, as described above with reference to example procedure 400, the system can then measure the target observable O. (2) The product O is measured using qubits in two registers. (2) S (2) The system then uses the first and second measurement results from different measurement repetitions to calculate the expected value of error mitigation for the target observable with respect to the noisy quantum state. Specifically, the system can use the second measurement result to calculate the expected value of the target observable, use the first measurement result to calculate the expected value of the cyclic shift operator, and divide the expected value of the target observable by the expected value of the cyclic shift operator.

[0152] This protocol does not require separate estimation of Tr(ρO). Furthermore, the measurement of O on two copies of ρ can be used, and the numerator and denominator of equation (8) can be estimated simultaneously, resulting in a relatively sample-efficient scheme.

[0153] For virtual distillation involving auxiliary bit-assisted measurements of M>2 copies of a noisy quantum state, S (N) For N>2, which is not Hermitian, the natural generalization of the above strategy still works as expected. Specifically, the cyclic shift operator S can be used. (N) The controlled version, for an expected value equal to Observables are sampled. Because the observables O are symmetric. (N) With S (N) It commutes, therefore it also commutes with the observable measured by this generalization of the exchange test. Thus, by first performing the higher-order exchange test, and then applying it to O... (N) By taking measurements, the expected value can be determined. Observable measurements are sampled.

[0154] Mitigating algorithm errors

[0155] Most error mitigation methods focus on reducing errors caused by defects in device implementation, such as decorrelation or control errors. Some of these techniques can be applied to algorithmic errors introduced during the noiseless implementation of randomized algorithms. The development of Hamiltonian simulations has led to the development of randomized evolution methods such as qDRIFT, randomized Trotter, and combinations thereof, which in some cases have advantages over their deterministic counterparts. Because these methods are randomized, they output mixed states rather than pure states even in the absence of noise. Furthermore, they rely on approximate parameters with a natural limit within which they converge to the pure state produced by exact evolution. As described below, virtual distillation applied to qDRIFT can suppress this deviation from exact evolution. For the specific model system considered below, it was found that virtual distillation can reduce the coherent spatiotemporal volume required to reach a certain accuracy threshold by 8 times or more compared to standard qDRIFT.

[0156] The qDRIFT method simulates the time evolution of a Hamiltonian H by constructing product formulas using randomized selection rules. These terms are randomly selected from H, with selection probabilities proportional to their interaction strengths in the Hamiltonian. The system then evolves forward in time under these Hamiltonian terms for a fixed time step. This process is repeated multiple times, generating product formulas that provide an approximation of the time evolution operator. When averaged over classical randomness (under the selection of interaction terms), qDRIFT generates quantum channels that approximate the exact evolution more closely than a single product formula. Importantly, unlike most deterministic Trotter methods, the scaling of this method does not explicitly depend on the number of terms in the Hamiltonian, but rather on the 1-norm of the coefficients.

[0157] Consider that it can be decomposed into H = ∑ i h i H i Hamiltonian, of which all h i Both can be achieved by absorbing the symbol into H. i H becomes a real number and a positive number. i The spectral norm is bounded by 1. Define λ = ∑ i h i The diamond norm distance between the qDRIFT channel and the real-time evolution is given by ∈=2λ. 2 t 2 The boundary is defined by η, where η is the number of qDRIFT selection steps performed for each instance of the qDRIFT channel and thus controls the amount of coherent evolution required. As η increases, the resulting quantum channel converges to a unitary state corresponding to the exact evolution.

[0158] The virtual distillation technique described so far can reduce the required coherent spatiotemporal volume by reducing the factor η required to achieve the same error in practice. Specifically, virtual distillation can influence the number of coherent steps η required to achieve the target accuracy. For this purpose, consider the Heisenberg Hamiltonian for each replication of up to 6 qubits. This Hamiltonian is given by the following equation.

[0159]

[0160] Among them, h i Let {-h, h} be the randomly chosen Z magnetic field strength, and apply periodic boundary conditions such that station N+1 is station 1. For the study described here, consider a time evolution length of t = N, and assume h = 1. A numerical study is performed on the number of coherent qDRIFT steps required to achieve a trace distance of 0.01 to the ideal state of evolution under such a Heisenberg model.

[0161] Figure 7 The results of this analysis are shown. Specifically, Figure 7 The reduction in qDRIFT coherence cost via virtual distillation in the Heisenberg model is shown. The number of coherent qDRIFT steps required to reach the target trace distance is shown with and without virtual distillation using two copies. It can be seen that the number of required steps is consistently reduced by at least 16-fold. When considering the additional overhead of using two copies, this is equivalent to an 8-fold reduction in the coherent spatiotemporal volume required to achieve the same error rate. These results demonstrate that the error mitigation techniques described herein can provide practical algorithmic improvements for real systems, particularly in NISQ systems. For example, by incorporating the error mitigation techniques described herein, algorithms can produce more accurate results by leveraging the reduced overhead (e.g., the overhead associated with quantum error-correcting codes).

[0162] The digital and / or quantum themes described in this specification, as well as the implementations of digital functional operations and quantum operations, can be implemented in digital electronic circuits, suitable quantum circuits (or more generally, quantum computing systems), in tangibly embodied 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 a combination of one or more of them. The term "quantum computing system" can include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0163] The implementation of the digital and / or quantum themes 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 a data processing device or for controlling the operation of a data processing device. 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 storage device, one or more qubits, or a combination of one or more of these. Alternatively or additionally, the program instructions can be encoded on artificially generated propagation signals capable of encoding digital and / or quantum information, such as machine-generated electrical, optical, or electromagnetic signals, which are generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing device.

[0164] The terms quantum information and quantum data refer to information or data carried, stored, or preserved by quantum systems, the smallest nontrivial system being a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, for example, having two or more energy levels. For example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational ground state is identified by the fundamental state and the first excited state; however, it should be understood that other settings, such as identifying computational states by higher-level excited states, are also possible. The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, 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 device can also be or further include special-purpose logic circuitry, such as FPGAs (Field-Programmable Gate Arrays), ASICs (Application-Specific Integrated Circuits), or quantum simulators (i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system). Specifically, a quantum simulator is a specialized quantum computer that does not have the capability to perform general-purpose quantum computing. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stack, database management system, operating system, or a combination of one or more of these.

[0165] Digital computer programs (also referred to or described as programs, software, software applications, modules, software modules, scripts, or code) can be written in any form of programming language, including compiled or interpreted languages ​​or declarative or procedural languages, and can be deployed in any form, including as standalone programs or as modules, components, subroutines, or other units suitable for digital computing environments. Quantum computer programs (also referred to or described as programs, software, software applications, modules, software modules, scripts, 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 (such as QCL or Quipper).

[0166] Digital and / or quantum computer programs may (but do not need to) correspond to files in a file system. Programs may be stored as a portion of a file containing 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 collaborative files (e.g., multiple files storing one or more modules, subroutines, or code sections). Digital and / or quantum computer programs may be deployed to execute on a single digital or quantum computer, or on multiple digital and / or quantum computers located in one location or distributed across multiple locations and interconnected via digital and / or quantum data communication networks. Typically, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum and digital data.

[0167] The processes and logical flows described in this specification can be executed by one or more programmable digital and / or quantum computers, operating where appropriate with one or more digital and / or quantum processors, to execute one or more digital and / or quantum computer programs to perform functions by manipulating input digital and quantum data and generating outputs. The processes and logical flows can also be executed by dedicated logic circuits, and the device can be implemented as dedicated logic circuits, such as FPGAs or ASICs or quantum simulators, or executed by a combination of dedicated logic circuits or quantum simulators and one or more programmable digital and / or quantum computers.

[0168] For a system to be "configured" to perform a specific operation or action by one or more digital and / or quantum computers, this means that the system has software, firmware, hardware, or a combination thereof installed on it, which, when in operation, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs, being configured to perform a specific operation or action means that one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause that device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause that device to perform the operation or action.

[0169] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs can be based on general-purpose or special-purpose digital and / or quantum processors or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or combinations thereof.

[0170] The fundamental components of a digital and / or quantum computer are a central processing unit (CPU) for executing instructions and one or more storage devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented or incorporated into dedicated logic circuitry or a quantum simulator. Typically, a digital and / or quantum computer will also include, or be operatively coupled to (to receive, transmit, or both digital and / or quantum data to, or both) one or more high-capacity 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, digital and / or quantum computers do not require such devices.

[0171] 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 memories, media, and storage devices, such as: semiconductor storage devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROMs and DVD-ROMs; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memories are devices capable of storing quantum data for long periods of time with high fidelity and efficiency, such as light-matter interfaces used for transmission and materials used to store and preserve quantum characteristics (e.g., superposition or quantum coherence) of quantum data.

[0172] Control of the various systems or portions thereof described in this specification may be implemented in digital and / or quantum computer program products, including 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 may be implemented as apparatus, methods, or systems, which may include one or more digital and / or quantum processing devices and memory storing executable instructions to perform the operations described in this specification.

[0173] While this specification contains many specific implementation details, these should not be construed as limiting the scope of the claims, but rather as descriptions of features possible in specific implementations. Some features described in this specification in the context of independent implementations may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented individually in multiple implementations or in any suitable sub-combination. Furthermore, although features may be described above as functioning in some combinations, and even initially claimed in this way, one or more features from the claimed combination may, in some cases, be removed from that combination, and the claimed combination may be for sub-combinations or variations thereof.

[0174] Similarly, although the operations are described in a specific order in the accompanying drawings, this should not be construed as requiring these operations to be performed in the specific order or sequence shown, or requiring all shown operations to be performed to obtain the desired result. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of the various system modules and components in the above implementation should not be construed 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.

[0175] Specific implementations of the subject matter have been described. Other embodiments are within the scope of the appended claims. For example, the actions described in the claims can be performed in different orders and the desired results can still be obtained. As an example, the processes depicted in the figures do not necessarily require the specific order or sequence shown to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.< / o>

Claims

1. A method for determining an expected value of error mitigation for a target observable with respect to a noisy quantum state, the method being performed by a system comprising a classical processor and a quantum computer, the method comprising: Multiple copies of a noisy quantum state are obtained by a quantum computer, wherein the noisy quantum state includes random errors caused by noise on the quantum computer; A quantum computer measures the tensor product of M copies of a noisy quantum state to calculate the expectation of a target observable about an entangled quantum state, where the entangled quantum state is composed of ρ M / Tr(ρ M The expression is given, where ρ represents the noise quantum state, Tr represents the trajectory operation, M≥1, and the eigenvalues ​​corresponding to the non-dominant eigenvectors of the noise quantum state are suppressed exponentially by M in the spectral decomposition of the entangled quantum state; A quantum computer measures the tensor product of M copies of a noisy quantum state to calculate the expectation value of the identity operator for the entangled quantum state; and The expected value of error mitigation for the target observable with respect to the noisy quantum state is determined by dividing the calculated expected value of the target observable with respect to the entangled quantum state by the expected value of the identity operator with respect to the entangled quantum state by a classical processor. The expected value of error mitigation for the target observable with respect to the noisy quantum state includes the expected value of the reconstruction of the target observable with respect to the approximately purified version of the noisy quantum state, and is approximately the expected value of the target observable calculated by the quantum computer in the absence of random errors and noise on the quantum computer.

2. The method as described in claim 1, wherein, Each copy of a noisy quantum state experiences noise of the same form and intensity.

3. The method as described in claim 1 or 2, wherein, Measuring the tensor product of M copies of a noisy quantum state to calculate the expectation of a target observable about the entangled quantum state includes: For each measurement repetition in the first number of measurement repetitions, measure i) the target observable and ii) the product of the cyclic shift operator of the tensor product of M copies of the noisy quantum state for each repetition to obtain the first measurement result; The expected value of the product of i) the target observable and ii) the cyclic shift operator of the tensor product of M replicates of the noisy quantum state is calculated using the first measurement results obtained from the first number of repeated measurements; and Normalize the expectation of i) the target observable and ii) the product of the cyclic shift operator of the tensor product of M copies of the noisy quantum state to obtain the expectation of the error mitigation of the target observable with respect to the noisy quantum state.

4. The method of claim 3, wherein, Normalizing the expected value of the target observable includes: For each measurement repetition in the second number of measurement repetitions, a cyclic shift operator is used to measure the tensor product of M copies of the noisy quantum state for each repetition to obtain the second measurement result; The expected value of the cyclic shift operator of the tensor product of M copies of the noisy quantum state is calculated using the second measurement results obtained from the second number of repeated measurements; and Divide the expectation of the product of i) the target observable and ii) the cyclic shift operator of the tensor product of M copies of the noisy quantum state by the expectation of the cyclic shift operator of the tensor product of M copies of the noisy quantum state.

5. The method of claim 1, wherein, Performing a measurement on the tensor product of M copies of a noisy quantum state includes performing parallel measurements.

6. The method as described in claim 1 or 2, wherein, M≥2, and the target observable acts on a single qubit.

7. The method of claim 6, wherein, Performing a measurement on the tensor product of M copies of a noisy quantum state includes: for each of multiple measurement repetitions, wherein the multiple measurement repetitions include K measurement repetitions, The diagonalization operator is applied to the tensor product of M copies of the noisy quantum state to obtain the evolving quantum state, wherein the diagonalization operator diagonalizes i) the product of the cyclic shift operator and ii) the symmetric version of the target observable with the cyclic shift operator; and The product of the cyclic shift operator with respect to the evolving quantum state and the target observable is measured to obtain the corresponding measurement result for the repetition of each qubit in the evolving quantum state.

8. The method of claim 7, further comprising: The expected value of the product of the cyclic shift operator and the target observable is calculated using the measurement results obtained from K repeated measurements; The expected value of the cyclic shift operator with respect to the evolving quantum state is calculated using the measurement results obtained from K repeated measurements; as well as Divide the expectation of the product of the cyclic shift operator with respect to the evolving quantum state and the target observable by the expectation of the cyclic shift operator with respect to the evolving quantum state.

9. The method of claim 7, wherein, The application of the cyclic shift operator couples the qubits in the first copy of the noisy quantum state to the corresponding qubits in each of the subsequent copies of the noisy quantum state.

10. The method of claim 1, wherein, Performing a measurement on the tensor product of M copies of a noisy quantum state includes performing a serial measurement.

11. The method as claimed in claim 1 or 2, wherein, M≥2, and the target observable acts on two or more qubits, the target observable comprising multiple tensor products of a qubit operator.

12. The method of claim 11, wherein, Performing a measurement on the tensor product of M copies of a noisy quantum state includes: For each tensor product of a one-qubit operator: For each measurement repetition of the first number of measurement repetitions: The first diagonalization operator is applied to the tensor product of M copies of the noisy quantum state to obtain the evolving quantum state, wherein the first diagonalization operator diagonalizes the tensor product of i) a qubit operator and ii) the product of a cyclic shift operator, and Measure the product of the tensor product of the cyclic shift operator and the one-qubit operator with respect to the evolving qubit to obtain the corresponding first measurement result of the first number of repetitions in the first number of repetitions of each qubit in the evolving quantum state; For each measurement repetition of the second number of measurements: The second diagonalization operator is applied to the tensor product of M copies of the noisy quantum state to obtain the second evolved quantum state, wherein the second diagonalization operator diagonalizes the cyclic shift operator, and The cyclic shift operator with respect to the second evolved quantum state is measured to obtain the corresponding second measurement result of the repetition of the second number of repetitions of each qubit in the second evolved quantum state.

13. The method of claim 12, further comprising: The expected value of the product of the target observable and the cyclic shift operator is calculated using the first measurement results obtained from the first number of repeated measurements. The expected value of the cyclic shift operator is calculated using the second measurement result obtained from the second number of repeated measurements. as well as Divide the expected value of the product of the target observable and the cyclic shift operator by the expected value of the cyclic shift operator.

14. The method as claimed in claim 1 or 2, wherein, Performing a measurement on the tensor product of M copies of a noisy quantum state to calculate the expected value of a target observable about an entangled quantum state includes performing an auxiliary bit-assisted measurement.

15. The method of claim 14, wherein, The target observable can be a single-qubit operator or a multi-qubit operator.

16. The method of claim 14, wherein, M≥2, and performing auxiliary bit-assisted measurements includes: for each measurement repetition in multiple measurement repetitions, Prepare two quantum registers, each of which includes a noisy quantum state and an auxiliary qubit prepared in the 0 state; Non-destructive measurements performed using cyclic shift operators include: Applying Hadamard gates to auxiliary qubits With the auxiliary qubit in state 1, apply the circular shift operator to the two quantum registers, and Measuring the auxiliary qubits in the X basis to obtain the corresponding first measurement result; and The qubits in two quantum registers are measured with respect to a symmetric version of the target observable to obtain the corresponding second measurement result.

17. The method of claim 16, further comprising: The expected value of the target observable quantity is calculated using the second measurement result; The expected value of the cyclic shift operator is calculated using the first measurement result; as well as Divide the expected value of the target observable by the expected value of the cyclic shift operator.

18. An apparatus comprising: One or more classic processors; as well as One or more quantum computing devices that communicate data with the one or more classical processors, wherein the one or more quantum computing devices include: One or more qubit registers, each qubit register comprising one or more qubits, and Multiple control devices are configured to operate the one or more qubit registers; The device is configured to perform the method as described in any one of claims 1 to 17.

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