Virtual extraction for quantum error mitigation

Abstract

In one aspect, a method includes obtaining multiple copies of a noisy quantum state, calculating an expectation value of the target observable for an entangled quantum state by measuring a tensor product of M copies of the noisy quantum state, where M≧1 and eigenvalues ​​corresponding to non-dominant eigenvectors of the noisy quantum state in a spectral decomposition of the entangled quantum state are exponentially suppressed in M, and using the calculated expectation value of the target observable for the entangled quantum state to determine the error-mitigated expectation value of the target observable for the noisy quantum state.

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of priority to U.S. Provisional Application No. 63 / 112,593, filed November 11, 2020, which is incorporated by reference in its entirety.

[0002] This specification relates to quantum computing. [Background technology]

[0003] Quantum error correction codes are used in quantum computing to protect quantum information from errors due to decoherence and other quantum noises. Quantum error correction codes spread information encoded in one logical qubit in a highly entangled state among several physical qubits. For example, the Shor code is a quantum error correction code that can correct phase and bit flip errors using a 9-qubit circuit that requires 8 auxiliary qubits to correct one qubit. Summary of the Invention [Means for solving the problem]

[0004] This specification describes techniques for mitigating errors in noisy quantum computations using virtual quantum state extraction.

[0005] In general, one innovative aspect of the subject matter described herein may be embodied in a method for determining an error-mitigated expectation value of a target observable with respect to a noisy quantum state, the method including obtaining multiple copies of the noisy quantum state, calculating an expectation value of the target observable with respect to the entangled quantum state by measuring a tensor product of M copies of the noisy quantum state, where M≧1 and eigenvalues ​​corresponding to non-dominant eigenvectors of the noisy quantum state in a spectral decomposition of the entangled quantum state are exponentially suppressed in M, and using the calculated expectation value of the target observable with respect to the entangled quantum state to determine the error-mitigated expectation value of the target observable with respect to the noisy quantum state.

[0006] Other implementations of this aspect include corresponding classical and quantum computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. One or more of the quantum and classical computer systems may be configured to perform a particular operation or action by installing software, firmware, hardware, or a combination thereof that, in operation, causes the system to perform the action. One or more computer programs may be configured to perform a particular operation or action by including instructions that, when executed by a data processing device, cause the device to perform the action.

[0007] The foregoing embodiments and other implementations can each optionally include one or more of the following features, alone or in combination: In some implementations, the noise experienced by each copy of the noisy quantum state has the same form and intensity.

[0008] In some implementations, the entangled quantum state is ρ M / Tr(ρM ), where ρ represents the noisy quantum state.

[0009] In some implementations, the method further includes measuring a tensor product of the M copies of the noisy quantum state to calculate an expectation value of the identity operator for the entangled quantum state, and determining the error-mitigated expectation value of the target observable for the noisy quantum state using the calculated expectation value of the target observable for the entangled quantum state includes i) dividing the expectation value of the target observable for the entangled quantum state by ii) the expectation value of the identity operator for the entangled quantum state.

[0010] In some implementations, measuring the tensor product of the M copies of the noisy quantum state to calculate an expectation value of the target observable for the entangled quantum state includes, for each first measurement iteration, measuring a product of i) the target observable and ii) a circular shift operator on the tensor product of the M copies of the noisy quantum state to obtain a first measurement result; using the first measurement result, calculating an expectation value of the product of i) the target observable and ii) the circular shift operator on the tensor product of the M copies of the noisy quantum state; and normalizing the expectation value of the product of i) the target observable and ii) the circular shift operator on the tensor product of the M copies of the noisy quantum state to obtain an error-mitigated expectation value of the target observable for the noisy quantum state.

[0011] In some implementations, normalizing the expectation value of the target observable includes, for each second measurement iteration, measuring a circular shift operator on a tensor product of M copies of the noisy quantum state to obtain a second measurement result, calculating the expectation value of the circular shift operator on the tensor product of M copies of the noisy quantum state using the second measurement result, and dividing the expectation value of the product of i) the target observable and ii) the circular shift operator on the tensor product of the M copies of the noisy quantum state by the expectation value of the circular shift operator on the tensor product of the M copies of the noisy quantum state.

[0012] In some implementations, performing measurements on a tensor product of the M copies of the noisy quantum state includes performing parallel measurements.

[0013] In some implementations, M≧2 and the target observable acts on a single qubit.

[0014] In some implementations, the step of performing measurements on the tensor product of the M copies of the noisy quantum state includes, for each of the K measurement iterations, applying a diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain an evolved quantum state, where the diagonalization operator diagonalizes a product of i) a circular shift operator and ii) a symmetric version of the target observable and the circular shift operator, and measuring the product of the target observable and the circular shift operator on the evolved quantum state to obtain a respective measurement result for each quantum bit in the evolved quantum state.

[0015] In some implementations, the method further includes using the measurement results to calculate an expectation value of a product of the target observable and a circular shift operator on the evolved quantum state, calculating an expectation value of the circular shift operator using the measurement results on the evolved quantum state, and dividing the expectation value of the product of the target observable and the circular shift operator on the evolved quantum state by the expectation value of the circular shift operator on the evolved quantum state.

[0016] In some implementations, a circular shift operator is applied to couple qubits in a first copy of the noisy quantum state to corresponding qubits in each other copy of the noisy quantum state.

[0017] In some implementations, performing measurements on the tensor product of the M copies of the noisy quantum state includes performing sequential measurements.

[0018] In some implementations, M≧2, the target observable acts on more than one qubit, and the target observable includes multiple tensor products of one qubit operators.

[0019] In some implementations, performing a measurement on the tensor product of the M copies of the noisy quantum state includes, for each tensor product of one qubit operator, applying a first diagonalization operator to the tensor product of the M copies of the noisy quantum state for each of the first measurement iterations to obtain an evolved quantum state, where the first diagonalization operator diagonalizes a product of i) the tensor product of the one qubit operator and ii) a circular shift operator; and applying a circular shift operator to the tensor product of the one qubit operator and the evolved quantum state. the tensor product of the M copies of the noisy quantum state to obtain a respective first measurement result for each quantum bit in the evolved quantum state; for each second measurement iteration, applying a second diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain a second evolved quantum state, where the second diagonalization operator diagonalizes the circular shift operator; and measuring the circular shift operator on the second evolved quantum state to obtain a respective second measurement result for each quantum bit in the second evolved quantum state.

[0020] In some implementations, the method further includes calculating an expectation of a product of the target observable and the circular shift operator using the first measurement result, calculating an expectation of the circular shift operator using the second measurement result, and dividing the expectation of the product of the target observable and the circular shift operator by the expectation of the circular shift operator.

[0021] In some implementations, measuring the tensor product of the M copies of the noisy quantum state to calculate the expectation value of the target observable with respect to the entangled quantum state includes performing an ancillary-assisted measurement.

[0022] In some implementations, the target observable includes any observable.

[0023] In some implementations, M≧2, and performing the ancilla-assisted measurement includes, for each of a plurality of measurement iterations, preparing two quantum registers, each quantum register including a noisy quantum state and an ancilla qubit prepared in a 0 state; performing a non-destructive measurement of a cyclic shift operator, including applying a Hadamard gate to the ancilla qubit and applying the cyclic shift operator to the two quantum registers in which the ancilla qubit is in a 1 state to measure the ancilla qubit in an X basis to obtain a respective 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 respective second measurement result.

[0024] In some implementations, the method includes calculating an expectation value of the target observable using the second measurement result, calculating an expectation value of the circular shift operator using the first measurement result, and dividing the expectation value of the target observable by the expectation value of the circular shift operator.

[0025] The subject matter described herein can be implemented in a particular way to realize one or more of the following advantages.

[0026] Performing meaningful computations using near-term quantum computers can be challenging due to the relatively high error rates of these devices. Correcting quantum errors promises to enable quantum computations with conveniently small noise levels, but the computational overhead is currently too high to be practical. The currently described error mitigation techniques improve the quality of noisy computations without the large overhead of traditional quantum error correction. This error mitigation approach can mitigate the effects of randomized errors arising from, for example, noise in near-term devices, as well as the effects of randomized errors inherent in randomized quantum algorithms implemented on error-free devices. In addition, this error mitigation approach is particularly suited for implementation on near-term devices, e.g., NISQ devices, and can therefore be applied to a wide range of technological applications, including molecular and materials simulations, optimization processes, and quantum neural networks.

[0027] The 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, drawings, and claims. [Brief description of the drawings]

[0028] [Figure 1] FIG. 1 is a block diagram of an example system that performs virtual sampling to mitigate errors when calculating expectation values ​​of observables for noisy quantum states. [Diagram 2] FIG. 1 illustrates an example quantum computing system. [Diagram 3] 1 is a flow diagram of an example process for determining an error-mitigated expectation value of a target observable with respect to a noisy quantum state. [Figure 4] 13 is a flow diagram of an example process for performing virtual extraction using measurements from diagonalization. [Diagram 5] 1 is a flow diagram of an example process for selecting a virtual extraction protocol. [Figure 6]1 is a flow diagram of an example process for virtual extraction using sequential measurements on two or more copies in a noisy quantum state, where a target observable acts on two or more qubits. [Figure 7] FIG. 1 illustrates an example hypothesis for numerically optimizing a diagonalization gate. [Figure 8] 13 is a graph showing qDRIFT coherent cost reduction with virtual extraction in the Heisenberg model. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0029] Like reference numbers and designations in the various drawings indicate like elements.

[0030] This specification describes techniques for mitigating errors in noisy quantum computation. In particular, the presently described techniques allow for efficient approximation of the computational output that would be generated in the absence of noise. Expectation values ​​of observables with respect to a nearly purified version of the noisy quantum state are reconstructed without explicitly preparing a purified quantum state. This approach is referred to herein as "virtual extraction", where the term "virtual" is used to emphasize that a purified version of the quantum state is not actually prepared (as in classical extraction schemes). More specifically, collective measurements of M copies of the noisy quantum state ρ are used to measure expectation values ​​of observables with respect to the entangled quantum state, given by

[0031]

number

[0032] Here, ρ = Σ i p i where |i〉〈i| is the spectral decomposition (e.g., density matrix) of the noisy quantum state ρ, and Tr represents the trace operation. In this method, the relative weights of the non-dominant eigenvectors

[0033]

number

[0034] is suppressed exponentially in M. This represents an improvement over other approaches that require the explicit preparation of a nearly purified state and typically only achieve a linear suppression in M. Thus, the virtual extraction is the trace of the product of the observable and M copies of the spectral decomposition of the noisy quantum state divided by the trace of M copies of the spectral decomposition of the noisy quantum state to approximate a corrected expectation value of the observable O (e.g., an expectation value with incoherent errors suppressed), e.g.,

[0035]

number

[0036] The resulting estimator converges exponentially faster toward the pure state closest to ρ as M increases. This is in contrast to other approaches that only achieve linear convergence to the corrected state. Thus, the presently described technique offers a technical improvement over these other approaches, since the corrected state can be achieved in a shorter time.

[0037] In this specification, the operations described are those that operate on multiple copies of the same, noisy state. It is assumed that the noise experienced by the individual copies has the same form and intensity. Even if this assumption is relaxed, the effective state corresponding to the product of the density matrices of the individual copies will still be measured, as long as the copies are not perturbed prior to virtual extraction. The letter N refers to the number of qubits in an individual system, and the letter M indicates the number of copies (sometimes referred to as subsystems). Superscripts with parentheses indicate operators that operate on multiple systems. For example, a circular shift operator among M copies is S (M) A bold superscript without parentheses indicates a copy that an operator / observable acts on, e.g. O1 indicates that observable O acts on subsystem 1. Non-bold font, i.e., superscripts without parentheses, indicate exponentiation as usual. Subscripts on operators generally indicate the qubits within the system that the operator acts on. The exception is when a subscript is used more generically as an index in a summation, which will be clear to one of skill in the art from the context and the presence of the Σ symbol.

[0038] Example operating environment FIG. 1 illustrates a block diagram 100 of an example system that performs virtual sampling to mitigate errors when calculating observable expectations for noisy quantum states. A quantum computing system 102 obtains multiple copies 104 of a noisy quantum state. For example, the quantum computing system 102 may include a quantum computer performing a quantum computation. Part of the quantum computation may include measuring an output quantum state to determine a property of the quantum state, e.g., an expectation value of a target observable, to obtain a result of the quantum computation. In the absence of noise, the output quantum state should be error-free. Thus, it should be possible to obtain an error-free expectation value by repeating the preparation and measurement of the output quantum state. However, for example, when the quantum computer is a NISQ device, the quantum computation performed by the quantum computer is likely to be a noisy quantum computation in practice. Thus, the output quantum state is generally a noisy quantum state, and the expectation value obtained by repeating the preparation and measurement of the output quantum state should be noisy. Thus, the quantum computer needs to correct the expectation value to obtain a more accurate and more meaningful result of the quantum computation.

[0039] To calculate the corrected expectation value, the quantum computer performs collective measurements of M copies of the noisy quantum state ρ to determine the expectation value of the target observable with respect to the entangled quantum state given by equation (1) above. That is, the quantum computer determines the expectation value given by equation (2) above.

[0040] To calculate the numerator and denominator of equation (2), the following equations are used:

[0041]

number

[0042] In formula (3), O i denotes an (arbitrary) observable O acting on subsystem i, and S (M) denotes the circular shift operator acting on the M-system, i.e.

[0043]

number

[0044] This identity can be proved by expanding the right hand side, keeping track of the progression of the indices. Without loss of generality, we can choose the index i to be 1, yielding

[0045]

number

[0046] Let O be the identity operator

[0047]

number

[0048] Setting it equal to gives an equivalent expression to the denominator of equation (2), i.e.,

[0049]

number

[0050] It becomes.

[0051] Therefore, to calculate the numerator and denominator of equation (2), the quantum computer

[0052]

number

[0053] and

[0054]

number

[0055] These operations are performed on the noisy quantum state, i.e., state

[0056]

number

[0057] and performing a circular shift operator OS on the target observable and the tensor product 106. (M) and applying a quantum circuit 108, which may be a quantum circuit designed to measure a product of the target observable with a circular shift operator on the tensor product 106, to the tensor product 106. After a number of measurement iterations 110, the quantum computer performs post-processing 112 of the obtained measurement results to obtain an expectation value of the product of the target observable with a circular shift operator on the tensor product of the M copies of the noisy quantum state, i.e.

[0058]

number

[0059] and the corresponding normalization factor, i.e.

[0060]

number

[0061] The normalization factor may correspond to the expectation value of the identity operator on the entangled quantum state. An error-mitigated estimate 114 of the expectation value of the observable may be obtained by dividing the expectation value o of the product of the target observable and the circular shift operator on the tensor product of M copies of the noisy quantum state by the normalization factor, as given by equation (2).

[0062] As discussed above, the quantum circuit 108 that the quantum computer applies to the tensor product 106 is a circular shift operator S (M) In some implementations, the quantum circuit 108 may include additional quantum gates to optimize the evaluation of the numerator and denominator of equation (2). For example, the example quantum circuit 116 further includes a single layer of two-qubit diagonalization gates 118 that may be applied in parallel, as described in more detail below with reference to FIG. 5. In general, the quantum circuit 108 may vary depending on various factors, such as the number of copies of the noisy state included in the tensor product 106, the type of the target observable, e.g., whether the target observable is a single-qubit or multi-qubit operator, and the availability of additional quantum computational resources for the quantum computer, e.g., whether ancillary-assisted measurements are feasible in the experimental device. A variation of virtual extraction is described in more detail below with reference to FIG. 5.

[0063] 2 depicts an example quantum computing system 200 that performs virtual sampling to mitigate errors when calculating expectation values ​​of observables for noisy quantum states. Example quantum computing system 200 is an example of a system implemented as classical computer programs on one or more classical computers and quantum computer programs on one or more quantum computing devices, at one or more locations, that can implement the systems, components, and techniques described herein.

[0064] Exemplary quantum computing system 200 includes a quantum computing device 202. Quantum computing device 202 may be used to perform the quantum computing operations described herein according to some implementations. 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, for example, a NISQ device. The components illustrated herein, their connections and relationships, and their functions are exemplary only and are not intended to limit the implementations of the invention described and / or claimed herein.

[0065] The exemplary quantum computing device 202 includes a qubit assembly 252 and a control and measurement system 204. The qubit assembly includes a plurality of qubits, e.g., qubits 206, that are used to perform algorithmic operations or quantum computations. Although the qubits shown in FIG. 2 are a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubit assembly 252 also includes adjustable coupling elements, e.g., couplers 208, that enable interactions between coupled qubits. In the schematic depiction of FIG. 2, each qubit is adjustably coupled to each of four adjacent qubits by a respective coupling element. However, this is an exemplary arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that enable coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.

[0066] Each qubit can be a physical two-level quantum system or device with levels representing logical values ​​of 0 and 1. The particular physical realization of the qubits and how they interact will depend on various factors, including the type of quantum computing device included in the example system 200 or the type of quantum computation the quantum computing device performs. For example, the qubits can be realized by atoms, molecules, or solid-state quantum systems, e.g., hyperfine atomic states, in an atomic quantum computer. As another example, the qubits can be realized by superconducting qubits or semiconducting qubits, e.g., superconducting transmon states, in a superconducting quantum computer. As another example, the qubits can be realized by nuclear spin states in an NMR quantum computer.

[0067] In some implementations, quantum computation can proceed by preparing an initial state, for example by performing quantum operations on the qubits to reset each qubit or obtain a target quantum state, and applying a sequence of unitary operators to the initial state. Applying the unitary operators to the quantum state may include applying a corresponding sequence of quantum logic gates to the qubits. Exemplary quantum logic gates include single-qubit gates, for example Pauli X, Pauli Y, Pauli Z (also referred to as X, Y, Z), two-qubit gates, for example the SWAP gate, Hadamard gates and ternary logic limit gates, X-control, Y-control, Z-control (also referred to as CX, CY, CZ), and gates encompassing three or more qubits (e.g., 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.

[0068] For example, in some implementations, the qubits in the qubit assembly 252 may be frequency tunable. In these examples, each qubit may have an associated operating frequency that may be adjusted by applying a voltage pulse through one or more drive lines coupled to the qubit. Exemplary operating frequencies include a qubit idle frequency, a qubit interaction frequency, and a qubit readout frequency. The different frequencies correspond to different operations that the qubit may perform. For example, setting the operating frequency to a corresponding idle frequency may result in the qubit not interacting strongly with other qubits and being in a state where it may be used to perform a single qubit gate. As another example, when the qubits interact in a fixed coupling via a coupler, the qubits may be configured to interact with each other by setting their respective operating frequencies to some gate-dependent frequency that is detuned from their common interaction frequency. In other cases, for example, when the qubits interact via tunable couplers, the qubits may be configured to interact with each other by setting parameters of the respective couplers to enable interaction between the qubits, and then setting the respective operating frequencies of the qubits to some gate-dependent frequency that is detuned from their common interaction frequency. Such interactions can be implemented to perform multi-qubit gates.

[0069] The type of control signal 210 used depends on the physical implementation of the qubit. For example, the control signal may include an RF or microwave pulse in an NMR or superconducting quantum computer system, or an optical pulse in an atomic quantum computer system.

[0070] Quantum computation may be completed by using a quantum observable, such as X or Z, to measure the state of the qubit using a respective control signal 210. A readout signal 212 resulting from the measurement represents the measurement result that is communicated to control and measurement system 204. Readout signal 212 may include an RF signal, a microwave signal, or an optical signal, depending on the physical format of the quantum computing device and / or qubit. For convenience, control signals 210 and readout signals 212 shown in FIG. 2 are depicted as addressing only selected elements of the qubit assembly (i.e., the top and bottom rows), although in operation, each element in qubit assembly 252 may be addressed.

[0071] Control and measurement system 204 is an example of a classical computer system that may be used to perform various operations on qubit assemblies 252, as described above, as well as other classical subroutines or calculations. Control and measurement system 204 includes one or more classical processors, e.g., classical processor 214, one or more storage devices, e.g., storage device 216, and one or more I / O units, e.g., I / O unit 218 connected by one or more data buses. Control and measurement system 204 may be programmed to send sequences of control signals 210 to the qubit assemblies, e.g., to perform a selected series of quantum gate operations, and to receive sequences of read signals 212 from the qubit assemblies, e.g., as part of performing a measurement operation.

[0072] The processor 214 is configured to process instructions for execution within the control and measurement system 204. In some implementations, the processor 214 is a single-threaded processor. In other implementations, the processor 214 is a multi-threaded processor. The processor 214 can process instructions stored in the storage device 216.

[0073] The storage device 216 stores information within the control and measurement system 204. In some implementations, the storage device 216 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the storage device 216 may include a storage device that can provide mass storage to the system 204, such as, for example, a hard disk device, an optical disk device, multiple computer devices, a storage device shared over a network (e.g., a cloud storage device), and / or some other mass storage device.

[0074] The input / output devices 218 provide input / output operations for the control and measurement system 204. The input / output devices 218 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers to send control signals 210 to and receive readout signals 212 from the qubit assemblies as appropriate for the physics of the quantum computer. In some implementations, the input / output devices 218 may also include one or more network interface devices, e.g., Ethernet cards, serial communication devices, e.g., RS-232 ports, and / or wireless interface devices, e.g., 802.11 cards. In some implementations, the input / output devices 218 may include drive devices configured to receive input data from and send output data to other external devices, e.g., keyboards, printers, and display devices.

[0075] An example control and measurement system 204 is shown in FIG. 2, although implementations of the subject matter and functional operations described herein can be implemented in other types of digital electronic circuitry, or computer software, firmware, or hardware, including the structures disclosed herein and their structural equivalents, or combinations thereof.

[0076] Exemplary system 200 further includes an exemplary classical processor 250. Classical processor 250 may be used to perform the classical computing operations described herein, according to some implementations. In FIG. 2, classical processor 250 is shown separate from quantum computing device 202, but in some implementations may be included in quantum computing device 202, such as processor 214.

[0077] Hardware Programming: Virtual Extraction 3 is a flow diagram of an example process 300 for determining an error-mitigated expectation value, 〈O〉, of a target observable O with respect to a noisy quantum state ρ. For convenience, process 300 is described as being performed by a system of one or more classical and quantum computing devices at one or more locations. For example, quantum computing device 200 of FIG. 2 , suitably programmed in accordance with this specification, can perform process 300.

[0078] The system obtains multiple noisy quantum states ρ (step 302). For example, the system may perform a quantum computation to prepare copies of the noisy quantum states or recover copies of the noisy quantum states from a quantum memory. In some implementations, the noise experienced by each copy of the noisy quantum state has the same form and intensity. The number of copies may be selected based on various factors, including the capabilities of the quantum device performing process 300, the type of target observable, and / or the target accuracy of the error-mitigated expectation of the output.

[0079] The system is, for example,

[0080]

number

[0081] 3. Calculate the expectation value of the target observable on the entangled quantum state by performing measurements on a tensor product of M≧1 copies of the noisy quantum state, where ...

[0082]

number

[0083] An entangled quantum state is a quantum state in which the eigenvalues ​​corresponding to the non-dominant eigenvectors of the noisy quantum state in the spectral decomposition of the entangled quantum state are exponentially suppressed in M. For example, an entangled quantum state is a quantum state ρ M It could be.

[0084] To perform a measurement, the system calculates for each of the first measurement iterations i) a target observable and ii) a noisy quantum state.

[0085]

number

[0086] Measure the product of M copies of , with the circular shift operator on the tensor product (e.g., O i S (M) (measure a first measurement result) to obtain a first measurement result. The system then uses the first measurement result to determine the target observable and

[0087]

number

[0088] The expected value of the product with the circular shift operator with respect to, e.g.

[0089]

number

[0090] can be calculated.

[0091] As explained above with reference to equation (3), a target observable,

[0092]

number

[0093] The expectation value of the product with the circular shift operator for is equal to the expectation value of the target observable for the entangled quantum state. Thus, in step 304, the system calculates the expectation value of the target observable for the entangled quantum state without having to explicitly prepare the entangled quantum state.

[0094] The system has a target observable and

[0095]

number

[0096] The system determines an error-mitigated expectation value of the target observable with respect to the noisy quantum state using the calculated expectation value of the product with the circular shift operator with respect to (step 306). For example, the system may determine an error-mitigated expectation value of the target observable with respect to

[0097]

number

[0098] The expected value of the product with the circular shift operator on

[0099]

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[0100] We can normalize this expectation by dividing it by the expectation of the circular shift operator with respect to . The normalized expectation provides an approximation to the error-reduced expectation of the target observable with respect to the noisy quantum state, as explained above with reference to equations (1)-(3).

[0101]

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[0102] To calculate the expectation value of the circular shift operator with respect to , 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 second measurement iteration, the system calculates the noisy quantum state

[0103]

number

[0104] The system then measures the cyclic shift operator on the tensor product of the M copies of to obtain a second measurement result. The system then uses the second measurement result to obtain the quantum state

[0105]

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[0106] The expected value of the circular shift operator with respect to, e.g.

[0107]

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[0108] Calculate.

[0109] The exemplary process 300 illustrates a general application of virtual extraction. However, in some implementations, the quantities of the numerator and denominator of Equation 2 may be evaluated in multiple separate ways. For example, in some implementations of the process, steps may be performed in parallel and / or in combination. An exemplary variation of the exemplary process 300 is described in detail below with reference to FIG. 5.

[0110] Programming the Hardware: Virtual Extraction Using Measurements by Diagonalization. 4 is a flow diagram of an example process 400 for performing virtual extraction using measurements with diagonalization. For convenience, process 400 is described as being performed by a system of one or more classical and quantum computing devices at one or more locations. For example, quantum computing device 200 of FIG. 2, suitably programmed according to this specification, can perform process 400.

[0111] The example process 400 is particularly suited for an M=2 implementation where the target observable O acts on a single qubit. For convenience, the example process 400 is described for the case where the target observable O is a Pauli Z operator acting on a single qubit. However, the example process 400 may also be applied to other single qubit observables, e.g., by applying an appropriate single qubit rotation prior to the virtual extraction procedure. For example, a Pauli X operator may be measured using a procedure for measuring the Pauli Z operator after applying a Hadamard gate to the appropriate qubit in each copy.

[0112] The system obtains multiple copies of the noisy quantum state p, step 402. Step 402 is similar to step 302 of example process 300, and the details will not be repeated for brevity.

[0113] The system is in a noisy quantum state, e.g.

[0114]

number

[0115] In step 304 of example process 300, the relationship in Equation (3) was used to calculate the required expectation value. In example process 400, rather than directly using the relationship in Equation (3), the system calculates the expectation value of the target observable with respect to the entangled quantum state by measuring the tensor product of two copies of

[0116]

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[0117] For the particular case where the target observable O is a Pauli Z operator acting on a single qubit,

[0118]

number

[0119] Then,

[0120]

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[0121] Equation (3) can be used to show that, using symmetric observables, [O (M) ,S (M) ]=0 (9) or in this case,

[0122]

number

[0123] It is.

[0124] S (2) and

[0125]

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[0126] Both S and S can be factorized into tensor products of operators acting separately on each pair of qubits, where the i-th pair consists of the i-th qubit of each system. Thus, S (2) and

[0127]

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[0128] and can be simultaneously diagonalized using operators that can be factorized with the same structure. The two-qubit unitary that performs this diagonalization for the i-th pair is

[0129]

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[0130] The matrix representation of this gate is given by

[0131]

number

[0132] Here, the choice of phase for the matrix elements can vary. Then, the operator B (2) can be defined as follows:

[0133]

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[0134] This unitary diagonalizes the individual coefficients that make up the observables as desired.

[0135]

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[0136] This diagonalization is particularly easy to implement when each qubit from the first copy of ρ is adjacent to a corresponding qubit from the second copy.

[0137] The procedure for measuring the observables necessary to estimate the numerator and denominator of equation (8) is then mitigated by applying a single layer of N two-qubit gates in parallel and measuring them on a computational basis (step 404). (2) But for all N values ​​of k,

[0138]

number

[0139] Since we diagonalize N operators Z k For all of,the,, the data necessary to estimate the error-mitigated expectation,values ​​are collected simultaneously.,By applying the appropriate single qubit rotation before,performing the virtual extraction, we can access any single qubit,observable on each qubit.

[0140] To develop some insight, we use the spectral decomposition of ρ

[0141]

number

[0142] and the resulting sum

[0143]

number

[0144] It is useful to consider two separate components of the state i=j: (2) In the +1 eigenspace of S (2) Measurement of probability

[0145]

number

[0146] gives the +1 eigenvalue of i. If i ≠ j, then

[0147]

number

[0148] is a coordinate superposition of symmetric and antisymmetric states

[0149]

number

[0150] For the components of this state, S (2) The measurement of is equal probability of +1 and -1 and 〈S (2) 〉=0. Combining these two cases gives the expected equality

[0151]

number

[0152] Gives S (2) O (2) Measurements follow a similar pattern.

[0153] It is interesting to contrast this behavior with the stabilizer theory of quantum error correction. In the stabilizer formalism, errors are detected by projecting the measurements into one or more symmetric -1 eigenspaces. In the presently described approach, errors are similarly supported in the symmetric eigenspaces where they are measured.

[0154] Returning to Figure 4, to perform measurements on the tensor product of two copies of the noisy quantum state to calculate the expectation value of the target observable on the entangled quantum state, the system applies, for each of multiple measurement iterations, i) a circular shift operator, and ii) a diagonalization operator that diagonalizes the product of a symmetric version of the target operator and the circular shift operator to the tensor product of the M copies of the noisy quantum state to obtain an evolved quantum state. The system then measures the product of the target observable and the circular shift operator on the evolved quantum state to obtain a respective measurement result for each qubit in the two copies of the noisy quantum state.

[0155] The system then uses the measurements to calculate the expectation given in equation (8) and also calculates the error-mitigated expectation of the target observable with respect to the noisy quantum state (step 406). For example, the system may use the measurements to calculate the expectation of the target observable and

[0156]

number

[0157] Calculate the expectation value of the product with the circular shift operator on and use the measurement result to

[0158]

number

[0159] Calculate the expectation value of the circular shift operator with respect to the target observable,

[0160]

number

[0161] The expected value of the product with the circular shift operator on

[0162]

number

[0163] We can divide by the expectation value of the circular shift operator with respect to

[0164] M=2 and O=Z i An exemplary algorithm for performing virtual extraction using measurements from diagonalization in the situation is given as follows: Input: K measurement repetitions, 2K copies of N qubits in a noisy quantum state ρ (presented two at a time). Output: 〈Z i Error-reduced estimate of

[0165]

number

[0166] . For each qubit i∈1,...,N, E i Set =0. Set D=0. for k∈K do 4. Perform any SWAP operations necessary to enable coupling of each qubit in the first copy of ρ with its corresponding qubit in the second copy. Between each qubit i in the first copy and the corresponding qubit in the second copy, there is a two-qubit gate defined by Eq. (10):

[0167]

number

[0168] applies. Both conditions are measured on a computational basis. The measurement result for the i-th qubit in the first copy of ρ

[0169]

number

[0170] and denote the measurement result for the i-th qubit in the second copy of ρ by

[0171]

number

[0172] It is expressed as: for i∈1,...,N do

[0173]

number

[0174] end for

[0175]

number

[0176] end for return {〈Z i 〉 corrected :=E i / D}

[0177] Program the hardware: Select a separate virtual extraction protocol 5 is a flow diagram of an example process 500 for selecting a virtual extraction protocol. For convenience, process 500 is described as being performed by a system of one or more classical and quantum computing devices at one or more locations. For example, quantum computing device 200 of FIG. 2, suitably programmed according to the present specification, can perform process 500.

[0178] The system determines whether an ancillary-assisted measurement is available on a valid experimental device (step 502). In response to determining that an ancillary-assisted measurement is not available, 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 extraction protocol using sequential measurements on two or more copies of the noisy quantum state (step 506).

[0179] Measuring observables in the context of supporting multiple qubits becomes challenging. The difficulty arises from the use of Equation 8, and in particular from the choice to use symmetric versions of the observables (defined in Equation 6). Using symmetric versions of multi-qubit observables means that it is not possible to perform the necessary diagonalization using separate unitary tensor products across each pair of qubits. As an example, the observable O=Z i Z j (This analysis is equally valid for other operators consisting of tensor products of single-qubit Pauli operators.) The product of a symmetric observable with a swap operator for M=2 is

[0180]

number

[0181] This operator cannot be factorized into a tensor product of operators with support on each pair of qubits, nor can it be diagonalized by operators that can be factorized in this way.

[0182] However, rather than using equation (8) to determine the corrected expectation value of O, the asymmetric form introduced in equation (2) can be used: O=Z i Z j Returning to our example where:

[0183]

number

[0184] Unlike the symmetric observable in equation (16), the operator

[0185]

number

[0186] can be factored into a tensor product over N pairs of qubits (one qubit from the first system with a corresponding qubit from the second system).

[0187]

number

[0188] is not a Hermitian matrix but a unitary one, so the quantity

[0189]

number

[0190] can be estimated by applying a circuit to diagonalize it and measuring it computationally.

[0191]

number

[0192] can be factored into a product of two-qubit operators, so a circuit to diagonalize this also works. The details of the diagonalization depend on the operators being measured.

[0193] In addition,

[0194]

number

[0195] is S (2) Since i does not commute with j, the numerator and denominator of (17) cannot be calculated simultaneously. It is also not possible to simultaneously measure the corrected expectation values ​​corresponding to different choices of i and j. More generally, it is not possible to simultaneously measure any single tensor product of one-qubit operators, regardless of the number of qubits they act on.

[0196] 6 is a flow diagram of an example process 600 for virtual extraction using sequential measurements on two or more copies in a noisy quantum state, where a target observable acts on two or more qubits. For convenience, the process 600 is described as being performed by a system of one or more classical and quantum computing devices at one or more locations. For example, a suitably programmed quantum computing device 200 of FIG. 2 according to the present specification can perform the process 600. Some of the steps described below are similar to those described with reference to the example process 300, and for the sake of brevity, the details will not be repeated.

[0197] The system obtains multiple copies of the noisy quantum state (step 602). The system decomposes the target observable into multiple tensor products, each involving a single qubit operator (step 604). The system performs sequential measurements on the tensor products of the M copies of the noisy quantum state (step 606). For each of the tensor products of the single qubit operators, the system performs a first of the measurement iterations. Each measurement iteration includes applying a first diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain a first evolved quantum state, where the first diagonalization operator diagonalizes a product of i) the tensor product of the one qubit operator and ii) the circular shift operator, and measures the product of the tensor product of the one qubit operator and the circular shift operator for the first evolved quantum state to obtain a respective first measurement result for each qubit in the first evolved quantum state (step 606a).

[0198] In addition, the system applies a second diagonalization operator to the tensor product of the M copies of the noisy quantum state for each second measurement iteration to obtain a second evolved quantum state. The system measures the circular shift operator on the second evolved quantum state to obtain a respective second measurement result for each qubit in the second evolved quantum state (step 606b).

[0199] The system uses the acquired measurements to calculate an error-mitigated expectation value of the target observable with respect to the noisy quantum state (step 608). For example, the system may use the first measurement to calculate the error-mitigated expectation value of the target observable with respect to the noisy quantum state (step 608).

[0200]

number

[0201] Calculate the expectation value of the product with the circular shift operator on

[0202]

number

[0203] Calculate the expectation value of the circular shift operator with respect to the target observable,

[0204]

number

[0205] The expected value of the product with the circular shift operator on

[0206]

number

[0207] We can divide by the expectation value of the circular shift operator with respect to

[0208] Returning to FIG. 5, in response to determining that the target observable is not a single qubit observable, the system executes a virtual extraction protocol using parallel measurements on two or more copies of the noisy quantum state (step 508).

[0209] For example, the system may perform virtual extraction using a diagonalization measure with M=2, as described above in the example process 400. As another example, the system may perform virtual extraction using a diagonalization measure with M>2 by generalizing the example process 400 to larger powers of ρ.

[0210] A circular shift operator S between M N-qubit systems (M) can be factorized into a tensor product of N M-qubit gates, similar to the swap operator. Specifically, this operator can be factorized into a tensor product of N single-qubit circular shift operators.

[0211]

number

[0212] is the operator S (M) Therefore,

[0213]

number

[0214] and S. (M) What is S (M) However, even if a matrix is ​​unitary but not Hermitian with M>2, it can be diagonalized at the same time.

[0215]

number

[0216] and S. (M) Since both x, y, and z can be factorized into tensor products over N M-tuples of qubits, the unitaries that diagonalize these operators can similarly be factorized into tensor products of M-qubit operators. Thus, the process 400 described above with respect to FIG. 4 can be applied.

[0217] Similar relations for determining the corrected expectation value of a multi-qubit observable described above with reference to step 506 apply to this generalized protocol for the two copy (M=2) case. The tools developed so far allow for the determination of Tr(Z k ρ M ) and Tr(ρ M ) can be estimated simultaneously. M ) is reconstructed for some multi-qubit Pauli operator P, then a generalization of equation (17) can be used, but measurements of the operator required for a particular P would be restricted to one at a time.

[0218] For the particular case of M=3, the corresponding circuit is

[0219]

number

[0220] and

[0221]

number

[0222] (allowing application of the example process 400 with M=3). FIG. 7 shows the diagonalization gate

[0223]

number

[0224] This shows an example hypothesis for numerically optimizing a gate (e.g., a gate with similar properties to the gate defined in equation (10)) that is

[0225]

number

[0226] and

[0227]

number

[0228] In Figure 7,

[0229]

number

[0230] represents the 2-qubit gate parameters for any 2-qubit gate of the i-th order. When the following formula is used, the parameters for the four 2-qubit gates allow for an error (measured in the Frobenius norm of the difference between the exact and approximate matrices) of approximately 5E-5.

[0231]

number

[0232] Returning to FIG. 5, in response to determining that an ancillary-assisted measurement is available in step 502, the system performs a virtual extraction for any observable using parallel measurements on any number M of copies of the noisy quantum state (step 510).

[0233] When performing virtual extraction using ancillary-assisted measurements, the form of the observables being measured is not restricted, and simultaneous measurements of operators acting on overlapping subsets of qubits are possible. Thus, virtual extraction using ancillary-assisted measurements is compatible with known techniques for efficiently measuring large groups of commutative operators.

[0234] The system uses a swap operator S to perform virtual extraction using ancillary-assisted measurements on M=2 copies of the noisy quantum state. (2) The numerator and denominator of Equation 8 can be approximated using non-destructive measurements of S. The system prepares (or otherwise obtains) two quantum registers for each of a number of measurement iterations, each register containing an ancilla qubit prepared in a noisy quantum state ρ and an ancilla qubit prepared in a 0 state. The system may use, for example, a swap test or a Hadamard test to determine whether S (2) Specifically, the system applies a Hadamard gate to the ancilla qubit, applies a circular shift operator to two registers where the ancilla qubit is required to be in one state, and measures the ancilla qubit in the X-base to obtain a first measurement result. (2)Since can be factorized into a tensor product of two-qubit swap gates, its controlled version can similarly be factorized into a series of N Fredkin gates (controlled swap gates). In some implementations, compiling this circuit may require some additional steps (such as evolving a single ancilla qubit into a GHZ state using a series of CNOT gates) to address connectivity limitations in upcoming devices.

[0235] Under the assumption that the target observable may be symmetric as described above with reference to example process 400, the system then (2) By measuring the qubits in two registers, we obtain the product O (2) S (2) to obtain respective second measurement results. The system can then use the first and second measurement results from the separate measurement iterations to calculate an error-mitigated expectation value of the target observable with respect to the noisy quantum state. In particular, the system can calculate an expectation value of the target observable using the second measurement result, calculate an expectation value of the circular shift operator using the first measurement result, and divide the expectation value of the target observable by the expectation value of the circular shift operator.

[0236] This protocol does not require separate estimation of Tr(ρO). Moreover, measurements of O for both 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.

[0237] For virtual extraction using ancillary-assisted measurements on M>2 copies of a noisy quantum state, S (N) is not Hermitian for N>2. However, natural generalizations to the above strategy still work as expected. Specifically, if the expectation

[0238]

number

[0239] To sample an observable equal to (N) A controlled version of the symmetric observable O may be used. (N) is S (N) and so it also commutes with the observables measured by this generalization of the swap test. Therefore, if the expectation is

[0240]

number

[0241] Observables that are (N) It can be sampled by measuring

[0242] Reducing algorithmic errors Most error mitigation methods focus on reducing errors due to imperfections in the implementation of the device, such as decoherence or control deviations. Some of these techniques may be applied to algorithmic errors incurred by randomized algorithmic implementations that are otherwise noise-free. Advances in Hamiltonian simulations have led to the evolution of randomized evolution methods, such as qDRIFT, random Trotta, and combinations of these, which in some cases have advantages over their deterministic counterparts. These methods, being randomized, output mixed states rather than pure states, even in the absence of noise. Furthermore, these methods rely on approximation parameters with natural constraints that converge to the pure states generated by the exact evolution. Applying virtual extraction to qDRIFT, as described below, can suppress this deviation from the exact evolution. For the particular model system considered below, it has been found that virtual extraction can reduce the coherent space-time volume required to reach a certain accuracy threshold by a factor of 8 or more compared to standard qDRIFT.

[0243] The qDRIFT method simulates the time evolution under a Hamiltonian H by constructing a product formula using randomized selection rules. Each term is randomly selected from H with a selection probability proportional to the strength of their interaction in the Hamiltonian. The system is then evolved under this Hamiltonian term for a fixed time step. This process is repeated multiple times to generate a product formula that provides an approximation to the time evolution operator. qDRIFT produces quantum channels that, when averaged over classical randomness (in the selection of interaction terms), more closely approximate the exact evolution than the individual product formulas. Importantly, the scaling of the method does not rely explicitly on the number of terms in the Hamiltonian, as most deterministic Trotta methods do, but rather on the 1-norm of the coefficients.

[0244] H=Σ i h i H i Consider a Hamiltonian that can be decomposed as follows (for all h i Change the sign to H i can be made a positive real number by absorbing H i The norm of the spectrum of is limited to 1. i h i Then, the diamond norm distance between the qDRIFT channel and the true time evolution is ∈=2λ 2 t 2 / η, where η is the number of qDRIFT selection steps performed to generate 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 one, which corresponds to the exact evolution.

[0245] The presently described virtual extraction technique can reduce the required coherent space-time volume by reducing this factor η required to achieve the same error in practice. In particular, virtual extraction can affect the number of coherent steps η required to reach a target accuracy. In this regard, consider a Heisenberg Hamiltonian with up to six qubits per copy. This Hamiltonian is

[0246]

number

[0247] is given by h i ∈{-h,h} is the randomly chosen Z field strength and periodic boundary conditions are applied such that N+1 is site 1. For the presently described investigation a time evolution length of t = N is considered and h = 1. Under such a Heisenberg model, the number of coherent qDRIFT steps required to achieve a trace distance from 0.01 to the ideal state of evolution was numerically investigated.

[0248] FIG. 8 shows the results of this analysis. In detail, FIG. 8 shows the coherent cost reduction of qDRIFT with virtual extraction in the Heisenberg model. The number of coherent qDRIFT steps required to reach the target trace distance with and without virtual extraction using two copies is shown. A consistent reduction of at least 16 times in the number of steps required is seen. When accounting for the overhead of using two copies, this becomes an 8 times reduction in the coherent space-time volume used to reach the same error rate. These results show that the use of the currently described error mitigation techniques allows for practical algorithmic improvements for real systems, especially in the NISQ regime. For example, by incorporating the currently described error mitigation techniques, algorithms can produce more accurate results with reduced overhead, such as that associated with quantum error correction codes.

[0249] Implementations of the digital and / or quantum subject matter and digital functional and quantum operations described herein may be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, in a quantum computing system, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware including the structures disclosed herein and their structural equivalents, or in one or more combinations thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0250] Implementations of the digital and / or quantum subject matter described herein may be embodied as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by or to control the operation of a data processing apparatus. The digital computer storage medium and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more quantum bits, or one or more combinations thereof. Alternatively, or in addition, the program instructions may be encoded on a signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, capable of encoding digital and / or quantum information, which is artificially generated and propagated, to encode the digital and / or quantum information for transmission to an appropriate receiving device for execution by the data processing apparatus.

[0251] The terms quantum information and quantum data refer to information or data carried by, held or stored in a quantum system, with the smallest non-trivial system being a qubit, a system that defines a unit of quantum information. The term "qubit" is understood to encompass all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, for example having more than two levels. By way of example, such systems may include atoms, electrons, photons, ions or superconducting qubits. In many implementations, the computational basis state is identified with a ground and a first excited state, but it is understood that other setups are possible in which the computational state is identified with a higher level excited state. The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and / or quantum processors or computers, and combinations thereof. The device may also be or further include special purpose logic circuitry, such as, for example, an FPGA (field programmable gate array), an ASIC (application specific integrated circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special purpose quantum computer that is not capable of performing general purpose quantum computations. In addition to hardware, the data processing device may optionally include code that generates an execution environment for digital computer programs and / or quantum computer programs, such as code that constitutes a processor firmware, a stack of protocols, a database management system, an operating system, or one or more combinations thereof.

[0252] A digital computer program may be referred to or described as a program, software, software application, module, software module, script, or code, and may be written in any type of programming language, including compiled, interpreted, declarative, or procedural languages, and may be distributed as a stand-alone program or in any form including modules, components, subroutines, or other units suitable for use in a digital computer environment. A quantum computer program may be referred to or described as a program, software, software application, module, software module, script, or code, and may be written in any type of programming language, including compiled, interpreted, declarative, or procedural languages, and then converted to a suitable quantum programming language, or written in a quantum programming language, such as QCL or Quipper.

[0253] The digital computer program and / or quantum computer program may correspond to a file in a file system, but this is not required. The program may be stored in a part of a file holding 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 integrated files, e.g. files storing one or more modules, subprograms, or code parts. The digital computer program and / or quantum computer program may be deployed to be executed on one digital computer or quantum computer, or on multiple digital computers and / or quantum computers located at one site or distributed across multiple sites and interconnected by a digital data communication network and / or a quantum data communication network. A quantum data communication network is understood to be a network that transmits quantum data using quantum systems, e.g. qubits. In general, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0254] The process and logic flows described herein may be performed by, and apparatus may be embodied as, one or more programmable digital computers and / or quantum computers, operating as appropriate with one or more digital and / or quantum processors, which execute one or more digital and / or quantum computer programs to operate on input digital and quantum data to perform functions and generate output. The process and logic flows may also be performed by, and apparatus may be embodied as, special purpose logic circuitry, e.g., FPGAs, ASICs, or quantum simulators, or a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0255] One or more digital and / or quantum computer systems "configured to" perform a particular operation or action refers to a system having installed thereon software, firmware, hardware, or a combination thereof that causes the system to perform the operation or action in operation. One or more digital computer programs and / or quantum computer programs configured to perform a particular operation or action refers to one or more programs that contain instructions that, when executed by a digital data processing device and / or a quantum data processing device, cause the device to perform the particular operation or action. A quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, causes the device to perform an operation or action.

[0256] A digital computer and / or quantum computer suitable for executing a digital computer program and / or a quantum computer program may be based on a general purpose or special purpose or both digital processor and / or quantum processor, or on another kind 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 a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, e.g. photonic, or a combination of these.

[0257] The essential elements of a digital computer and / or quantum computer are a central processing device for carrying out or executing instructions and one or more storage devices for storing instructions and digital and / or quantum data. The central processing device and storage devices may be supplemented with or incorporated into dedicated logic circuits or quantum simulators. In general, a digital computer and / or quantum computer also includes or is operatively coupled to one or more mass storage devices, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information, for storing digital and / or quantum data, for communicating or transferring digital and / or quantum data to and from these mass storage devices. However, a digital computer and / or quantum computer need not have such devices.

[0258] 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 including, by way of example, semiconductor memory devices such as, for example, EPROM, EEPROM, and flash memory devices, magnetic disks such as, for example, internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems of, for example, trapped atoms or trapped electrons. A quantum memory is understood to be a device capable of storing quantum data with high fidelity and efficiency for long periods of time, for example, an optical interface, where light is used for the transmission, as well as for the storage and protection of the quantum characteristics of the quantum data, such as superposition or quantum coherence.

[0259] Control of the various systems or portions thereof described herein may 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 herein may be implemented as apparatus, methods, or systems that may each include one or more digital and / or quantum processing devices and memory devices that store executable instructions for performing the operations described herein.

[0260] Although this specification includes details of many specific implementations, these should not be interpreted as limitations on the scope of what may be claimed, but as descriptions of features that may be specific to a particular implementation. Certain features described in the context of individual implementations in this specification may also be implemented in combination in one implementation. Conversely, various features described in the context of one implementation may also be implemented separately in multiple implementations or in any suitable subcombination. Moreover, although each feature is described above as working in a combination and in some cases originally claimed as such, one or more features of a claimed combination may in some cases be realized from the combination, and the claimed combination may be directed to a subcombination or a variation of the subcombination.

[0261] Similarly, although operations are shown in the figures in a particular order, this should not be understood as requiring that such operations be performed in the particular order or sequential order shown, or that all of the illustrated operations be performed, to achieve desirable results. In certain environments, multitasking and parallel processing may be advantageous. Moreover, it should be understood that the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and that the program components and systems described may generally be integrated together in a single software product or packaged into multiple software products.

[0262] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order or sequence shown to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. [Explanation of symbols]

[0263] 102 Quantum Computing System 104 Multiple copies of noisy quantum states 106 Tensor products of noisy quantum states 108 Quantum circuit 110 Measurement repeats 112 Post-processing 114 Error-reduced estimation of the expectation of an observable 116 Example Quantum Circuits 202 Quantum Computing Devices 204 Control and Measurement Systems 206 qubits 208 Combiner 210 Control Signal 212 Read signal 214 processor 216 Storage Devices 218 I / O Unit 250 Classical Processors 252 qubit assembly

Claims

1. 1. A method for determining an error-mitigated expectation value of a target observable with respect to a noisy quantum state, the method being performed by a system comprising one or more classical processors and a quantum computer, the quantum computer having a plurality of qubits; obtaining, by the quantum computer, multiple copies of the noisy quantum state; calculating, by the quantum computer, an expectation value of the target observable with respect to an entangled quantum state by measuring a tensor product of M copies of the noisy quantum state, the entangled quantum state being given by ρ M / Tr(ρ M ), where ρ represents a noisy quantum state and M≧1, and where eigenvalues ​​corresponding to non-dominant eigenvectors of the noisy quantum state in a spectral decomposition of the entangled quantum state are exponentially suppressed in M; measuring the tensor product of M copies of the noisy quantum state to calculate the expectation value of the identity operator on the entangled quantum state; determining, by the one or more classical processors, the error-mitigated expectation value of the target observable with respect to the noisy quantum state using the calculated expectation value of the target observable with respect to the entangled quantum state, wherein determining the error-mitigated expectation value of the target observable with respect to the noisy quantum state using the calculated expectation value of the target observable with respect to the entangled quantum state comprises the steps of: i) dividing the expectation value of the target observable with respect to the entangled quantum state by ii) the expectation value of the identity operator with respect to the entangled quantum state; The method includes:

2. The method of claim 1 , wherein the noise experienced by each copy of the noisy quantum state has the same form and intensity.

3. measuring a tensor product of M copies of the noisy quantum state to calculate the expectation value of the target observable with respect to the entangled quantum state, For each first measurement iteration, measuring a product of i) the target observable and ii) a circular shift operator on a tensor product of M copies of the noisy quantum state to obtain a first measurement result for each iteration; using the first measurement to calculate an expectation value of the product of i) the target observable and ii) a circular shift operator on the tensor product of M copies of the noisy quantum state; normalizing the expectation value of i) the product of the target observable and ii) a circular shift operator on the tensor product of M copies of the noisy quantum state to obtain the error-mitigated expectation value of the target observable with respect to the noisy quantum state; The method according to claim 1 or 2, comprising:

4. Normalizing the expectation of the target observable comprises: measuring the cyclic shift operator on a tensor product of M copies of the noisy quantum state for each second measurement iteration to obtain a second measurement result; using the second measurement to calculate the expectation value of the cyclic shift operator with respect to a tensor product of M copies of the noisy quantum state; Dividing the expectation value of the product of i) the target observable and ii) the circular shift operator on the tensor product of M copies of the noisy quantum state by the expectation value of the circular shift operator on the tensor product of M copies of the noisy quantum state; 4. The method of claim 3, comprising:

5. 3. The method of claim 1 or 2, wherein M≧2 and the target observable acts on a single quantum bit.

6. The step of performing measurements on a tensor product of M copies of the noisy quantum state comprises, for each of K measurement iterations: applying a diagonalization operator to a tensor product of M copies of the noisy quantum state to obtain an evolved quantum state, the diagonalization operator diagonalizing i) a circular shift operator, and ii) a product of a symmetric version of the target observable and the circular shift operator; measuring a product of the target observable and the circular shift operator on the evolved quantum state to obtain a respective measurement result for each qubit in the evolved quantum state; 6. The method of claim 5, comprising:

7. using the measurement to calculate an expectation value of the product of the target observable and the cyclic shift operator on the evolved quantum state; using the measurements to calculate the expectation value of the cyclic shift operator with respect to the evolved quantum state; dividing the expectation value of the product of the target observable and the circular shift operator on the evolved quantum state by the expectation value of the circular shift operator on the evolved quantum state; 7. The method of claim 6, further comprising:

8. 8. The method of claim 6 or 7, wherein applying the circular shift operator couples qubits in a first copy of the noisy quantum state to corresponding qubits in each other copy of the noisy quantum state.

9. 3. The method of claim 1 or 2, wherein M≧2 and the target observable acts on two or more quantum bits and includes multiple tensor products of one-qubit operators.

10. performing a measurement on a tensor product of M copies of the noisy quantum state, For each tensor product of one-qubit operators, For each of the first measurement repetitions, applying a first diagonalization operator to a tensor product of the M copies of the noisy quantum state to obtain an evolved quantum state, the first diagonalization operator diagonalizing a product of i) the tensor product of the one-qubit operator and ii) a circular shift operator; measuring a product of the tensor product of the one-qubit operator and the cyclic shift operator on the evolved quantum state to obtain a respective first measurement result for each qubit in the evolved quantum state for each iteration; For each second measurement repeat, applying a second diagonalization operator to the tensor product of the M copies of the noisy quantum state to obtain a second evolved quantum state, wherein the second diagonalization operator diagonalizes the cyclic shift operator; measuring the cyclic shift operator on the second evolved quantum state to obtain for each iteration a respective second measurement result on each qubit in the second evolved quantum state; 10. The method of claim 9, comprising:

11. using the first measurement to calculate an expectation of a product of the target observable and the circular shift operator; calculating an expectation value of the cyclic shift operator using the second measurement; dividing the expectation of the product of the target observable and the circular shift operator by the expectation of the circular shift operator; 11. The method of claim 10, further comprising:

12. 3. The method of claim 1 or 2, wherein measuring a tensor product of M copies of the noisy quantum state to calculate the expectation value of the target observable with respect to the entangled quantum state comprises performing an ancillary-assisted measurement.

13. M≧2, and the step of performing an ancillary-assisted measurement includes, for each of a plurality of measurement iterations, providing two quantum registers, each including the noisy quantum state and an ancilla qubit prepared in a 0 state; performing a non-destructive measurement of a cyclic shift operator, applying a Hadamard gate to the ancilla qubit; applying the cyclic shift operator to the two quantum registers in which the ancilla qubits are in the 1 state; measuring the ancilla qubits on an X basis to obtain first respective measurements; and measuring qubits in the two quantum registers with respect to symmetric versions of the target observable to obtain respective second measurements; 13. The method of claim 12, comprising:

14. calculating an expectation of the target observable using the second measurement; calculating an expectation value of the cyclic shift operator using a first measurement; Dividing the expectation of the target observable by the expectation of the circular shift operator; 14. The method of claim 13, further comprising:

15. one or more classical processors; and one or more quantum computing devices in data communication with the one or more classical processors, the quantum computing hardware comprising: one or more qubit registers, each of which comprises one or more qubits; a plurality of control devices configured to operate the one or more qubit registers; Equipped with 15. An apparatus, the apparatus being configured to perform the method according to any one of claims 1 to 14.

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