Quantum computing device, quantum computing method, and program

Abstract

To provide a technique which can reduce an error of an expected value due to noise.SOLUTION: A quantum computing device according to one aspect includes: an approximation calculation unit which calculates a plurality of approximate states respectively representing quantum states for approximating a ground state of a given Hamiltonian on the basis of a quantum circuit which may be affected by noise; a formulation unit which formulates an expression for obtaining a quantum state which minimizes an expected value of the Hamiltonian on the basis of the plurality of approximate states; and an expected value calculation unit which calculates a minimum expected value of the Hamiltonian by a generalized subspace method with the plurality of approximate states as bases, on the basis of the expression.SELECTED DRAWING: Figure 5

Description

Technical Field

[0001] The present disclosure relates to a quantum computing device, a quantum computing method, and a program.

Background Art

[0002] In order to solve practical problems with a quantum computer, it is essential to reduce the influence of noise in the quantum computer as much as possible and perform accurate quantum calculations. One method for realizing accurate quantum calculations is known as a quantum error correction code. With a quantum error correction code, it is possible to make the influence of noise arbitrarily small by increasing the size of the code. However, constructing a quantum error correction code requires many qubits, and with current technology, it is difficult to prepare such a large number of qubits. Therefore, various methods have been proposed that can suppress the influence of noise without requiring many qubits, and these are collectively called quantum error suppression methods (Non-Patent Documents 1 and 2).

[0003] Among quantum error suppression methods, one of the most practical methods is a method called PEC (Probabilistic Error Cancellation) (Non-Patent Document 3). This method sacrifices the number of measurements and effectively realizes the inverse transformation of noise by post-classical processing, thereby making it effectively possible to calculate the expected value of a physical quantity in a quantum circuit without noise. However, this method requires the noise model in the quantum circuit to be completely identified by quantum process tomography (Non-Patent Document 4).

Prior Art Documents

Non-Patent Documents

[0004]

Non-Patent Document 1

[0005] In PEC, it is necessary to completely identify the noise model, and for this purpose, quantum process tomography must be performed. While executing quantum process tomography requires estimating an exponential number of expected values, in realistic estimation of expected values, only a finite number of measurements can be made, so errors occur in identifying the noise model. For this reason, quantum process tomography for identifying the noise model can only be performed incompletely, and thus it is impossible to completely remove the influence of noise by PEC.

[0006] This disclosure has been made in view of the above points, and provides a technique capable of suppressing the error of the expected value due to noise.

Means for Solving the Problem

[0007] A quantum computing device according to an aspect of this disclosure includes an approximation calculation unit that calculates a plurality of approximate states each representing a quantum state approximating the ground state of a given Hamiltonian based on a quantum circuit that can be affected by noise, a formulation unit that formulates an equation for obtaining a quantum state that minimizes the expected value of the Hamiltonian based on the plurality of approximate states, and an expected value calculation unit that calculates the minimum expected value of the Hamiltonian by a generalized subspace method using the plurality of approximate states as a basis, based on the equation.

Advantages of the Invention

[0008] A technique capable of suppressing the error of the expected value due to noise is provided.

Brief Description of the Drawings

[0009]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Mode for Carrying Out the Invention

[0010] Hereinafter, an embodiment of the present invention will be described.

[0011] <Problem Setting> Consider the problem of obtaining the ground energy (expected value) of the following Hamiltonian H.

[0012]

Number

[0013] Assume that there is a quantum circuit that takes the above Hamiltonian H as input and outputs a quantum state (hereinafter also referred to as an approximate state) that approximates the ground state of the Hamiltonian H by a method such as the VQE (Variational Quantum Eigensolver) method. Also, since this quantum circuit is affected by noise as it is, it is assumed that quantum error suppression is performed by executing PEC. However, in the quantum process tomography when identifying the noise model, it is allowed that an error caused by a finite number of measurements occurs in the estimation of the noise. That is, it is allowed that the characterization of the noise is incomplete when executing PEC, in other words, PEC is executed under an incomplete quantum process tomography.

[0014] At this time, it is assumed that D quantum circuits to which PEC is applied can obtain D approximate states {ρ i | i = 1, ···, D} output by these D quantum circuits respectively.

[0015] <Proposed Method> Under the above problem settings, in what follows, a method for suppressing the error of PEC based on the characterization of incomplete noise will be described using a quantum error suppression method called the Generalized Quantum Subspace Expansion (GSE) (Reference 1) that does not require noise information. GSE is a method that integrates and generalizes quantum error suppression methods called the virtual distillation method (References 2 and 3) and the subspace expansion method (References 4 to 7). In GSE, an effective physical state can be (virtually) prepared based on non-physical states, enabling the estimation of the expected value under the physical state. In the proposed method, the state in which PEC is executed under incomplete quantum process tomography, that is, {ρ i |i = 1, ···, D} is used as the basis of GSE.

[0016] <Formulation> In the proposed method, the purpose is to effectively obtain a quantum state ρ EM that minimizes the expected value of the Hamiltonian H according to the following equation (1).

[0017]

Equation

[0018] Hereinafter, a quantum computing device 10 that calculates D approximate states {ρ i |i = 1, ···, D} and, after formulating the above equation (1), calculates the quantum state ρ EM using this equation (1) will be described.

[0019] <Configuration Example of Quantum Computing Device 10> A configuration example of the quantum computing device 10 according to this embodiment will be described with reference to FIG. 1. FIG. 1 is a diagram showing a configuration example of the quantum computing device 10 according to this embodiment.

[0020] As shown in FIG. 1, the quantum computing device 10 according to this embodiment includes a control device 100 and a quantum processor 200.

[0021] The control device 100 transmits a control signal to the quantum processor 200 and acquires a calculation result from the quantum processor 200. Thereby, quantum calculation is performed. The control device 100 is realized by, for example, a classical computer or the like.

[0022] The quantum processor 200 constitutes a two-level quantum system called a qubit (physical qubit), and performs physical operations such as initialization, gate operation (unitary transformation), and measurement on the physical qubit according to a control signal from the control device 100. The quantum system for realizing the qubit is not particularly limited, and any quantum system may be used. For example, a quantum system realized by a superconducting circuit, an ion trap, a photon, a quantum dot, or the like can be used.

[0023] <Hardware configuration example of the control device 100> A hardware configuration example of the control device 100 according to this embodiment will be described with reference to FIG. 2. FIG. 2 is a diagram showing a hardware configuration example of the control device 100 according to this embodiment.

[0024] As shown in FIG. 2, the control device 100 according to this embodiment includes an input device 101, a display device 102, an external I / F 103, a communication I / F 104, a RAM (Random Access Memory) 105, a ROM (Read Only Memory) 106, an auxiliary storage device 107, and a processor 108. These hardware components are communicably connected to each other via a bus 109.

[0025] The input device 101 is, for example, a keyboard, a mouse, a touch panel, physical buttons, etc. The display device 102 is, for example, a display, a display panel, etc. Note that the control device 100 may not have at least one of the input device 101 and the display device 102, for example.

[0026] The external I / F 103 is an interface with an external device such as the recording medium 103a. Examples of the recording medium 103a include a CD (Compact Disc), a DVD (Digital Versatile Disk), an SD memory card (Secure Digital memory card), a USB (Universal Serial Bus) memory card, etc.

[0027] The communication I / F 104 is an interface for transmitting and receiving various signals to and from the quantum processor 200. The RAM 105 is a volatile semiconductor memory (storage device) that temporarily holds programs and data. The ROM 106 is a non-volatile semiconductor memory (storage device) that can hold programs and data even when the power is turned off. The auxiliary storage device 107 is a non-volatile storage device (storage device) such as an HDD (Hard Disk Drive), an SSD (Solid State Drive), a flash memory, etc. The processor 108 is an arithmetic device such as a CPU (Central Processing Unit), for example.

[0028] Note that the hardware configuration shown in FIG. 2 is an example, and the hardware configuration of the control device 100 is not limited to this. For example, the control device 100 may have a plurality of auxiliary storage devices 107 and a plurality of processors 108, may not have a part of the illustrated hardware, or may have various hardware other than the illustrated hardware.

[0029] <Functional configuration example of the control device 100> A functional configuration example of the control device 100 according to the present embodiment will be described with reference to FIG. 3. FIG. 3 is a diagram showing a functional configuration example of the control device 100 according to the present embodiment.

[0030] As shown in FIG. 3, the control device 100 according to the present embodiment includes a ground state approximation unit 110, a formulation unit 120, an optimal solution calculation unit 130, and a state calculation unit 140. Each of these units is realized, for example, by a process executed by the processor 108 for one or more programs installed in the control device 100.

[0031] The ground state approximation unit 110 performs PEC on a quantum circuit that outputs an approximate state of the ground state with the input of the Hamiltonian H by a method such as the VQE method while allowing an incomplete quantum process tomography, and then performs PEC on the D quantum circuits subjected to PEC. Calculate D approximate states {ρ i |i = 1, ···, D}. When calculating each approximate state ρ i , the quantum processor 200 is also used.

[0032] The formulation unit 120 formulates the above formula (1) using the D approximate states {ρ i |i = 1, ···, D} calculated by the ground state approximation unit 110.

[0033] The optimal solution calculation unit 130 calculates c = (c1, ···, c D ) that minimizes the expected value of the Hamiltonian H according to the above formula (1) as the optimal solution c (0) =(c1 (0) , ···, c D (0) ). Here, the optimal solution calculation unit 130 includes a first matrix element calculation unit 131, a second matrix element calculation unit 132, and a solution calculation unit 133.

[0034] The first matrix element calculation unit 131 calculates the element G ij of the (i, j) component of the matrix G for all pairs of (i, j) according to the following formula (2). Note that 1 ≦ i ≦ D and 1 ≦ j ≦ D.

[0035] [Number] Here, Tr represents the trace, and k is a label for identifying each Pauli operator. At this time, the first matrix element calculation unit 131 applies the virtual distillation method to the quantum states ρ i , ρ j and performs a measurement on the Pauli operator P k to calculate the output value of P k ρ i ρ j in the above formula (2). Note that the part of the formula for which the trace is taken, that is, P k ρ i ρ j in the above formula (2) is calculated by the quantum processor 200. On the other hand, the trace Tr[·], the multiplication of the weight h k , and the sum over k are calculated by the processor 108. The trace Tr[P k ρ i ρ j can be calculated by repeatedly measuring the output value of P k ρ i ρ j by the quantum processor 200 a sufficient number of times and taking the average of those output values.

[0036] The second matrix element calculation unit 132 calculates the element S ij of the (i, j) component of the matrix S for all pairs of (i, j) according to the following formula (3). Note that 1 ≤ i ≤ D and 1 ≤ j ≤ D.

[0037] [Number] At this time, the second matrix element calculation unit 132 applies the virtual distillation method to the quantum states ρ i , ρ j and performs a measurement to calculate the output value of ρ i ρ j in the above formula (3). Note that the part of the formula for which the trace is taken, that is, ρ iρ j The part of j is calculated by the quantum processor 200. The trace Tr[·] is calculated by the processor 108. The trace Tr[ρ i ρ j can be calculated by repeatedly measuring the output values of ρ i ρ j by the quantum processor 200 a sufficient number of times and taking the average of those output values.

[0038] The solution calculation unit 133 solves the generalized eigenvalue equation G·c = E·S·c to calculate c (0) corresponding to the minimum eigenvalue E = E (0) as the optimal solution. Note that the minimum eigenvalue E (0) becomes the error-suppressed energy expectation value.

[0039] The state calculation unit 140 calculates the quantum state ρ (0) using the optimal solution c EM That is, the state calculation unit 140 calculates the quantum state ρ (0) by setting c = c EM in the above formula (1).

[0040] Also, the state calculation unit 140 outputs the quantum state ρ EM to a predetermined output destination. Note that the output destination of the quantum state ρ EM is not limited to a specific output destination and can be any output destination. For example, it can be the storage area of the auxiliary storage device 107, a display device 102 such as a program or a display, or another device or equipment communicably connected to the control device 100.

[0041] <Calculation process of the quantum state ρ EM > Hereinafter, the calculation process of the quantum state ρ EM will be described with reference to FIG. 4. FIG. 4 is a flowchart showing an example of the calculation process of the quantum state. Note that step S101 in FIG. 4, or steps S101 to S102, may be executed in advance.

[0042] The ground state approximation unit 110 performs PEC on a quantum circuit that outputs an approximate state of the ground state with the Hamiltonian H as input by, for example, the VQE method while allowing imperfect quantum process tomography, and then calculates D approximate states {ρ i |i = 1, ···, D} using the D quantum circuits on which PEC has been performed (step S101). Note that the quantum processor 200 is also used when calculating each approximate state ρ i .

[0043] The formulation unit 120 formulates the above formula (1) using the D approximate states {ρ i |i = 1, ···, D} calculated in step S101 above (step S102).

[0044] The optimal solution calculation unit 130 calculates, as the optimal solution c = (c1, ···, c D ), the one that minimizes the expected value of the Hamiltonian H according to the above formula (1) (step S103). The details of this step will be described later. (0) =(c1 (0) , ···, c D (0) ).

[0045] The state calculation unit 140 calculates the quantum state ρ (0) using the optimal solution c EM according to the above formula (1) (step S104). This quantum state ρ EM is output to a predetermined output destination in advance.

[0046] <Calculation process of the optimal solution c (0) > Hereinafter, the calculation process of the optimal solution c (0) in step S103 of FIG. 4 will be described with reference to FIG. 5. FIG. 5 is a flowchart showing an example of the calculation process of the optimal solution c (0) .

[0047] The first matrix element calculation unit 131 of the optimal solution calculation unit 130 calculates, for all pairs of (i, j), the element G ijCalculate it according to the above formula (2) (step S201). As a result, G ij A D×D matrix G with (i,j) components is obtained.

[0048] For all pairs of (i,j), the second matrix element calculation unit 132 of the optimal solution calculation unit 130 calculates the element S of the (i,j) component of the matrix S ij Calculate it according to the above formula (3) (step S202). As a result, S ij A D×D matrix S with (i,j) components is obtained.

[0049] The solution calculation unit 133 of the optimal solution calculation unit 130 solves the generalized eigenvalue equation G·c = E·S·c to calculate c (0) corresponding to the minimum eigenvalue E = E (0) as the optimal solution (step S203). This optimal solution c (0) corresponds to the local solution in the trial wave function of the above formula (1). Note that the generalized eigenvalue method may be solved by a known method, and it can be easily solved using an existing library or the like.

[0050] <Summary> As described above, even if the characterization of noise (that is, the identification of the noise model) is incomplete when the quantum computer 10 according to the present embodiment executes PEC on the quantum circuit that outputs the approximation of the ground state of the Hamiltonian H, the quantum circuit can estimate the expected value in the physical state by executing the GSE based on the approximate state output by the quantum circuit. Therefore, according to the quantum computer 10 according to the present embodiment, powerful quantum error suppression is possible even under the condition of allowing incomplete quantum process tomography.

[0051] Also, in PEC, if the noise strength is overestimated too much, the noise and the inverse transformation of the effective noise do not cancel each other out, generating a non-physical state that does not effectively satisfy the positive value property. As a result, an error may occur in the expected value of the physical quantity. On the other hand, in the quantum computing device 10 according to the present embodiment, since the expected value under a physical state can be estimated by executing GSE, the error can be suppressed.

[0052] In the above embodiment, the quantum state ρ was calculated and output in step S104 of FIG. 4. However, the present invention is not limited to this. For example, up to the optimal solution c in step S203 of FIG. 5, EM it is not calculated, and only the minimum eigenvalue E (0) is calculated, and this minimum eigenvalue E (0) may be output. (0)

[0053] The present invention is not limited to the specifically disclosed above embodiments, and various modifications, changes, combinations with known technologies, etc. are possible without departing from the description of the claims.

[0054] [References] Reference 1: N. Yoshioka, H. Hakoshima, Y. Matsuzaki, Y. Tokunaga, Y. Suzuki, and S. Endo, Generalized Quantum Subspace Expansion, Phys. Rev. Lett. 129, 020502 (2022). Reference 2: W. J. Huggins, S. McArdle, T. E. O'Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual Distillation for Quantum Error Mitigation, Phys. Rev. X 11, 041036 (2021). ​Reference 3: B. Koczor, Exponential Error Suppression for Near-Term Quantum Devices, Phys. Rev. X 11, 031057 (2021). Reference 4: J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong, Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States, Phys. Rev. A 95, 042308 (2017). Reference 5: J. R. McClean, Z. Jiang, N. C. Rubin, R. Babbush, and H. Neven, Decoding Quantum Errors with Subspace Expansions, Nat. Commun. 11, 636 (2020). Reference 6: T. Takeshita, N. C. Rubin, Z. Jiang, E. Lee, R. Babbush, and J. R. McClean, Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources, Phys. Rev. X 10, 011004 (2020). Reference 7: N. Yoshioka, T. Sato, Y. O. Nakagawa, Y. Ohnishi, and W. Mizukami, Variational Quantum Simulation for Periodic Materials, Phys. Rev. Res. 4, 013052 (2022).

Explanation of Symbols

[0055] 10 Quantum computing device 100 Control device 101 Input device 102 Display device 103 External I / F 103a Recording medium 104 Communication I / F 105 RAM 106 ROM 107 Auxiliary storage device 108 Processor 109 Bus 110 Ground state approximation unit 120 Formulation unit 130 Optimal solution calculation unit 131 First matrix element calculation unit 132 Second matrix element calculation unit 133 Solution calculation unit 140 State calculation unit 200 Quantum processor

Claims

1. An approximation calculation unit that calculates a plurality of approximate states each representing a quantum state that approximates the ground state of a given Hamiltonian based on a quantum circuit that can be affected by noise; A formulation unit that formulates an equation for obtaining a quantum state that minimizes the expected value of the Hamiltonian based on the plurality of approximate states; An expected value calculation unit that calculates the minimum expected value of the Hamiltonian by a generalized subspace method using the plurality of approximate states as a basis based on the equation; A quantum computing device having the above.

2. The quantum circuit is a quantum circuit that takes the Hamiltonian as an input and outputs the approximate state, The approximation calculation unit, Performs quantum error suppression on the quantum circuit by PEC that allows noise model identification by incomplete quantum process tomography, The quantum computing device according to claim 1, wherein the plurality of approximate states are calculated by the quantum circuit subjected to the quantum error suppression.

3. The formulation unit, Formulates, as an equation for obtaining a quantum state that minimizes the expected value of the Hamiltonian, an equation that virtually constructs a quantum state representing a physical state using any two of the plurality of approximate states and the same number of complex numbers as the number of the plurality of approximate states. The quantum computing device according to claim 1 or 2.

4. Let the Hamiltonian be \(H\), and the plurality of approximate states be \(\rho\) １ , ···, \(\rho\) Ｄ , let the complex numbers be \(c\) １ , ···, \(c\) Ｄ Then,[[]]END]] The formulation unit, c ｉ ＊ c ｊ ρ ｉ ρ ｊ The sum with respect to 1 ≤ i ≤ D and 1 ≤ j ≤ D is formulated as the above formula, The expected value calculation unit, Tr[Hρ ｉ ρ ｊ to form a matrix G with (i, j) components, and Tr[ρ ｉ ρ ｊ to form a matrix S with (i, j) components, and solve the generalized eigenvalue equation G·c = E·S·c (where c = (c １ , ···, c Ｄ )) to calculate the minimum eigenvalue of the said generalized eigenvalue equation as the minimum expected value of the said Hamiltonian, the quantum computing device according to claim 3.

5. c = c corresponding to the minimum eigenvalue （０） A solution calculation unit that calculates this as an optimal solution, The optimal solution c = c （０） and a state calculation unit that calculates a quantum state representing the physical state based on the formula and the like, the quantum computing device according to claim 4 having the same.

6. An approximation calculation procedure for calculating a plurality of approximate states each representing a quantum state that approximates the ground state of a given Hamiltonian based on a quantum circuit that can be affected by noise; A formulation procedure for formulating an equation for obtaining a quantum state that minimizes the expected value of the Hamiltonian based on the plurality of approximate states; An expected value calculation procedure for calculating the minimum expected value of the Hamiltonian by a generalized subspace method using the plurality of approximate states as a basis based on the equation; A quantum computing method executed by a computer having the above.

7. An approximation calculation procedure for calculating a plurality of approximate states each representing a quantum state that approximates the ground state of a given Hamiltonian based on a quantum circuit that can be affected by noise; A formulation procedure for formulating an equation for obtaining a quantum state that minimizes the expected value of the Hamiltonian based on the plurality of approximate states; An expected value calculation procedure for calculating the minimum expected value of the Hamiltonian by a generalized subspace method using the plurality of approximate states as a basis based on the equation; A program that causes a computer to execute.

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