Estimation device, estimation method, and program

Abstract

An estimation device according to one aspect of the present disclosure is for estimating an expected value of a physical quantity under a state of a quantum system that has evolved over time. The estimation device comprises: a selection unit that selects a plurality of error rates of physical noise when a Trotterized quantum circuit that approximates the time evolution of the quantum system is executed by a quantum processor, and a plurality of optimal numbers of Trotter steps of the Trotterized quantum circuit; a calculation unit that calculates a plurality of expected values related to the physical quantity by executing, by the quantum processor, a plurality of Trotterized quantum circuits respectively corresponding to the plurality of error rates and the optimal numbers of Trotter steps; and an extrapolation unit that performs one-dimensional extrapolation using the plurality of expected values to calculate the estimated value of the expected value of the physical quantity.

Description

Estimation device, estimation method, and program

[0001] The present disclosure relates to an estimation device, an estimation method, and a program.

[0002] To perform accurate quantum computation, it is necessary to suppress errors caused by physical noise within the quantum computer and errors caused by quantum algorithms. For this reason, an error suppression method that suppresses both of these errors has been proposed for quantum algorithms, called the Trotter expansion (Non-Patent Document 1).

[0003] S. Endo, Q. Zhao, Y. Li, S. Benjamin, and X. Yuan, "Mitigating Algorithmic Errors in a Hamiltonian Simulation", Phys. Rev. A 99, 012334 (2019).

[0004] However, since many expected values ​​are required to construct an estimator for estimating the expected value of a physical quantity, the estimation accuracy may be reduced.

[0005] The present disclosure has been made in consideration of the above points, and aims to estimate an expected value of a physical quantity with high accuracy.

[0006] An estimation device according to one aspect of the present disclosure is an estimation device that estimates the expectation value of a physical quantity under the state of a time-evolved quantum system, and includes: a selection unit that selects a plurality of error rates of physical noise when a Trotter expansion quantum circuit that approximates the time evolution of the quantum system is executed by a quantum processor, and a plurality of optimal Trotter numbers for the Trotter expansion quantum circuit; a calculation unit that calculates a plurality of expectation values ​​for the physical quantity by executing a plurality of Trotter expansion quantum circuits that respectively correspond to the plurality of error rates and the optimal Trotter numbers by the quantum processor; and an extrapolation unit that calculates an estimate of the expectation value of the physical quantity by performing one-dimensional extrapolation using the plurality of expectation values.

[0007] The expected value of the physical quantity can be estimated with high accuracy.

[0008] FIG. 1 is a diagram showing an example of noise occurring in a Trotter expansion quantum circuit. FIG. 2 is a diagram showing an example of the configuration of a quantum computing device according to the present embodiment. FIG. 3 is a diagram showing an example of the hardware configuration of a control device according to the present embodiment. FIG. 4 is a diagram showing an example of the functional configuration of a control device according to the present embodiment. FIG. 5 is a flowchart showing an example of an expected value estimation process according to the present embodiment. FIG. 6 is a diagram showing data points for comparison with a conventional method. FIG. 7 is a diagram (part 1) showing a comparative example with a conventional method. FIG. 8 is a diagram (part 2) showing a comparative example with a conventional method.

[0009] Hereinafter, an embodiment of the present invention will be described in detail with reference to the drawings.

[0010] <Conventional Error Suppression Methods and Their Background> To solve practical problems using quantum computers, it is necessary to minimize the influence of physical noise within the quantum computer and perform accurate quantum computations. On the other hand, the influence of physical noise can be minimized by increasing the code size using quantum error correction codes (also known as quantum error correction codes) (References 1 and 2). However, constructing quantum error correction codes requires a huge number of quantum bits, and it is difficult to prepare such a large number of quantum bits with current technology. For this reason, various methods that can suppress the influence of physical noise without consuming a huge number of quantum bits have been investigated, and these methods are collectively referred to as quantum error suppression methods or quantum error suppression methods (or simply "error suppression methods") (References 3 and 4).

[0011] Among the quantum error suppression techniques, one of the most practical is error extrapolation (References 5-7).

[0012] Error extrapolation is a technique for estimating the expected value of a physical quantity when the error rate is zero by measuring multiple expected values ​​with different error rates using a quantum computer and then performing polynomial or exponential extrapolation using these expected values. It has been reported that error extrapolation can be used to perform time evolution simulations of the two-dimensional transverse magnetic field Ising model on a 127-qubit quantum computer, a scale that would be difficult to simulate naively on a classical computer (Reference 8). Demonstration of quantum supremacy through simulations of the time evolution of quantum systems will continue to be pursued, and research into quantum error suppression methods to improve the computational accuracy of demonstration experiments is also expected to be pursued.

[0013] A quantum algorithm called the Trotter expansion (References 9 and 10) is often used in the time evolution of quantum systems. It is known that errors caused by the Trotter expansion algorithm occur even in the absence of physical noise. Therefore, when using a Trotter expansion quantum circuit in the presence of physical noise, it is necessary to suppress not only errors caused by physical noise but also errors caused by the Trotter expansion algorithm. In response to this, an error suppression method has been proposed that suppresses both errors caused by physical noise and errors caused by the Trotter expansion algorithm (Non-Patent Document 1).

[0014] The error suppression method proposed in Non-Patent Document 1 suppresses both errors caused by physical noise and errors caused by the Trotter expansion algorithm by serially performing extrapolation related to physical noise and extrapolation related to the Trotter number. The algorithm of the error suppression method proposed in Non-Patent Document 1 is outlined in the following steps a to c.

[0015] Step a: Select a plurality of different Trotter numbers, and prepare (construct) Trotter expansion quantum circuits corresponding to the plurality of Trotter numbers.

[0016] Step b: For each of the multiple Trotter expansion quantum circuits prepared in step a above, the following steps (a) to (b) are carried out. As a result, an expected value is obtained for each of the multiple Trotter expansion quantum circuits in which the influence of errors caused by physical noise is reduced. Note that, hereinafter, errors caused by physical noise are also referred to as physical errors, and their error rate is also referred to as the physical error rate.

[0017] (a) Calculate the expected value of a physical quantity under each of a plurality of physical error rates.

[0018] (b) Using the multiple expected values ​​obtained in (a) above, one-dimensional extrapolation is performed on the physical error rate.

[0019] Step c: Using the multiple expected values ​​obtained in step b above, perform one-dimensional extrapolation of the Trotter number, thereby obtaining an expected value that is less affected by errors caused by the Trotter expansion algorithm.

[0020] <Problems with Conventional Error Suppression Methods> The error suppression method proposed in Non-Patent Document 1 has a problem in that the number of expected values ​​required to estimate the expected values ​​of physical quantities (i.e., the number of expected values ​​required in (i) of the above step b and the number of expected values ​​required in the above step c) is large.

[0021] Typically, when performing only one-dimensional extrapolation of the physical error rate, the required expected values ​​are also prepared on one-dimensional parameters. That is, expected values ​​in a quantum circuit that generates physical noise can be obtained for each of multiple different physical error rates. However, in the method proposed in Non-Patent Document 1, one-dimensional extrapolation of the physical error rate is performed on data points (hereinafter, the expected values ​​required to construct an estimator of a physical quantity will also be simply referred to as data points), and then one-dimensional extrapolation of the Trotter number is performed. Therefore, in the method proposed in Non-Patent Document 1, expected values ​​must be prepared on two-dimensional parameters, namely, the physical error rate and the Trotter number.

[0022] On the other hand, since the number of measurements that can be performed on a quantum computer is finite, if the number of required expected values ​​is large, the number of measurements that can be performed to obtain one expected value will be reduced. Therefore, if the number of required expected values ​​is large, the variance of each expected value will increase, and the accuracy of the expected value estimation may decrease.

[0023] <Proposed Method> Below, we propose a one-dimensional extrapolation method that can efficiently suppress both errors caused by physical noise and errors caused by the Trotter expansion algorithm by preparing data points with the optimal Trotter number selected for each of several different physical error rates. The proposed method makes it possible to efficiently suppress both types of errors with fewer data points than the conventional method proposed in Non-Patent Document 1, and therefore makes it possible to estimate the expected value of a physical quantity with high accuracy.

[0024] <Relationship between the optimal number of trotters and the physical error rate> First, as the basis of the proposed method, we set the optimal number of trotters as M ｏｐｔ , the number of quantum bits is n, and the physical error rate is p ｎ Then, the optimal number of trotters M ｏｐｔ The physical error rate p ｎ Dependence on M ｏｐｔ ∝(1 / √(p ｎ ))

[0025] Hamiltonian H = Σ ｉ H ｉ (However, H ｉ is the time evolution operator U generated by ｅｘａｃｔ =exp(-iHt)=exp(-iΣ ｉ H ｉ t), where t represents time.

[0026] First-order Trotter expansion Π ｊ exp(-iH ｊ t) = [Π ｊ exp(-iH ｊ t / M)] Ｍ (where M is the number of trotters) ｅｘａｃｔBy approximating U, we construct a Trotter expansion quantum circuit and simulate the dynamics of the Hamiltonian H. ＴＲ :=Π ｊ exp(-iH ｊ t / M), U ＴＲ Let us call this one layer of the Trotter expansion. ＴＲ By the definition of (U ＴＲ ) Ｍ =Π ｊ exp(-iH ｊ Note that t) is satisfied.

[0027] U ｅｘａｃｔ Using the above, the Trotter expansion quantum circuit can be expressed as follows:

[0028] Here, E(m) has a maximum singular value of 1 / M ｍ－１ is an error term that depends on

[0029] The above error term E(m) satisfies the following:

[0030] Here, ||A|| ｏｐ is the maximum singular value of operator A. As a result, when M → ∞, the Trotter expansion quantum circuit is U ｅｘａｃｔ It can be seen that it coincides exactly with

[0031] However, when physical noise (hereinafter simply referred to as noise) occurs in a Trotter expansion quantum circuit, the depth of the quantum circuit increases as the Trotter number M increases, and the circuit is therefore significantly affected by the noise. For this reason, there is a trade-off relationship in which if the Trotter number M is too large, the calculation accuracy actually deteriorates. Due to this trade-off relationship, for a Trotter expansion quantum circuit in a situation where noise occurs, it is necessary to select an optimal Trotter number M that minimizes the error. ｏｐｔ In the following, we consider the time evolution operator U ｅｘａｃｔ The quantum process of π ｅｘａｃｔ = [U ｅｘａｃｔ ], one layer U of noiseless Trotter expansion quantum circuit ＴＲ The quantum process of π ＴＲ = [UＴＲ ], one layer U of the Trotter expansion quantum circuit with noise ＴＲ The quantum process of ε ＴＲ Here, for an operator A representing a unitary operation and a quantum state ρ, [A](ρ) = AρA † is.

[0032] In this case, the ideal time evolution operator U ｅｘａｃｔ The quantum process of π ｅｘａｃｔ and the quantum process ε of the noisy Trotter expansion quantum circuit ＴＲ Ｍ Distance D (π ｅｘａｃｔ , ε ＴＲ Ｍ ) can be evaluated from above as shown in equation (1) below:

[0033] Here, any two quantum processes π １ and π ２ and the quantum state ρ, D(π １ , π ２ ) is defined below.

[0034] Here, ||A|| represents the trace norm, and ||A|| = Tr[√(A † A)], i.e., for a properly chosen quantum state ρ, the quantum process π １ and π ２ The trace norm of １ , π ２ ) is defined as

[0035] It is known that for positive constants a and b, the following holds (Reference 11):

[0036] Therefore, the above formula (1) can be evaluated by the following formula (2).

[0037] This allows the distance D(π ｅｘａｃｔ , ε ＴＲ Ｍ ) is the optimal number of trotters M ｏｐｔ is M ｏｐｔ =√(a / b).

[0038] Here, we consider a situation in which Global Depolarizing Noise occurs as physical noise in each layer of the Trotter expansion for a Trotter expansion quantum circuit. Hereinafter, we will refer to Global Depolarizing Noise as N ｇｌｏｂ．ｄｅｐｏ． In this case, the Trotter expansion quantum circuit with noise is expressed as shown in Figure 1. Note that the Global Depolarizing Noise N ｇｌｏｂ．ｄｅｐｏ． is expressed as follows for the quantum state ρ:

[0039] Here, I is a unitary operation expressed as an identity matrix.

[0040] Although the simplest global depolarizing noise has been assumed as physical noise, it has been theoretically and numerically confirmed that in quantum circuits with deep depth, local physical noise can be approximately regarded as global depolarizing noise (References 12 and 13). Therefore, if the quantum circuit has deep depth, the content described below is considered to be similarly applicable to any physical noise.

[0041] As a noise model, it is more realistic to set local stochastic noise that occurs each time a quantum gate that acts on about one or two qubits is used. For this reason, it is better to give a relational expression for the physical error rate of the local stochastic noise. Therefore, in the following, the physical error rate p ｎ and the physical error rate p of local stochastic noise will be described.

[0042] Suppose there is a quantum gate G with n quantum bits that operates at a depth of d, and that quantum gate G is composed of one or two quantum gates with a time of O(nd), and each time one or two quantum gates operate, local stochastic noise of one or two quantum bits occurs with a physical error rate p. In this case, the probability that local stochastic noise occurs in at least one location after operating the quantum gate G is approximately ndp. Therefore, if the influence of these local stochastic noises is represented by one of the global depolarizing noises, the physical error rate p of the global depolarizing noise is ｎ is considered to be as follows:

[0043] That is, the physical error rate p ｎ It is thought that it can be approximated by O(ndp).

[0044] Below, we will explain how to determine the optimal Trotter number M by appropriately evaluating the distance when global depolarizing noise occurs in a quantum circuit. ｏｐｔ The second term in the parentheses on the right-most side of the above equation (1) is expressed by the following equation (3).

[0045] Moreover, the term on the rightmost side of the above equation (3) can be expressed by the following equation (4).

[0046] This results in ε ＴＲ and π ＴＲ Distance D (ε ＴＲ , π ＴＲ ) is p ｎ Therefore, D(ε ＴＲ , π ＴＲ ) ∝ p ｎ Using M ｏｐｔ =√(a / b) can be expressed by the following equation (5).

[0047] Here, c is a positive constant. Also, floor(·) represents the floor function. ｏｐｔ If is not an integer equal to or greater than 1, the value of c is reset.

[0048] Therefore, the value of c should be appropriately selected to minimize the physical error rate p ｎ where M is an integer greater than or equal to 1 ｏｐｔ By setting and constructing a Trotter expansion quantum circuit, it is thought that it will be possible to suppress both errors caused by physical noise and errors caused by the Trotter expansion algorithm using fewer data points.

[0049] <Theoretical configuration of one-dimensional extrapolation method according to the proposed method> Below, we will explain the theoretical configuration of one-dimensional extrapolation method that can efficiently suppress both errors caused by physical noise and errors caused by the Trotter expansion algorithm.

[0050] Using the above formulas (1) to (5), the physical error rate p ｎ Under the assumption that is small, the physical error rate p ｎ and the number of trotters M ｏｐｔ The output state of the Trotter expansion quantum circuit ρ(p ｎ ) can be expressed as follows:

[0051] where ρ ０ is the input state, Δ=Σ ｉ＜ｊ [H ｉ , H ｊ ]ρ ０ U ｅｘａｃｔ † +h.c. Also, ρ ｅｘａｃｔ : = [U ｅｘａｃｔ ](ρ ０ ) where h.c. is the Hermitian conjugate, which is the Hermitian conjugate of the entire first term, i.e., {Σ ｉ＜ｊ [H ｉ , H ｊ ]ρ ０ U ｅｘａｃｔ †} † Represents.

[0052] Therefore, ρ(p ｎ ) is expressed as follows: ｎ ) is expected to be expanded in polynomials.

[0053] Here, Aｉ are the real expansion coefficients.

[0054] Thus, as in the usual polynomial extrapolation (References 3 and 7), we obtain m+1 expected values ​​{〈A〉(λ ｉ p ｎ ) |i=0,...,m} (where 1=λ ０ <λ １ <...<λ ｍ ) and √(p ｎ ), we can obtain the following equations (7) and (8) as estimators of the expected values ​​by extrapolation:

[0055] The estimators shown in equations (7) and (8) make it possible to estimate the expected value of the physical quantity A with high accuracy using a small number of data points.

[0056] <<Algorithm of the Proposed Method>> The proposed method is executed according to the algorithm shown in the following steps 1 to 4.

[0057] Step 1: m+1 parameters (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) where M ｏｐｔ （ｉ） = floor(c / √(p ｎ （ｉ） )) to set the optimal Trotter number. In addition, when the error rate of the local stochastic noise is p, the total number of qubits is n, and the depth per layer of the Trotter expansion quantum circuit is d, p ｎ （ｉ） = ndp. An appropriate positive constant is set as c, but M ｏｐｔ （ｉ） If m is 0, then c is reselected. Note that the value of m can be set to a smaller value than in the conventional method.

[0058] Step 2: m+1 (p ｎ （ｉ） , M ｏｐｔ （ｉ）) and construct m+1 Trotter expansion quantum circuits corresponding to each of the

[0059] Step 3: For each i ∈ {0,...,m}, (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) using a Trotter expansion quantum circuit corresponding to the physical quantity A, the expected value 〈A〉 (p ｎ （ｉ） ) is calculated. This gives m+1 expected values ​​{〈A〉(p ｎ （ｉ） ) |i=0,...,m} is obtained. ｎ （ｉ） , M ｏｐｔ （ｉ） ) for the Trotter expansion quantum circuit corresponding to the input state ρ ０ By measuring the physical quantity A multiple times in the output state when inputting ｎ （ｉ） ) can be obtained. In this case, since the value of m is smaller than that of the conventional method, it is possible to increase the number of measurements required to estimate one expected value, and it is possible to obtain a highly accurate expected value (p ｎ （ｉ） ) can be obtained.

[0060] Step 4: λ in the above equation (8) ｉ o p ｎ （ｉ） The extrapolation coefficient g is calculated by substituting ｉ Calculate the expected value of m+1 obtained in step 3 above, {〈A〉(p ｎ （ｉ） ) |i=0,...,m}, and use the above formula (7) to find the ｉ p ｎ ) to (p ｎ （ｉ） ) and the error term O(p ｎ （ｍ＋１）／２ ) is removed from the equation, and the expected value of physical quantity A is １Ｄ ｅｘｔｒａ． That is, the expected value of the physical quantity A is calculated using the following equations (9) and (10): １Ｄ ｅｘｔｒａ． Calculate.

[0061] As a result, the expected value of the physical quantity A can be estimated with high accuracy by １Ｄ ｅｘｔｒａ． is obtained.

[0062] The quantum computing device 10 that estimates the expected value of the physical quantity A using the above proposed method will be described below.

[0063] <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. 2. Fig. 2 is a diagram showing an example of the configuration of the quantum computing device 10 according to this embodiment.

[0064] As shown in FIG. 2, the quantum computing device 10 according to this embodiment includes a control device 100 and a quantum processor 200.

[0065] The control device 100 transmits a control signal to the quantum processor 200 and obtains a calculation result from the quantum processor 200. In this way, quantum calculation is performed. The control device 100 is realized by, for example, a classical computer or the like.

[0066] The quantum processor 200 configures a quantum two-level system called a quantum bit (physical quantum bit), and performs physical operations such as initialization, gate operation (unitary transformation), and measurement on the physical quantum bit in response to a control signal from the control device 100. The quantum system for realizing the quantum bit 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 may be used.

[0067] <Example of Hardware Configuration of Control Device 100> An example of a hardware configuration of the control device 100 according to this embodiment will be described with reference to Fig. 3. Fig. 3 is a diagram showing an example of the hardware configuration of the control device 100 according to this embodiment.

[0068] 3, 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 random access memory (RAM) 105, a read only memory (ROM) 106, an auxiliary storage device 107, and a processor 108. Each of these pieces of hardware is connected to each other via a bus 109 so as to be able to communicate with each other.

[0069] The input device 101 is, for example, a keyboard, a mouse, a touch panel, a physical button, etc. The display device 102 is, for example, a display, a display panel, etc. Note that the control device 100 does not necessarily have to include at least one of the input device 101 and the display device 102, for example.

[0070] The external I / F 103 is an interface with an external device such as a 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), and a USB (Universal Serial Bus) memory card.

[0071] 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 stores programs and data. The ROM 106 is a non-volatile semiconductor memory (storage device) that can store 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), or a flash memory. The processor 108 is an arithmetic device such as a CPU (Central Processing Unit).

[0072] 3 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 multiple auxiliary storage devices 107 or multiple processors 108, may not have some of the hardware shown in the figure, or may have various hardware other than the hardware shown in the figure.

[0073] <Example of functional configuration of control device 100> An example of the functional configuration of the control device 100 according to this embodiment will be described with reference to Fig. 4. Fig. 4 is a diagram showing an example of the functional configuration of the control device 100 according to this embodiment.

[0074] 4, the control device 100 according to this embodiment includes a parameter selection unit 110, a quantum circuit construction unit 111, an expected value calculation unit 112, an extrapolation unit 113, and an output unit 114. Each of these units is realized by, for example, processing in which one or more programs installed in the control device 100 are executed by the processor 108 or the like.

[0075] The parameter selection unit 110 executes step 1 of the proposed method. That is, the parameter selection unit 110 selects m+1 parameters (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) (i=0, . . . , m). The parameter selection unit 110 selects each parameter (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) or a parameter stored in a storage area such as the auxiliary storage device 107 may be selected. The value of m is set in advance, but can be a smaller value than in the conventional method.

[0076] The quantum circuit construction unit 111 executes step 2 of the proposed method. That is, the quantum circuit construction unit 111 selects the m+1 parameters (p ｎ （ｉ） , M ｏｐｔ （ｉ）) are constructed. ｎ （ｉ） , M ｏｐｔ （ｉ） ) is a Trotter expansion quantum circuit corresponding to C （ｉ） Let's say.

[0077] The expectation calculation unit 112 executes step 3 of the proposed method. That is, the expectation calculation unit 112 calculates the Trotter expansion quantum circuit C （ｉ） Using the above, the quantum processor 200 calculates the expected value of the physical quantity A (p ｎ （ｉ） ) is calculated. More specifically, the expectation calculation unit 112 calculates the Trotter expansion quantum circuit C （ｉ） For input state ρ ０ By measuring the physical quantity A multiple times in the output state when inputting ｎ （ｉ） ) is calculated. This gives m+1 expected values ​​{〈A〉(p ｎ （ｉ） ) |i=0, . . . , m} is obtained.

[0078] The extrapolation unit 113 executes step 4 of the proposed method. That is, the extrapolation unit 113 calculates the m+1 expected values ​​{〈A〉(p ｎ （ｉ） ) |i=0,...,m}, the expected value of the physical quantity A is calculated by the above equations (9) and (10). １Ｄ ｅｘｔｒａ． As a result, by one-dimensional extrapolation according to the proposed method, we can obtain a highly accurate estimate of the expected value of the physical quantity A, １Ｄ ｅｘｔｒａ． is obtained.

[0079] The output unit 114 outputs the expected value calculated by the extrapolation unit 113. １Ｄ ｅｘｔｒａ． The output is output to a predetermined output destination. The output destination is not limited to a specific output destination, and can be any output destination. For example, the output destination can be a storage area of ​​the auxiliary storage device 107, a program, a display device 102 such as a display, or another device or equipment connected to the control device 100 so as to be able to communicate with the control device 100.

[0080] <Example of Expected Value Estimation Process> An example of the expected value estimation process according to this embodiment will be described with reference to Fig. 5. Fig. 5 is a flowchart showing an example of the expected value estimation process according to this embodiment.

[0081] The parameter selection unit 110 selects m+1 parameters (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) (i=0, . . . , m) is selected (step S101).

[0082] The quantum circuit construction unit 111 selects the m+1 parameters (p ｎ （ｉ） , M ｏｐｔ （ｉ） ) corresponding to each of m+1 Trotter expansion quantum circuits C （ｉ） is constructed (step S102).

[0083] The expectation calculation unit 112 calculates the Trotter expansion quantum circuit C constructed in step S102 for each i∈{0, . . . , m}. （ｉ） Using the above, the quantum processor 200 calculates the expected value of the physical quantity A (p ｎ （ｉ） ) is calculated (step S103).

[0084] The extrapolation unit 113 calculates the m+1 expected values ​​{〈A〉(p ｎ （ｉ） ) |i=0,...,m}, the expected value of the physical quantity A is calculated by the above equations (9) and (10). １Ｄ ｅｘｔｒａ． is calculated (step S104).

[0085] The output unit 114 outputs the expected value obtained in step S104. １Ｄ ｅｘｔｒａ． is output to a predetermined output destination (step S105).

[0086] <Comparison with Conventional Methods> An experiment conducted to compare the proposed method with conventional methods will be described below. In this experiment, a time evolution simulation of a one-dimensional transverse-field Ising model was performed. In addition to a case where error suppression was not performed (hereinafter referred to as "without error suppression"), two comparisons were performed: a case where polynomial extrapolation was performed on the Trotter number after polynomial extrapolation on the physical error rate using the method described in Non-Patent Document 1 (hereinafter referred to as Conventional Method 1); and a case where exponential extrapolation was performed on the physical error rate after polynomial extrapolation on the Trotter number using the method described in Non-Patent Document 1 (hereinafter referred to as Conventional Method 2).

[0087] In this experiment, the proposed method was １ ~P ３ The three data points, P shown in FIG. 6 for Conventional Methods 1 and 2 １ ~P ９ In FIG. 6, "Trotter steps" indicates the number of trotters, and "Error rate" indicates the physical error rate p ｎ , and "Absolute error" represent the absolute error between the expected value under the exact time evolution state and the expected value under each Trotter number and physical error rate, respectively.

[0088] In this case, as shown in Fig. 7, the proposed method is able to reduce the standard deviation of the estimated value (estimated value of the expected value of the physical quantity) compared to conventional methods 1 and 2. Furthermore, as shown in Fig. 8, the proposed method is able to reduce the error between the estimated value and the exact value compared to conventional methods 1 and 2. Therefore, it can be said that the proposed method is able to estimate expected values ​​with smaller variance and smaller error compared to conventional methods 1 and 2.

[0089] <Summary> As described above, the quantum computing device 10 according to this embodiment can efficiently suppress, with a small number of data points, both errors caused by physical noise in the quantum processor 200 and errors caused by the algorithm of the Trotter expansion quantum circuit. This makes it possible to increase the number of measurements that can be assigned to one data point compared to conventional methods, and as a result, it becomes possible to estimate the expected value of a desired physical quantity with high accuracy.

[0090] The present invention is not limited to the above-described specifically disclosed embodiments, and various modifications, changes, and combinations with known technologies are possible without departing from the scope of the claims.

[0091] [References] Reference 1: PW Shor, Scheme for Reducing Decoherence in Quantum Computer Memory, Phys. Rev. A 52, R2493 (1995). Reference 2: A. Steane, Multiple-Particle Interference and Quantum Error Correction, Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 452, 19960136 (1996). Reference 3: Z. Cai, R. Babbush, SC Benjamin, S. Endo, WJ Huggins, Y. Li, JR McClean, and TE O'Brien, Quantum Error Mitigation, arXiv:2210.00921. Reference 4: S. Endo, Z. Cai, SC Benjamin, and X. Yuan, Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation, J. Phys. Soc. Jpn. 90, 032001 (2021). Reference 5: K. Temme, S. Bravyi, and JM Gambetta, Error Mitigation for Short-Depth Quantum Circuits, Phys. Rev. Lett. 119, 180509 (2017). Reference 6: Y. Li and SC Benjamin, Efficient Variational Quantum Simulator Incorporating Active Error Minimization, Phys. Rev.Reference 8: Y. Kim et al., Evidence for the Utility of Quantum Computing before Fault Tolerance, Nature 618, 7965 (2023). Reference 9: M. Suzuki, Generalized Trotter's Formula and Systematic Approximants of Exponential Operators and Inner Derivations with Applications to Many-Body Problems, Commun. Math. Phys. 51, 183 (1976). Reference 10: S. Lloyd, Universal Quantum Simulators, Science 273, 1073 (1996). Reference 11: G. C. Knee and W. J. Munro, Optimal Trotterization in Universal Quantum Simulators under Faulty Control, Phys. Rev. A 91, (2015). Reference 12: D. Qin, Y. Chen, and Y. Li, Error Statistics and Scalability of Quantum Error Mitigation Formulas, Npj Quantum Inf. 9, 1 (2023). Reference 13: K. Tsubouchi, T. Sagawa, and N. Yoshioka, Universal Cost Bound of Quantum Error Mitigation Based on Quantum Estimation Theory, Phys. Rev. Lett. 131, 210601 (2023).

[0092] REFERENCE SIGNS LIST 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 Parameter selection unit 111 Quantum circuit construction unit 112 Expected value calculation unit 113 Extrapolation unit 114 Output unit 200 Quantum processor

Claims

1. An estimation device that estimates the expectation value of a physical quantity in the state of a time-evolved quantum system, comprising: a selection unit that selects a plurality of error rates of physical noise when a Trotter expansion quantum circuit that approximates the time evolution of the quantum system is executed by a quantum processor, and a plurality of optimal Trotter numbers for the Trotter expansion quantum circuit; a calculation unit that calculates a plurality of expectation values ​​for the physical quantity by executing a plurality of Trotter expansion quantum circuits that correspond respectively to the plurality of error rates and the optimal Trotter numbers by the quantum processor; and an extrapolation unit that calculates an estimate of the expectation value of the physical quantity by performing one-dimensional extrapolation using the plurality of expectation values.

2. The selection unit selects the i-th error rate as p ｎ （ｉ） (where n is the number of quantum bits), the error rate p ｎ （ｉ） The optimal number of trotters corresponding to M ｏｐｔ （ｉ） = floor(c / √(p ｎ （ｉ） )) (where c is a positive constant and floor(·) is a floor function), the plurality of error rates are determined as p ｎ （ｉ） and the optimum number of trotters M ｏｐｔ （ｉ） The estimation device according to claim 1 , wherein the estimation device selects:

3. The selection unit selects the optimum number of trotters M ｏｐｔ （ｉ） The estimation device according to claim 2 , wherein the positive constant c is selected so that c is an integer greater than or equal to 1.

4. The selection unit sets the error rate to p when the error rate of local stochastic noise per layer of the Trotter expansion type quantum circuit is p and the depth per layer is d. ｎ （ｉ） By determining that p = ndp, the plurality of error rates are ｎ （ｉ） and the optimum number of trotters M ｏｐｔ （ｉ） The estimation device according to claim 2 or 3, wherein the estimation device selects:

5. The calculation unit calculates the error rate p for each i when i∈{0,...,m}. ｎ （ｉ） and the optimum number of trotters M ｏｐｔ （ｉ） 5. The estimation device according to claim 4, wherein the quantum processor executes a Trotter expansion quantum circuit corresponding to:

6. The extrapolation unit calculates the error rate p ｎ （ｉ） (i∈{0,...,m}) to calculate each extrapolation coefficient g ｉ Calculate the i-th expected value and the extrapolation coefficient g ｉ 6. The estimation device according to claim 5, wherein the sum of products of and with respect to i is calculated as an estimate of the expected value of the physical quantity.

7. A method for estimating the expectation value of a physical quantity in the state of a time-evolved quantum system, the method comprising: a selection step of selecting a plurality of error rates of physical noise when a Trotter expansion quantum circuit that approximates the time evolution of the quantum system is executed by a quantum processor, and a plurality of optimal Trotter numbers for the Trotter expansion quantum circuit; a calculation step of calculating a plurality of expectation values ​​for the physical quantity by executing a plurality of Trotter expansion quantum circuits corresponding to the plurality of error rates and the optimal Trotter numbers, respectively, by the quantum processor; and an extrapolation step of calculating an estimate of the expectation value of the physical quantity by one-dimensional extrapolation using the plurality of expectation values.

8. A program for estimating the expectation value of a physical quantity in the state of a time-evolved quantum system, comprising: a selection procedure for selecting multiple error rates of physical noise when a Trotter expansion quantum circuit that approximates the time evolution of the quantum system is executed by a quantum processor, and multiple optimal Trotter numbers for the Trotter expansion quantum circuit; a calculation procedure for calculating multiple expectation values ​​for the physical quantity by executing multiple Trotter expansion quantum circuits corresponding to the multiple error rates and the optimal Trotter numbers, respectively, on the quantum processor; and an extrapolation procedure for calculating an estimate of the expectation value of the physical quantity by performing one-dimensional extrapolation using the multiple expectation values.

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