Estimation device and estimation method

Abstract

An estimation device according to one aspect disclosed herein estimates the expected value of a physical quantity under a state of a time-evolved quantum system, and includes an estimation unit that estimates the expected value of the physical quantity by using an estimator that estimates the expected value of the physical quantity under a physical state in which an error caused by physical noise when the time evolution of the quantum system is calculated by a quantum processor and an error caused by an algorithm approximating the time evolution can be controlled.

Description

Estimation device and estimation method

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

[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 called the Trotter expansion has been proposed to suppress both types of errors in quantum algorithms.

[0003] P. W. Shor, "Scheme for Reducing Decoherence in Quantum Computer Memory," Phys. Rev. A 52, R2493 (1995).A. Steane, "Multiple-Particle Interference and Quantum Error Correction," Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 452, 19960136 (1996).Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O'Brien, "Quantum Error Mitigation," arXiv:2210.00921.S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation," J. Phys. Soc. Jpn. 90, 032001 (2021).K. Temme, S. Bravyi, and J. M. Gambetta, "Error Mitigation for Short-Depth Quantum Circuits," Phys. Rev. Lett. 119, 180509 (2017).Y. Li and S. C. Benjamin, "Efficient Variational Quantum Simulator Incorporating Active Error Minimization," Phys. Rev. X 7, 021050 (2017).S. Endo, S. C. Benjamin, and Y. Li, "Practical Quantum Error Mitigation for Near-Future Applications," Phys. Rev. X 8, 031027 (2018).Y. Kim et al., "Evidence for the Utility of Quantum Computing before Fault Tolerance," Nature 618, 7965 (2023).S. Anand, K. Temme, A. Kandala, and M. Zaletel, "Classical Benchmarking of Zero Noise Extrapolation beyond the Exactly-Verifiable Regime," arXiv:2306.17839.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).S. Lloyd, "Universal Quantum Simulators," Science 273, 1073 (1996).S. Endo, Q. Zhao, Y. Li, S. Benjamin, and X. Yuan, “Mitigating Algorithmic Errors in a Hamiltonian Simulation," Phys. Rev. A 99, 012334 (2019). N. Yoshioka, H. Hakoshima, Y. Matsuzaki, Y. Tokunaga, Y. Suzuki, and S. Endo, "Generalized Quantum Subspace Expansion," Phys. Rev. Lett. 129, 020502 (2022).

[0004] However, conventional error suppression methods cannot guarantee that the effective state corresponding to the estimator for estimating the expected value of a physical quantity is a physical state, and therefore the expected value may be estimated under a non-physical state. This can result in a large bias in the expected value estimation, resulting in a decrease in its accuracy.

[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 an expected value of a physical quantity under the state of a time-evolved quantum system, and includes an estimation unit that estimates the expected value of the physical quantity using an estimator that estimates the expected value of the physical quantity under a physical state that can suppress errors caused by physical noise when the time evolution of the quantum system is calculated by a quantum processor and errors caused by an algorithm that approximates the time evolution.

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

[0008] 1 is a diagram showing an example of the configuration of a quantum computing device according to the present embodiment; FIG. 2 is a diagram showing an example of the hardware configuration of a control device according to the present embodiment; FIG. 3 is a diagram showing an example of the functional configuration of a control device according to the present embodiment; FIG. 4 is a flowchart showing an example of an expected value estimation process according to the present embodiment; FIG. 5 is a diagram (part 1) showing a comparative example with a conventional method; FIG. 6 is a diagram showing an example of settings for comparison with a conventional method; FIG. 7 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> In order to solve practical problems using a quantum computer, it is necessary to reduce the influence of physical noise within the quantum computer as much as possible 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) (Non-Patent Documents 1 and 2). However, constructing a quantum error correction code 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") (Non-Patent Documents 3 and 4).

[0011] Among the quantum error suppression techniques, one of the most practical is the error extrapolation method (Non-Patent Documents 5 to 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 performing polynomial or exponential extrapolation using these expected values. It has been reported that error extrapolation can be used to perform a time evolution simulation of a two-dimensional transverse magnetic field Ising model on a 127-qubit quantum computer, a scale that is difficult to simulate naively on a classical computer (Non-Patent Document 8). Furthermore, quantum circuits have been proposed that are more difficult to simulate on a classical computer than the quantum circuit proposed in Non-Patent Document 8 (Non-Patent Document 9). Therefore, it is expected that demonstrations of quantum supremacy through simulations of the time evolution of quantum systems will continue to be pursued, and along with this, improvements in accuracy using quantum error suppression techniques will also be sought.

[0013] A quantum algorithm called the Trotter expansion (Non-Patent Documents 10 and 11) 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 12).

[0014] In the error suppression method proposed in Non-Patent Document 12, extrapolation regarding physical noise and extrapolation regarding the Trotter number are performed serially to suppress both errors caused by physical noise and errors caused by the Trotter expansion algorithm.

[0015] <Problems with Conventional Error Suppression Methods> The problem with the error suppression method proposed in Non-Patent Document 12 is that there is no guarantee that the effective state corresponding to the estimator for estimating the expected value of a physical quantity is physical. This can result in a large bias in the expected value estimation, reducing the accuracy of the expected value estimation. Here, when an operator X satisfies X≧0 and Tr(X)=1, X is called a physical state. Note that the term "estimator" refers to a method, algorithm, etc. for estimating the expected value of a physical quantity, etc.

[0016] Below, we will explain why a large bias can occur in expectation value estimation under unphysical conditions, and then explain how the method proposed in Non-Patent Document 12 effectively estimates expectation value under unphysical conditions.

[0017] <<Why large bias can occur in expectation value estimation under non-physical conditions>> The expectation value we want to estimate through error suppression is ideally the expectation value under physical conditions, where there are no errors due to physical noise or algorithm-related errors. Therefore, intuitively, it is thought that bias can be suppressed if the effective state corresponding to the expectation value estimator is physical.

[0018] The fact that bias can be suppressed by estimating the expectation value under a physical state can also be understood from the following discussion: For a general operator X and a physical quantity O that is a Hermitian operator, the following inequality holds:

[0019] |Tr(XO)|≦||X|| ｏｐ Tr(|O|) where ||X|| ｏｐ is the operator norm, and |X| = √(X † X) is the largest eigenvalue.

[0020] In particular, X is a physical state X ｐｈｙｓ When ||X ｐｈｙｓ || ｏｐ Since ≦1 holds, |Tr(X ｐｈｙｓ O) | ≦ Tr(| O |). Therefore, the physical state X ｐｈｙｓ The expected value of the physical quantity O under ｐｈｙｓ It can be seen that it does not depend on .

[0021] On the other hand, X is a non-physical state X ｕｎｐｈｙｓ If |Tr(X ｕｎｐｈｙｓ O) |≦||X ｕｎｐｈｙｓ || ｏｐ Tr(|O|). Therefore, X ｕｎｐｈｙｓ has a large negative eigenvalue, ||X ｕｎｐｈｙｓ || ｏｐ also becomes larger, and the non-physical state X ｕｎｐｈｙｓ It can be seen that the expected value of the physical quantity O under the condition can be larger than Tr(|O|). Therefore, it can be seen that a large bias can occur under an effectively unphysical condition.

[0022] In fact, Non-Patent Document 13 shows that in performance evaluation of basis energy estimation, expectation value estimation under effectively physical states enables more accurate estimation. Non-Patent Document 13 compares the performance of generalized quantum subspace expansion, which performs expectation value estimation under effectively physical states, with exponential extrapolation, which performs expectation value estimation under non-physical states. As a result, it is shown that the larger the physical noise, the lower the estimated value becomes compared to the true energy with exponential extrapolation, while the generalized quantum subspace expansion can estimate a value closer to the true value.

[0023] <<Expected Value Estimation by the Method Proposed in Non-Patent Document 12>> In the method proposed in Non-Patent Document 12, the expected value of the physical quantity O is estimated by an algorithm configured by the following steps 1 to 4.

[0024] Step 1: D different numbers of trotters, M ｉ (i=1,...,D), and select D Trotter numbers M ｉ Prepare (construct) a Trotter expansion quantum circuit corresponding to each of the above.

[0025] Step 2: For each of the D Trotter expansion type quantum circuits, perform the following steps (a) to (c).

[0026] (A) The number of trotters is M ｉ In the Trotter expansion quantum circuit, the error rate of physical noise is p ｊCalculate the expectation value of the physical quantity O under

[0027] (b) Repeat the above (a) and obtain different error rates p ｊ L expected values ​​of the physical quantity under

[0028] (c) Using the results of (b) above, polynomial or exponential extrapolation is performed from L expected values ​​at different physical noise error rates to obtain results in which errors related to physical noise are suppressed. ｍｉｔｉｇａｔｅｄ ｐｈｙｓ (M ｉ ) is written as

[0029] Step 3: D different O obtained in step 2 above ｍｉｔｉｇａｔｅｄ ｐｈｙｓ (M ｉ ) to obtain the Trotter number M ｉ This result is obtained by polynomial extrapolation with respect to ｅｓｔ Let's say.

[0030] Step 4: O ｅｓｔ is output as the expected value of the physical quantity O.

[0031] Here, when polynomial extrapolation is used in the extrapolation of physical errors in the above procedure 2 (c), O ｅｓｔ can be written as follows:

[0032] where c(M ｉ , p ｊ ) is determined by extrapolation to the physical noise and the Trotter number, and Σ ｉ，ｊ c (M ｉ , p ｊ ) = 1. Also, ρ(M ｉ , p ｊ ) is the number of trotters M ｉ and error rate p ｊ This is the physical state when the corresponding Trotter expansion quantum circuit is executed and the time evolution is simulated.

[0033] Therefore, the expected value estimator O shown in Equation 1 above ｅｓｔ The corresponding effective state is expressed as follows:

[0034] ρ ｅｓｔis the physical state ρ(M ｉ , p ｊ ), but in general, ρ ｅｓｔ ≧0, and ρ ｅｓｔ is a non-physical state.

[0035] In addition, when exponential extrapolation is used in the extrapolation of physical errors in step 2(c) above, it is similarly shown that there is no guarantee that the effective state corresponding to the expected value estimator is physical.

[0036] <Proposed Method> Below, we propose a method that combines an extrapolation method based on multiple error sources with an extrapolation method based on the algorithmic error of the Trotter expansion to suppress both errors caused by physical noise and errors caused by the algorithm. Unlike the method proposed in Non-Patent Document 12, the proposed method effectively estimates the expected value under the physical conditions shown below.

[0037] Here, E ０，ｉ ＋ is a real coefficient, and the error rate p ｉ and the reciprocal of the Trotter number ε ｉ = 1 / M ｉ It is determined by ρ ＥＭ is ρ ＥＭ ≧0 and Tr(ρ ＥＭ ) = 1, it is a physical state. ２ Note that the sum is taken over all combinations of (i, j).

[0038] <Details of the proposed method> Hamiltonian H = Σ ｋ h ｋ P ｋ (However, h ｋ is a real weight, P ｋ is the Pauli operator.) We consider the problem of finding the expectation value of the following physical quantity O under a time-evolved state generated by

[0039] Here, o α is a real number.

[0040] Time evolution U of Hamiltonian H over time T ｅｘａｃｔ =exp(-iHT)={exp(-iHT / M)} Ｍ Trotter Deployment U Ｔｒｏｔｔｅｒ ={Π ｋ exp(-ih ｋ P ｋ T / M) Ｍ Here, M is the number of divisions in the Trotter expansion. For simplicity, we will use the first-order Trotter expansion below, but please note that the proposed method also applies when using higher-order Trotter expansions.

[0041] Trotter Deployment U Ｔｒｏｔｔｅｒ The error that occurs in the quantum circuit is generated in the Trotter expansion U Ｔｒｏｔｔｅｒ There are two types of errors: errors caused by physical noise acting on the U ｅｘａｃｔ and U Ｔｒｏｔｔｅｒ The error O(T ２ / M) (i.e., errors caused by the Trotter expansion algorithm). In order to suppress these two types of errors, the proposed method uses the state ρ ＥＭ effectively creates

[0042] E ＋ = (E ｉ，ｊ ＋ ) In this case, E ＋ is p ｉ and ε ｉ Hereinafter, a method for constructing the matrix E will be described.

[0043] When the expectation value 〈O〉(p,ε) of the physical quantity O is expanded to the Lth order by Taylor expansion, the following is obtained:

[0044] However, A ｉｊ is expressed as follows:

[0045] In other words, A ｉｊ is the differential coefficient of the Taylor series of two variables.

[0046] For simplicity, the case where L=2 is considered below. In this case, the above equation 5 becomes: 〈O〉(p,ε)=〈O〉(0,0)+A ０１ ε+A １０ p+A １１ pε+A ０２ ε ２ +A ２０ p ２ can be written as:

[0047] Therefore, D expected values ​​〈O〉(p １ , ε １ ),...,〈O〉(p Ｄ , ε Ｄ ) to the second order, the following equations can be obtained:

[0048] Ea=r where E, a, and r are expressed as follows.

[0049] This allows the matrix E to be determined, and the pseudo-inverse matrix of this matrix E is E ＋ For simplicity, the case where L=2 has been described above, but the matrix E can also be determined in the same way for a general L-th order.

[0050] As a result, the state ρ ＥＭ The expectation value of the physical quantity O under

[0051] This is the estimator of the proposed method. Note that the expected value <O> shown in the above equation (8) ｅｓｔ (To be precise, the expected value of the physical quantity O) ＥＭ Note that it is not necessary to provide

[0052] Hereinafter, the expected value of the physical quantity O is calculated by the estimator shown in the above equation 8. ｅｓｔ A quantum computing device 10 for estimating the following will be described.

[0053] <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 an example of the configuration of the quantum computing device 10 according to this embodiment.

[0054] As shown in FIG. 1, a quantum computing device 10 according to this embodiment includes a control device 100 and a quantum processor 200.

[0055] 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.

[0056] 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.

[0057] <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. 2. Fig. 2 is a diagram showing an example of the hardware configuration of the control device 100 according to this embodiment.

[0058] 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 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.

[0059] 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.

[0060] 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.

[0061] 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).

[0062] 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 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.

[0063] <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. 3. Fig. 3 is a diagram showing an example of the functional configuration of the control device 100 according to this embodiment.

[0064] As shown in FIG. 3 , the control device 100 according to this embodiment includes a parameter selection unit 110, a quantum circuit construction unit 111, an order setting unit 112, a trace calculation unit 113, a pseudo-inverse matrix calculation unit 114, an expectation calculation unit 115, a determination unit 116, and an output unit 117. These units are realized, for example, by a process in which one or more programs installed in the control device 100 are executed by a processor 108 or the like. The control device 100 according to this embodiment also includes a parameter storage unit 118. The parameter storage unit 118 is realized, for example, by a storage area of ​​the auxiliary storage device 107 or the like. However, the parameter storage unit 118 may also be realized, for example, by a storage area of ​​a storage device (e.g., a storage device provided in a database server) communicatively connected to the control device 100.

[0065] The parameter selection unit 110 selects the parameter stored in the parameter storage unit 118 (p ｉ , M ｉ ) among D (p ｉ , M ｉ ) is selected, where D is a predetermined integer of 1 or more.

[0066] The quantum circuit construction unit 111 constructs D (p ｉ , M ｉ ) and construct D Trotter expansion quantum circuits corresponding to the respective

[0067] The order setting unit 112 sets the value of the order L when Taylor-seriesing the expected value <O>(p, ε) of the physical quantity O. At this time, the order setting unit 112 starts from L=2 and adds 1 to the value of L until the determining unit 116 determines that a predetermined termination condition is satisfied.

[0068] The trace calculation unit 113 calculates the expected value <O> shown in the above equation 8 using the quantum processor 200. ｅｓｔ Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) and Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) to calculate.

[0069] The pseudo-inverse matrix calculation unit 114 calculates the D (p ｉ , M ｉ ) to construct a matrix E, and then calculate the pseudo-inverse matrix E ＋ Calculate.

[0070] The expected value calculation unit 115 calculates the Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) and Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) and the pseudo inverse matrix E calculated by the pseudo inverse matrix calculation unit 114 ＋ Using the above, the expected value <O> shown in the above equation 8 is ｅｓｔ Calculate.

[0071] The determination unit 116 determines whether a predetermined termination condition is satisfied. Here, the termination condition may be, for example, L>2 and <0> ｅｓｔ The change in the value of <O> is less than the threshold value. ｅｓｔ The change in the calculated value of 〈O〉 at a certain order L is ｅｓｔ <O> ｅｓｔ （Ｌ） For example, |〈O〉 ｅｓｔ （Ｌ） -〈O〉 ｅｓｔ （Ｌ－１） |This is what I mean.

[0072] The output unit 117 outputs <O> when the determination unit 116 determines that the termination condition is satisfied. ｅｓｔ is output as an expected value of the physical quantity O 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.

[0073] The parameter storage unit 118 stores the error rate p ｉand the number of trotters M ｉ The pair (p ｉ , M ｉ ) is stored as a parameter.

[0074] <Expected Value Estimation Process> An example of the expected value estimation process according to this embodiment will be described with reference to Fig. 4. Fig. 4 is a flowchart showing an example of the expected value estimation process according to this embodiment.

[0075] The parameter selection unit 110 selects the parameter stored in the parameter storage unit 118 (p ｉ , M ｉ ) among D (p ｉ , M ｉ ) is selected (step S101).

[0076] The quantum circuit construction unit 111 constructs D (p ｉ , M ｉ ) are constructed (step S102).

[0077] The degree setting unit 112 sets the value of the degree L as L←2 (step S103).

[0078] The trace calculation unit 113 calculates ２ For all combinations of (i, j), Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) is calculated (step S104). ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) can be expressed as follows:

[0079] Therefore, the trace calculation unit 113 calculates Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) can be calculated.

[0080] Step 1-1: The trace calculation unit 113 controls the quantum processor 200 to calculate (p ｉ , M ｉ) corresponding to the physical state ρ(p ｉ , ε ｉ ) and (p ｊ , M ｊ ) corresponding to the physical state ρ(p ｊ , ε ｊ ) and the virtual distillation method (References 1-3) is applied to α By taking measurements on P α ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) output value.

[0081] Step 1-2: The trace calculation unit 113 calculates Tr(P α ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ After calculating the weight o α The sum for α is calculated by multiplying Tr(P α ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) is the P by the quantum processor 200 α ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) can be calculated by repeatedly measuring the output value of the test piece a sufficient number of times and averaging the output values.

[0082] The trace calculation unit 113 calculates ２ For all combinations of (i, j), Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ The trace calculation unit 113 calculates Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) can be calculated.

[0083] Step 2-1: The trace calculation unit 113 controls the quantum processor 200 to calculate (p ｉ , M ｉ ) corresponding to the physical state ρ(p ｉ , ε ｉ ) and (p ｊ , M ｊ ) corresponding to the physical state ρ(p ｊ , ε ｊ ) and ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) output value.

[0084] Step 2-2: The trace calculation unit 113 calculates Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) is calculated. ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) is the quantum computation of ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ ) can be calculated by repeatedly measuring the output value of the test piece a sufficient number of times and averaging the output values.

[0085] The pseudo-inverse matrix calculation unit 114 calculates the D (p ｉ , M ｉ ) and then construct a matrix E, and then calculate the pseudo-inverse matrix E ＋ = (E ｉ，ｊ ＋ ) is calculated (step S106).

[0086] The expected value calculation unit 115 calculates the Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) and Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ, ε ｊ )) and the pseudo-inverse matrix E calculated in step S106 above. ＋ Using the above, the expected value <O> shown in the above equation 8 is ｅｓｔ is calculated (step S107).

[0087] The determination unit 116 determines whether L>2 and <O> ｅｓｔ It is determined whether the amount of change in becomes less than the threshold value (step S108).

[0088] In step S108 above, L>2 and <O> ｅｓｔ If it is not determined that the amount of change in L is less than the threshold, the order setting unit 112 updates the value of the order L by setting L←L+1 (step S109). After the value of the order L is updated, the process returns to step S104, and steps S104 to S107 are executed using the updated value of the order L.

[0089] On the other hand, if L>2 in step S108 and <O> ｅｓｔ When it is determined that the amount of change in <O> at the current value of the degree L is less than the threshold, the output unit 117 ｅｓｔ is output as an expected value of the physical quantity O to a predetermined output destination (step S110).

[0090] <Comparison with Conventional Methods> Comparative Example 1 One result of comparing the proposed method with conventional methods will be described with reference to FIG. 5. FIG. 5 is a diagram (part 1) showing a comparison with a conventional method. Note that the example shown in FIG. 5 is the result of simulating the time evolution of a one-dimensional transverse magnetic field Ising model. Furthermore, the method described in Non-Patent Document 12 was adopted as the conventional method. Furthermore, the trace distance represents the closeness between the effective state and the exact state, and it can be said that the closer it is to 0, the better the error suppression.

[0091] As shown in Figure 5, the proposed method has a smaller tracing distance than the conventional method, and therefore can be said to be able to suppress errors (i.e., both errors caused by physical noise and errors caused by the algorithm) compared to the conventional method.

[0092] Comparative Example 2 Another result of comparing the proposed method with the conventional method will be described with reference to Figs. 6 and 7. Fig. 6 is a diagram showing an example of settings for comparison with the conventional method. Fig. 7 is a diagram (part 2) showing a comparative example with the conventional method. In Comparative Example 2, as in Comparative Example 1, the time evolution of a one-dimensional transverse magnetic field Ising model was simulated.

[0093] For the nine noisy data points shown in Figure 6, we compared the proposed method with a case where no error suppression method was used (hereinafter referred to as "without error suppression"), a case where polynomial extrapolation of the physical error rate followed by polynomial extrapolation of the Trotter number (hereinafter referred to as "conventional method 1"), and a case where exponential extrapolation of the physical error rate followed by polynomial extrapolation of the Trotter number (hereinafter referred to as "conventional method 2"). Conventional methods 1 and 2 are both described in Non-Patent Document 12. The absolute error in Figure 6 is the absolute error between the expected value under a strict time evolution state and the expected value under each Trotter number and error rate.

[0094] In this case, as shown in Figure 7, the proposed method has the smallest estimation error compared to the conventional method, and therefore can be said to be able to estimate the expected value with the smallest estimation error. Note that without error suppression, the data point with the smallest absolute error was used when calculating the estimation error.

[0095] <Summary> As described above, the quantum computing device 10 according to this embodiment can suppress both errors caused by physical noise in the quantum processor 200 and errors caused by the algorithm of the Trotter expansion quantum circuit. Moreover, while conventional methods perform expected value estimation under effectively non-physical conditions, which can lead to bias, the quantum computing device 10 according to this embodiment performs expected value estimation under effectively physical conditions, thereby enabling more accurate expected value estimation.

[0096] 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.

[0097] [References] Reference 1: WJ Huggins, S. McArdle, TE O'Brien, J. Lee, NC Rubin, S. Boixo, KB Whaley, R. Babbush, and JR McClean, Virtual Distillation for Quantum Error Mitigation, Phys. Rev. X 11, 041036 (2021). Reference 2: B. Koczor, Exponential Error Suppression for Near-Term Quantum Devices, Phys. Rev.

[0098] 10 Quantum computing device 100 Control device 101 Force input device 102 Display device 103 External I / F 103a Recording medium 104 Communication I / F 105 RAM 106 ROM 107 Auxiliary memory device 108プロセッサ109 バス110 パラメータ selection department 111 Quantum circuit construction part 112 Number of times setting part 113 トレース calculation part 114 Suspected inverse column calculation part 115 Expected value calculation part 116 Determination part 117 Output part 118 Parameter memory part

Claims

1. An estimation device for estimating an expected value of a physical quantity under the state of a time-evolved quantum system, comprising: an estimation unit that estimates the expected value of the physical quantity using an estimator that estimates the expected value of the physical quantity under a physical state that can suppress errors caused by physical noise when the time evolution of the quantum system is calculated by a quantum processor and errors caused by an algorithm that approximates the time evolution.

2. The algorithm for approximating the time evolution is the Trotter expansion, and the estimator is defined as follows: i∈{1, ..., D} (where D is a predetermined integer of 2 or more), the physical quantity is O, and the error rate of the physical noise in the quantum circuit implementing the Trotter expansion is p ｉ , the number of Trotters in the Trotter expansion is M ｉ , the error rate p ｉ Under the above Trotter number M ｉ The state when the time evolution of the quantum system is simulated by the quantum circuit of ｉ , ε ｉ ), the error rate p ｉ and the Trotter number M ｉ The pseudo-inverse matrix of the matrix E determined by ＋ = (E ｉ，ｊ ＋ ), then E for (i, j) ∈ ({1, ..., D}, {1, ..., D}) ０，ｉ ＋ E ０，ｊ ＋ Tr(Oρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ )) is the numerator, and E for (i, j) ∈ ({1, ..., D}, {1, ..., D}) ０，ｉ ＋ E ０，ｊ ＋ Tr(ρ(p ｉ , ε ｉ ) ρ(p ｊ , ε ｊ 2. The estimation device according to claim 1, wherein the value is expressed as a value having a denominator of the sum of 3. The physical quantity is O = o １ P １ +...+o Ｌ P Ｌ (However, α is a real number, P α 3. The estimation device according to claim 1 or 2, wherein L is expressed as the Pauli operator), and the estimation unit estimates the expected value of the physical quantity sequentially while adding one to the value of L one by one starting from L=2, and when L>2 and an amount of change between the expected value of the physical quantity estimated when L−1 and the expected value of the physical quantity estimated when L becomes less than a predetermined threshold, the expected value of the physical quantity estimated when L is set to L as a final estimation result.

4. An estimation device for estimating an expected value of a physical quantity under the state of a time-evolved quantum system, comprising: an estimation procedure executed by a computer to estimate the expected value of the physical quantity using an estimator that estimates the expected value of a physical quantity under a physical state that can suppress errors caused by physical noise when the time evolution of the quantum system is calculated by a quantum processor and errors caused by an algorithm that approximates the time evolution.

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