Quantum information processing method, classical computer, hybrid system, and quantum information processing program

Abstract

To efficiently obtain an expectation value of an observable by using a quantum computer.SOLUTION: A classic computer 110, when a state |ψ> of N quantum bits is expressed by linear coupling of a plurality of calculation grounds, acquires measured values of the plurality of calculation grounds measured by a quantum computer 120. Then, the classic computer 110: on the basis of the measured values of the plurality of calculation grounds, selects R measured values in which the number of times of appearance is large or the R measured values in which the number of times of appearance is a predetermined value or larger from the plurality of measured values; on the basis of the number of times of appearance of the selected R measured values, calculates a weighting factor fr with respect to the R calculation grounds |zr> (r=1,..., R) according to the number of times of appearance; and performs approximate calculation of an expectation value <ψ|O|ψ> of an observable O, by a weighted sum of a plurality of transition matrix elements <zr|O|zr'> which sets a combination of the weighting factor fr and a weighting factor fr' as a weighting.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The disclosed technology relates to a quantum information processing method, a classical computer, a hybrid system, and a quantum information processing program. [Background technology]

[0002] Variational quantum algorithms for performing quantum computation using Noisy Intermediate-Scale Quantum (NISQ) devices are known (see, for example, Non-Patent Documents 1 and 2). To perform a variational quantum algorithm, it is necessary to calculate the expectation value of the physical quantity to be observed (hereinafter, also simply referred to as "observable") using a quantum computer.

[0003] When calculating the expectation value of an observable using a quantum computer, a state is generated by the quantum computer and then the state is measured. The expectation value of the observable is then calculated based on the measurement result of the state by the quantum computer. Note that a series of processes consisting of one state generation by the quantum computer and the measurement of that state is also called a "shot."

[0004] When applying the variational quantum algorithm to quantum chemical calculations, the number of shots required to sufficiently reduce the statistical fluctuation of the state measurement may be excessive. For example, Non-Patent Document 3 considers a quantum chemical calculation problem to obtain the energy of an actual molecule, and shows that a huge number of shots are required to reduce the statistical fluctuation more than the energy accuracy required in the quantum chemical field, and it takes several days to obtain the expected value of the energy just once. For this reason, for example, Non-Patent Document 4 attempts to reduce the total number of shots by measuring multiple Pauli operators simultaneously. [Prior art documents] [Non-patent literature]

[0005] [Non-Patent Document 1] Peruzzo, A., McClean, J., Shadbolt, P. et al. “A variational eigenvalue solver on a photonic quantum processor”. Nat Commun 5, 4213 (2014). [Non-Patent Document 2] “Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation”, Suguru Endo, Zhenyu Cai, Simon C. Benjamin, Xiao Yuan, Journal of the Physical Society of Japan, 90, 032001 (2021) [Non-Patent Document 3] Gonthier, JF, Radin, MD, Buda, C., Doskocil, EJ, Abuan, CM, and Romero, J. “Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver”. [Non-Patent Document 4] Rubin, NC, Babbush, R., and McClean J., “Application of fermionic marginal constraints to hybrid quantum algorithms”. New J. Phys. 20, 053020 (2018). Summary of the Invention [Problem to be solved by the invention]

[0006] The disclosed technology has been made in consideration of the above-mentioned circumstances, and aims to provide a quantum information processing method, a classical computer, a hybrid system, and a quantum information processing program that can efficiently obtain the expected value of an observable using a quantum computer. [Means for solving the problem]

[0007] In order to achieve the above object, a quantum information processing method according to one embodiment of the present disclosure is a quantum information processing method executed by a classical computer in a hybrid system including a classical computer and a quantum computer, and when a state |ψ> of N quantum bits is expressed by a linear combination of a plurality of computational bases, measurement values ​​of each of a plurality of computational bases measured by the quantum computer are acquired, and R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value are selected from the plurality of measurement values ​​based on the measurement values ​​of each of the plurality of computational bases, and R computational bases |z corresponding to the occurrence count are selected based on the occurrence counts of the selected R measurement values. r >(r=1, ,R) weighting factor f r Calculate the weighting factor f r and the weighting factor f r’ The transition matrix elements are weighted by the combination of <z r |O|z r’ > is a quantum information processing method in which the expectation value <ψ|O|ψ> of an observable O is approximated by a weighted sum of <ψ|O|ψ>, and the process is performed by a classical computer. Effect of the Invention

[0008] The disclosed technology has the effect of efficiently obtaining the expected value of an observable using a quantum computer. [Brief description of the drawings]

[0009] [Figure 1] 1 is a diagram illustrating an example of a schematic configuration of a hybrid system 100 according to an embodiment of the present invention. [Diagram 2]FIG. 1 is a schematic block diagram of a classical computer 110, a control device 121, and a computer functioning as a user terminal 130. [Diagram 3] FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment. [Figure 4] FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0010] Hereinafter, embodiments of the disclosed technology will be described in detail with reference to the drawings.

[0011] <Hybrid system 100 according to the embodiment>

[0012] Fig. 1 shows a hybrid system 100 according to an embodiment. The hybrid system 100 of the present embodiment includes a classical computer 110, a quantum computer 120, and a user terminal 130. As shown in Fig. 1, the classical computer 110, the quantum computer 120, and the user terminal 130 are connected via a computer network, such as an Internet Protocol (IP) network, for example.

[0013] In the hybrid system 100 of this embodiment, the quantum computer 120 performs a predetermined quantum computation in response to a request from the classical computer 110, and outputs the computation result of the quantum computation to the classical computer 110. The classical computer 110 outputs the computation result according to the quantum computation to the user terminal 130. In this way, the hybrid system 100 as a whole executes a predetermined computation process.

[0014] The classical computer 110 is configured by having a communication unit 111 such as a communication interface, a processing unit 112 such as a processor or a CPU (Central processing unit), and an information storage unit 113 including a storage device or storage medium such as a memory or a hard disk, and by executing a program for performing each process. The classical computer 110 may include one or more devices or servers. The program may include one or more programs, and may be recorded in a computer-readable storage medium to be a non-transient program product.

[0015] As an example, quantum computer 120 generates an electromagnetic wave to be irradiated to at least any one of quantum bits in quantum bit group 123 based on information transmitted from classical computer 110. Then, quantum computer 120 executes a quantum circuit by irradiating at least any one of quantum bits in quantum bit group 123 with the generated electromagnetic wave.

[0016] 1, the quantum computer 120 includes a control device 121 that communicates with the classical computer 110, an electromagnetic wave generating device 122 that generates electromagnetic waves in response to a request from the control device 121, and a group of quantum bits 123 that receives electromagnetic wave irradiation from the electromagnetic wave generating device 122. The electromagnetic wave generating device 122 and the group of quantum bits 123 in the quantum computer 120 are also QPUs (Quantum Processing Units). Note that in this embodiment, the term "quantum computer" does not mean that no operations are performed using classical bits, but refers to a computer that includes operations using quantum bits.

[0017] The control device 121 is a classical computer that performs operations using classical bits, and alternatively performs some or all of the processes described in this specification as being performed by the classical computer 110. For example, the control device 121 may store or determine a quantum circuit in advance, and generate quantum gate information for executing the quantum circuit U(θ) in the set of quantum bits 123 in response to receiving a parameter θ of the quantum circuit U(θ).

[0018] The user terminal 130 is a classical computer that performs operations using classical bits. The user terminal 130 accepts information input by a user and executes processing according to the information.

[0019] The classical computer 110, the control device 121, and the user terminal 130 can be realized, for example, by a computer 50 shown in Fig. 2. The computer 50 includes a central processing unit (CPU) 51, a memory 52 as a temporary storage area, and a non-volatile storage unit 53. The computer 50 also includes an input / output interface (I / F) 54 to which an external device, an output device, etc. are connected, and a read / write (R / W) unit 55 that controls reading and writing of data from and to a recording medium. The computer 50 also includes a network I / F 56 that is connected to a network such as the Internet. The CPU 51, the memory 52, the storage unit 53, the input / output I / F 54, the R / W unit 55, and the network I / F 56 are connected to one another via a bus 57.

[0020] The hybrid system 100 of the embodiment efficiently calculates the expectation value <ψ|O|ψ> of the observable O. Hereinafter, the prerequisites for the processing executed by the hybrid system 100 of the embodiment will be described.

[0021] [1. Problem setting and background]

[0022] Consider the state |ψ> of N quantum bits generated on a quantum computer. The state |ψ> is a vertical vector having 2 to the power of N components. In this embodiment, it is possible to express |ψ>=U|0> using a specific quantum circuit U, and it is possible to repeatedly generate the state |ψ> on a quantum computer. Note that |0> represents the initialized state of N quantum bits.

[0023] The quantum information processing method executed by the hybrid system 100 of this embodiment is a method for efficiently measuring the expectation value of an observable O on a quantum computer, as shown in the following equation (1).

[0024]

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[0025] [2. Conventional methods for measuring expectations and their issues]

[0026] The conventional method for measuring the expectation of an observable O consists of the following steps:

[0027] (1) Decomposition of an observable O into Pauli operators An observable O on N qubits can be expressed as a linear combination of the following Pauli operators:

[0028]

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[0029] The observable O is expressed by the Pauli operator P as shown in the following equation (2): i It is expressed as a linear combination of

[0030]

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[0031] Here, I, X, Y, and Z are the identity operator and the Pauli matrix of one quantum bit. i is the expansion coefficient, and M is the total number of Pauli operators. Based on the above equation (2), the expectation value 〈ψ|O|ψ〉 of the operator observable O is expressed by the following equation (3).

[0032]

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[0033] (2) Measurement of the expectation value of the Pauli operator Next, the terms on the right side of the above equation (3) are calculated by a quantum computer. i |ψ> is measured. <ψ|P i |ψ> is the expectation value of the Pauli operator. The expectation value of the Pauli operator <ψ|P i |ψ> can be obtained by a standard measurement operation on a quantum computer. Specifically, we generate the state |ψ> once on a quantum computer and measure it with the Pauli operator P i When a measurement operation is performed according to the above, the result will be either +1 or -1. The probability is determined by |ψ> and P i is determined by a combination of

[0034] State i The l obtained by generating i The average of the measurements is expected to be <ψ|P i |ψ>. Note that l i Each measurement result has a value of either +1 or -1.

[0035] For example, if a state is generated 100 times, +1 is obtained 30 times, and -1 is obtained 70 times, then the expected value is estimated to be (30-70) / 100=-0.4.

[0036] Note that since the measurement results are probabilistic, the estimated expected value <ψ|P i There is statistical fluctuation in |ψ>. This statistical fluctuation is the number of trials l to get +1 or -1.i For 1 / √l i As mentioned above, in the field of quantum computers, the series of processes of generating a state and measuring it is also called a "shot." For this reason, in the following, i is also simply referred to as the "number of shots."

[0037] (3) Addition of expectation values ​​of Pauli operators Next, the expected value obtained in (2) above is <ψ|P i Based on the value of |ψ>, the expected value <ψ|O|ψ> of the observable O is calculated by calculating the above equation (3).

[0038] Thus, although the NISQ device cannot directly measure the expectation value of the observable O, <ψ|O|ψ>, it can measure the expectation value of the Pauli operator, <ψ|P i Since |ψ> is easily measurable, the expectation value of the Pauli operator <ψ|P i |ψ>, the expectation value <ψ|O|ψ> of the observable O is calculated.

[0039] [3. Issues] When using a variational quantum algorithm to perform quantum chemical calculations using an NISQ device, the total number of shots required to sufficiently reduce the statistical fluctuations of the measurement is l1+···+l M In many cases, M is too large. In this case, quantum chemical calculations may not be completed within a realistic time. In particular, in systems with a large number of quantum bits, M = O(N 4 ) Pauli operators, it is necessary to measure the expectation values ​​of the observables O. Therefore, in this embodiment, a method for efficiently calculating the expectation value of the observable O is proposed.

[0040] [4. Method for Measuring Expected Value in the Present Embodiment] The expectation value of the observable O in the state |ψ> is expressed by the following equation (4).

[0041]

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[0042] In general, the state |ψ> is expressed as a linear combination of basis vectors belonging to the complete system. There are various possibilities for the complete system, but in the following, in order to concretely proceed with the discussion, we use the complete system as the computational basis |n> (n=0,1,...,2 N -1) is used. The state |ψ> is expanded as shown in the following equation (5) using the computational basis |n>.

[0043]

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[0044] Here, the expansion coefficient α n is a complex number calculated by the inner product of the state |ψ> and the computational basis |n>.

[0045]

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[0046] α in the above formula (6) n is also called the probability amplitude, and its absolute value squared, |α n | 2 represents the probability. Therefore, if a state |ψ> is prepared on a quantum computer and a standard measurement operation is performed on that state |ψ>, the measurement result will be n=0,1,...,2 N -1 is probabilistically obtained. And |α n | 2 represents the probability that the measurement result will be a particular value n.

[0047] Conversely, by preparing a state |ψ> on a quantum computer and repeating the operation of projecting and measuring to the computational basis |n>, a frequency distribution of the measurement result n can be obtained, and the true probability |α n | 2 It is possible to estimate the value of

[0048] Here, the probability amplitude α n We can rewrite the expectation value 〈ψ|O|ψ〉 using the computational basis |n〉(n=0,1,...,2 N -1) forms a complete system, the following equation (7) is derived.

[0049]

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[0050] In the last line of the above formula (7), α n = 0 or α m The above formula (7) does not include the term |α m | 2 |α n | 2 With weights as<m|O|n> / (α m α n * ) gives the expectation value at the state |ψ> of the observable O. As explained below,<m|O|n> can be computed efficiently by a classical computer. m α n * can be measured using a quantum computer. Therefore, |α n | 2 and α m α n * was calculated using a classical computer.<m|O|n> By combining these, we obtain the expectation value of the observable O, 〈ψ|O|ψ〉.

[0051] Specifically, the transition matrix elements corresponding to the combination of the computational basis |n> and the computational basis |m><m|O|n> is the Pauli operator P of the observable O given by the above equation (2). i By decomposition using the formula, it is expanded as follows.

[0052]

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[0053] In the above formula <m|P i |n> can be efficiently calculated by a classical computer. Note that being efficiently calculated by a classical computer means that it can be calculated in linear time with respect to the number of quantum bits. And the transition matrix element<m|O|n> was calculated by a classical computer. <m|P i It is obtained by adding |n>.

[0054] Next, α m α n * When m=n, α m α n * =|α n | 2 Therefore, as mentioned above, it is possible to measure using a quantum computer by sampling in the computational basis |n> of |ψ>. On the other hand, when m ≠ n, a different measurement operation on the quantum computer is required. To explain this point, the following transformation is necessary.

[0055]

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[0056] Here, A m,n ,B m,n is expressed by the following equation.

[0057]

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[0058] In addition, Am,n ,B m,n can be estimated using a quantum computer by projective measurement. m,n ,B m,n and weights |α n | 2 By combining with α m α n * It becomes possible to calculate the value of

[0059] In general, α m α n * Since α has a complex phase, different terms cancel each other out or reinforce each other in the sum of the above equation (7). m α n * is called the "interference weight."

[0060] Obtained as above, |α n | 2 , A m,n ,B m,n , and<m|O|n> By combining the values ​​of , the expectation value 〈ψ|O|ψ〉 is calculated according to the following equation (10).

[0061]

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[0062] In addition, 2 N ×2 N Since an exponentially large number of terms must be added up, it may seem that computational costs will be difficult. However, depending on the type of problem to be solved, such as quantum chemical calculations, it is empirically possible to greatly reduce the number of terms to be added up.

[0063] For example, in quantum chemical problems, expectation value calculations are often performed for eigenstates of the Hamiltonian, such as the ground state or low-energy excited states, and it is known that in many systems, the probability amplitude corresponding to the Hartree-Fock state dominates such states, with other components being relatively small.

[0064] In addition, it is mathematically known that due to the existence of conserved quantities in the target system (e.g., the conservation of the number of electrons and the conservation of spin, etc.), the eigenstates have finite amplitudes only in a limited part of the entire basis. For this reason, the weight |α n | 2 has a significant value only for certain elements of n, and for other elements it is zero or a value that is statistically negligible in estimating the expectation value in a quantum computer. Therefore, when taking the sum for m and n, such bases can be ignored from the beginning. Therefore, the transition matrix element<m|O|n> All we need to do is evaluate the following:

[0065] Also, for such a pair (m,n), the interference weight α m α n * When calculating A for a part (m,n), m,n ,B m,n It turns out that measuring is sufficient.

[0066] To verify this, for any l (where α l ≠ 0), consider the following identity (11):

[0067]

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[0068] In the above equation (11), for example, if l=0, the interference weight α0α n * and weight |α0| 2If the value of is known, then all other interference weights α m α n * The value of can be obtained from the above equation (11). Therefore, α0α n * In order to calculate the value of A using the above formula (8), 0,n ,B 0,n If you measure the other A m,n ,B m,n It can be seen that no measurement is necessary.

[0069] To summarize the above points, in this embodiment, the probability amplitude α n (n=0,1,...,2 N -1), the expected value <ψ|O|ψ> of the observable O is expressed as follows:

[0070]

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[0071] In this embodiment, the weight |α n | 2 and interference weight α m α n * was calculated by a classical computer.<m|O|n> By combining these, we approximate the expectation value <ψ|O|ψ> of the observable O.

[0072] In quantum chemistry problems, which are important applications of this method, the state |ψ> has a bias with respect to the calculation basis |n> in many interesting cases, so the weight |α n | 2 takes a significant non-zero value only for a limited number of n, and we determined that this greatly reduces the number of terms to be added in the sum of equation (10) above.

[0073] [Advantages of the proposed method over conventional methods]

[0074] In this embodiment, when measuring the expectation value <ψ|O|ψ> of the observable O, the observable O is treated as a Pauli operator P i Instead of expanding and measuring the state |ψ>, we use a computational basis |n>(n=0,1,...,2 N -1) was performed to avoid the problem of "measurement of multiple Pauli operators". 0,n ,B 0,n However, for biased states |ψ> that often appear in quantum chemical calculations, the weight |α n | 2 Since it is necessary to consider only certain (hereinafter referred to as R) computational bases |n> for which |n takes a significant value, the number of quantities to be measured can be relatively small.

[0075] Therefore, when the method of this embodiment is used, the measurement on the quantum computer is as follows.

[0076] (1) Calculate the state |ψ> by repeatedly sampling in the basis |n>. n | 2 Measure. (2) For a given n, A 0,n Depending on n, R-1 types of measurements are required. (3) For a given n, B 0,n Depending on n, R-1 types of measurements are required.

[0077] Therefore, according to the method of this embodiment, a total of 1+2(R-1) types of measurements are required. In contrast, in the conventional method, M types of measurements are required, corresponding to the number of Pauli operators that appear in the expansion of the observable O. Considering the Hamiltonian of quantum chemistry, the number of Pauli operators is M=O(N 4 ), which is very large, in the method of this embodiment, it is empirically known that R is much smaller than this, so the number of types of measurements required is greatly reduced. Therefore, when measuring the expectation value <ψ|H|ψ> of the Hamiltonian H, the total number of shots required to sufficiently reduce statistical fluctuations can also be greatly reduced.

[0078] [Description of the algorithm corresponding to this method]

[0079] The overall flow of the algorithm executed by the hybrid system 100 of this embodiment is as follows.

[0080] 1. The quantum computer 120 generates a state |ψ> of N quantum bits. Then, the quantum computer 120 samples the state |ψ> in the computational basis |n> and obtains one of the values ​​0, 1, ..., as the measurement value. The quantum computer 120 repeats the generation of the state |ψ> and the measurement of the computational basis |n> L times to obtain a sequence of measurement values ​​{x}=x (1) ,x (2) ,···,x (L) get.

[0081] 2. The classical computer 110 selects R measurement values ​​with a high occurrence frequency from the sequence of measurement values ​​{x}. Alternatively, the classical computer 110 may select R measurement values ​​with an occurrence frequency equal to or greater than a predetermined value from the sequence of measurement values ​​{x}. Next, the classical computer 110 selects a sequence of values ​​{z}=z, which is arranged in order of occurrence frequency based on each of the selected R measurement values. (1) ,z (2) ,...,z (R) Next, the classical computer 110 sets the number of occurrences T r (r=1, . . . , R). Then, the classical computer 110 calculates the weighting coefficient f r Calculate.

[0082]

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[0083] As shown in the following formula, the weighting factor f r is the weight |α n | 2 It is also an approximation of

[0084]

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[0085] 3. The classical computer 110 calculates the transition matrix element based on {z}. <z r |O|z r’ ,R; r'=1,,R.

[0086] 4. The quantum computer 120 is in the state |φ A,r Projection operator |φ for > A,r ><φ A,r By measuring |, the projection operator |φ A,r ><φ A,r The classical computer 110 obtains the measurement result of |φ A,r ><φ A,r Based on the measurement results of |, the projection operator |φ A,r ><φ A,r Expected value of |A r (r = 2, , R). Note that the state |φ A,r > is expressed by the following formula: Furthermore, L' measurements are taken for each r.

[0087]

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[0088] 5. Quantum computer 120 is in the state |φ B,r Projection operator |φ for > B,r ><φ B,r By measuring |, the projection operator |φ B,r ><φ B,r The classical computer 110 obtains the measurement result of |φ B,r ><φ B,r Based on the measurement results of |, the projection operator |φ B,r ><φ B,r Expected value of |B r (r = 2, , R). Note that the state |φB,r > is expressed by the following formula: Furthermore, L'' measurements are taken for each r.

[0089]

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[0090] 6. The classical computer 110 calculates the weighting factor f r (r=1, ,R) and the expected value A r (r=2, ,R) and the expected value B r (r=2, , R) according to the following equation (13), the interference weight g r Approximate calculation is performed.

[0091]

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[0092] In addition, the interference weight g r can also be expressed by the following equation:

[0093]

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[0094] The classical computer 110 calculates the other component of the interference weight, α, according to the following equation, which is obtained from equation (11) above: zr α zr’ * Calculate.

[0095]

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[0096] 7. The classical computer 110 calculates the interference weight g r α corresponds to z1 α zr *and the other component of the interference weight, α zr α zr’ * and, <z r |O|z r’ >, we approximately calculate the expectation value <ψ|O|ψ> of the observable O according to the following equation (14).

[0097]

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[0098] [Method of estimating the expectation value of the projection operator using a quantum computer]

[0099] Next, we explain how to estimate the expectation value of the projection operator using a quantum computer. The projection operator is P = P 2 In particular, when P=|φ><φ|, P is said to be a projection operator for state |φ>. The expectation value of the projection operator P in state |ψ> is expressed by the following equation (15).

[0100]

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[0101] A method for estimating the probability p in the above equation (15) will be described below. First, we prepare a quantum circuit U that converts the state |φ> into a state |n> represented by a single component of the computational basis. Note that any quantum state |Φ> of N qubits can be expressed by a computational basis |k> (k=0,1,...,2 N -1), this is expressed by the following equation (16).

[0102]

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[0103] Here, |φ> can be expressed by only a single component of the computational basis, meaning that only one of α0, α1, … in the above formula (16) is 1, and the others are all 0. The state |φ>, state |n>, and quantum circuit U are expressed by the following formula (17).

[0104]

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[0105] Such a quantum circuit U can exist. If the state |φ> is originally represented by a single component of the computational basis, such as |n>, then there is no need for conversion, and it is sufficient to consider a circuit that does nothing (U=I). By combining the above formula (15), which defines the probability p, with the above formula (17), we obtain the following formula (18).

[0106]

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[0107] Because the computational basis is the simultaneous eigenstate of the Z operator for each quantum bit, and because of the principles of quantum mechanics (e.g., Born's rule), we can see that the probability p is the probability that when a quantum bit is measured in the Z basis on the state U|ψ>, the measurement result is n.

[0108] This probability is estimated as follows. First, the state |ψ> is generated. Then, the quantum circuit U is applied to this state. Finally, a measurement is performed using the Z basis to obtain the measurement result. This is considered as a series of processes, and the measurement results are collected by repeating this process l times. Then, the number l' of times that result n was obtained out of l measurements is found, and the ratio l' / l is taken as the estimate of p. The state |φ>, which is represented by only two components of the computational basis, is expressed by the following equation (19).

[0109]

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[0110] In the method of this embodiment, there are many cases where the expected value of the projection operator for the state |φ> in the above formula (19) is estimated. In such cases, it is preferable to prepare a quantum circuit U expressed by the following formula (20).

[0111]

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[0112] Although there is no unique method for constructing a quantum circuit that operates as expressed by the above formula (20), it can be constructed using various techniques.

[0113] [Operation of the Hybrid System 100 of the Embodiment]

[0114] Next, a specific operation of the hybrid system 100 according to the embodiment will be described. In each device of the hybrid system 100, the processes shown in Figs.

[0115] The hybrid system 100 approximately calculates the expectation value <ψ|O|ψ> of the observable O represented by the following equation (A1), using the following equation (A2).

[0116]

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[0117] The state of N qubits |ψ> is expressed by multiple expansion coefficients α n and multiple computational bases |n>(n=0,1,...,2 N -1). Specifically, the state |ψ> of N qubits is expressed by a linear combination of multiple computational bases |n>.

[0118]

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[0119] First, in step S100, the user terminal 130 transmits to the classical computer 110 computation target information, which is information on a computation target, and computation method information, which is information on a computation method, both input by the user.

[0120] The calculation target information includes, for example, information on an observable O corresponding to a physical quantity of the calculation target. The calculation method information includes, for example, information on a quantum circuit and information on the number of measurement shots. The information on the quantum circuit includes a weighting coefficient f r A quantum circuit U that generates the state |ψ>, which is used to calculate f Information about the structure of the interference weighting coefficient g r The quantum circuit U used to calculate g The information on the number of measurement shots includes L, which indicates the number of measurements of a calculation basis |n>, which will be described later, and L' and L'', which indicate the number of measurements for calculating the expected value of a projection operator, which will be described later.

[0121] Next, in step S102, the classical computer 110 receives the computation target information and computation method information transmitted from the user terminal 130. Then, in step S102, the classical computer 110 receives the quantum circuit U f Based on the information about the structure of the quantum circuit U f In step S102, the classical computer 110 determines the number of measurement shots based on L, which represents the number of measurements of the calculation basis |n> in the calculation method information.

[0122] In step S104, the classical computer 110 transmits various information necessary for quantum computation to the quantum computer 120. Specifically, the classical computer 110 transmits the quantum circuit U determined in step S102 to the quantum computer 120. f The structure and the number of measurement shots, as well as the calculation method information and calculation target information received in step S102, are transmitted to the quantum computer 120.

[0123] In step S106, the control device 121 receives the various information transmitted from the classical computer 110 in step S104.

[0124] In step S108, the control device 121 causes the quantum computer 120 to execute quantum computation according to the various information received in step S106. The quantum computer 120, under the control of the control device 121, repeats the measurement of the computation basis |n> of the above formula (A3) L times to obtain a series of measurement values ​​of the computation basis |n>, {x}=x (1) ,x (2) ,···,x (L) Get the.

[0125] Specifically, the quantum computer 120 generates an electromagnetic wave to be irradiated to at least one of the quantum bits in the quantum bit group 123 in response to the control of the control device 121. Then, the quantum computer 120 irradiates the generated electromagnetic wave to at least one of the quantum bits in the quantum bit group 123, and generates a quantum circuit U that generates a state |ψ>. f Quantum circuit U f The gate operation of each quantum gate included in is converted into a corresponding electromagnetic wave waveform, and the generated electromagnetic wave is irradiated to the quantum bit group 123 by the electromagnetic wave generating device 122. Then, the quantum computer 120 outputs the measurement result obtained by the measurement.

[0126] In step S110, the control device 121 transmits the measurement result obtained in step S108 to the classical computer 110.

[0127] In step S112, the classical computer 110 receives the measurement result transmitted from the control device 121 in step S110. Next, the classical computer 110 receives the measurement result, which is a sequence of measurement values ​​of the calculation basis |n> {x}=x (1) ,x (2) ,···,x (L) Based on this, R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value are selected from the sequence of measurement values ​​{x}.

[0128] In step S112, the classical computer 110 calculates a value {z}=z (1) ,z (2) ,...,z (R) Next, the classical computer 110 sets the number of occurrences T of the R measurements included in {z}. r (r=1, , R). Then, the classical computer 110 calculates |α zr | 2 The weighting factor f r Calculate.

[0129]

number

[0130] In step S114, the classical computer 110 calculates the computation basis |z r > and the computational basis |z r’ > Multiple transition matrix elements for each combination with <z r |O|z r’ ,R; r'=1,,R.

[0131] In step S116, the classical computer 110 selects the quantum circuit U g Based on the information about the structure of the quantum circuit U gIn step S116, the classical computer 110 determines the number of measurement shots based on L′ and L″, which represent the number of measurements for calculating the expected value of the projection operator in the calculation method information.

[0132] 4, the classical computer 110 transmits various information necessary for quantum computation to the quantum computer 120. Specifically, the classical computer 110 transmits the quantum circuit U determined in step S116 to the quantum computer 120. g Structure and number of measurement shots of and N-qubit state |φ A,r Projection operator |φ for > A,r ><φ A,r Information about | and the N-qubit state |φ B,r Projection operator |φ for > B,r ><φ B,r The quantum circuit U g is the projection operator |φ A,r ><φ A,r Expectation value of | and projection operator |φ B,r ><φ B,r A quantum circuit is a circuit for obtaining the expected value of |. Note that for these quantum circuits, a quantum circuit is required for each r for A, and a quantum circuit is required for each r for B. Therefore, 2(R-1) types of quantum circuits are required.

[0133] In step S120, the control device 121 receives the various information transmitted from the classical computer 110 in step S118.

[0134] In step S122, the control device 121 causes the quantum computer 120 to execute quantum calculations according to the various information received in step S120. The quantum computer 120, under the control of the control device 121, calculates a projection operator |φ A,r ><φ A,r By repeating the measurement of |L' times, the projection operator |φ A,r ><φ A,rIn addition, the quantum computer 120 acquires the measurement result of the projection operator |φ B,r ><φ B,r By repeating the measurement of |L'' times, the projection operator |φ B,r ><φ B,r Obtain the measurement results for |.

[0135] Specifically, the quantum computer 120 generates an electromagnetic wave to be irradiated to at least one of the quantum bits in the quantum bit group 123 in response to the control of the control device 121. Then, the quantum computer 120 irradiates the generated electromagnetic wave to at least one of the quantum bits in the quantum bit group 123, and generates a quantum circuit U g Quantum circuit U g The gate operation of each quantum gate included in is converted into a corresponding electromagnetic wave waveform, and the generated electromagnetic wave is irradiated to the quantum bit group 123 by the electromagnetic wave generating device 122. Then, the quantum computer 120 outputs the measurement result obtained by the measurement.

[0136] In step S124, the control device 121 transmits the measurement result obtained in step S122 to the classical computer 110.

[0137] In step S126, the classical computer 110 receives the measurement result transmitted from the control device 121 in step S124. Next, the classical computer 110 calculates the projection operator |φ A,r ><φ A,r Based on the measurement results of |, the projection operator |φ A,r ><φ A,r Expected value of |A r In addition, the classical computer 110 calculates the projection operator |φ B,r ><φ B,r Based on the measurement results of |, the projection operator |φ B,r ><φ B,r Expected value of |B r Calculate.

[0138] In step S128, the classical computer 110 calculates the weighting factor f r and the expected value A calculated in step S126 r and expected value B r Based on this, α in the above formula (A2) is calculated according to the following formula (A5). z1 α zr * The interference weight g r Approximate calculation is performed.

[0139]

number

[0140] In step S130, the classical computer 110 calculates the weighting factor f1 calculated in step S112 and the interference weight g calculated in step S128. r Based on this, the other component of the interference weight, α zr α zr’ * Approximate calculation is performed.

[0141]

number

[0142] In step S132, the classical computer 110 calculates the interference weight g r α corresponds to z1 α zr * and α calculated in step S130 zr α zr’ * and calculated in step S114. <z r |O|z r’ >, the expectation value <ψ|O|ψ> of the observable O is approximately calculated according to the above equation (A2).

[0143] In step S134, the classical computer 110 transmits to the user terminal 130 the expected value <ψ|O|ψ> of the observable O, which is the calculation result obtained in step S132.

[0144] In step S136, the user terminal 130 receives the calculation result transmitted from the classical computer 110.

[0145] As described above, the classical computer in the hybrid system of the embodiment acquires the measurement values ​​of each of the multiple computational bases measured by the quantum computer when the state |ψ> of N quantum bits is expressed by a linear combination of multiple computational bases. Then, based on the measurement values ​​of each of the multiple computational bases, the classical computer selects R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value from the multiple measurement values, and selects R computational bases |z according to the occurrence count based on the occurrence count of the selected R measurement values. r >(r=1, ,R) weighting factor f r Then, the classical computer calculates the weighting factor f r and the weighting factor f r’ The transition matrix elements are weighted by the combination of <z r |O|z r’ >, we approximate the expectation value <ψ|O|ψ> of the observable O. This allows us to efficiently obtain the expectation value of the observable O using a quantum computer.

[0146] Furthermore, by appropriately dividing roles between classical and quantum computers, the expectation value of the observable O can be obtained efficiently. EXAMPLES

[0147] Next, an example will be described. In this example, a quantum chemical calculation problem is taken as an example, and a numerical comparison result between the method of this embodiment and the conventional method is shown. Specifically, using the Hamiltonian H of various molecules and their ground states |ψ>, the magnitude of statistical fluctuations in the measurement of the expectation value <ψ|H|ψ> is theoretically evaluated (see the method described later), and the statistical fluctuations are calculated to be within the accuracy (10 -3 We calculated the total number of shots required to achieve a quantum bit count of 100 s or less for the two methods. The results are shown in Table 1. For H2, which has a small number of quantum bits, the total number of shots is reduced slightly, but for LiH and H2O, the method reduces the total number of shots significantly.

[0148] Note that R is the number of significant calculation bases included in the sum of the above formula (14) in this method, and specifically, it was numerically obtained as the smallest natural number R that satisfies the following inequality (21).

[0149]

number

[0150] When R as shown in the above formula (21) is used, the energy expectation value (E IS ) is the exact value obtained by the FCI method (E FCI ) with the accuracy required in quantum chemistry (the difference between the two is 10 -3 Hartree (see below)

[0151] [Table 1]

[0152] The above table shows the magnitude of statistical fluctuations in measurements of the expectation value <ψ|H|ψ> of the Hamiltonian H for various molecules in the ground state |ψ>. -3The total number of shots required to achieve Hartree or less is shown. For each molecule, N is the number of qubits, M is the total number of Pauli operators included in the Hamiltonian, and R is the number of significant computational bases included in the above equation (14) of the sum of our method.

[0153] E IS -E FCI is the energy expectation value (E IS ) and the exact value obtained by the FCI method (E FCI ) (unit: Hartree). FCI The quantum chemistry calculation software PySCF was used to calculate the . The known STO-3G basis set was used in the calculation.

[0154] [Estimation of statistical error in measuring the expectation value <ψ|O|ψ> of an observable O] Statistical fluctuations occur in the measured value of the expectation value 〈ψ|O|ψ〉 of the observable O. Here, we explain the theoretical estimation of the magnitude of the fluctuations with respect to the number of shots.

[0155] The measurements required in this method are L projection measurements of |ψ> onto |n>, A r (r=2, ,R) r Each measurement is performed L'' times to obtain (r = 2, , R). After quantifying the statistical error from each measurement, the statistical error of the expected value 〈ψ|O|ψ〉 is derived using the error propagation formula.

[0156] [Projection measurement of |ψ> onto |n>] |ψ> is expressed as |n>(n=0,1,...,2 N -1) L projection measurements are taken. In this case, |n> is l n The probability that |n> occurs in one measurement is p n =|<n|ψ> | 2 The number of occurrences of each n is l0, l1, ..., l n_max (n_max=2 N-1) as a random variable, its probability distribution is given by the following multinomial distribution:

[0157]

number

[0158] l n is the actual observed value, and the corresponding random variable is L n Its expectation and variance are expressed by the following equations.

[0159]

number

[0160] In addition, L n There are the following restrictions regarding

[0161]

number

[0162] For this reason, L n and L m The covariance of (n≠m) becomes nonzero.

[0163]

number

[0164] In addition, p n =|<n|ψ> | 2 If the estimator of is given by the following equation, its variance and covariance are given by the following equations:

[0165]

number

[0166] The observed number of occurrences of |n>n By using p n As an estimate of p n =l n / L is obtained, whose standard deviation is given by:

[0167]

number

[0168] As mentioned above, the following formula was introduced, which is the following p n gives an approximation of

[0169]

number

[0170] Therefore, f r The statistical error of can be approximated as follows:

[0171]

number

[0172] Similarly, the covariance can be approximated as follows:

[0173]

number

[0174] (A r and B r Measurement of First, consider the following measurement:

[0175]

number

[0176] For each r, we iterate over the projection measurements in a basis (and a set of orthonormal bases) of the state |ψ> to obtain A r It is possible to estimate the state |ψ> as a certain basis, which is expressed by the following equation.

[0177]

number

[0178] Since each measurement result is either the above base or something else, the number of times the above base appears is described by a binomial distribution. If L' measurements are taken and the above base appears l' times, then A r The estimate of A r = l' / L'. The standard deviation is expressed by the following formula:

[0179]

number

[0180] Since independent measurements must be made for each r, the covariance is zero. B r can be similarly measured and estimated.

[0181]

number

[0182] If we measure L'' times for each r, then B r The standard deviation of is expressed by the following formula:

[0183]

number

[0184] [ <o>statistical error] According to the above formula (14), the expected value <o>We constructed an estimator for the measured quantities fr(r=1,...,R), A r , B r It can also be considered as a function of (r=2,...,R).

[0185]

number

[0186] Therefore, by applying the error propagation formula, <o>The statistical error of σ <o>< / o> can be estimated as follows:

[0187]

number

[0188] The technology of the present disclosure is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the spirit and scope of the present invention.

[0189] For example, in the above embodiment, information may be transmitted and received in any manner between the classical computer 110 and the quantum computer 120. For example, transmission and reception of parameters of a quantum circuit and transmission and reception of measurement results between the classical computer 110 and the quantum computer 120 may be performed sequentially each time a predetermined calculation is completed, or may be performed after all calculations are completed.

[0190] In the above embodiment, the calculation target information is transmitted from the user terminal 130 to the classical computer 110, and the classical computer 110 executes a calculation according to the calculation target information. However, the present invention is not limited to this. The user terminal 130 may transmit the calculation target information to the classical computer 110 or a storage medium or storage device accessible to the classical computer 110 via a computer network such as an IP network, or may store the calculation target information in a storage medium or storage device and hand it over to the operator of the classical computer 110, who may input the calculation target information to the classical computer 110 using the storage medium or storage device.

[0191] In addition, in the above embodiment, the quantum circuit is executed by irradiating electromagnetic waves. However, the present invention is not limited to this, and the quantum circuit may be executed by a different method.

[0192] In the above embodiment, the quantum computer 120 executes quantum computation, but the present invention is not limited to this. For example, quantum computation may be executed by a classical computer that mimics the behavior of a quantum computer.

[0193] In the above embodiment, the process executed by the quantum computer 120 may be executed by the classical computer 110. Alternatively, in the above embodiment, the process executed by the classical computer 110 may be executed by the quantum computer 120. For example, in the above embodiment, the classical computer 110 calculates a computational basis |z r > and the computational basis |z r’ > multiple transition matrix elements for each combination with <z r |O|z r’ However, the present invention is not limited to this. For example, the quantum computer 120 may calculate a plurality of transition matrix elements <z r |O|z r’ >(r=1, ,R;r'=1, ,R).

[0194] Furthermore, in the above embodiment, it is assumed that the classical computer 110 and the quantum computer 120 are managed by different organizations, but the classical computer 110 and the quantum computer 120 may be managed as a single entity by the same organization. In this case, it is not necessary to transmit quantum computing information from the classical computer 110 to the quantum computer 120 and to transmit measurement results from the quantum computer 120 to the classical computer 110. In this case, it is also possible that the control device 121 of the quantum computer 120 plays the role of the classical computer 110 in the above description.

[0195] It should be noted that in the above embodiment, unless there is a statement such as "based only on XX", "depending only on XX", or "only in the case of XX", it is assumed in this specification that additional information may be taken into consideration. As an example, the statement "do b if a" does not necessarily mean "always do b if a" unless expressly stated otherwise.

[0196] In addition, in the above embodiments, when expressions such as "optimize" or "optimized parameters" are used, it should be noted that these expressions of "optimization" mean to approach an optimal state. Therefore, when trying to obtain parameters that minimize a certain function, it should be noted that the parameters obtained by optimizing the function may be a local solution rather than a global solution that minimizes the function.

[0197] Furthermore, even if there is an aspect of a method, program, terminal, device, server or system (hereinafter referred to as a "method, etc.") that performs an operation different from that described in this specification, each aspect of the disclosed technology is directed to an operation that is the same as any of the operations described in this specification, and the existence of an operation different from that described in this specification does not make the method, etc. outside the scope of each aspect of the technology of the present disclosure.

[0198] Furthermore, although the present specification has been described as an embodiment in which the program is pre-installed, the program may also be provided by being stored on a computer-readable recording medium.

[0199] Furthermore, each component of the hybrid system of this embodiment does not have to be realized by a single computer or server, but may be realized in a distributed manner across multiple computers connected by a network.

[0200] For example, the processing performed by the classical computer of the above embodiments may be distributed among multiple classical computers connected by a network. Or, for example, the processing performed by the quantum computer of each of the above embodiments may be distributed among multiple quantum computers connected by a network. In this case, a hybrid system is configured by at least one classical computer and at least one quantum computer.

[0201] For example, when a hybrid system is configured with multiple classical computers and multiple quantum computers, and a state |ψ> of N quantum bits is expressed by a linear combination of multiple computational bases, one or more quantum computers among the multiple quantum computers measure each of the multiple computational bases to obtain measurement values ​​of the multiple computational bases. Then, one or more classical computers among the multiple classical computers obtain the measurement values ​​of the multiple computational bases measured by the quantum computer. Next, based on the measurement values ​​of each of the multiple computational bases, one or more classical computers among the multiple classical computers select R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value from the multiple measurement values, and select R computational bases |z according to the occurrence count based on the occurrence count of the selected R measurement values. r >(r=1, ,R) weighting factor f r Next, one or more classical computers among the multiple classical computers or one or more quantum computers among the multiple quantum computers calculate the computational basis |z r > and the computational basis |z r’ > multiple transition matrix elements for each combination with <z r |O|z r’ Then, one or more of the classical computers calculates a weighting factor f r and the weighting factor f r’ The transition matrix elements are weighted by the combination of <z r |O|z r’ >, we approximate the expectation value <ψ|O|ψ> of the observable O. [Explanation of symbols]

[0202] 100 Hybrid System 110 Classical Computers 111 Communications Department 112 Processing section 113 Information storage section 120 Quantum Computer 121 Control device 122 Electromagnetic wave generation device 123 qubit group 130 User terminal< / o> < / o> < / o>

Claims

1. A quantum information processing method executed by a classical computer in a hybrid system including a classical computer and a quantum computer, When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, Obtaining measurements of each of a plurality of computational bases measured by the quantum computer; Based on the measurement values ​​of each of the multiple calculation bases, R measurement values ​​having a high occurrence frequency or R measurement values ​​having a predetermined value or more of occurrence frequency are selected from the multiple measurement values, and based on the occurrence frequency of the selected R measurement values, R calculation bases |z ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate Weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of A quantum information processing method in which the processing is carried out by a classical computer.

2. The state |ψ〉 of the N qubits is expressed by a plurality of computational bases and an expansion coefficient α ｎ (n=0,1,...,2 Ｎ -1) and based on multiple expansion coefficients α ｎ It is expressed by a linear combination of computational bases with weights, When the expected value <ψ|O|ψ> is approximated, R calculation bases |z ｒ >Weighting factor f for (r=1, . . . , R) ｒ and the computational basis |z ｒ >Expansion coefficient α ｚｒ and the computational basis |z ｒ’ >Expansion coefficient α ｚｒ’ α is calculated based on ｚｒ α ＊ ｚｒ’ and several transition matrix elements <z ｒ |O|z ｒ’ 〉, the expectation value <ψ|O|ψ> of the observable O is approximately calculated according to the following formula (A1):

2. The quantum information processing method according to claim 1. [0010] (A1)

3. A quantum information processing method executed by a hybrid system including a classical computer and a quantum computer, comprising: When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, A quantum computer measures each of a plurality of computational bases to obtain measurement values ​​of the plurality of computational bases; A classical computer obtains measurements of the multiple computational bases measured by the quantum computer; A classical computer selects R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value from the plurality of measurement values ​​based on the measurement values ​​of each of the plurality of calculation bases, and calculates R calculation bases |z according to the occurrence count based on the occurrence count of the selected R measurement values. ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate A classical or quantum computer calculates the computational basis |z ｒ > and the computational basis |z ｒ’ For each combination of ｒ |O|z ｒ’ Calculate > A classical computer calculates the weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of Quantum information processing methods.

4. When the expectation value <ψ|O|ψ> of the observable O represented by the following formula (A2) is approximately calculated using the following formula (A1), The quantum computer expands the N-qubit state |ψ〉 into a number of expansion coefficients α ｎ and multiple computational bases |n> (n=0, 1, ..., 2 Ｎ −1), the measurement of the calculation basis |n> is repeated L times to obtain a series of measurements of the calculation basis |n> {x}=x （１） , x （２） , ..., x （Ｌ） Get A classical computer selects R measurement values ​​that have a high frequency of occurrence or R measurement values ​​that have a frequency of occurrence equal to or greater than a predetermined value from the measurement value series {x} based on the measurement value series {x}, and calculates a value {z}=z （１） , z （２） , ..., z （Ｒ） and the number of occurrences T of the R measured values ​​included in {z} is set. ｒ (r=1, . . . , R) is calculated, and |α ｚｒ | ２ The weighting factor f ｒ Calculate A classical computer or a quantum computer calculates a transition matrix element <z of the following formula (A1) based on the {z}. ｒ |O|z ｒ’ Calculate (r=1,...,R; r'=1,...,R), The quantum computer has an N-qubit state |φ Ａ，ｒ > is the projection operator |φ Ａ，ｒ ><φ Ａ，ｒ By measuring |, the projection operator |φ Ａ，ｒ ><φ Ａ，ｒ | is measured, and the N-qubit state |φ Ｂ，ｒ > is the projection operator |φ Ｂ，ｒ ><φ Ｂ，ｒ By measuring |, the projection operator |φ Ｂ，ｒ ><φ Ｂ，ｒ Obtain the measurement results of | A classical computer calculates the projection operator |φ Ａ，ｒ ><φ Ａ，ｒ Based on the measurement result of |, the projection operator |φ Ａ，ｒ ><φ Ａ，ｒ Expected value A of | ｒ Calculate the projection operator |φ Ｂ，ｒ ><φ Ｂ，ｒ Based on the measurement result of |, the projection operator |φ Ｂ，ｒ ><φ Ｂ，ｒ Expected value B of | ｒ Calculate The classical computer calculates the weighting factor f ｒ and the expected value A ｒ and the expected value B ｒ Based on this, according to the following formula (A5), α ｚ１ α ｚｒ ＊ The interference weight g ｒ The weighting coefficient f ｒ and the interference weight g ｒ Based on this, according to the following formula (A6), α ｚｒ α ｚｒ’ ＊ The interference weight g ｒ The above α ｚ１ α ｚｒ ＊ and the α ｚｒ α ｚｒ’ ＊ and the transition matrix element <z ｒ |O|z ｒ’ and approximating the expected value <ψ|O|ψ> according to the following formula (A1): The quantum information processing method according to claim 3. [0025] (A1) 【number】 (A2) 【number】 (A3) 【number】 (A4) 【number】 (A5) 【number】 【number】 (A6)

5. A quantum information processing program to be executed by a classical computer in a hybrid system including a classical computer and a quantum computer, When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, Obtaining measurements of each of a plurality of computational bases measured by the quantum computer; Based on the measurement values ​​of each of the multiple calculation bases, R measurement values ​​having a high occurrence frequency or R measurement values ​​having a predetermined value or more of occurrence frequency are selected from the multiple measurement values, and based on the occurrence frequency of the selected R measurement values, R calculation bases |z ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate Weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of A quantum information processing program that allows processing to be performed by a classical computer.

6. A classical computer in a hybrid system including a classical computer and a quantum computer, When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, Obtaining measurements of each of a plurality of computational bases measured by the quantum computer; Based on the measurement values ​​of each of the multiple calculation bases, R measurement values ​​having a high occurrence frequency or R measurement values ​​having a predetermined value or more of occurrence frequency are selected from the multiple measurement values, and based on the occurrence frequency of the selected R measurement values, R calculation bases |z ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate Weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of Classical computer.

7. A hybrid system including a classical computer and a quantum computer, When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, A quantum computer measures each of a plurality of computational bases to obtain measurement values ​​of the plurality of computational bases; A classical computer obtains measurements of the multiple computational bases measured by the quantum computer; A classical computer selects R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value from the plurality of measurement values ​​based on the measurement values ​​of each of the plurality of calculation bases, and calculates R calculation bases |z according to the occurrence count based on the occurrence count of the selected R measurement values. ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate A classical or quantum computer calculates the computational basis |z ｒ > and the computational basis |z ｒ’ For each combination of ｒ |O|z ｒ’ Calculate > A classical computer calculates the weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of Hybrid system.

8. 1. A hybrid system including a plurality of classical computers and a plurality of quantum computers, When the state of N qubits |ψ〉 is expressed by a linear combination of multiple computational bases, One or more quantum computers among the plurality of quantum computers obtain measurements of the plurality of computational bases by measuring each of the plurality of computational bases; one or more classical computers among the plurality of classical computers obtain measurements of the plurality of computational bases measured by the quantum computer; One or more of the multiple classical computers selects R measurement values ​​that occur frequently or R measurement values ​​whose occurrence count is equal to or greater than a predetermined value from the multiple measurement values ​​based on the measurement values ​​of each of the multiple calculation bases, and selects R calculation bases |z according to the occurrence count based on the occurrence count of the selected R measurement values. ｒ >Weighting factor f for (r=1, . . . , R) ｒ Calculate One or more classical computers among the plurality of classical computers or one or more quantum computers among the plurality of quantum computers are ｒ > and the computational basis |z ｒ’ For each combination of ｒ |O|z ｒ’ Calculate > One or more of the classical computers have a weighting factor f ｒ and the weighting factor f ｒ’ A plurality of transition matrix elements <z ｒ |O|z ｒ’ approximating the expectation value of the observable O, <ψ|O|ψ>, by a weighted sum of Hybrid system.

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