WO2023022014A1 - Quantum information processing method, classical computer, hybrid system, and quantum information processing program - Google Patents

Abstract

A classical computer according to the present invention obtains measurement values of a plurality of computational bases which have been measured by a quantum computer, when a state |ψ> of N quantum bits is expressed by a linear combination of the plurality of computational bases. On the basis of the measurement values of the plurality of computational bases, the classical computer selects, from among the measurement values, R measurement values of which the number of appearances is high or R measurement values of which the number of appearances is equal to or higher than a given value. The classical computer calculates, on the basis of the number of appearances of the selected R measurement values, a weighting factor fr for R computational bases |zr> (r=1, ..., R) in accordance with the number of appearances, and then approximates an expected value <ψ|O|ψ> of an observable O by the weighted sum of a plurality of transition matrix elements <zr|O|zr'> where the weight is a combination of the weighting factor fr and a weighting factor fr'.

Description

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

The disclosed technology relates to quantum information processing methods, classical computers, hybrid systems, and quantum information processing programs.

Variational quantum algorithms for performing quantum computations using Noisy Intermediate-Scale Quantum (NISQ) devices are known (e.g. Peruzzo, A., McClean, J., Shadbolt, P. et al. "A variational eigenvalue solver on a photonic quantum processor". Nat Commun 5, 4213 (2014).), Reference 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)). In order to execute the variational quantum algorithm, it is necessary to calculate the expected value of the physical quantity to be observed (hereinafter also simply referred to as "observable") with a quantum computer.

　When calculating the expected value of an observable using a quantum computer, the state is measured after the state is generated by the quantum computer. Then, the expected value of the observable is calculated based on the measurement result of the state by the quantum computer. Note that a series of processes including generation of a single state by a quantum computer and measurement of the corresponding state is also referred to as a “shot”.

When applying variational quantum algorithms to quantum chemical calculations, the number of shots required to sufficiently reduce statistical fluctuations in state measurements may be excessive. For example, Reference 3 (Gonthier, J. F., Radin, M. D., Buda, C., Doskocil, E. J., Abuan, C. M., and Romero, J. “Identifying challenges toward practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver”.) considers the quantum chemical calculation problem to obtain the energy of the actual molecule, and aims to reduce the statistical fluctuations beyond the accuracy of the energy required in the field of quantum chemistry. requires a huge number of shots, and it takes several days just to obtain the expected energy value once. For this reason, for example, Reference 4 (Rubin, N. C., Babbush, R., and McClean J., “Application of fermionic marginal constraints to hybrid quantum algorithms”. New J. Phys. 20, 053020 (2018).) attempts to reduce the total number of shots by simultaneously measuring multiple Pauli operators.

The disclosed technology has been made in view of the above circumstances, and provides a quantum information processing method, a classical computer, a hybrid system, and quantum information that can efficiently obtain the expected value of an observable using a quantum computer. It aims at providing a processing program.

In order to achieve the above object, a quantum information processing method according to one aspect 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, comprising: When the state |ψ> of the N qubits is represented by a linear combination of , the measured value of each of the plurality of computational bases measured by the quantum computer is obtained, and based on the measured value of each of the plurality of computational bases , from a plurality of measured values, select R measured values with a large number of occurrences or R measured values with a number of occurrences equal to or greater than a predetermined value, and based on the number of occurrences of the selected R measured values, the number of occurrences Calculate weighting factors f _r for R calculation bases |z _r > (r=1 _, . . . , _R ) according to A quantum information processing method in which a classical computer executes a process of approximating an expected value <ψ|O|ψ> of an observable O by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r′ > be.

According to the disclosed technology, it is possible to obtain the expected value of observables efficiently using a quantum computer.

It is a figure showing an example of a schematic structure of hybrid system 100 of this embodiment. 1 is a schematic block diagram of a computer functioning as classical computer 110, controller 121, and user terminal 130. FIG. It is a figure which shows an example of the sequence which the hybrid system 100 of this embodiment performs. It is a figure which shows an example of the sequence which the hybrid system 100 of this embodiment performs.

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

<Hybrid system 100 according to the embodiment>

FIG. 1 shows a hybrid system 100 according to an embodiment. The hybrid system 100 of this embodiment includes a classical computer 110, a quantum computer 120, and a user terminal 130. 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 as an example, as shown in FIG.

In the hybrid system 100 of this embodiment, the quantum computer 120 performs a predetermined quantum calculation in response to a request from the classical computer 110 and outputs the calculation result of the quantum calculation to the classical computer 110. The classical computer 110 outputs calculation results according to quantum calculation to the user terminal 130 . As a result, the hybrid system 100 as a whole performs predetermined calculation processing.

The classical computer 110 includes a communication unit 111 such as a communication interface, a processing unit 112 such as a processor and a CPU (Central processing unit), and an information storage unit 113 including a storage device or storage medium such as a memory and a hard disk. It is configured by executing a program for processing. Note that classical computer 110 may include one or more devices or servers. Also, the program may include one or more programs, and may be recorded on a computer-readable storage medium to be a non-transitory program product.

As an example, the quantum computer 120 generates electromagnetic waves for irradiating at least one quantum bit in the quantum bit group 123 based on information transmitted from the classical computer 110 . The quantum computer 120 executes a quantum circuit by irradiating at least one of the quantum bits in the quantum bit group 123 with the generated electromagnetic wave.

In the example of FIG. 1, the quantum computer 120 includes a controller 121 that communicates with the classical computer 110, an electromagnetic wave generator 122 that generates electromagnetic waves in response to a request from the controller 121, and an electromagnetic wave generator 122 that emits electromagnetic waves. and a qubit group 123 that receives The electromagnetic wave generator 122 and the quantum bit group 123 in the quantum computer 120 are also QPUs (Quantum Processing Units). In this embodiment, the term "quantum computer" does not mean that it does not perform computations using classical bits, but refers to computers that include computations using quantum bits.

The control device 121 is a classical computer that operates with classical bits, and alternatively performs some or all of the processing described herein as being performed in the classical computer 110 . For example, the control device 121 stores or determines a quantum circuit in advance, and executes the quantum circuit U(θ) in the quantum bit group 123 in response to receiving the parameter θ of the quantum circuit U(θ). of quantum gate information may be generated.

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

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

The hybrid system 100 of the embodiment efficiently calculates the expected value <ψ|O|ψ> of the observable O. Prerequisites for the processing executed by the hybrid system 100 of the embodiment will be described below.

[1. Problem setting and background]

Consider the state |ψ> of N qubits generated on a quantum computer. The state |ψ> is a vertical vector having 2 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. |0> represents the state of the initialized N qubits.

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

(1)

[2. Conventional expected value measurement methods and issues]

The conventional method for measuring the expected value of observable O consists of the following steps.

(1) Decomposition of Observable O into Pauli Operators Observable O on N qubits is represented by a linear combination of the following Pauli operators.

Observable O is represented by a linear combination of Pauli operators P _i as shown in Equation (2) below.

(2)

where I, X, Y, Z are the identity operator and the Pauli matrix of one qubit. Also, c _i in the above equation (2) is the expansion coefficient, and M is the total number of Pauli operators. Based on the above equation (2), the expected value <ψ|O|ψ> of the observable O which is the operator is represented by the following equation (3).

(3)

(2) Measurement of Expected Value of Pauli Operator Next, each term <ψ|P _i |ψ> on the right side of the above equation (3) is measured by a quantum computer. <ψ|P _i |ψ> is the expected value of the Pauli operator. The expectations of the Pauli operators <ψ|P _i |ψ> are obtained by standard measurement operations in quantum computers. Specifically, when the state |ψ> is generated once on a quantum computer and a measurement operation is performed according to the Pauli operator P _i , either +1 or −1 results are obtained stochastically. The probability is determined by the combination of |ψ> and P _i .

The average of l _i measurement results obtained by generating the state l _i times is used as the expected value <ψ|P _i |ψ>. Note that each of the l _i measurement results is either +1 or -1.

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

Since the measurement result is probabilistic, the estimated expected value <ψ|P _i |ψ> has statistical fluctuations. This statistical fluctuation is reduced by 1/√l _i for the number of trials l _i to obtain +1 or -1. As described above, in the field of quantum computers, a series of processes of generating states and performing measurements is also called a "shot". For this reason, l _i is also simply referred to as the “shot number” below.

(3) Addition processing of the expected value of the Pauli operator Next, the above equation (3) is calculated based on the value of the expected value <ψ|P _i |ψ> obtained in the above (2). , the expected value <ψ|O|ψ> of the observable O is computed.

Thus, the NISQ device cannot directly measure the expected value <ψ|O|ψ> of the observable O, but the expected value of the Pauli operator <ψ|P _i |ψ> can be easily measured. Therefore, the expected value <ψ|O|ψ> of the observable O is calculated based on the expected value <ψ|P _i |ψ> of the Pauli operator.

[3. Theme]
When the variational quantum algorithm is used when performing quantum chemical calculations using the NISQ device, the total number of shots l ₁ + _. There are many. In this case, the quantum chemical calculation may not be completed within a realistic time. In particular, in a system with a large number of qubits N, it is necessary to measure the expected values of a very large number (M=O(N ⁴ )) of Pauli operators. Therefore, in this embodiment, a method for efficiently calculating the expected value of the observable O is proposed.

[4. Method of measuring expected value in the present embodiment]
The expected value based on the state |ψ> of the observable O is represented by the following equation (4).

(4)

In general, the state |ψ> is represented as a linear combination of basis vectors belonging to the complete system. Although there are various possibilities as candidates for the complete system, in order to advance the discussion concretely, the calculation basis |n> (n=0, ¹ , . use. The state |ψ> is expanded by the calculation basis |n> as shown in the following equation (5).

(5)

Here, the expansion coefficient α _n is a complex number calculated by the inner product of the state |ψ> and the calculation basis |n>.

(6)

α _n in the above equation (6) is also called probability amplitude, and |α _n | ² , which is the square of its absolute value, represents probability. Therefore, when a state |ψ> is prepared on a quantum computer and a standard measurement operation is performed on the state |ψ>, n=0, 1, . . . , 2 ^N − Any value of 1 is obtained stochastically. |α _n | ² represents the probability that the measurement result will be a certain value n.

Conversely, the frequency distribution of the measurement result n can be obtained by repeating the operation of preparing the state |ψ> on the quantum computer and projecting it onto the calculation basis |n>, and from that frequency distribution the true It becomes possible to estimate the value of |α _n | ² , which is the probability of .

We can now rewrite the expected value <ψ|O|ψ> using the probability amplitude α _n , which is a quantity associated with the state |ψ>. When the following expression representing that the calculation basis |n> (n=0, 1, . . . , 2 ^N −1) forms a complete system, the following expression (7) is derived.

(7)

It should be noted that the last line of the above equation (7) does not include a term that satisfies α _n =0 or α _m =0. The above equation (7) is obtained by _adding < _m |O|n>/( _αmαn ^* ) with |αm| ² | _αn | ² as a weight, to obtain the state |ψ> This means that the expected value at is obtained. As will be explained below, <m|O|n> can be efficiently computed by classical computers. Also, α _m α _n ^* can be measured using a quantum computer. Therefore, by combining |α _n | ² and α _m α _n ^* measured using quantum computers with <m|O|n> calculated using classical computers, the expectation The value <ψ|O|ψ> is obtained.

Specifically, the transition matrix element <m|O|n> corresponding to the combination of the calculation basis |n> and the calculation basis |m> is the Pauli operator of the observable O given by the above equation (2). Decomposition by P _i expands as follows.

<m|P _i |n> in the above equation can be efficiently calculated by a classical computer. In addition, being able to be efficiently calculated by a classical computer means being able to be calculated in linear time with respect to the number of quantum bits. Then, the transition matrix elements <m|O|n> are obtained by summing <m|P _i |n> calculated by the classical computer.

Next, α _m α _n ^* will be described. In the case of m=n, α _m α _n ^* = |α _n | ^2. Therefore, as described above, it is possible to measure |ψ> using a quantum computer by sampling with the calculation basis |n>. . On the other hand, if m≠n, another measurement operation is required on the quantum computer. To explain this point, the following transformations are required.

(8)

Here, A _m,n and B _m,n are represented by the following equations.

(9)

Note that A _m,n and B _m,n can be estimated using a quantum computer by projection measurement. Thus, by combining A _m,n , B _m,n with the separately measured weights |α _n | ² , it is possible to calculate the value of α _m α _n ^* .

In general, since α _m α _n ^* has a complex phase, different terms cancel each other out and construct each other in the sum of the above equation (7). For this reason, α _m α _n ^* is hereinafter referred to as an “interference weight”.

By combining the values of |α _n | ² , A _m,n , B _m,n , and <m|O|n> obtained as described above, the expected value <ψ|O|ψ> is calculated.

(10)

In addition, since 2 ^N ×2 ^N exponentially large number of terms are to be added, it seems that a difficulty in terms of computational cost arises. However, empirically, it can be seen that the number of terms to be added is greatly reduced depending on the type of problem to be solved, such as quantum chemical calculation.

For example, in quantum chemistry problems, expectation calculations are often performed on eigenstates of the Hamiltonian, such as the ground state or low-energy excited states. are known to be major and other components to be relatively small.

In addition, it is mathematically possible that the eigenstate has a finite amplitude only in a limited portion of the entire basis due to the existence of conserved quantities in the target system (for example, the conservation of the number of electrons and the conservation of the spin). know. For this reason, the weight |α _n | ² has a significant value only for certain elements with respect to n, and is zero or a value so small as to be statistically negligible in estimating the expected value in a quantum computer for other elements. Therefore, such a basis can be ignored from the beginning when summing over m and n. Therefore, it follows that the transition matrix element <m|O|n> needs to be evaluated only for certain computation basis pairs (m,n) that give non-negligible weights.

In addition, when determining the interference weight α _m α _n ^* for such a pair (m, n), A _{m, n} and B _{m, n} for a part of (m, n) are measured. It turns out that it is enough.

To see this, consider the following identity (11), which holds for any l, where α _l ≠0.

(11)

In the above equation (11), if l ₌ 0 _, for example, if the ^values of the interference weight α ₀ α _n ^* and the weight |α ₀ ^| It can be obtained from the above formula (11). Therefore, in order to obtain the value of α ₀ α _n ^* by the above equation (8), if only A _0,n and B _0,n are measured, the other A _m,n and B _m,n It turns out that the measurement of is unnecessary.

To summarize the above points, in this embodiment, _the expected value of the observable O < ^ψ | O|ψ> is represented by the following equation.

Then, in this embodiment, by combining the weights |α _n | ² and the interference weights α _m α _n ^* measured by the quantum computer with <m|O|n> calculated by the classical computer, the observable Approximate the expected value of O <ψ|O|ψ>.

In that case, in the problem of quantum chemistry, which is an important application target of this method, in many interesting cases, the state |ψ> has a bias with respect to the calculation basis |n>, so the weight |α _n | We have made the insight that ² only takes non-zero significant values for some limited number of n, and have identified that the sum of equation (10) above greatly reduces the number of terms to add.

[Superiority of proposed method over conventional method]

In this embodiment, when measuring the expected value <ψ|O|ψ> of the observable O, instead of expanding the observable O by the Pauli operator P _i and measuring it, the state |ψ> is calculated Expansions on the basis |n>(n=0, ¹ , . As a result, although it is necessary to measure A _0,n , B _0,n, etc. in this embodiment, the weight |α _n ^| Since it is sufficient to consider only certain specific (hereinafter referred to as R number of) calculation bases |n> that take a value of , the number of quantities to be measured can be relatively small.

Therefore, when using the method of this embodiment, the measurement on the quantum computer is as follows.

(1) Measure ^the weights |α _n |
(2) Measure A _0,n for a particular n. Depending on n, R-1 measurements are required.
(3) Measure B _0,n for a particular n. Depending on n, R-1 measurements are required.

Therefore, according to the method of this embodiment, a total of 1+2(R−1) types of measurements are required. On the other hand, in the conventional method, M types of measurements are required corresponding to the number of Pauli operators appearing in the expansion of the observable O. Considering the Hamiltonian of quantum chemistry, the number of Pauli operators is as large as M=O(N ⁴ ). Therefore, the number of types of measurements required is also greatly reduced. Therefore, when measuring the expected value <ψ|H|ψ> of the Hamiltonian H, it is possible to greatly reduce the total number of shots required to sufficiently reduce statistical fluctuations.

[Explanation of algorithm corresponding to this method]

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

1. Quantum computer 120 generates N qubit states |ψ>. Then, the quantum computer 120 samples the state |ψ> with the calculation basis |n>, and obtains any value of 0, 1, . The quantum computer 120 repeats the generation of the state |ψ> and the measurement of the calculation basis |n> L times to obtain a series of measured values {x}=x ⁽¹⁾ , x ⁽²⁾ , . . . , x ^(L) is obtained.

2. The classical computer 110 selects R measurements with high frequency of occurrence from the series {x} of measurements. Note that the classical computer 110 may select R measured values whose frequency of appearance is equal to or higher than a predetermined value from the sequence {x} of measured values. Next, the classical computer 110 generates a sequence of values {z}=z ⁽¹⁾ , z ⁽²⁾ , . . . , z ^(R) is set. Next, the classical computer 110 calculates the number of occurrences T _r (r=1, . . . , R) of the R measurements contained in {z}. Classical computer 110 then calculates the weighting factor f _r according to Equation (12) below.

(12)

Note that, as shown in the following equation, the weighting factor f _r is also an approximate value of the weight |α _n | ² appearing in the above equation (10).

3. Classical computer 110 computes the transition matrix elements <z _r |O|z _r′ >(r=1, . . . , R; r′=1, . . . , R) based on {z} .

4. Quantum computer 120 calculates the projection operator |φ _A,r ><φ A by measuring the projection operator |φ _{A,r >} <φ _{A,r | , r} | _Classical computer ₁₁₀ computes the _expected value A _{r (} r= 2, . . . , R). The state |φ _A,r > is represented by the following equation. Also, L' measurements are taken for each r.

5. Quantum computer 120 calculates the projection operator |φ _B,r ><φ B by measuring the projection operator |φ _{B,r >} <φ _{B ,r} | _{, r} | _Classical computer 110 computes the _expected value B _r ( _{r =} 2, . . . , R). The state |φ _B,r > is represented by the following equation. Also, L'' measurements are taken for each r.

6. The classical computer 110 calculates the weight coefficient f _r (r=1, . . . , R), the expected value A _r ( _r =2, . , R), the interference weight _gr is approximated according to the following equation (13).

(13)

Note that the interference weight _gr is also expressed by the following equation.

Classical computer 110 calculates the other component of the interference weight α _zr α _zr' ^* according to the following equation derived from equation (11) above.

 

7. Classical computer 110 computes the following based on α _z1 α _zr ^* corresponding to the interference weight g _r , the other component α _zr α _zr' ^* of the interference weight, and <z _r |O|z _r' > The expected value <ψ|O|ψ> of the observable O is approximately calculated according to the equation (14).

(14)

[Method of estimating expected value of projection operator using quantum computer]

Next, a method of estimating the expected value of the projection operator using a quantum computer will be described. A projection operator is a linear operator that satisfies P= ^P2 . In particular, we say that P is a projection operator over a state |φ> when P=|φ><φ|. The expected value of the projection operator P in the state |ψ> is represented by the following equation (15).

(15)

A method for estimating the probability p in the above equation (15) will be described below.
First, a quantum circuit U is prepared for transforming the state |φ> into the state |n> represented by a single component of the calculation basis. An arbitrary quantum state |Φ> of N qubits is represented by the following equation (16) using a calculation basis |k> (k=0, 1, . . . , 2 ^N −1).

(16)

Here, |φ> can be represented by only a single component of the calculation basis if only one of α ₀ , α ₁ , . . . Meaning. The state |φ>, the state |n>, and the quantum circuit U are represented by the following equation (17).

(17)

Such a quantum circuit U can exist. If the state |φ> is initially represented by a single component of the calculation basis such as |n>, there is no need for conversion, and a circuit (U=I) that does nothing can be considered. The following equation (18) is obtained by combining the above equation (15) and the above equation (17), which are the definitions of the probability p.

(18)

Due to the fact that the computational basis is the simultaneous eigenstate of the Z operator per qubit, and the principles of quantum theory (e.g., Born's rule), the probability p is measured on the state U|ψ> qubit in the Z basis is the probability of obtaining the measurement result n when

Estimation of this probability is performed as follows. First, the state |ψ> is generated. Then, the quantum circuit U is caused to act on that state. Finally, a Z-basis measurement is performed to obtain the measurement result. The process up to this point is regarded as a series of processes, and the measurement results are collected by repeating this process l times. Then, the number l′ of the l measurements that yielded the result n is obtained, and the ratio l′/l is taken as the estimated value of p. Note that the state |φ> represented by only two components of the calculation basis is represented by the following equation (19).

(19)

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

(20)

Although the configuration method of the quantum circuit that acts as represented by the above formula (20) is not unique, it can be configured by various methods.

[Operation of Hybrid System 100 of Embodiment]

Next, specific operations of the hybrid system 100 of the embodiment will be described. In each device of hybrid system 100, each process shown in FIGS. 3 and 4 is executed.

The hybrid system 100 approximates the expected value <ψ|O|ψ> of the observable O represented by the following equation (A1) using the following equation (A2).

(A1)

(A2)

The state |ψ> of the N qubits has a plurality of expansion coefficients α _n and a plurality of calculation bases |n> (n=0, 1, . . . , 2 ^N −1) according to the following equation (A3). and are represented by Specifically, the state |ψ> of N qubits is represented by a linear combination of a plurality of calculation bases |n>.

(A3)

First, in step S100, the user terminal 130 transmits, to the classical computer 110, calculation target information, which is information about the calculation target, and calculation method information, which is information about the calculation method, input by the user.

The calculation target information includes, for example, information related to the observable O corresponding to the physical quantity to be calculated. The calculation method information includes, for example, information on the quantum circuit, information on the number of measurement shots, and the like. Note that the information about the quantum circuit includes information about the structure of the quantum circuit _Uf that generates the state |ψ>, which is used to calculate the weighting factor _fr , which will be described later, _and It contains information about the structure of the quantum circuit _Ug used for . Further, the information on the number of measurement shots includes L representing the number of measurements of the calculation basis |n> described later, and L′ and L″ representing the number of measurements for calculating the expected value of the projection operator described later. include.

Next, in step S<b>102 , the classical computer 110 receives the calculation target information and the calculation method information transmitted from the user terminal 130 . Then, in step S102, the classical computer 110 determines the structure of the quantum circuit _Uf based on the information regarding the structure of the quantum circuit _Uf in the calculation method information. Also, in step S102, the classical computer 110 determines the number of measurement shots based on L representing the number of measurements of the calculation basis |n> in the calculation method information.

In step S<b>104 , the classical computer 110 transmits various information necessary for quantum computation to the quantum computer 120 . Specifically, the classical computer 110 transmits to the quantum computer 120 the structure of the quantum circuit _Uf and the number of measurement shots determined in step S102, and the calculation method information and calculation target information received in step S102.

At step S106, the control device 121 receives various information transmitted from the classical computer 110 at step S104.

In step S108, the control device 121 causes the quantum computer 120 to perform quantum computation according to the various information received in step S106. The quantum computer 120 repeats the measurement of the calculation basis |n> of the above formula (A3) L times according to the control by the control device 121, thereby obtaining a series of measured values of the calculation basis |n> {x}=x ^{( 1)} Get , x ⁽²⁾ , . . . , x ^(L) .

Specifically, the quantum computer 120 generates electromagnetic waves for irradiating at least one of the quantum bits in the quantum bit group 123 under the control of the control device 121 . Then, the quantum computer 120 irradiates at least one of the quantum bits in the quantum bit group 123 with the generated electromagnetic wave, and executes a quantum circuit U _f that generates the state |ψ>. The gate operation of each quantum gate included in the quantum circuit _Uf is converted into a corresponding electromagnetic wave waveform, and the generated electromagnetic wave is applied to the quantum bit group 123 by the electromagnetic wave generator 122 . The quantum computer 120 then outputs the measurement result obtained by the measurement.

In step S110, the control device 121 transmits the measurement results obtained in step S108 to the classical computer 110.

At step S112, the classical computer 110 receives the measurement results sent from the controller 121 at step S110. Next ^, the classical ^computer 110 calculates ^the measured value of the series {x}, R measured values with a large number of appearances or R measured values with a number of appearances greater than or equal to a predetermined value are selected.

Also, in step S112, the classical computer 110 sets values {z}=z ⁽¹⁾ , z ^{(2 )} , . Next, the classical computer 110 calculates the number of occurrences T _r (r=1, . . . , R) of the R measurements contained in {z}. Then, the classical computer 110 calculates a weighting factor f _r that is an approximation of |α _zr | ² in the above formula (A1) according to the following formula (A4).

(A4)

In step S114, classical _computer 110 generates _a plurality of Compute the transition matrix elements <z _r |O|z _r′ >(r=1, . . . , R; r′=1, .

In step S116, the classical computer 110 determines the structure of the quantum circuit _Ug based on the information regarding the structure of the quantum circuit _Ug in the calculation method information. Also, in step S116, the classical computer 110 determines the number of measurement shots based on L' and L'' representing the number of measurements for calculating the expected value of the projection operator in the calculation method information.

In step S118 shown in FIG. 4, the classical computer 110 transmits various information necessary for quantum computation to the quantum computer 120. FIG. Specifically, the classical computer 110 calculates the structure of the quantum circuit U _g determined in step S116, the number of measurement shots, and the projection operator |φ _A,r >< for the state |φ _A,r > of the N qubits. Information about φ _A,r | and information about the projection operator |φ _{B, r} ><φ _B,r | Note that the quantum circuit U _g is a quantum circuit for obtaining the expected value of the projection operator |φ _{A,r >} <φ _{A ,r} | circuit. Regarding these quantum circuits, A requires a quantum circuit for every r, and B requires a quantum circuit for every r. Therefore, 2(R-1) types of quantum circuits are required.

At step S120, the control device 121 receives various information transmitted from the classical computer 110 at step S118.

In step S122, the control device 121 causes the quantum computer 120 to perform quantum computation according to the various information received in step S120. Quantum computer 120 repeats the measurement of projection operator |φ _A,r ><φ _A,r | L′ times under the control of control device 121 to obtain projection operator |φ _A,r ><φ Obtain the measurement result of _A,r |. In addition, the quantum computer 120 repeats the measurement of the projection operator |φ _B,r ><φ _B,r | L″ times in accordance with the control by the control device 121 so that the projection operator |φ _B,r ><obtain the measurement result of φ _B,r |.

Specifically, the quantum computer 120 generates electromagnetic waves for irradiating at least one of the quantum bits in the quantum bit group 123 under the control of the control device 121 . Then, the quantum computer 120 irradiates at least one of the quantum bits in the quantum bit group 123 with the generated electromagnetic wave, and executes the quantum circuit _Ug . The gate operation of each quantum gate included in the quantum circuit _Ug is converted into a corresponding electromagnetic wave waveform, and the electromagnetic wave generator 122 irradiates the quantum bit group 123 with the generated electromagnetic wave. The quantum computer 120 then outputs the measurement result obtained by the measurement.

In step S124, the control device 121 transmits the measurement results obtained in step S122 to the classical computer 110.

At step S126, the classical computer 110 receives the measurement results sent from the controller 121 at step S124. Next, the classical computer 110 computes the expected value A _r of the projection operator |φ _{A,r >} <φ _{A ,r} | to calculate Further, the classical computer 110 calculates the expected value B _r of the projection operator |φ _B,r ><φ _B,r | based on the measurement result of the projection operator |φ _B,r ><φ _B,r | calculate.

In step S128, _{the classical} computer ₁₁₀ performs the above-described An interference weight g _r corresponding to α _z1 α _zr ^* in Equation (A2) is approximately calculated.

(A5)

At step S130, the classical computer 110, based on the weighting factor _f1 calculated at step S112 and the interference weight _gr calculated at step S128, according to the following equation (A6), the other component of the interference weight Approximate α _zr α _zr' ^* .

(A6)

At step S132, the classical computer 110 calculates α _z1 α _zr ^* corresponding to the interference weight g _r calculated at step S128, α _zr α _zr′ ^* calculated at step S130, and < Based on z _r |O|z _r′ >, the expected value <ψ|O|ψ> of the observable O is approximately calculated according to the above equation (A2).

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.

In step S136, the user terminal 130 receives the calculation results sent from the classical computer 110.

As described above, the classical computer of the hybrid system of the embodiment performs a plurality of calculations measured by the quantum computer when the state |ψ> of N qubits is represented by a linear combination of a plurality of calculation bases. Obtain a measurement of each of the bases. Then, the classical computer selects R measured values having a large number of occurrences or R measured values having a predetermined number of occurrences or more from the plurality of measured values based on the measured values of each of the plurality of calculation bases. , based on the number of occurrences of the R selected measurements, calculate the weighting factors f _r for the R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences. Then _, the classical computer _calculates the _expected value of the observable _O <ψ|O|ψ> is approximated. Thereby, the expected value of the observable O can be obtained efficiently using a quantum computer.

In addition, the expected value of the observable O can be efficiently obtained by appropriate division of roles between the classical computer and the quantum computer.

Next, examples will be described. In this example, taking the problem of quantum chemical calculation as an example, the results of numerical comparison between the method of this embodiment and the conventional method will be shown. Specifically, using the Hamiltonian H of various molecules and their ground state |ψ>, the magnitude of statistical fluctuations in the measurement of the expected value <ψ|H|ψ> is theoretically evaluated (using the method described later See), the total number of shots required to reduce the statistical fluctuations to the accuracy (10 ⁻³ Hartree) or less required in quantum chemistry was obtained for the two methods. The results are shown in Table 1 below. It can be seen that the total number of shots is slightly reduced for H ₂ with a small number of qubits, but for LiH and H ₂ O, the total number of shots is significantly reduced by this method.

Note that R is the number of significant calculation bases included in the sum of the above formula (14) of this method. Specifically, the minimum natural number R that satisfies the following inequality (21) is calculated numerically.

(21)

Using R as shown in the above formula (21), the expected energy value (E _IS ) obtained by this method is the exact value (E _FCI ) obtained by the FCI method, and the precision required in quantum chemistry ( It was confirmed that the difference between the two was reproduced at 10 ^-3 Hartree or less).

The above table shows the number of shots required to reduce the size of the statistical fluctuations to 10 ⁻³ Hartree or less in the measurement of the expected value <ψ|H|ψ> at the ground state |ψ> of the Hamiltonian H of various molecules. is shown. For each numerator, N is the number of qubits and M is the total number of Pauli operators contained in the Hamiltonian. R is the number of significant computational bases included in equation (14) above for the sum of our approach.

E _IS −E _FCI is the difference (in Hartree) between the expected energy value (E _IS ) obtained by this method and the exact value (E _FCI ) obtained by the FCI method. The quantum chemistry calculation software PySCF was used to determine the _EFCI . The known STO-3G basis functions were used in performing this calculation.

[Estimation of statistical error in the measurement of the expected value <ψ|O|ψ> of the observable O]
Statistical fluctuations occur in the measured value of the expected value <ψ|O|ψ> of the observable O. Here, a theoretical estimation of the magnitude of fluctuation with respect to the number of shots will be described.

The measurements required in this method are L projection measurements of |ψ> onto |n>, L′ measurements to obtain A _r (r=2, . . . , R), and B _r ( L″ measurements each to obtain r=2, . . . , R). After quantifying the statistical error from each measurement, the error propagation formula derives the statistical error of the expected value <ψ|O|ψ>.

[Projective measurement of |ψ> onto |n>]
Suppose |ψ> is projected L times with |n> (n=0, 1, . . . , 2 ^N −1). At this time, suppose that |n> appears _ln times. The probability that |n> appears in one measurement is p _n =|<n|ψ>| ² . The number of occurrences l ₀ , l ₁ , . . . , l _{n_max} (n_max=2 ^N −1) are regarded as random variables, their probability distributions are given by the following multinomial distributions.

_ln is an actually observed value, and the corresponding random variable is denoted by _Ln . Its expected value and variance are expressed by the following equations.

Note that _Ln has the following restrictions.

Therefore, the covariance of L _n and L _m (n≠m) is nonzero.

If the estimator of p _n =|<n|ψ> ^|

 

Using the observed number of |n> occurrences l _n gives an estimate of _{p n =} l _n /L, whose standard deviation is given by:

As mentioned above, we have introduced the following equation, which gives the following approximation of _pn .

 

Therefore, the statistical error of f _r can be approximated by:

(22)

Similarly, the covariance can be approximated as follows.

(23)

(Measurement of A _r and B _r )
First, consider the following measurements.

For each r, A _r can be estimated by repeating the projective measurement on a basis (and a set of basis orthonormal to it) of the state |ψ>. A certain basis of the state |ψ> is represented by the following equation.

Since each measurement result is a binary choice of whether the basis appears or not, the number of times the basis appears is described by the binomial distribution. Assuming L′ measurements are made and the above basis occurs l′ times, an estimate of A _r is given by A _r =l′/L′. Moreover, the standard deviation is represented by the following equation.

(24)

The covariance is zero because we need to make independent measurements for each r. _Br expressed by the following equation can be similarly measured and estimated.

Assuming L'' measurements for each r, the standard deviation of _Br is given by the following equation.

(25)

[Statistical error of <O>]
We have constructed an estimator of the following expected value <O> according to equation (14) above, which is the measured quantity fr (r=1, . . . , R), A _r , B _r (r=2, . , R).

Therefore, by applying the error propagation formula, the statistical error σ _<O> of <O> can be estimated as follows.

It should be noted that the technology of the present disclosure is not limited to the above-described embodiments, and various modifications and applications are possible without departing from the gist of the present invention.

For example, in the above embodiment, information may be transmitted and received between the classical computer 110 and the quantum computer 120 in any way. For example, the transmission and reception of quantum circuit parameters and the 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 all calculations may be performed. Sending and receiving may occur after completion.

Further, in the above-described embodiment, an example is described in which calculation target information is transmitted from the user terminal 130 to the classical computer 110, and the classical computer 110 executes calculation according to the calculation target information. isn't it. The user terminal 130 may transmit the calculation target information to the classical computer 110 or a storage medium or storage device accessible by the classical computer 110 via a computer network such as an IP network, but the information may not be stored in the storage medium or storage device. may be passed to the operator of the classical computer 110, and the operator may input the calculation target information to the classical computer 110 using the storage medium or storage device.

Also, in the above embodiment, the case where the quantum circuit is executed by irradiation of electromagnetic waves has been described as an example, but the invention is not limited to this, and the quantum circuit may be executed by a different method.

Also, in the above embodiment, the case where the quantum computer 120 executes quantum computation has been described as an example, but the present invention is not limited to this. For example, quantum computation may be performed by a classical computer that mimics the behavior of a quantum computer.

Further, in the above embodiment, the classical computer 110 may execute the processing executed by the quantum computer 120 . Alternatively, in the above embodiment, the quantum computer 120 may execute the processing that the classical computer 110 executes. For example, in the _{above embodiment, the classical computer 110 generates a plurality of transition matrix elements <z r |} O|z _{r′ >} (r= 1,...
·, R; r′=1, . . . , R) has been described as an example; For example, even if quantum computer 120 calculates a plurality of transition matrix elements <z _r |O|z _r′ > (r=1, . . . , R; r′=1, . good.

Also, 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 together by the same organization. . In this case, transmission of quantum computation information from the classical computer 110 to the quantum computer 120 and transmission of measurement results from the quantum computer 120 to the classical computer 110 are unnecessary. Also, in this case, it is conceivable that the controller 121 of the quantum computer 120 plays the role of the classical computer 110 in the above description.

It should be noted that, in the above embodiment, if there is no description of "only" such as "based only on XX", "only in response to XX", or "only in the case of XX", in this specification Note that it is assumed that , may also consider additional information. As an example, the statement "when a, do b" does not necessarily mean "when a, do b", unless explicitly stated.

In addition, in the above embodiments, when expressions such as "optimize" or "optimized parameters" are used, these expressions of "optimization" mean approaching an optimal state. Please note that Therefore, when trying to obtain parameters that minimize a certain function, the parameters obtained by optimizing the function may not be the global solution that minimizes the function, but the local solution. Note that it is assumed

In addition, even if there is an aspect in which some method, program, terminal, device, server, or system (hereinafter "method, etc.") performs operations different from those described in this specification, each aspect of the disclosed technology is The existence of operations that are the same as any of the operations described herein and that differ from the operations described herein indicate that the methods, etc. is not outside the scope of

It should be noted that the processing executed by the CPU reading the software (program) in the above embodiment may be executed by various processors other than the CPU. In this case, the processor is PLD (Programmable Logic Device) whose circuit configuration can be changed after manufacturing such as FPGA (Field-Programmable Gate Array), and ASIC (Application Specific Integrated Circuit) to execute specific processing. A dedicated electric circuit or the like, which is a processor having a specially designed circuit configuration, is exemplified. Also, each process may be executed by one of these various processors, or a combination of two or more processors of the same or different type (for example, a plurality of FPGAs, a combination of a CPU and an FPGA, etc.). ) can be run. More specifically, the hardware structure of these various processors is an electric circuit in which circuit elements such as semiconductor elements are combined.

Also, in the specification of the present application, an embodiment in which the program is pre-installed has been described, but it is also possible to store the program in a computer-readable recording medium and provide it. Specifically, in each of the above-described embodiments, the program has been pre-stored (installed) in the storage, but the present invention is not limited to this. Programs are stored on non-transitory storage media such as CD-ROM (Compact Disk Read Only Memory), DVD-ROM (Digital Versatile Disk Read Only Memory), and USB (Universal Serial Bus) memory. may be provided in the form Also, the program may be downloaded from an external device via a network.

Also, each component of the hybrid system of this embodiment does not have to be implemented by a single computer or server, but may be implemented by being distributed among multiple computers connected by a network.

For example, the processing executed by the classical computers of the above embodiments may be distributed and processed by a plurality of classical computers connected by a network. Alternatively, for example, the processing executed by the quantum computers of the above embodiments may be distributed and processed by a plurality of quantum computers connected by a network. In this case, at least one or more classical computers and at least one or more quantum computers constitute a hybrid system.

For example, when a hybrid system is configured by a plurality of classical computers and a plurality of quantum computers, and the state |ψ> of N qubits is represented by a linear combination of a plurality of calculation bases, one of the plurality of quantum computers One or more quantum computers obtain measurements of a plurality of computational bases by measuring each of the plurality of computational bases. Then, one or more classical computers among the plurality of classical computers acquire the measurement values of the plurality of calculation bases measured by the quantum computer. Next, one or more classical computers out of the plurality of classical computers, based on the measured values of each of the plurality of calculation bases, R measured values having a large number of occurrences from the plurality of measured values or the number of occurrences is a predetermined value R number of measured values are selected, and based on the number of occurrences of the selected R number of measured values, R number of calculation bases |z _r >(r=1, . . . R) to calculate the weighting factor f _r . Next, one or more classical computers among the plurality of classical computers or one or more quantum computers among the plurality of quantum computers are provided for each combination of the calculation basis |z _r > and the calculation basis |z _r′ > Compute the transition matrix elements <z _r |O|z _r′ > of . Then, one or more classical computers among the plurality of classical computers have a plurality of transition matrix elements <z _r |O|z _r′ > weighted by a combination of the weighting factor f _r and the weighting factor f _r′. The expected value <ψ|O|ψ> of the observable O is approximated by the weighted sum.

The disclosure of Japanese Patent Application No. 2021-133701 filed on August 18, 2021 is incorporated herein by reference in its entirety. All publications, patent applications and technical standards mentioned herein are to the same extent as if each individual publication, patent application and technical standard were specifically and individually noted to be incorporated by reference. incorporated herein by reference.

Claims (8)

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 |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   Acquiring a measurement value of each of a plurality of calculation bases measured by a quantum computer,
   Based on the measured values of each of the plurality of calculation bases, R measured values with a large number of appearances or R measured values with a number of appearances greater than or equal to a predetermined value are selected from the plurality of measured values, and R selected based on the number of occurrences of the measured values of, calculate a weighting factor f _r for R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences;
   _{The expected} value of observable O <ψ|O _{| ψ} approximating >,
   A quantum information processing method in which processing is performed by a classical computer.
2. The state |ψ> of the N qubits has a plurality of expansions based on a plurality of calculation bases and expansion coefficients α _n (n=0, 1, . . . , 2 ^N −1) for each of the plurality of calculation bases. is represented by a linear combination of calculation bases weighted by the coefficients α _n ,
   When approximating the expected value <ψ|O|ψ>, a weighting factor f _r for R calculation bases |z _r > (r=1, . . . , R) and a calculation base |z _r > α _zr α ^* zr′ calculated based on the expansion coefficient α _zr for and the expansion coefficient α _{zr′ for} the calculation basis |z _r′ > and the plurality of transition matrix elements <z _r |O|z _r′ > based on and approximately calculate the expected value <ψ|O|ψ> of the observable O according to the following formula (A1),
   The quantum information processing method according to claim 1.
   

   

   
   
   (A1)
3. A quantum information processing method executed by a hybrid system including a classical computer and a quantum computer,
   When the state |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   A quantum computer obtains measurements of a plurality of calculation bases by measuring each of the plurality of calculation bases,
   A classical computer obtains measurements of multiple computational bases measured by a quantum computer,
   A classical computer, based on the measured values of each of the plurality of calculation bases, selects R measured values having a large number of occurrences or R measured values having a number of occurrences equal to or greater than a predetermined value from the plurality of measured values, and selects Based on the number of occurrences of the R measured values obtained, calculate a weighting factor f _r for the R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences;
   A classical computer or a quantum computer calculates a plurality of transition matrix elements <z _r |O|z _r' > for each combination of computation basis |z _r > and computation basis |z _r' >;
   A classical computer computes the _{expected value} of observable _{O <} ψ approximating |O|ψ>,
   Quantum information processing method.
4. When the expected value <ψ|O|ψ> of the observable O represented by the following equation (A2) is approximated using the following equation (A1),
   A quantum computer converts the state |ψ> of N qubits into a plurality of expansion coefficients α _n and a plurality of calculation bases |n> (n=0, 1, . . . , 2 ^N −1 according to the following equation (A3) ) and by repeating the measurement of the calculation basis |n> L times, the series {x}=x ⁽¹⁾ , x ⁽²⁾ , . , x ^(L) , and
   A classical computer, based on the series {x} of the measured values, selects the R measured values having a large number of occurrences or the R measured values having a number of occurrences equal to or greater than a predetermined value in the series {x} of the measured values. select the measured value, set {z}=z ⁽¹⁾ , z ^{( 2} ) , . , ^and calculate the number of occurrences T _r (r=1, _. calculate _r ,
   A classical computer or a quantum computer performs transition matrix element <z _r |O|z _r′ > (r=1, . . . , R; r′=1 , ..., R),
   A quantum computer measures the projection operator |φ _{A , r} ><φ _{A,r | , r} | and measuring the projection operator _| φ _{B, r} ><φ _B,r | _,r ><φ _B,r |
   A classical computer calculates the expected value A _r of the projection operator |φ _A,r ><φ _A,r | based on the measurement result of the projection operator |φ _A,r ><φ _A,r | and calculating the expected value B _r of the projection operator |φ _B,r ><φ _B,r | based on the measurement result of the projection operator |φ _B,r ><φ _B,r |
   A classical computer approximates an interference weight g _r corresponding to α _z1 α _zr ^* according to the following equation (A5) based on the weighting factor f _r , the expected value _Ar , and the expected value B _r Based on the weighting factor f _r and the interference weight g _r , α _zr α _zr′ ^* is approximated according to the following equation (A6), and the α _z1 corresponding to the interference weight _gr The _{expected value} < ^ψ _{| O |} ^ψ approximating >,
   The quantum information processing method according to claim 3.
   

   

   
   
   (A1)
   

   

   
   
   (A2)
   

   

   
   
   (A3)
   

   

   
   
   (A4)
   

   

   
   
   (A5)
   

   

   
   
   

   

   
   
   (A6)
5. A quantum information processing program to be executed by a classical computer of a hybrid system including a classical computer and a quantum computer,
   When the state |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   Acquiring a measurement value of each of a plurality of calculation bases measured by a quantum computer,
   Based on the measured values of each of the plurality of calculation bases, R measured values with a large number of appearances or R measured values with a number of appearances greater than or equal to a predetermined value are selected from the plurality of measured values, and R selected based on the number of occurrences of the measured values of, calculate a weighting factor f _r for R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences;
   _{The expected} value of observable O <ψ|O _{| ψ} approximating >,
   A quantum information processing program for making a classical computer perform processing.
6. A classical computer of a hybrid system comprising a classical computer and a quantum computer,
   When the state |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   Acquiring a measurement value of each of a plurality of calculation bases measured by a quantum computer,
   Based on the measured values of each of the plurality of calculation bases, R measured values with a large number of appearances or R measured values with a number of appearances greater than or equal to a predetermined value are selected from the plurality of measured values, and R selected based on the number of occurrences of the measured values of, calculate a weighting factor f _r for R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences;
   _{The expected} value of observable O <ψ| _O | _ψ approximating >,
   classical computer.
7. A hybrid system comprising a classical computer and a quantum computer,
   When the state |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   A quantum computer obtains measurements of a plurality of calculation bases by measuring each of the plurality of calculation bases,
   A classical computer obtains measurements of multiple computational bases measured by a quantum computer,
   A classical computer, based on the measured values of each of the plurality of calculation bases, selects R measured values having a large number of occurrences or R measured values having a number of occurrences equal to or greater than a predetermined value from the plurality of measured values, and selects Based on the number of occurrences of the R measured values obtained, calculate a weighting factor f _r for the R calculation bases |z _r >(r=1, . . . , R) according to the number of occurrences;
   A classical computer or a quantum computer calculates a plurality of transition matrix elements <z _r |O|z _r' > for each combination of computation basis |z _r > and computation basis |z _r' >;
   A classical computer computes the _{expected value} of observable _{O <} ψ approximating |O|ψ>,
   hybrid system.
8. A hybrid system comprising a plurality of classical computers and a plurality of quantum computers,
   When the state |ψ> of N qubits is represented by a linear combination of multiple computational bases,
   One or more quantum computers of the plurality of quantum computers obtain measurements of the plurality of calculation bases by measuring each of the plurality of calculation bases,
   One or more classical computers of the plurality of classical computers obtain measurements of a plurality of computational bases measured by a quantum computer;
   One or more classical computers out of the plurality of classical computers, based on the measured values of each of the plurality of calculation bases, R measured values having a large number of occurrences from the plurality of measured values, or the number of occurrences is a predetermined value or more Select R measurements, and based on the number of occurrences of the R selected measurements, for R computational bases |z _r >(r=1, . Calculate the weighting factor f _r ,
   One or more classical computers among the plurality of classical computers or one or more quantum computers among the plurality of quantum computers have a plurality of transition matrices for each combination of the calculation basis |z _r > and the calculation basis |z _r′ > Compute the element <z _r |O|z _r' >,
   weighted sum of a plurality of transition matrix elements <z _r |O|z _r' > where one or more classical computers among the number of classical computers weight a combination of weighting factor f _r and weighting factor f _r' Approximate the expected value <ψ|O|ψ> of the observable O by
   hybrid system.

Images