JP7630826B2 - Quantum information processing method, classical computer, hybrid system, and quantum information processing program - Google Patents

Description

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

A variational quantum algorithm for performing quantum computation using a Noisy Intermediate-Scale Quantum (NISQ) device is 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, simply referred to as "observable") using a quantum computer.

When calculating the expected value of an observable using a quantum computer, a state is generated by the quantum computer and then that state is measured. The expected value of the observable is then calculated based on the results of measuring the state by the quantum computer. Note that a series of processes consisting of generating one state by the quantum computer and measuring that state is also called a "shot."

When applying variational quantum algorithms to quantum chemical calculations, the number of shots required to sufficiently reduce the statistical fluctuations in state measurements may be excessive. For example, Non-Patent Document 3 considers a quantum chemical calculation problem to find the energy of an actual molecule, and shows that a huge number of shots are required to reduce the statistical fluctuations more than the energy accuracy required in quantum chemistry, and it takes several days just to obtain the expected value of the energy once. For this reason, for example, Non-Patent Document 4 attempts to reduce the total number of shots by measuring multiple Pauli operators simultaneously.

Peruzzo, A., McClean, J., Shadbolt, P. et al. “A variational eigenvalue solver on a photonic quantum processor”. Nat Commun 5, 4213 (2014). “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) Gonthier, J. F., Radin, M. D., Buda, C., Doskocil, E. J., Abuan, C. M., and Romero, J. “Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver”. Rubin, N. C., Babbush, R., and McClean J., “Application of fermionic marginal constraints to hybrid quantum algorithms”. New J. Phys. 20, 053020 (2018).

The disclosed technology has been made in consideration of the above 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.

In order to achieve the above-mentioned object, a quantum information processing method of 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, in which, when a state |ψ〉 of N quantum bits is expressed by a linear combination of multiple computational bases, measurement values of each of the multiple computational bases measured by the quantum computer are acquired, and based on the measurement values of each of the multiple computational bases, 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 multiple measurement values, and based on the occurrence counts of the selected R measurement values, weighting coefficients f _r for R computational bases |z _r 〉 (r = 1, ..., R) corresponding to the occurrence _counts are calculated, and an approximate calculation of an expectation _value <_{ψ|O|ψ> of an observable O is performed by a weighted sum of multiple transition matrix elements <z r} |O|z _r' >, where the combination of weighting coefficient f r and weighting coefficient f r' is used as weight.

The disclosed technology has the effect of making it possible to efficiently obtain the expected value of an observable using a quantum computer.

1 is a diagram illustrating an example of a schematic configuration of a hybrid system 100 according to an embodiment of the present invention. 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. FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment. FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment.

Below, an embodiment of the disclosed technology will be described in detail with reference to the drawings.

<Hybrid system 100 according to the embodiment>

Figure 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. As shown in Figure 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.

In the hybrid system 100 of this embodiment, the quantum computer 120 performs a specified 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 the calculation result according to the quantum calculation to the user terminal 130. In this way, the specified calculation process is executed by the hybrid system 100 as a whole.

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. Note that the classical computer 110 may include one or more devices or servers. The program may include one or more programs, and may be recorded on a computer-readable storage medium to form a non-transient program product.

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

In the example of FIG. 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 of the quantum computer 120 are also QPUs (Quantum Processing Units). Note that in this embodiment, "quantum computer" does not mean that no operations are performed using classical bits, but refers to a computer that includes operations using quantum bits.

The control device 121 is a classical computer that performs calculations using classical bits, and instead performs some or all of the processing 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 quantum bit group 123 in response to receiving a parameter θ of the quantum circuit U(θ).

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

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 external devices and output devices are connected, and a read/write (R/W) unit 55 that controls the 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.

The hybrid system 100 of the embodiment efficiently calculates the expected value <ψ|O|ψ> of the observable O. Below, we will explain the assumptions for the processing performed by the hybrid system 100 of the embodiment.

[1. Problem setting and background]

Consider the state |ψ> of N quantum bits generated on a quantum computer. The state |ψ> is a vertical vector with 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. Note that |0> represents the initialized state of N quantum bits.

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).

(1)

[2. Conventional methods for measuring expectations and their challenges]

The conventional method for measuring the expectation value of an observable O consists of the following steps:

(1) Decomposition of an observable O into Pauli operators An observable O on N quantum bits is expressed as a linear combination of the following Pauli operators.

The observable O is expressed by a linear combination of Pauli operators P _i as shown in the following equation (2).

(2)

Here, I, X, Y, and Z are the identity operator and the Pauli matrix of one quantum bit. In addition, c _i in the above formula (2) is the expansion coefficient, and M is the total number of Pauli operators. Based on the above formula (2), the expectation value <ψ|O|ψ> of the operator, the observable O, is expressed by the following formula (3).

(3)

(2) Measurement of the expectation value of the Pauli operator Next, each term <ψ|P _i |ψ> on the right side of the above formula (3) is measured by a quantum computer. <ψ|P _i |ψ> is the expectation value of the Pauli operator. The expectation value <ψ|P _i |ψ> of the Pauli operator is obtained by a standard measurement operation in a quantum computer. Specifically, when a state |ψ> is generated once on a quantum computer and a measurement operation according to the Pauli operator P _i is performed, a result of either +1 or -1 is probabilistically obtained. The probability is determined by the combination of |ψ> and P _i .

The average of the l _i measurement results obtained by generating the state l _i times is used as the expected value <ψ|P _i |ψ>, where each of the l _i measurement results has a value of either +1 or −1.

For example, if a state is generated 100 times, +1 is obtained 30 times, and -1 is obtained 70 times, the 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 smaller by 1/√l _i with respect to the number of trials l _i to obtain +1 or -1. As mentioned above, in the field of quantum computers, a series of processes of generating a state and performing a measurement is also called a "shot." For this reason, hereinafter, l _i will also be simply called the "number of shots."

(3) Addition process of expectation values of Pauli operators Next, the expectation value <ψ|P _i |ψ> of the observable O is calculated by calculating the above equation (3) based on the value of the expectation value <ψ|O|ψ> obtained in (2) above.

In this way, although a NISQ device cannot directly measure the expectation value <ψ|O|ψ> of the observable O, the expectation value <ψ|P _i |ψ> of the Pauli operator can be easily measured, so the expectation value <ψ| _O |ψ> of the observable O is calculated based on the expectation value <ψ|P i |ψ> of the Pauli operator.

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 fluctuation of the measurement, l ₁ + ... +l _M , often becomes excessively large. 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 quantum bits N, it is necessary to measure the expectation values of a very large number of Pauli operators (M = O (N ⁴ )). Therefore, in this embodiment, a method for efficiently calculating the expectation value of the observable O is proposed.

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

(4)

In general, the state |ψ〉 is expressed as a linear combination of basis vectors belonging to a complete system. There are various possibilities for the complete system, but in the following, in order to concretely advance the discussion, a computational basis |n〉 (n=0, 1, ..., 2 ^N -1) is used as the complete system. The state |ψ〉 is expanded as shown in the following formula (5) using the computational basis |n〉.

(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)

In the above formula (6), α _n is also called the probability amplitude, and its absolute value squared, |α _n | ² , represents the probability. Therefore, when a state |ψ〉 is prepared on a quantum computer and a standard measurement operation is performed on the state |ψ〉, one of the values n=0, 1, ..., 2 ^N -1 is probabilistically obtained as the measurement result. And |α _n | ² represents the probability that the measurement result will be a certain value n.

Conversely, by repeating the operation of preparing a state |ψ〉 on a quantum computer and projecting and measuring it onto the computational basis |n〉, a frequency distribution of the measurement result n can be obtained, and the value of |α _n | ² , which is the true probability, can be estimated from the frequency distribution.

Here, the expectation value <ψ|O|ψ> can be rewritten using the probability amplitude α _n , which is a quantity associated with the state |ψ>. When the following equation, which indicates that the computational basis |n> (n=0, 1, ..., 2 ^N -1) forms a complete system, is used, the following equation (7) is derived.

(7)

In addition, the last line of the above formula (7) does not include a term where α _{n =} ⁰ or α _m =0. The above formula (7) means that the expected value of the observable O in the state | _ψ ^> is obtained by adding up <m|O|n>/(α _m α _n ^* ) with |α m | 2 |α n | 2 as a weight. As described below, <m|O|n> can be efficiently calculated by a classical computer. Also, α _m α _n ^* can be measured using a quantum computer. Therefore, the expected value <ψ|O|ψ> of the observable O is obtained by combining |α _n | ² and α _m α _n ^* measured using a quantum computer with <m|O|n> calculated using a classical computer.

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

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 a linear time with respect to the number of quantum bits. The transition matrix element <m|O|n> is obtained by adding up the <m|P _i |n> calculated by the classical computer.

Next, α _m α _n ^* will be explained. When m = n, α _m α _n ^* = |α _n | ² , so as described 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 a quantum computer is required. To explain this point, the following transformation is required.

(8)

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

(9)

In addition, A _m,n and B _m,n can be estimated by projective measurement using a quantum computer. Therefore, it is possible to calculate the value of α _m α _n ^* by combining A _m,n and B _m,n with the separately measured weight |α _n | ² .

In general, since α _m α _n ^* has a complex phase, different terms cancel or reinforce each other in the sum of the above equation (7), and therefore 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 according to the following equation (10).

(10)

It may seem that there may be a problem in terms of computational cost because an exponentially large number of terms, 2 ^N × 2 ^N , are to be added up. However, it is empirically known that the number of terms to be added up can be greatly reduced depending on the type of problem to be solved, such as quantum chemical calculations.

For example, in quantum chemistry 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 for such states, with other components being relatively small.

In addition, it is mathematically known that the eigenstate has a finite amplitude only in a limited part of the entire basis 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.). For this reason, the weight |α _n | ² has a significant value only for certain elements with respect to n, and for other elements, it is zero or a value so small that it can be statistically ignored in estimating the expected value in a quantum computer. Therefore, when taking the sum for m and n, such a basis can be ignored from the beginning. Therefore, it is sufficient to evaluate the transition matrix element <m|O|n> only for a certain pair of calculation bases (m, n) that gives a weight that cannot be ignored.

Furthermore, when determining the interference weight α _m α _n ^* for such a pair (m, n), it is sufficient to further measure A _m,n and B _m,n for a portion of the (m, n) pairs.

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

(11)

In the above formula (11), for example, if l=0, then if the values of the interference weight α ₀ α _n ^* and the weight |α ₀ | ² are known, the values of all other interference weights α _m α _n ^* can be obtained from the above formula (11). Therefore, in order to obtain the value of α ₀ α _n ^* using the above formula (8), it is clear that as long as A _0,n and B _0,n are measured, there is no need to measure the other A _m,n and B _m,n .

To summarize the above points, in this embodiment, the expected value <ψ|O|ψ> of the observable O is expressed by the following equation using the probability amplitude α _n (n=0, 1, . . . , 2 ^N −1) of the state |ψ>.

In this embodiment, the expectation value <ψ|O|ψ> of the observable O is approximately calculated by combining the weight |α _n | ² and the interference weight α _m α _n ^* measured by the quantum computer with <m|O|n> calculated by the classical computer.

In addition, at that time, the inventors made the insight that in quantum chemistry problems, which are important applications of this method, in many interesting cases, the state |ψ〉 has a bias with respect to the computational basis |n〉, and therefore the weight |α _n | ² takes a significant non-zero value only for a limited number of n, and specified that the number of terms to be added in the sum of the above equation (10) is greatly reduced.

[Advantages of the proposed method over conventional methods]

In this embodiment, when measuring the expectation value <ψ|O|ψ> of the observable O, instead of expanding the observable O with the Pauli operator P _i and measuring it, the state |ψ> is expanded in the computational basis |n> (n=0, 1, ..., 2 ^N -1) to avoid the problem of "measurement of a large number of Pauli operators". As a result, although measurements of A _0,n , B _0,n, etc. are required in this embodiment, for biased states |ψ> that often appear in quantum chemical calculations, it is only necessary to consider certain (hereinafter, R) computational bases |n> in which the weight |α _n | ² takes a significant value, and therefore 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) Calculation of state |ψ〉 Measure weights |α _n | ² by repeatedly sampling in basis |n〉.
(2) Measure A _0,n for a specific n. Depending on n, R-1 measurements are required.
(3) Measure B _0,n for a specific 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. 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 very large, M=O(N ⁴ ), whereas in the method of this embodiment, it is empirically known that R is much smaller than this, so the types of measurements required are also 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.

[Description of the algorithm corresponding to this method]

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

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) .

2. The classical computer 110 selects R measurement values with a high occurrence frequency from the measurement value series {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 measurement value series {x}. Next, the classical computer 110 sets a sequence of values {z}=z ⁽¹⁾ , z ⁽²⁾ , ..., z ^(R) arranged in descending order of occurrence frequency based on each of the selected R measurement values. Next, the classical computer 110 calculates the occurrence count T _r (r=1, ..., R) of the R measurement values included in {z}. Then, the classical computer 110 calculates a weighting coefficient f _r according to the following formula (12).

(12)

As shown in the following equation, the weighting coefficient f _r is also an approximation of the weight |α _n | ² appearing in the above equation (10).

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

4. The quantum computer 120 measures the projection operator |φ _{A,r ＞} ＜φ _{A,r | for the state |φ A,r} ＞ of the N quantum bits to obtain the measurement result of the projection operator |φ _A,r ＞＜φ _{A, r} |. The classical computer 110 calculates the expectation value A _{r (r} =2,...,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 |. The state |φ _A,r ＞ is expressed by the following formula. Also, L' measurements are performed for each r.

5. The quantum computer 120 measures the projection operator |φ _{B,r ＞} ＜φ _{B,r | for the state |φ B,r} ＞ of the N quantum bits to obtain the measurement result of the projection operator |φ _B,r ＞＜φ _B,r |. The classical computer 110 calculates the expectation value _{B r (r} =2 _{, ..., 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 |. Note that the state |φ _B,r ＞ is expressed by the following formula. Also, L'' measurements are performed for each r.

6. The classical computer 110 approximates the interference weights g r by combining the weighting coefficients f _r (r=1,...,R), the expectation values A _r (r=2,...,R), and the expectation values B _r ( _r =2,...,R) according to the following equation (13):

(13)

The interference weight g _r can also be expressed by the following equation.

The classical computer 110 calculates the other components of the interference weights α _zr α _zr′ ^* according to the following equation, which is derived from equation (11) above:

7. The classical computer 110 approximately calculates the expectation value < _ψ |O| _ψ > of the observable O based on α _z1 α _zr ^* corresponding to the interference weight g _r , other components of the interference weight α _zr α _zr' ^* , and <z r |O|z r'> according to the following equation (14).

(14)

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

Next, a method for estimating the expectation value of a projection operator using a quantum computer will be described. A projection operator is a linear operator that satisfies P = ^P2 . 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 formula (15).

(15)

In the following, a method for estimating the probability p in the above equation (15) will be described.
First, a quantum circuit U is prepared that converts a state |φ〉 into a state |n〉 represented by a single component of a computational basis. Note that an arbitrary quantum state |Φ〉 of N quantum bits is expressed by the following formula (16) using a computational basis |k〉 (k=0, 1, ..., 2 ^N -1).

(16)

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

(17)

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).

(18)

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 the measurement result n will be obtained when the quantum bit is measured in the Z basis on the state U | ψ 〉.

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 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).

(19)

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).

(20)

The construction of a quantum circuit that operates as expressed by the above formula (20) is not unique, but it can be constructed using various techniques.

[Operation of the hybrid system 100 according to the embodiment]

Next, the specific operation of the hybrid system 100 of the embodiment will be described. Each device of the hybrid system 100 executes the processes shown in Figures 3 and 4.

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

(A1)

(A2)

The state |ψ> of N quantum bits is expressed by a plurality of expansion coefficients α _n and a plurality of computational bases |n> (n=0, 1, ..., 2 ^N -1) according to the following formula (A3). Specifically, the state |ψ> of N quantum bits is expressed by a linear combination of a plurality of computational bases |n>.

(A3)

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

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 information on the structure of a quantum circuit U _f that generates a state |ψ〉, which is used to calculate a weighting coefficient f _r described later, and information on the structure of a quantum circuit U _g that is used to calculate an interference weighting coefficient g _r described later. The information on the number of measurement shots includes L, which represents the number of measurements of a calculation basis |n〉, which will be described later, and L' and L'', which represent the number of measurements for calculating an expectation value of a projection operator, which will be described later.

Next, in step S102, 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 on 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 S104, 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 and the number of measurement shots of the quantum circuit _Uf determined in step S102, and the computation method information and computation target information received in step S102.

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

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 computational basis |n> of the above formula (A3) L times to obtain a series of measurement values of the computational basis |n> {x}=x ⁽¹⁾ , x ⁽²⁾ , ..., x ^(L) of the computational basis |n>.

Specifically, the quantum computer 120 generates electromagnetic waves to be irradiated to at least any 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 waves to at least any one of the quantum bits in the quantum bit group 123, and executes the quantum circuit _Uf 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 waves are 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.

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

In step S112, the classical computer 110 receives the measurement results transmitted from the control device 121 in step S110. Next, the classical computer 110 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 measurement value series {x}, based on the measurement value series {x}=x ⁽¹⁾ , x ⁽²⁾ , ..., x ^(L) of the computational basis |n>, which is the measurement result.

In step S112, the classical computer 110 sets the R measured values in ascending order as {z}=z ⁽¹⁾ , z ⁽²⁾ , ..., z ^(R) . Next, the classical computer 110 calculates the number of occurrences T _r (r=1, ..., R) of the R measured values included in {z}. Then, the classical computer 110 calculates a weighting coefficient f _r , which is an approximation of |α _zr | ² in the above formula (A1), according to the following formula (A4).

(A4)

In step S114, the classical computer 110 calculates multiple transition matrix elements <zr|O| _zr '> ( _r =1, ..., R; r'=1, ..., R) for each combination of the computational basis | _zr > and the computational basis |zr _' > in the above formula (A2) based on {z} set in step S112.

In step S116, the classical computer 110 determines the structure of the quantum circuit _Ug based on information on 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'' that represent 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. Specifically, the classical computer 110 transmits to the quantum computer 120 the structure and number of measurement shots of the quantum circuit U _g determined in step S116, information on the projection operator |φ _{A, r} ＞＜φ _{A,r | for the state |φ A,r} ＞ of the N quantum bits, and information on the projection operator |φ _{B ,} r ＞＜φ _{B,r | for the state |φ B,r} ＞ of the N quantum bits. 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 | and the expected value of the projection operator |φ _B,r ＞＜φ _B,r |. 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. For this reason, 2(R-1) types of quantum circuits are required.

In step S120, the control device 121 receives various information sent from the classical computer 110 in step S118.

In step S122, the control device 121 causes the quantum computer 120 to execute quantum computation according to the various information received in step S120. The quantum computer 120 repeats the measurement of the projection operator |φ _A,r ＞＜φ _A,r | L′ times in accordance with the control of the control device 121 to obtain the measurement result of the projection operator |φ _A,r ＞＜φ _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 of the control device 121 to obtain the measurement result of the projection operator |φ _B,r ＞＜φ _B,r |.

Specifically, the quantum computer 120 generates electromagnetic waves to be irradiated to at least any 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 waves to at least any one of the quantum bits in the quantum bit group 123 to execute 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 generated electromagnetic waves are 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.

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

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 an 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 |. In addition, the classical computer 110 calculates an 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 |.

In step S128, the classical computer 110 approximately calculates an interference weight g _r corresponding to α z1 α zr ^* in the above equation (A2) based on the weighting coefficient f _r calculated in step _S112 and the expected values A _r and B _r calculated in step _S126 , according to the following equation (A5).

(A5)

In step S130, the classical computer 110 approximately calculates the other components α zr α zr' * of the interference weight based on the weighting coefficient f ₁ calculated in step S112 and the interference weight g _r calculated in step _S128 according to the following equation ( _A6 ⁾ .

(A6)

In step S132, the classical computer 110 approximates the expectation value <ψ|O|ψ> of the observable O according to the above equation (A2) based on α _z1 α _zr ^* corresponding to the interference weight g _r calculated in step S128, α _zr α _zr' ^* calculated in step S130, and <z _r |O|z _r '> calculated in step S114.

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

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

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 that occur more than a predetermined value from the multiple measurement values, and calculates weighting coefficients f r for R computational bases |z _r > (r = 1, ..., R) according to the number of occurrences based on the number of occurrences of the selected _R measurement values. Then, the classical computer approximately calculates the expected value < _ψ |O|ψ> of the observable O by a weighted sum of multiple transition matrix elements <z _r |O|z _r' >, with the combination of the weighting coefficient _{f r} and the weighting coefficient f r' as the weight. This makes it possible to efficiently obtain the expected value of the observable O using a quantum computer.

In addition, by appropriately dividing up the roles between classical and quantum computers, the expectation value of the observable O can be obtained efficiently.

Next, an example will be described. In this example, a problem of quantum chemical calculation is taken as an example, and a numerical comparison result between the method of this embodiment and the conventional method is shown. Specifically, the magnitude of statistical fluctuation in the measurement of the expectation value <ψ|H|ψ> is theoretically evaluated using the Hamiltonian H of various molecules and its ground state |ψ> (see the method described later), and the total number of shots required to make the statistical fluctuation equal to or less than the accuracy required in quantum chemistry (10 ⁻³ Hartree) 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 quantum bits, but the total number of shots is significantly reduced for LiH and H ₂ O by this method.

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 determined as the smallest natural number R that satisfies the following inequality (21).

(21)

It has been confirmed that, when R as shown in the above formula (21) is used, the energy expectation value (E _IS ) calculated by this method reproduces the exact value (E _FCI ) calculated by the FCI method with the accuracy required by quantum chemistry (the deviation between the two is 10 ⁻³ Hartree or less).

The above table shows the sum of the number of shots required to reduce the magnitude of statistical fluctuations in the measurement of the expectation value <ψ|H|ψ> of the Hamiltonian H of various molecules to 10 ⁻³ Hartree or less. For each molecule, N is the number of quantum bits, and M is the total number of Pauli operators included in the Hamiltonian. R is the number of significant calculation bases included in the above equation (14) of the sum of this method.

E _IS -E _FCI is the difference (unit: Hartree) between the energy expectation value (E _IS ) calculated by this method and the exact value (E _FCI ) calculated by the FCI method. The quantum chemistry calculation software PySCF was used to calculate E _FCI . The known STO-3G basis set was used in this calculation.

[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, a theoretical estimate of the magnitude of the fluctuations 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 L″ measurements to obtain B _r (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.

[Projection measurement of |ψ〉 onto |n〉]
Suppose that |ψ〉 is projected and measured L times with |n〉 (n = 0, 1, ..., 2 ^N -1). In this case, suppose that |n〉 appears l _n times. The probability that |n〉 appears in one measurement is p _n = |<n|ψ〉| ^2. If the number of occurrences of each n, l ₀ , l ₁ , ..., l _{n_max} (n_max = 2 ^N -1), is regarded as a random variable, its probability distribution is given by the following multinomial distribution.

While _ln is an actually observed value, the corresponding random variable is represented as _Ln . Its expectation and variance are expressed by the following equations.

Note that there are the following restrictions regarding _Ln .

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

When the estimated quantity of p _n =|<n|ψ>| ² is expressed by the following equation, its variance and covariance are given by the following equations.

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

As mentioned above, the following equation was introduced, which gives an approximation of p _n as follows:

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

(22)

The covariance can also be approximated similarly as follows:

(23)

(Measurement of _Ar and B _r )
First, consider the following measurement:

For each r, A _r can be estimated by repeating the projection measurement in a basis of state |ψ〉 (and a set of orthonormal bases thereto), where the basis of state |ψ〉 is expressed by the following equation:

Since each measurement result is either the above-mentioned basis or something else, the number of times the above-mentioned basis appears is described by a binomial distribution. If L' measurements are made and the above-mentioned basis appears l' times, the estimate of A _r is given by A _r = l'/L'. Its standard deviation is expressed by the following formula:

(24)

Since independent measurements must be made for each r, the covariance is zero. Similar measurements and estimates can be made for B _r , which is given by:

If measurements are taken L'' times for each r, the standard deviation of B _r is expressed by the following formula:

(25)

[Statistical error of <O>]
The following estimator of the expectation value <O> has been constructed using the above equation (14), but this quantity can also be regarded as a function of the measurements fr (r=1, . . . , R), A _r , and B _r (r=2, . . . , R).

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

The technology disclosed herein is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the spirit and scope of the 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 manner. For example, the transmission and reception of quantum circuit parameters and 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.

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 then inputs the calculation target information to the classical computer 110 using the storage medium or storage device.

In addition, in the above embodiment, a quantum circuit is executed by irradiating electromagnetic waves, but this is not limited to the above, and the quantum circuit may be executed by a different method.

In the above embodiment, the quantum computer 120 performs quantum computation, 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.

In addition, in the above embodiment, the classical computer 110 may execute the process executed by the quantum computer 120. Alternatively, in the above embodiment, the quantum computer 120 may execute the process executed by the classical computer 110. For example, in the above embodiment, the classical computer 110 calculates a plurality of transition matrix elements < _zr |O| _{zr '} >(r=1,...,R;r'=1,...,R) for each combination of the computational base | _zr > and the computational base |zr'>, but this is not limited to this. For example, the quantum computer 120 may calculate a plurality of transition matrix elements < _zr |O|zr _' >(r=1,...,R;r'=1,...,R).

In addition, 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 unnecessary 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 conceivable that the control device 121 of the quantum computer 120 will take on the role of the classical computer 110 in the above description.

Please note that in the above embodiment, unless there is a statement such as "based only on XX", "depending only on XX", or "in the case of XX only", 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.

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.

Furthermore, even if there is an aspect of a method, program, terminal, device, server, or system (hereinafter, "method, etc.") that performs an operation different from that described in this specification, each aspect of the disclosed technology is directed to an operation identical to 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 disclosed technology.

In addition, although the present specification has described an embodiment in which the program is pre-installed, the program can also be provided by storing it on a computer-readable recording medium.

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.

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

For example, when a hybrid system is configured by a plurality of classical computers and a plurality of quantum computers, and a state |ψ> of N quantum bits is expressed by a linear combination of a plurality of computational bases, one or more quantum computers among the plurality of quantum computers measure each of the plurality of computational bases to obtain the measurement values of the plurality of computational bases. Then, one or more classical computers among the plurality of classical computers obtain the measurement values of the plurality of computational bases measured by the quantum computer. Next, based on the measurement values of each of the plurality of computational bases, one or more classical computers among the plurality of classical computers select R measurement values that occur frequently or R measurement values that occur a predetermined value or more from the plurality of measurement values, and calculate a weighting factor f _r for R computational bases |z _r > (r = 1, ..., R) according to the number of occurrences based on the number of occurrences of the selected R measurement values. Next, one or more classical computers among the plurality of classical computers or one or more quantum computers among the plurality of quantum computers calculate a plurality of transition matrix elements <z _r |O|z r _' > for each combination of computational base |z _r > and computational base |z _{r '} >. Then, one or more of the multiple classical computers approximately calculates the expectation value <ψ|O|ψ> of the observable _O by a weighted sum of multiple transition matrix elements < _zr |O|zr _' >, where the combination of weighting coefficients fr and fr _' is used as weights.

Reference Signs List 100 Hybrid system 110 Classical computer 111 Communication unit 112 Processing unit 113 Information storage unit 120 Quantum computer 121 Control device 122 Electromagnetic wave generating device 123 Quantum bit group 130 User terminal

Claims (8)

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 that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value are selected from the multiple measurement values, and based on the occurrence counts of the selected R measurement values, weighting coefficients f _r for the R calculation bases |z _r > (r = 1, ..., R) corresponding to the occurrence counts are calculated;
The expectation value <ψ|O|ψ> of the observable O is approximately calculated by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r′ >, where the combination of the weighting coefficient _{f r and} the weighting coefficient f r′ is used as the weight.
A quantum information processing method in which the processing is carried out by a classical computer. The state |ψ〉 of the N quantum bits is expressed by a linear combination of a plurality of computational bases with a plurality of expansion coefficients α _n (n=0, 1, . . . , 2 ^N −1) for each of the plurality of computational bases, the plurality of computational bases being weighted _;
When approximately calculating the expectation value <ψ|O|ψ>, an expectation value <ψ| _O |ψ> of an observable O is approximately calculated according to the following formula (A1) based on a weighting coefficient f _r for _R calculation bases | _{z r >} (r=1, . . . , R), an expansion coefficient _{α zr for} the calculation base ^| z _{r >} and an expansion coefficient α zr' for the calculation base |z _r' >, and a plurality of transition matrix elements <z r |O|z r'>.
2. The quantum information processing method according to claim 1.

(A1) 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, from the plurality of measurement values, R measurement values that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value, based on the measurement values of each of the plurality of calculation bases, and calculates weighting coefficients f _r for the R calculation bases |z _r > (r = 1, ..., R) according to the occurrence count, based on the occurrence counts of the selected R measurement values;
A classical computer or a quantum computer calculates a plurality of transition matrix elements <z _r |O|z _{r '} > for each combination of a computational basis |z _r > and a computational basis |z _{r '} >;
A classical computer approximately calculates an expectation value <ψ|O|ψ> of an observable O by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r' >, where the _weights are combinations of weighting coefficients _f r and f r'.
Quantum information processing methods. 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),
A quantum computer repeats the measurement of a computational basis |n> L times when expressing an N-qubit state |ψ> using a plurality of expansion coefficients α _n and a plurality of computational bases |n> (n=0, 1, . . . , 2 ^N −1) according to the following formula (A3), thereby obtaining a series of measurement values of the computational basis |n>, {x}=x ⁽¹⁾ , x ⁽²⁾ , . . . , x ^(L) ;
a classical computer selects, based on the sequence of measurement values {x}, R measurement values that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value from the sequence of measurement values {x}, sets values {z}=z ⁽¹⁾ , z ⁽²⁾ , ..., z ^(R) in which the R measurement values are arranged in descending order, calculates the number of occurrences T _r (r=1, ..., R) of the R measurement values included in {z}, and calculates a weighting coefficient f _r that is an approximation of |α _zr | ² according to the following formula (A4):
A classical computer or a quantum computer calculates transition matrix elements < _zr |O|zr _' > (r=1, ..., R; r'=1, ..., R) of the following formula (A1) based on the {z},
a quantum computer measures a projection operator |φ _{A, r} ＞＜φ _{A,r | for a state |φ A,r} ＞ of N quantum bits to obtain a measurement result of the projection operator |φ _A,r ＞＜φ _A,r |, and measures a projection operator |φ _{B ,r} ＞＜φ _B,r | for a state |φ B,r ＞ of N quantum bits to obtain a measurement result of the projection operator |φ _B,r ＞＜φ _B,r |;
a classical computer calculates an expectation value A r of the projection operator |φ _A,r ＞＜φ _A,r | based on a measurement result of the projection operator |φ _A,r ＞＜φ _A,r |, and calculates an expectation value _{B r} of the projection operator |φ _B,r ＞＜φ _B,r | based on a measurement result of the projection operator |φ _B,r ＞＜φ _B,r |;
a classical computer approximately calculates an interference weight g _r corresponding to α _z1 α _zr ^* according to the following formula (A5) based on the weighting coefficient f _r , the expectation value A _r , and the expectation value B _r , approximately calculates α _zr α zr' ^* according to the following formula (A6) based on the weighting coefficient f _r and the interference weight g _r , and approximately calculates the expectation value < _ψ |O|ψ> according to the following formula (A1) based on _the α _z1 α _zr ^* corresponding to the interference weight _{g r} , the α _zr α _zr' ^* , and the transition matrix element <z r |O|z r'>;
The quantum information processing method according to claim 3.

(A1)

(A2)

(A3)

(A4)

(A5)


(A6) 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 that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value are selected from the multiple measurement values, and based on the occurrence counts of the selected R measurement values, weighting coefficients f _r for the R calculation bases |z _r > (r = 1, ..., R) corresponding to the occurrence counts are calculated;
The expectation value <ψ|O|ψ> of the observable O is approximately calculated by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r′ >, where the combination of the weighting coefficient _{f r and} the weighting coefficient f r′ is used as the weight.
A quantum information processing program that allows processing to be performed by a classical computer. 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 that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value are selected from the multiple measurement values, and based on the occurrence counts of the selected R measurement values, weighting coefficients f _r for the R calculation bases |z _r > (r = 1, ..., R) corresponding to the occurrence counts are calculated;
The expectation value <ψ|O|ψ> of the observable O is approximately calculated by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r′ >, where the combination of the weighting coefficient _{f r and} the weighting coefficient f r′ is used as the weight.
Classical computer. 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, from the plurality of measurement values, R measurement values that occur frequently or R measurement values whose occurrence count is equal to or greater than a predetermined value, based on the measurement values of each of the plurality of calculation bases, and calculates weighting coefficients f _r for the R calculation bases |z _r > (r = 1, ..., R) according to the occurrence count, based on the occurrence counts of the selected R measurement values;
A classical computer or a quantum computer calculates a plurality of transition matrix elements <z _r |O|z _{r '} > for each combination of a computational basis |z _r > and a computational basis |z _{r '} >;
A classical computer approximately calculates an expectation value <ψ|O|ψ> of an observable O by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r' >, where the _weights are combinations of weighting coefficients _f r and f r'.
Hybrid system. 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, based on the measurement values of each of the multiple calculation bases, 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 calculates weighting coefficients f r for the R calculation bases |z _r > (r = 1, ..., R) according to the occurrence count based on the occurrence counts of the selected _R measurement values;
One or more classical computers among the plurality of classical computers or one or more quantum computers among the plurality of quantum computers calculate a plurality of transition matrix elements <z _r |O|z r _' > for each combination of a computational basis |z _r > and a computational basis |z _{r '} >;
one or more classical computers among the number of classical computers calculates an approximation of an expectation value <ψ| _O |ψ _> of the observable O by a weighted sum of a plurality of transition matrix elements <z _r |O|z _r′ >, the weights being a combination of a weighting factor f r and a weighting factor f r′;
Hybrid system.