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

Abstract

This classic computer transmits a quantum circuit for preparing quantum states to a quantum computer. With respect to each of a first quantum bit group including n quantum bits and a second quantum bit group including n quantum bits different from the quantum bits in the first quantum bit group, the quantum computer sets n pairs of the quantum bits in the first quantum bit group and the quantum bits in the second quantum bit group, acquires a result of Bell measurement by executing the Bell measurement for a tensor product of the quantum states by using each of the n pairs and the quantum circuit, and transmits the result of Bell measurement to the classic computer. The classic computer calculates an expectation value of each of a plurality of Pauli products in the quantum states on the basis of the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators on the basis of a combination of arbitrary Pauli products among the plurality of Pauli products.

Description

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

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

[0002] When performing any computational process using a quantum computer, it is important to efficiently perform the task of "measuring the expectation value of an operator." For example, when performing quantum chemical calculations using a quantum computer, the ground state energy can be obtained by measuring the expectation value of a Hamiltonian operator for the ground state. Usually, when measuring the expectation value of a complex operator such as a Hamiltonian operator using a relatively shallow quantum circuit, it is necessary to repeatedly run multiple types of quantum circuits.

[0003] Note that an operator can be expressed as a linear sum of any Pauli products, which are operators expressed as a direct product of Pauli operators. Reference 1 (Hsin-Yuan Huang, Richard Kueng, John Preskill, "Information-theoretic bounds on quantum advantage in machine learning", Phys. Rev. Lett. 126, 190505 (https: / / journals.aps.org / prl / abstract / 10.1103 / PhysRevLett.126.190505)) and Reference 2 (Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven, "Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning", Quantum 4, 276 (2020) (https: / / quantum-journal.org / papers / q-2020-06-04-276 / )) describe a method for calculating values ​​related to any Pauli product using a quantum circuit with 2n qubits.

[0004] For example, Literature 1 discloses a technique for calculating the absolute value of the expectation value of an arbitrary Pauli product using a quantum circuit having 2n quantum bits, and Literature 2 discloses a technique for calculating the expectation value of an arbitrary Pauli product using a quantum circuit having 2n quantum bits.

[0005] In order to obtain the expectation value of the operator, it is necessary to appropriately process the expectation value of the Pauli product, but neither Reference 1 nor Reference 2 discloses a method for doing so.

[0006] The disclosed technology has been made in consideration of the above circumstances, and provides a quantum information processing program, a classical computer, and a hybrid system that can efficiently obtain the expected value of an operator.

[0007] In order to achieve the above object, the quantum information processing method 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 the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer sets n pairs of qubits in a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, and performs Bell measurements on a tensor product of the quantum state using each of the n pairs and the quantum circuit to obtain results of the Bell measurements, and sends the results of the Bell measurements to the classical computer, and the classical computer calculates, based on the results of the Bell measurements, an expectation value for each of a plurality of Pauli products in the quantum state, and calculates an expectation value for each of a plurality of operators based on any combination of Pauli products among the plurality of Pauli products.

[0008] According to the disclosed technology, it is possible to obtain the expected value of the operator efficiently.

[0009] 1 is a diagram illustrating an example of a schematic configuration of a hybrid system 100 according to an embodiment. FIG. 2 is a schematic block diagram of a classical computer 110, a control device 121, and a computer functioning as a user terminal 130. FIG. 3 is a diagram for explaining Bell measurement. FIG. 4 is a diagram illustrating an example of a sequence executed by the hybrid system 100 according to an embodiment. FIG. 5 is a diagram illustrating an example of a sequence executed by the hybrid system 100 according to an embodiment.

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

[0011] <Hybrid System 100 According to First Embodiment> Fig. 1 shows a hybrid system 100 according to the first embodiment. The hybrid system 100 of this embodiment includes a classical computer 110, which is an example of an information processing device, a quantum computer 120, and a user terminal 130. As shown in Fig. 1, the classical computer 110, the quantum computer 120, and the user terminal 130 are connected via a computer network, such as an IP network, for example.

[0012] In the hybrid system 100 of this embodiment, the quantum computer 120 performs a predetermined quantum computation in response to a request from the classical computer 110, and outputs the computation result of the quantum computation to the classical computer 110. The classical computer 110 outputs the computation result of the quantum computation to the user terminal 130. In this way, the predetermined computation process is executed by the hybrid system 100 as a whole.

[0013] The classical computer 110 includes a communication unit 111 such as a communication interface, a processing unit 112 such as a processor or a central processing unit (CPU), and an information storage unit 113 including a storage device or storage medium such as a memory or a hard disk, and is configured to execute programs for performing each process. The classical computer 110 may include one or more devices or servers. The program may include one or more programs, and may be recorded on a computer-readable storage medium to form a non-transitory program product.

[0014] As an example, quantum computer 120 generates an electromagnetic wave to be irradiated onto at least one of quantum bits in quantum bit group 123 based on information transmitted from classical computer 110. Then, quantum computer 120 executes a quantum circuit by irradiating at least one of quantum bits in quantum bit group 123 with the generated electromagnetic wave.

[0015] 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 requests 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, the term "quantum computer" does not mean a computer that does not perform any operations using classical bits, but refers to a computer that includes operations using quantum bits or a computer that simulates the behavior of a quantum computer. A computer that simulates the behavior of a quantum computer is, for example, a classical computer that performs quantum simulations.

[0016] The control device 121 is a classical computer that performs operations using classical bits, and alternatively performs some or all of the processing described in this specification as being performed by the classical computer 110.

[0017] The user terminal 130 is a classical computer that performs calculations using classical bits. The user terminal 130 receives information input by a user and executes processing according to the information.

[0018] 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 serving 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, output devices, etc. are connected, and a read / write (R / W) unit 55 that controls reading and writing of data from and to a recording medium. The computer 50 also includes a network I / F 56 that is connected to a network such as the Internet. The CPU 51, the memory 52, the storage unit 53, the input / output I / F 54, the R / W unit 55, and the network I / F 56 are connected to one another via a bus 57.

[0019] [Outline of the present embodiment] When performing some computational processing using a quantum computer, it is important to efficiently measure the expected value of an operator. However, in order to measure the expected value of an operator, it is necessary to use multiple types of quantum circuits.

[0020] As shown in the following formula (1), the operator O ｉ Each of the coefficients c ｋ （ｉ） and the Pauli product P ｋ （ｉ） Based on this, it can be calculated by the following formula (1): where i is an index for identifying the operator, and k is an index for identifying the Pauli product.

[0021] (1)

[0022] Therefore, the Pauli product P ｋ （ｉ） If we can obtain the expected value of the operator O ｉ The expected value of can also be calculated.

[0023] Therefore, the hybrid system 100 of this embodiment acquires expectation values ​​of multiple Pauli products by performing Bell measurements on two quantum states, and then calculates the expectation value of the target operator by selecting an arbitrary Pauli product required for calculating the expectation value of the operator from the expectation values ​​of the multiple Pauli products.

[0024] When performing Bell measurements on two identical quantum states, the square of the expectation value of the Pauli product is obtained. In this case, the expectation value of the Pauli product is not obtained, so the sign of the expectation value of the Pauli product must be obtained separately and the expectation value of the Pauli product itself must be calculated based on the sign and the square root of the square of the expectation value of the Pauli product.

[0025] Therefore, the hybrid system 100 of the first embodiment calculates the square of the expectation value of the Pauli product based on the result of the Bell measurement by the quantum computer 120, and also acquires the sign of the expectation value of the Pauli product. The hybrid system 100 of the first embodiment calculates the expectation value of the Pauli product based on the acquired sign and the square root of the square of the expectation value of the Pauli product. Then, the hybrid system 100 of the first embodiment calculates the expectation value P ｋ （ｉ） and the previously obtained coefficient c ｋ （ｉ） Based on this, according to the above formula (1), the operator O ｉ The expected value of is calculated as follows. A specific explanation is given below. Note that the tensor product will also be written as (×) below.

[0026] In the following, a quantum state ρ will be considered. In this embodiment, the quantum state ρ represents either of two types of quantum states: a mixed state and a pure state. When the quantum state ρ is a pure state, the quantum state ρ can be described as in the following equation. In this case, the quantum state ρ is described using the pure state |ψ>. Note that ψ below also represents a quantum state.

[0027] ρ = |ψ><ψ|

[0028] In addition, in this embodiment, the expectation value of the operator O in the quantum state ρ is denoted as Tr(ρO). Tr(·) represents a trace for a matrix, and generally, the expectation value of the operator O in the quantum state ρ can be denoted as Tr(ρO). For this reason, in this embodiment, the expectation value of the operator O in the quantum state ρ is denoted as Tr(ρO).

[0029] When the quantum state ρ is described as in the above equation, the following equation holds:

[0030] Tr(ρO)=<ψ|O|ψ>

[0031] [Bell Measurement] Fig. 3 is a diagram for explaining Bell measurement. Fig. 3 shows a quantum circuit U for preparing a quantum state ρ, a first group of qubits Q1, a second group of qubits Q2, a Hadamard gate H, and a CNOT gate 10. Although the example in Fig. 3 shows four qubits, the number of qubits can be expanded to n.

[0032] As shown in FIG. 3, in Bell measurement, for each of a first quantum bit group Q1 containing n quantum bits and a second quantum bit group Q2 containing n quantum bits different from the quantum bits in the first quantum bit group Q1, n pairs of quantum bits in the first quantum bit group Q1 and quantum bits in the second quantum bit group Q2 are set.

[0033] Then, as shown in Fig. 3, a Bell measurement result is obtained by performing a Bell measurement on the tensor product of the quantum state ρ using each of the n pairs of quantum bits. At this time, as shown in Fig. 3, a Bell measurement result of the tensor product ρ(×)ρ of the quantum state ρ and the quantum state ρ is obtained between the quantum bit q1 and the quantum bit q2. The Bell measurement is a projection measurement onto a Bell basis, and the following Bell basis |Φ is obtained as a result of the Bell measurement. １ ± >,|Φ ２ ± > is obtained. Note that |0> in the following formula ρ indicates that the corresponding quantum bit belongs to the quantum state ρ.

[0034] (2)

[0035] The above Bell basis |Φ １ ± >,|Φ ２ ± > is the Pauli product X of the Pauli operators X, Y, and Z ρ (×) X ρ , Y ρ (x) Y ρ , Z ρ (×)Z ρ Therefore, we perform Bell measurements on all n pairs of qubits and then combine the measurements into Nｓｈｏｔ By repeating this process, we can obtain the squared expectation values ​​of all kinds of Pauli products |<Tr(ρO)>| ２ Therefore, the square of the expectation value of any Pauli product |<Tr(ρO)>| ２ By combining the square root of and its sign, it is possible to calculate the expectation value of any kind of operator.

[0036] [Operation of Hybrid System 100 of First Embodiment] A specific operation of the hybrid system 100 of the first embodiment will be described. Each device of the hybrid system 100 executes the processes shown in FIG.

[0037] First, in step S100, the user terminal 130 transmits to the classical computer 110 computation target information, which is information about the computation target, and computation method information, which is information about the computation method, both input by the user.

[0038] The calculation target information includes, for example, a set of calculation target operators {O ｉ} and information on the sign of the expectation value of the Pauli product. The calculation method information includes, for example, information on a state preparation method for preparing the quantum state ρ, information on the initial state of the quantum state ρ, and the number of measurement shots N ｓｈｏｔ etc. The state preparation method for preparing the quantum state ρ includes, for example, information about a quantum circuit U for preparing the quantum state ρ. For example, when the quantum state ρ is a pure state ρ=|ψ〉<ψ|, the quantum circuit U is a quantum circuit that can set the quantum state |ψ〉 by acting on the initial state |i〉 of the quantum computer 120. That is, U|i〉=|ψ〉.

[0039] Next, in step S102, the classical computer 110 receives the computation target information and computation method information transmitted from the user terminal 130.

[0040] 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 information included in the computation method information on a state preparation method for preparing the quantum state ρ, information on the initial state of the quantum state ρ, and the number of measurement shots N ｓｈｏｔ etc. are sent to the quantum computer 120.

[0041] In step S106, the control device 121 of the quantum computer 120 receives the various information transmitted from the classical computer 110 in step S104.

[0042] In step S108, the control device 121 causes the quantum computer 120 to perform quantum computations according to the various information received in step S106. Specifically, as shown in Fig. 3, the control device 121 of the quantum computer 120 sets n pairs of quantum bits in the first quantum bit group Q1 and quantum bits in the second quantum bit group Q2 for each of a first quantum bit group Q1 including n quantum bits and a second quantum bit group Q2 including n quantum bits different from the quantum bits in the first quantum bit group Q1. Then, the control device 121 outputs a control signal to the quantum computer 120 to perform a Bell measurement on the tensor product ρ(×)ρ of the quantum state ρ using each of the n pairs and the quantum circuit received in step S108.

[0043] The quantum computer 120, under the control of the control device 121, calculates N ｓｈｏｔ The measurement result is obtained by performing Bell measurements. The Bell measurement is a sampling of the tensor product ρ(×)ρ of the quantum state ρ, and the sampling means repeatedly projecting and measuring the tensor product ρ(×)ρ of the quantum state ρ onto a Bell basis. As a result, the control device 121 of the quantum computer 120 obtains the Bell basis |Φ １ ± >,|Φ ２ ± > Obtain the number of times it is observed.

[0044] Specifically, the quantum computer 120 generates an electromagnetic wave to be irradiated onto at least one of the quantum bits in the quantum bit group 123 in accordance with the control of the control device 121. The quantum computer 120 then irradiates the generated electromagnetic wave onto at least one of the quantum bits in the quantum bit group 123, executing a quantum circuit that generates the tensor product ρ(×)ρ of the quantum state ρ. The gate operation of each quantum gate included in the quantum circuit is converted into a corresponding electromagnetic wave waveform, and the generated electromagnetic wave is irradiated onto the quantum bit group 123 by the electromagnetic wave generating device 122. The quantum computer 120 then outputs the measurement result obtained by the measurement. Note that, in this embodiment, an example will be described in which the quantum circuit is executed by irradiating an electromagnetic wave, but the present invention is not limited to this, and the quantum circuit may be executed by a different method.

[0045] In step S110, the control device 121 calculates the Bell basis |Φ, which is the result of the Bell measurement obtained in step S108. １ ± >,|Φ ２ ± > is sent to the classical computer 110.

[0046] In step S112, the classical computer 110 receives the results of the Bell measurement sent from the control device 121 in step S110.

[0047] In step S114, the classical computer 110 calculates a plurality of Pauli products P ｋ （ｉ） The sign of each expected value s ｋ （ｉ） In addition, multiple Pauli products P ｋ （ｉ） There are various methods for obtaining the signs of the expected values ​​of the above.

[0048] For example, the classical computer 110 may calculate multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) to obtain the sign of the expectation value. ｋ（ｉ） If we only need to obtain the sign of each expectation value of ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is often not a problem even if the calculation accuracy is low. For this reason, for example, the classical computer 110 employs a known calculation method that allows for relatively high-speed calculations, and calculates a plurality of Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is calculated approximately, the sign of the expected value s ｋ （ｉ） Get.

[0049] Alternatively, for example, the classical computer 110 may calculate the expectation value Tr(ρ′P ｋ （ｉ） ) to obtain multiple Pauli products P ｋ （ｉ） As mentioned above, the Pauli products P ｋ （ｉ） If we only need to obtain the sign of each expectation value of ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） For this reason, the classical computer 110 uses a known calculation method that allows for relatively high-speed calculations to calculate the expectation value Tr(ρ′P ｋ （ｉ） ) to obtain multiple Pauli products P ｋ （ｉ） The sign of each expected value s ｋ （ｉ） Get.

[0050] Alternatively, for example, the classical computer 110 may calculate multiple Pauli products P ｋ （ｉ） The sign of each expected value s ｋ （ｉ） may be determined in advance.

[0051] In step S116, the classical computer 110 calculates |Φ １ ± > and |Φ ２ ± Based on the number of observations of ｋ （ｉ） The squared value of each expected value (Tr(ρP ｋ （ｉ） )) ２ Calculate.

[0052] Next, in step S118, the classical computer 110 calculates the multiple Pauli products P calculated in step S116. ｋ （ｉ） The squared value of each expected value (Tr(ρP ｋ （ｉ） )) ２ By calculating the square root of ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) |

[0053] It should be noted that the measurement by the quantum computer 120 may contain an error, and the square of the expected value may be calculated as a negative value.

[0054] Therefore, the classical computer 110 of this embodiment calculates a plurality of Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | is calculated. In the following formula, the square of the expected value (Tr(ρP ｋ （ｉ） )) ２ is represented as v.

[0055] Then, in step S118, the classical computer 110 calculates the multiple Pauli products P obtained in step S114. ｋ （ｉ） Each symbol s ｋ （ｉ） and the multiple Pauli products P obtained in step S116 ｋ（ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is calculated.

[0056]

[0057] Then, in step S120, the classical computer 110 calculates the set of operators to be calculated {O ｉ}, where O is a number of operators included in ｉ The expected value of each of ｉ Specifically, the classical computer 110 calculates the expectation value Tr(ρP ｋ （ｉ） ) based on a combination of any Pauli product of multiple operators O ｉ The expected value of each of ｉ >Calculate.

[0058] Note that the calculation process in step S118 calculates the expectation values ​​of all kinds of Pauli products. Therefore, in step S120, the classical computer 110 calculates the expectation values ​​of the multiple operators O from the expectation values ​​of each of the multiple Pauli products calculated in step S118. ｉ The expectation value of the Pauli product required for calculating is selected, and based on the selected Pauli product expectation value, multiple operators O are calculated according to the following formula: ｉ The expected value of each of ｉ > is calculated. ｋ （ｉ） are predetermined by known techniques.

[0059]

[0060] In step S122, the classical computer 110 calculates the plurality of operators O obtained in step S120. ｉ The expected value of each of ｉ The calculation result of (1) is transmitted to the user terminal 130.

[0061] In step S124, the user terminal 130 receives the results sent from the classical computer 110.

[0062] As described above, the classical computer in the hybrid system of the first embodiment transmits a quantum circuit for preparing a quantum state to the quantum computer. Next, the quantum computer sets n pairs of qubits in the first qubit group and qubits in the second qubit group for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group. Next, the quantum computer performs Bell measurements on the tensor product of the quantum state using each of the n pairs and the quantum circuit to obtain results of the Bell measurements and transmits the results of the Bell measurements to the classical computer. Next, the classical computer calculates the expectation values ​​of each of a plurality of Pauli products in the quantum state based on the results of the Bell measurements. Then, the classical computer calculates the expectation values ​​of each of a plurality of operators based on any combination of Pauli products among the plurality of Pauli products. This allows the expectation values ​​of the operators to be obtained efficiently.

[0063] More specifically, the classical computer in the hybrid system of the first embodiment uses |Φ １ ± > and |Φ ２ ± Based on the number of observations of >, multiple Pauli products P ｋ （ｉ） The squared value of each expected value (Tr(ρP ｋ （ｉ） )) ２ Next, the classical computer calculates the squared expected value (Tr(ρP ｋ （ｉ） )) ２ By calculating the square root of ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) |. Then, the classical computer calculates the multiple Pauli products P ｋ （ｉ）and the sign of the predetermined expectation value for each of the Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and the classical computer calculates a plurality of predetermined coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and based on multiple operators O ｉ This allows us to efficiently obtain the expectation of the operator.

[0064] Even when quantum computations are performed using currently available quantum computers, noise occurs during the computation process, which can lead to errors in the computation results. To directly and accurately calculate the expected value of an operator using such a quantum computer requires the quantum computer to execute a quantum circuit of an impossibly large size. In contrast, by having a quantum computer calculate only the values ​​necessary to calculate the Pauli product, while having a classical computer execute the Pauli product expectation value and the operator expectation value, it is possible to efficiently and accurately obtain the expected value of the operator.

[0065] Therefore, the classical computer of this embodiment pre-calculates the expectation values ​​of all Pauli products based on the results of Bell measurements obtained by the quantum computer, and selects from the Pauli product expectation values ​​an arbitrary Pauli product expectation value required to calculate the expectation value of the operator. The classical computer then calculates the expectation value of the operator to be calculated based on the selected arbitrary Pauli product expectation value. Note that when calculating the Pauli product expectation value, the classical computer also obtains the sign of the Pauli product expectation value. This makes it possible to efficiently and accurately obtain the expectation value of the operator. Furthermore, by temporarily storing the measurement results obtained by the quantum computer in the classical computer, it becomes possible to easily calculate the expectation value of another operator that needs to be calculated later, without using a quantum computer.

[0066] For example, when the hybrid system 100 of the first embodiment calculates the expectation value of the Hamiltonian operator H using a variational quantum eigensolver (VQE), the following calculation process can be performed. In this case, the quantum state ρ becomes a quantum state ρ(θ) with a parameter θ. The parameter θ collectively represents a plurality of parameters.

[0067] When the hybrid system 100 calculates the expectation value of the Hamiltonian operator H using VQE, the classical computer 110 and the quantum computer 120 calculate the parameter θ such that Tr(ρ(θ)H), which is the expectation value of the Hamiltonian operator H, is minimized. ＊ In this case, ρ(θ ＊ ) is calculated as the ground state, and Tr(ρ(θ ＊ )H) is calculated as the ground state energy.

[0068] In this case, the classical computer 110 reuses the sign of the expectation value obtained in the previous step in the repeated calculations in VQE as the sign of the expectation value in the next step or later in a predetermined period, thereby obtaining a plurality of Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ）) is calculated. ＊ By updating the code at a fixed frequency rather than every time during the search for , the cost of code estimation can be reduced.

[0069] The Hamiltonian operator H is expressed by the following equation: ｋ represents the coefficient, and P ｋ represents the Pauli product.

[0070]

[0071] When the parameter θ is obtained, the Pauli product P ｋ Expected value of < P ｋ The sign of > is s ｋ It should be noted that this symbol s ｋ is predetermined by the method described above.

[0072] The parameter θ ＊ While the parameter θ is being searched, the parameter θ is shifted by a predetermined amount. １，＋ For example, θ = (θ １ , θ ２ , ..., θ ｌ ) and θ １，＋ =(θ １ +π / 2, θ ２ , ..., θ ｌ ) is assumed. The optimal parameter θ ＊ While the parameter θ is being searched, the parameter θ is changed successively. Each time the parameter θ is changed, the Pauli product P ｋ Expected value of < P ｋ > is also calculated and its sign s ｋ is also calculated, the classical computer 110 calculates the code s ｋ ｐ is also stored in the storage unit. Note that p is the parameter θ after the change. １，＋ For example, the parameter θ １，＋ If so, the sign is s ｋ １，＋ It is written as follows.

[0073] The optimal parameter θ by VQE ＊When searching for , the parameter θ is changed little by little according to the optimization algorithm to minimize the energy. However, every time the parameter θ is changed, the Pauli product P ｋ Expected value of < P ｋ > symbol s ｋ On the other hand, if the change in the parameter θ is small, the sign s ｋ ｐ is expected to be constant.

[0074] Therefore, for example, the classical computer 110 sets the number of steps Ts for updating the code, and updates the code in a predetermined manner only when the number of steps is a multiple of Ts, thereby reducing the calculation cost for estimating the code.

[0075] <Hybrid system 100 according to second embodiment> Next, a hybrid system according to a second embodiment will be described. Note that the configuration of the hybrid system 100 according to the second embodiment is the same as that of the first embodiment, and therefore the same reference numerals are used and the description thereof will be omitted.

[0076] The second embodiment differs from the first embodiment in that the sign of the expectation value of the Pauli product is not estimated, but the expectation value of the Pauli product itself is estimated.

[0077] Specifically, the classical computer 110 of the second embodiment calculates |Φ １ ± > and |Φ ２ ± Based on the number of observations of >, multiple Pauli products P ｋ （ｉ） The value for each expected value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) in the quantum state σ. ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) is calculated. Then, the classical computer 110 of the second embodiment calculates a plurality of Pauli products Pｋ （ｉ） The value for each expected value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) in the quantum state σ as the Pauli product P ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) to obtain the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is calculated. This allows us to calculate multiple operators O without obtaining the sign of the expectation value of the Pauli product. ｉ It is possible to calculate the expected values ​​of each of the following. A specific description will be given below. Note that hereinafter, the quantum state σ is also referred to as an auxiliary state σ.

[0078] In the second embodiment, the Bell basis |Φ １ ± >,|Φ ２ ± >The number of observations is obtained.

[0079] (3)

[0080] The above Bell basis |Φ １ ± >,|Φ ２ ± > is the Pauli product X of the Pauli operators X, Y, and Z ρ (×) X σ , Y ρ (x) Y σ , Z ρ (×)Z σ Therefore, we perform Bell measurements on all n pairs of qubits and then combine the measurements into N ｓｈｏｔ By repeating this process, we can obtain the expected value of any Pauli product (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) is obtained.

[0081] The classical computer 110 of the second embodiment calculates the expectation value of the Pauli product (Tr(σP ｋ （ｉ） )) is calculated separately. Then, the classical computer 110 of the second embodiment calculates the value related to the expectation value of the Pauli product (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) is the expectation value of the Pauli product in the auxiliary state σ (Tr(σP ｋ （ｉ） )) to obtain the value related to the expectation value of the Pauli product for the quantum state ρ (Tr(ρP ｋ （ｉ） This gives the expectation values ​​of all kinds of Pauli products, and by combining the expectation values ​​of any Pauli products, it is possible to calculate the expectation values ​​of all kinds of operators.

[0082] [Operation of Hybrid System 100 of Second Embodiment] A specific operation of the hybrid system 100 of the second embodiment will be described. Each device of the hybrid system 100 executes the processes shown in FIG.

[0083] First, in step S200, the user terminal 130 transmits to the classical computer 110 computation target information, which is information about the computation target, and computation method information, which is information about the computation method, both input by the user.

[0084] The calculation method information of the second embodiment further includes, in addition to the calculation method information of the first embodiment, information on a state preparation method for preparing the auxiliary state σ and information on the initial state of the auxiliary state σ.

[0085] Regarding the auxiliary state σ, the expectation value Tr(σP) for any Pauli product P must be able to be efficiently calculated by the classical computer 110. In addition, the expectation values ​​of all Pauli products included in the operator O to be calculated must be non-zero. For example, σ = |φ><φ| and |φ>=|φ １ >(×)・・・|φ １ >(However, |φ １This condition is satisfied if |φ^i_ki> (i = 1, ..., m) is a quantum state of one qubit and is not an eigenstate of the Pauli operators X, Y, and Z). More generally, σ = |φ><φ|, and |φ> = |φ^1_k1>(×)|φ^2_k2> ... |φ^m_km>. In this case, the auxiliary state σ is a product state of quantum states with qubit numbers k1, k2, ..., km. Here, state |φ^i_ki> (i = 1, ..., m) is a quantum state with ki qubits. Here, assume that k1 + k2 + ... + km = n. For example, k1, ..., km can be preset by the user. n is the total number of qubits used to calculate the auxiliary state σ. If k1, ..., km are sufficiently small, it is possible to efficiently calculate the expectation value of any Pauli product P using a classical computer.

[0086] Next, in step S202, the classical computer 110 receives the computation target information and computation method information transmitted from the user terminal 130.

[0087] In step S204, the classical computer 110 transmits various pieces of information necessary for quantum computation to the quantum computer 120. Specifically, the classical computer 110 transmits, to the quantum computer 120, information on a state preparation method for preparing the quantum state ρ, information on the initial state of the quantum state ρ, information on a state preparation method for preparing the auxiliary state σ, information on the initial state of the auxiliary state σ, and the number of measurement shots N ｓｈｏｔ etc. are sent to the quantum computer 120.

[0088] In step S204, the classical computer 110 transmits information about a state preparation method for preparing the auxiliary state σ to the quantum computer 120, and the quantum computer 120 determines the auxiliary state σ in accordance with the information about the state preparation method for preparing the auxiliary state σ. However, the classical computer 110 may also determine the auxiliary state σ by computation. For example, the classical computer 110 may determine the expected value Tr(ρP ｋ （ｉ） ) is calculated by performing an optimization calculation to minimize the objective function that represents the variance of the calculation results of Tr(ρP ｋ（ｉ） ), an optimal auxiliary state σ that reduces the variance of the calculation results may be determined in advance, and information on the auxiliary state σ may be transmitted to the quantum computer 120. The quantum computer 120 uses the auxiliary state σ determined by the classical computer 110 to perform the calculation process described below.

[0089] In step S206, the control device 121 of the quantum computer 120 receives the various information transmitted from the classical computer 110 in step S204.

[0090] In step S208, the control device 121 causes the quantum computer 120 to perform quantum computations according to the various information received in step S206. Specifically, as shown in Fig. 3, the control device 121 of the quantum computer 120 sets n pairs of quantum bits in the first quantum bit group Q1 and quantum bits in the second quantum bit group Q2 for each of a first quantum bit group Q1 including n quantum bits and a second quantum bit group Q2 including n quantum bits different from the quantum bits in the first quantum bit group Q1. Then, the control device 121 outputs a control signal to the quantum computer 120 to perform a Bell measurement on the tensor product ρ(×)σ of the auxiliary state σ and the quantum state ρ using each of the n pairs and the quantum circuit received in step 108.

[0091] The quantum computer 120, under the control of the control device 121, calculates N ｓｈｏｔ The measurement result is obtained by performing Bell measurements. The Bell measurement is a sampling of the tensor product ρ(×)σ of the quantum state ρ, and the sampling means repeatedly projecting and measuring the tensor product ρ(×)σ of the quantum state ρ onto a Bell basis. As a result, the control device 121 of the quantum computer 120 obtains the Bell basis |Φ １ ± >,|Φ ２ ± > Obtain the number of observations.

[0092] In step S210, the control device 121 calculates the Bell basis |Φ, which is the result of the Bell measurement obtained in step S208. １ ± >,|Φ ２ ± > is sent to the classical computer 110.

[0093] In step S212, the classical computer 110 receives the results of the Bell measurement sent from the control device 121 in step S210.

[0094] In step S214, the classical computer 110 calculates the Pauli product P ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) is calculated.

[0095] In step S216, the classical computer 110 calculates |Φ １ ± > and |Φ ２ ± Based on the number of observations of ｋ （ｉ） Expectation value of and the Pauli product P in the auxiliary state σ ｋ （ｉ） The product of the expected value of (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) as a function of the expectation value of the Pauli product.

[0096] Next, in step S218, the classical computer 110 calculates the Pauli product P ｋ （ｉ） Expectation value of and the Pauli product P in the auxiliary state σ ｋ （ｉ） The product of the expected value of (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) in the quantum state ρ as the Pauli product P ｋ （ｉ） The expected value Tr(σP ｋ（ｉ） ) to obtain the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is calculated.

[0097] Then, in step S220, the classical computer 110 calculates the set of operators to be calculated {O ｉ}, where O is a number of operators included in ｉ The expected value of each of ｉ Specifically, the classical computer 110 calculates the expectation value Tr(ρP ｋ （ｉ） ) based on a combination of any Pauli product of multiple operators O ｉ The expected value of each of ｉ >Calculate.

[0098] In step S222, the classical computer 110 calculates the plurality of operators O obtained in step S220. ｉ The expected value of each of ｉ The calculation result of (1) is transmitted to the user terminal 130.

[0099] In step S224, the user terminal 130 receives the results sent from the classical computer 110.

[0100] As described above, the classical computer in the hybrid system of the second embodiment calculates |Φ １ ± > and |Φ ２ ± Based on the above, multiple Pauli products P ｋ （ｉ） The value for each expected value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) is calculated by a classical computer. ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) is calculated. Then, the classical computer calculates the expected value (Tr(ρPｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) to the expected value Tr(σP ｋ （ｉ） ) to obtain the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) is calculated by a classical computer using a plurality of predetermined coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and based on multiple operators O ｉ Calculate the expected value of each of the following. This allows you to efficiently obtain the expected value of the operator. You can also efficiently obtain the expected value of the operator without taking the sign of the expected value.

[0101] The technology of the present disclosure is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the spirit of the present invention.

[0102] For example, in the above embodiment, any method may be used to transmit and receive information between the classical computer 110 and the quantum computer 120. For example, 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.

[0103] Furthermore, in the above embodiment, an example has been described in which calculation target information is transmitted from the user terminal 130 to the classical computer 110, and the classical computer 110 executes calculations according to the calculation target information, but 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 into the classical computer 110 using the storage medium or storage device.

[0104] Furthermore, in the above embodiment, the quantum circuit is executed by irradiating electromagnetic waves, but the present invention is not limited to this, and the quantum circuit may be executed by a different method.

[0105] 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 simulates the behavior of a quantum computer.

[0106] Furthermore, although the above embodiment assumes that the classical computer 110 and the quantum computer 120 are managed by different organizations, 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 play the role of the classical computer 110 in the above description.

[0107] It should be noted that in the above embodiments, unless there is a description of "only" such as "based on XX only," "depending only on XX," or "in the case of XX only," this specification assumes that additional information may also be taken into consideration. For example, the description "in the case of a, do b" does not necessarily mean "in the case of a, always do b," unless explicitly stated otherwise.

[0108] Furthermore, when expressions such as "optimize" or "optimized parameters" are used in the above embodiments, it should be noted that these expressions of "optimization" mean approaching 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.

[0109] Furthermore, even if there is an aspect of a method, program, terminal, device, server, or system (hereinafter referred to as a "method, etc.") that performs an operation different from that described in this specification, each aspect of the disclosed technology is directed to an operation that is identical to one 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.

[0110] Furthermore, although the present specification has been described as an embodiment in which the program is pre-installed, the program may also be provided by being stored on a computer-readable recording medium.

[0111] In the above embodiment, the processing performed by the CPU after reading the software (program) may be performed by various processors other than the CPU. Examples of processors in this case include programmable logic devices (PLDs) (such as field-programmable gate arrays (FPGAs)) whose circuit configuration can be changed after manufacture, and dedicated electrical circuits such as application-specific integrated circuits (ASICs) that are processors with circuit configurations specifically designed to perform specific processing. Each processing may be performed by one of these various processors, or by a combination of two or more processors of the same or different types (e.g., multiple FPGAs, or a combination of a CPU and an FPGA). The hardware structure of these various processors is, more specifically, an electrical circuit that combines circuit elements such as semiconductor devices.

[0112] In addition, although the above embodiments have described the case where the program is pre-stored (installed) in storage, the present invention is not limited to this. The program may be provided in a form stored in a non-transitory storage medium such as a CD-ROM (Compact Disk Read Only Memory), a DVD-ROM (Digital Versatile Disk Read Only Memory), or a USB (Universal Serial Bus) memory. The program may also be downloaded from an external device via a network.

[0113] Furthermore, each process of this embodiment may be implemented by a computer or server equipped with a general-purpose processor and a storage device, and each process may be executed by a program. This program is stored in a storage device, and may be recorded on a recording medium such as a magnetic disk, optical disk, or semiconductor memory, or may be provided via a network. Of course, any other components do not have to be implemented by a single computer or server, and may be distributed across multiple computers connected via a network.

[0114] 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.

[0115] For example, the processing performed by the classical computer of the above embodiments may be distributed among multiple classical computers connected by a network. Alternatively, the processing performed by the quantum computer of each of the above embodiments may be distributed among multiple quantum computers connected by a network. In this case, a hybrid system is formed by at least one classical computer and at least one quantum computer.

[0116] For example, in a hybrid system including multiple classical computers and multiple quantum computers, one or more of the multiple classical computers transmit a quantum circuit for preparing a quantum state to the quantum computer. Next, the one or more quantum computers among the multiple quantum computers establish n pairs of qubits in the first qubit group and qubits in the second qubit group for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, perform Bell measurements on the tensor product of the quantum state using each of the n pairs, and transmit the Bell measurement results to the classical computer. Then, the one or more classical computers among the multiple classical computers calculate expectation values ​​of each of a plurality of Pauli products in the quantum state based on the results of the Bell measurements, and calculate expectation values ​​of each of a plurality of operators based on any combination of Pauli products among the plurality of Pauli products.

[0117] All publications, patent applications, and technical standards mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent application, or technical standard was specifically and individually indicated to be incorporated by reference.

[0118] (Supplementary Notes) The following supplementary notes are provided regarding aspects of the present disclosure.

[0119] (Supplementary Note 1) A quantum information processing method executed by a classical computer in a hybrid system including a classical computer and a quantum computer, wherein the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer: for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, sets n pairs of qubits in the first qubit group and qubits in the second qubit group, performs Bell measurements on a tensor product of the quantum state using each of the n pairs and the quantum circuit to obtain results of the Bell measurements and sends the results of the Bell measurements to the classical computer, the classical computer calculates, based on the results of the Bell measurements, an expectation value for each of a plurality of Pauli products in the quantum state, and calculates an expectation value for each of a plurality of operators based on any combination of Pauli products among the plurality of Pauli products.

[0120] (Appendix 2) Multiple operators O ｉ (i is an index for identifying the operator) ｋ （ｉ） and the Pauli product P ｋ （ｉ） (k is an index for identifying the Pauli product) and can be calculated by the following formula (A): (A) The result of the Bell measurement on the tensor product of the quantum state ρ is expressed by the following equation (B1): (B1) The classical computer calculates |Φ １ ± > and |Φ ２ ± Based on the above, multiple Pauli products P in the quantum state ρ ｋ （ｉ） The squared value of each expected value (Tr(ρP ｋ （ｉ） )) ２ Calculate the squared value (Tr(ρP ｋ（ｉ） )) ２ By calculating the square root of ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and calculate the multiple Pauli products P ｋ （ｉ） and a plurality of Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and calculating the predetermined number of coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and according to the above formula (A), a plurality of operators O ｉ 2. The quantum information processing method of claim 1, further comprising: calculating an expectation value for each of

[0121] (Supplementary Note 3) The quantum computer has a plurality of Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) to obtain the sign of the expectation value, and the classical computer calculates the sign of the expectation value obtained by the quantum computer and a plurality of Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） 3. The quantum information processing method of claim 2, wherein

[0122] (Supplementary Note 4) The classical computer calculates an expectation value Tr(ρ′P ｋ（ｉ） ) and calculate the calculated expected value Tr(ρ′P ｋ （ｉ） ) and the sign of the multiple Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） 3. The quantum information processing method of claim 2, wherein

[0123] (Supplementary Note 5) The operator is a Hamiltonian operator H, a variational quantum eigensolver (VQE) is used to calculate the expectation value of the Hamiltonian operator H, the quantum state ρ is a quantum state ρ(θ) with a parameter θ, and the classical computer calculates the parameter θ such that Tr(ρ(θ)H), which is the expectation value of the Hamiltonian operator H, is minimized. ＊ In the repeated calculations in the VQE when searching for , the sign of the expectation value obtained in the previous step is reused as the sign of the expectation value in the next step or later in a predetermined period, thereby obtaining a plurality of Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） 3. The quantum information processing method of claim 2, wherein

[0124] (Appendix 6) Multiple operators O ｉ (i is an index for identifying the operator) ｋ （ｉ） and the Pauli product P ｋ （ｉ） (k is an index for identifying the Pauli product) and can be calculated by the following formula (A): (A) The result of the Bell measurement on the tensor product of the quantum state ρ and the quantum state σ is expressed by the following equation (B2): (B2) The classical computer calculates |Φ １ ± > and |Φ ２ ± Based on the above, multiple Pauli products P ｋ （ｉ） The value for each expected value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) and calculate the Pauli product P ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) and calculate the value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) to the expected value Tr(σP ｋ （ｉ） ) to obtain the multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and calculating the predetermined number of coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr(ρP ｋ （ｉ） ) and according to the above formula (A), a plurality of operators O ｉ 2. The quantum information processing method of claim 1, further comprising: calculating an expectation value for each of

[0125] (Supplementary Note 7) The quantum information processing method according to Supplementary Note 6, wherein the quantum state σ is a product state of quantum states of quantum bit numbers k1, k2, ..., km, where k1 + k2 + ... + km = n, and n is the total number of quantum bits for calculating the quantum state σ.

[0126] (Supplementary Note 8) The classical computer calculates the expected value Tr(ρP ｋ （ｉ）7. The quantum information processing method according to claim 6, further comprising: performing an optimization calculation to minimize an objective function that represents a variance in the calculation results of the quantum state σ; and transmitting information about the quantum state σ to the quantum computer.

[0127] (Supplementary Note 9) A quantum information processing program executed by a classical computer in a hybrid system including a classical computer and a quantum computer, the quantum information processing program causing a classical computer to execute processes in which the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer sets n pairs of qubits in a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, for each of the first qubit group and the second qubit group, the second qubit group including n qubits different from the qubits in the first qubit group, performs a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sends the result of the Bell measurement to the classical computer, the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among the plurality of Pauli products.

[0128] (Supplementary Note 10) A classical computer in a hybrid system including a classical computer and a quantum computer, wherein the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer: for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, sets n pairs of qubits in the first qubit group and qubits in the second qubit group, performs a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sends the result of the Bell measurement to the classical computer, the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

[0129] (Supplementary Note 11) A hybrid system including a classical computer and a quantum computer, wherein the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer: for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, sets n pairs of qubits in the first qubit group and qubits in the second qubit group, performs a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sends the result of the Bell measurement to the classical computer, the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

[0130] (Supplementary Note 12) A hybrid system including a plurality of classical computers and a plurality of quantum computers, wherein one or more of the plurality of classical computers send a quantum circuit for preparing a quantum state to the quantum computer, wherein the one or more quantum computers of the plurality of quantum computers: for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, set n pairs of qubits in the first qubit group and qubits in the second qubit group, perform Bell measurements on a tensor product of the quantum state using each of the n pairs to obtain results of the Bell measurements and send the results of the Bell measurements to the classical computer, wherein the one or more classical computers of the plurality of classical computers calculate, based on the results of the Bell measurements, an expectation value for each of a plurality of Pauli products in the quantum state, and calculate an expectation value for each of a plurality of operators based on any combination of Pauli products among the plurality of Pauli products.

Claims

1. A quantum information processing method executed by a classical computer in a hybrid system including a classical computer and a quantum computer, comprising: the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer; the quantum computer sets n pairs of a quantum bit in a first quantum bit group containing n quantum bits and a quantum bit in a second quantum bit group containing n quantum bits different from the quantum bits in the first quantum bit group, performs a Bell measurement on a tensor product of a quantum state using each of the n pairs and the quantum circuit to obtain a result of the Bell measurement, and sends the result of the Bell measurement to the classical computer; the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement; and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

2. Multiple Operators O ｉ (i is an index for identifying the operator) ｋ （ｉ） and the Pauli product P ｋ （ｉ） (k is an index for identifying the Pauli product) and the following formula (A) can be used to calculate the (A) The result of the Bell measurement on the tensor product of the quantum state ρ is expressed by the following equation (B1): (B1) The classical computer calculates |Φ １ ± > and |Φ ２ ± Based on the above, multiple Pauli products P ｋ （ｉ） The squared value of each expectation value (Tr(ρP ｋ （ｉ） )) ２ Calculate the squared value (Tr(ρP ｋ （ｉ） )) ２ By calculating the square root of ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) |, and calculating the multiple Pauli products P ｋ （ｉ） and a plurality of Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） ) and calculating the predetermined number of coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） ) according to the above formula (A), ｉ The quantum information processing method according to claim 1 , further comprising the step of: calculating an expectation value for each of 3. The quantum computer is ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） ) to obtain the sign of the expectation value, and the classical computer calculates the sign of the expectation value obtained by the quantum computer and a plurality of Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） 3. The quantum information processing method according to claim 2, further comprising calculating 4. The classical computer calculates an expectation value Tr(ρ'P ｋ （ｉ） ) and calculate the calculated expected value Tr(ρ′P ｋ （ｉ） ) and the sign of the multiple Pauli products P ｋ （ｉ） The absolute value of each expected value |Tr(ρP ｋ （ｉ） ) | and based on the multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） 3. The quantum information processing method according to claim 2, further comprising calculating 5. The operator is a Hamiltonian operator H, and a variational quantum eigensolver (VQE) is used to calculate the expectation value of the Hamiltonian operator H, the quantum state ρ is a quantum state ρ(θ) with a parameter θ, and the classical computer calculates the parameter θ such that Tr(ρ(θ)H), which is the expectation value of the Hamiltonian operator H, is minimized. ＊ In the repeated calculation in the VQE when searching for, the sign of the expectation value obtained in the previous step is reused as the sign of the expectation value in the next step or later in a predetermined period, thereby obtaining a plurality of Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） 3. The quantum information processing method according to claim 2, further comprising calculating 6. Multiple Operators ｉ (i is an index for identifying the operator) ｋ （ｉ） and the Pauli product P ｋ （ｉ） (k is an index for identifying the Pauli product) and the following formula (A) can be used to calculate the (A) The result of the Bell measurement for the tensor product of the quantum state ρ and the quantum state σ is expressed by the following equation (B2): (B2) The classical computer calculates |Φ １ ± > and |Φ ２ ± Based on the above, multiple Pauli products P ｋ （ｉ） The value for each expected value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) and calculate the Pauli product P ｋ （ｉ） The expected value Tr(σP ｋ （ｉ） ) and calculate the value related to the expectation value (Tr(ρP ｋ （ｉ） )) (Tr(σP ｋ （ｉ） )) to the expected value Tr(σP ｋ （ｉ） ) to obtain multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） ) and calculating the predetermined number of coefficients c ｋ （ｉ） and multiple Pauli products P ｋ （ｉ） Each expected value Tr (ρP ｋ （ｉ） ) according to the above formula (A), ｉ The quantum information processing method according to claim 1 , further comprising the step of: calculating an expectation value for each of 7. The quantum information processing method according to claim 6, wherein the quantum state σ is a product state of quantum states with quantum bit numbers k1, k2, ..., km, where k1 + k2 + ... + km = n, and n is the total number of quantum bits for calculating the quantum state σ.

8. The classical computer calculates the expected value Tr(ρP ｋ （ｉ） 7. The quantum information processing method according to claim 6, further comprising: determining the quantum state σ in advance by performing an optimization calculation to minimize an objective function that represents a variance in a calculation result of a step (a) of the quantum information processing method according to claim 6, and transmitting information on the quantum state σ to the quantum computer.

9. A quantum information processing program executed by a classical computer in a hybrid system including a classical computer and a quantum computer, comprising: the classical computer sending a quantum circuit for preparing a quantum state to the quantum computer; the quantum computer setting n pairs of a quantum bit in the first quantum bit group and a quantum bit in the second quantum bit group for each of a first quantum bit group including n quantum bits and a second quantum bit group including n quantum bits different from the quantum bits in the first quantum bit group, performing a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sending the result of the Bell measurement to the classical computer; the classical computer calculating an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement; and calculating an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

10. A classical computer in a hybrid system including a classical computer and a quantum computer, wherein the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer sets n pairs of a quantum bit in the first quantum bit group and a quantum bit in the second quantum bit group, for each of a first quantum bit group including n quantum bits and a second quantum bit group including n quantum bits different from the quantum bits in the first quantum bit group, performs a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sends the result of the Bell measurement to the classical computer, the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

11. A hybrid system including a classical computer and a quantum computer, wherein the classical computer sends a quantum circuit for preparing a quantum state to the quantum computer, the quantum computer sets n pairs of a quantum bit in the first quantum bit group and a quantum bit in the second quantum bit group, for each of a first quantum bit group including n quantum bits and a second quantum bit group including n quantum bits different from the quantum bits in the first quantum bit group, performs a Bell measurement on a tensor product of the quantum state using each of the n pairs to obtain a result of the Bell measurement and sends the result of the Bell measurement to the classical computer, the classical computer calculates an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculates an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

12. A hybrid system including a plurality of classical computers and a plurality of quantum computers, wherein one or more classical computers of the plurality of classical computers send a quantum circuit for preparing a quantum state to the quantum computer, wherein the one or more quantum computers of the plurality of quantum computers: for each of a first qubit group including n qubits and a second qubit group including n qubits different from the qubits in the first qubit group, set n pairs of qubits in the first qubit group and qubits in the second qubit group, perform a Bell measurement on a tensor product of a quantum state using each of the n pairs to obtain a result of the Bell measurement and send the result of the Bell measurement to the classical computer, wherein the one or more classical computers of the plurality of classical computers calculate an expectation value of each of a plurality of Pauli products in the quantum state based on the result of the Bell measurement, and calculate an expectation value of each of a plurality of operators based on an arbitrary combination of Pauli products among each of the plurality of Pauli products.

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