JP7499540B2 - Quantum information processing method for open quantum systems, classical computer, quantum computer, quantum information processing program, and data structure - Google Patents

Description

The disclosed technology relates to quantum information processing methods, classical computers, quantum computers, quantum information processing programs, and data structures for open quantum systems.

Conventionally, variational quantum eigensolver (VQE) (hereinafter simply referred to as "VQE") is known (see, for example, Non-Patent Document 1). VQE approximately calculates the smallest eigenvalue of a Hamiltonian by sequentially updating the parameters of a quantum circuit.

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik and J. L.O’Brien, "A variational eigenvalue solver on a photonic quantum processor.", Nature Communications,5, article number: 4213, 2014.

However, the VQE disclosed in the above-mentioned non-patent document 1 is based on the premise of an isolated quantum system, and does not take open quantum systems into consideration.

The disclosed technology has been made in consideration of the above circumstances, and aims to provide a quantum information processing method, classical computer, quantum computer, quantum information processing program, and data structure for an open quantum system that can calculate the steady state and physical quantities in the steady state in the open quantum system.

In order to achieve the above object, the quantum information processing method for an open quantum system according to the first aspect of the present disclosure is a quantum information processing method executed by a hybrid system including a classical computer and a quantum computer, in which the classical computer calculates a Lindebradian based on a Hamiltonian and a dissipation operator, the classical computer calculates an objective function according to the Lindebradian, the classical computer determines the structure of a first quantum circuit for obtaining a stationary state of the open quantum system, the classical computer outputs the objective function, the structure of the first quantum circuit, and initial values of parameters of the first quantum circuit, and the quantum computer calculates a quantum calculation using a variational-quantum-Eigensolver (VQE) according to the objective function, the structure of the first quantum circuit, and the initial values of the parameters output from the classical computer. a classical computer performs a quantum calculation according to the information representing the physical quantity of the object to be measured, the second quantum circuit, and the initial value of the measurement quantum bit, the classical computer performs a quantum calculation according to the information representing the physical quantity of the object to be measured output from the classical computer, the second quantum circuit, and the initial value of the measurement quantum bit, the quantum computer outputs a measurement result of the quantum calculation, and the classical computer outputs a calculation result of the physical quantity of the object to be measured according to the measurement result output from the quantum computer.

The classical computer of the second aspect of the present disclosure is a quantum information processing method for an open quantum system in which, when determining the structure of the first quantum circuit, the classical computer determines the structure of the first quantum circuit so as to satisfy conditions (II) and (III) expressed in the following formula (A), and, when outputting the second quantum circuit, the classical computer outputs the second quantum circuit that satisfies conditions (I), (II), and (III) expressed in the following formula (A).

(A)

where ρ^ denotes the density matrix of the mixed state in an open quantum system, ρ^ ^† denotes the Hermitian conjugate of ρ^, and |ψ> denotes a quantum state. Moreover, the above (II) holds for any |ψ>.

The matrix representing the first quantum circuit of the third aspect of the present disclosure includes a matrix for generating a distribution of eigenvalues and a matrix for performing a basis transformation.

The parameters of the first quantum circuit of the fourth aspect of the present disclosure include a parameter θ that controls the distribution of eigenvalues and a parameter φ of a matrix for performing a basis transformation.

In the fifth aspect of the present disclosure, when setting the initial value of the measurement quantum bit, the classical computer calculates a probability λ _q (θ ^' ) according to the parameter θ ^' generated by the quantum computer, and sets the initial value of the measurement quantum bit according to the calculated probability λ _q (θ ^' ).

The second quantum circuit of a sixth aspect of the present disclosure is a unitary circuit U(φ ^' ) of the first quantum circuit, in which the parameter φ ^' generated by the quantum computer is reflected.

A seventh aspect of the present disclosure repeats setting the initial value of the measurement quantum bit by the classical computer and executing the quantum computation by the quantum computer according to the initial value of the measurement quantum bit until the calculation result of the quantum computation by the quantum computer converges.

The classical computer and the quantum computer of the eighth aspect of the present disclosure are connected via a computer network, and the classical computer and the quantum computer transmit and receive information via the computer network.

A ninth aspect of the present disclosure is a quantum information processing method for an open quantum system, in which a classical computer calculates a Lindebrandian based on a Hamiltonian and a dissipation operator, calculates an objective function according to the Lindebrandian, determines the structure of a first quantum circuit for determining a stationary state of an open quantum system, outputs the structure of the first quantum circuit and initial values of parameters of the first quantum circuit, sets initial values of measurement quantum bits according to parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure of the first quantum circuit and the initial values of the parameters output from the quantum computer, outputs information representing a physical quantity of the measurement target, a second quantum circuit for measuring the physical quantity of the measurement target in a stationary state, and the initial values of the measurement quantum bits, and outputs a calculation result of the physical quantity of the measurement target according to the information representing the physical quantity of the measurement target output from the quantum computer, the second quantum circuit being the measurement quantum circuit, and the initial values of the measurement quantum bits.

A tenth aspect of the present disclosure is a quantum information processing method for an open quantum system that executes a process in which a quantum computer executes a quantum calculation using a variational quantum eigensolver (VQE) in accordance with an objective function corresponding to a Lindbrandian calculated based on a Hamiltonian and a dissipation operator output from a classical computer, a structure of a first quantum circuit for determining a stationary state of an open quantum system, and initial values of parameters of the first quantum circuit, generates parameters of the first quantum circuit, outputs the parameters of the first quantum circuit, executes a quantum calculation in accordance with information representing a physical quantity of a measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a stationary state, and initial values of measurement quantum bits set in accordance with the parameters of the first quantum circuit, and outputs a measurement result of the quantum calculation.

The eleventh aspect of the present disclosure is a classical computer that executes the following process: calculates the Lindebrandian based on a Hamiltonian and a dissipation operator; calculates an objective function according to the Lindebrandian; determines the structure of a first quantum circuit for determining the stationary state of an open quantum system; outputs the structure of the first quantum circuit and initial values of parameters of the first quantum circuit; sets initial values of measurement quantum bits according to parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure of the first quantum circuit and the initial values of the parameters output from the quantum computer; outputs information representing the physical quantity of the object to be measured, a second quantum circuit for measuring the physical quantity of the object to be measured in the stationary state, and the initial values of the measurement quantum bits; and outputs a calculation result of the physical quantity of the object to be measured according to the measurement result of the quantum calculation according to the information representing the physical quantity of the object to be measured output from the quantum computer, the second quantum circuit that is the measurement quantum circuit, and the initial values of the measurement quantum bits.

A twelfth aspect of the present disclosure is a quantum computer that executes a process of performing a quantum computation using a VQE (Variational-Quantum-Eigensolver) in accordance with an objective function output from a classical computer according to a Lindbrandian calculated based on a Hamiltonian and a dissipation operator, the structure of a first quantum circuit for determining a stationary state of an open quantum system, and initial values of parameters of the first quantum circuit, generating parameters of the first quantum circuit, outputting the parameters of the first quantum circuit, performing a quantum computation in accordance with information representing a physical quantity of a measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a stationary state, and initial values of measurement quantum bits set in accordance with the parameters of the first quantum circuit, and outputting a measurement result of the quantum computation.

A thirteenth aspect of the present disclosure is a quantum information processing program for causing a classical computer to execute a process of: calculating a Lindebrandian based on a Hamiltonian and a dissipation operator; calculating an objective function according to the Lindebrandian; determining a structure of a first quantum circuit for determining a stationary state of an open quantum system; outputting the structure of the first quantum circuit and initial values of parameters of the first quantum circuit; setting initial values of measurement quantum bits according to parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure of the first quantum circuit and the initial values of the parameters output from the quantum computer; outputting information representing a physical quantity of the measurement target, a second quantum circuit for measuring the physical quantity of the measurement target in a stationary state, and the initial values of the measurement quantum bits; and outputting a calculation result of the physical quantity of the measurement target according to the information representing the physical quantity of the measurement target output from the quantum computer, the second quantum circuit being the measurement quantum circuit, and the initial values of the measurement quantum bits.

A fourteenth aspect of the present disclosure is a quantum information processing program for causing a quantum computer to execute a process that executes a quantum calculation using a variational-quantum-Eigensolver (VQE) according to an objective function according to a Lindbrandian calculated based on a Hamiltonian and a dissipation operator, output from a classical computer, a structure of a first quantum circuit for determining a stationary state of an open quantum system, and initial values of parameters of the first quantum circuit, generates parameters of the first quantum circuit, outputs the parameters of the first quantum circuit, executes a quantum calculation according to information representing a physical quantity of a measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a stationary state, and initial values of measurement quantum bits set according to the parameters of the first quantum circuit, and outputs a measurement result of the quantum calculation.

A fifteenth aspect of the present disclosure is a data structure relating to the configuration of a quantum circuit used in quantum computation by a quantum computer, the data structure being information for determining the stationary state of an open quantum system, including information representing a matrix for generating a distribution of eigenvalues and a matrix for performing a basis transformation, and used in a process of generating parameters of the quantum circuit by quantum computation by the quantum computer using a Variational-Quantum-Eigensolver (VQE) based on the quantum circuit and initial values of parameters of the quantum circuit.

The disclosed technology has the effect of being able to calculate the stationary state and the physical quantities at the stationary state in an open quantum system.

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. 1 is a diagram illustrating an example of a quantum circuit according to an embodiment of the present invention. FIG. 1 is a diagram illustrating an example of a quantum circuit according to an embodiment of the present invention. FIG. 13 is a diagram illustrating an example of D ^∼ (θ) constituting the quantum circuit of the present embodiment. FIG. 13 is a diagram illustrating an example of D ^∼ (θ) constituting the quantum circuit of the present embodiment. FIG. 2 is a diagram illustrating an example of U(φ) constituting the quantum circuit of the present embodiment. FIG. 2 is a diagram illustrating an example of a calculation process of the hybrid system 100. FIG. 13 is a diagram showing a simulation result in an example. FIG. 13 is a diagram showing a simulation result in an example.

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

<Hybrid system 100 according to this embodiment>

Figure 1 shows a hybrid system 100 according to this embodiment. The hybrid system 100 according to 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 IP (Internet Protocol) 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 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. Note that in this embodiment, the term "quantum computer" does not mean that no operations are performed using classical bits, but refers to a computer that includes operations using quantum bits.

The control device 121 is a classical computer that performs calculations using classical bits, and 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 in the quantum bit group 123 in response to receiving parameters of the quantum circuit.

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 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 this embodiment calculates the steady state of an open quantum system. Below, we will explain the overview of this embodiment and the prerequisites.

[Overview of the hybrid system 100 of this embodiment]

In this embodiment, we propose VQE (hereinafter simply referred to as "dVQE") in a dissipative system, which is an open quantum system. dVQE calculates a non-equilibrium steady state (NESS) (hereinafter also simply referred to as "steady state" or simply "NESS") in an open quantum system using a quantum circuit represented by variational parameters (hereinafter simply referred to as "parameters").

In this embodiment, when calculating the steady state of an open quantum system, quantum computation is performed using a quantum circuit that has certain physical requirements. Specifically, when attempting to obtain the steady state of an open quantum system described by n quantum bits, the quantum circuit of this embodiment is expressed by 2n quantum bits.

Figure 3 shows an explanatory diagram for explaining the quantum circuit used in this embodiment. As shown in Figure 3, the quantum circuit of this embodiment is represented by n quantum bits 70 corresponding to the physical space (also called the left space) and n quantum bits 72 corresponding to the auxiliary space (also called the right space). Also, as shown in Figure 3, the quantum circuit of this embodiment includes a portion 74 representing an eigenvalue distribution and a portion 76 representing a basis transformation.

In this embodiment, first, the parameters of a quantum circuit represented by 2n quantum bits are optimized using VQE. This results in a steady state of an open quantum system. After the parameters of the quantum circuit are optimized by VQE, in this embodiment, n quantum bits out of the 2n quantum bits are excluded. Then, in this embodiment, the physical quantities of the steady state of an open quantum system described by n quantum bits are obtained. This will be explained in detail below.

[1 Lindbrandian representation of open quantum systems]

An open quantum system is a system that has interactions with the external environment. Open quantum systems also follow non-unitary time evolution. In contrast, isolated quantum systems follow unitary time evolution.

In this embodiment, the steady state of an open quantum system whose non-unitary time evolution is described by the Lindbrandian is obtained by quantum computation. The Lindbrandian has two properties imposed on the operator of the time evolution in an open quantum system. One is the property of completely positive trace preserving map (CPTP), and the other is Markov property.

Specifically, the time evolution of the mixed state ρ^(t) is given by the well-known Lindblad equation (see, for example, references 1 and 2 below).

Reference 1: G. Lindblad. , "On the generators of quantum dynamical semigroups.", Comm. Math. Phys.,48(2):119-130, 1976.
Reference 2: Kristan Temme, Fernando Pastawski, and Michael J Kastoryano. "Hypercontractivity of quasifree quantum semigroups." Journal of Physics A: Mathematical and Theoretical, 47(40):405303,2014.

The action Lρ^(t) of the Lindbrandian L on the mixed state ρ^(t) in a finite-dimensional Hilbert space is expressed by the following equation (1). Note that the "^" attached to a symbol indicates that the symbol is a matrix.

(1)

The space in which the Lindbrandian operates is given as follows:

Finite-dimensional Hilbert spaces are expressed by the following formula:

The space in which the Hamiltonian H^ of the system acts is given as follows. The Hamiltonian H^ of the system describes the unitary part of the time evolution.

When the hyperoperator D[Γ^ _k ] in the above formula is applied to the mixed state ρ^(t), it is expressed by the following formula (2): where "†" represents a Hermitian conjugate.

(2)

Dissipation with amplitude γ _k in the second term on the right-hand side of the above equation (1) plays a role in destroying the coherence of the pure state, and the space in which the corresponding superoperator acts is given as follows:

In addition, Γ^ _k in the above equation (2) is the kth jump operator and determines the details of dissipation.

[1.1 Vector representation of the Lindblad equation]

The time-independent Lindebrandian has at least one steady state that satisfies equation (3) below.

(3)

In the above equation (3), _ρ̂SS represents the density matrix of the NESS (see, for example, Reference 3 below).

Reference 3: A. Rivas and S.F. Huelga. "Open Quantum Systems: An Introduction.", Springer Briefs in Physics., Springer Berlin Heidelberg, 2011.

In order to use known numerical calculation methods, in this embodiment, the density "matrix" ρ^ of the mixed state in an open quantum system is mapped to a "vector" |ρ> in a space called the "operator space." The space in which the density matrix ρ^ acts is given as follows:

The space of vectors |ρ> is defined by the following equation:

The new representation of the state is called a vector representation and is expressed by the following equation (4).

(4)

Note that C in the above formula (4) is a normalization coefficient and is expressed by the following formula.

The normalization methods used in the two representations are different. In the matrix representation, normalization is performed so that the diagonal sum becomes 1, as shown in the following equation. On the other hand, in the vector representation, normalization is performed so that the L2 norm of |ρ> becomes 1.

The space corresponding to the symbol |i>, which represents a ket, is called the physical space, and the space corresponding to the symbol <j|, which represents a bra, is called the auxiliary space. Alternatively, the space corresponding to the symbol |i>, which represents a ket, is also called the left space. Also, the space corresponding to the symbol <j|, which represents a bra, is also called the right space.

These two subspaces are distinguished by the subscripts in the following formula:

,

The elements of these two subspaces are expressed, for example, by the following formula:

,

Using the mapping defined by equation (4) above, the operator A^ acting on the density matrix ρ^ from the left is mapped to an operator in left space (or auxiliary space) as follows: The operator B^ acting on the density matrix ρ^ from the right is mapped to an operator in right space (or physical space) as follows:

(5)


(6)


(7)

To obtain the Lindblad equation in vector representation, the above equation (5) which represents the mapping is applied to the above equation (1), and the following equations (8) to (10) are derived.

(8)


(9)


(10)

The Lindbrandian L^ vector representation operates on the space expressed by the following formula:

When using the vector representation, the problem of finding the NESS reduces to a standard linear algebra problem. Specifically, the solution to the NESS is obtained by solving the equation given by the non-Hermitian operator in Equation (11) below.

(11)

Other eigenstates whose eigenvalues are not zero generally satisfy the following equations (12) to (13).

(12)


(13)

Here, |ρ _i > is the vector representation of the i-th eigenmode at the eigenvalues that satisfy the following equation (see, for example, References 3 and 4), and λ _i is the i-th eigenvalue. In this case, |ρ _i > denotes the damping mode.

Reference 4: H.P. Breuer and F. Petruccione., "The Theory of Open Quantum Systems.", Oxford University Press, 2002.

In addition, when the following formula (B1) is satisfied, the following formula (B2) is also satisfied, and damping modes having complex eigenvalues always appear in pairs. Note that λ ^* _i is the complex conjugate of _λi .

(B1)


(B2)

[2 Calculation method for dVQE]

This embodiment uses a variational method to obtain the steady state of an open quantum system. 2n quantum bits are used to optimize the quantum circuit of this embodiment. On the other hand, n quantum bits are used to measure the physical quantity in the steady state of the open quantum system. This change in the number of quantum bits corresponds to a switch from a vector representation to a matrix representation of the density matrix.

The details are explained below.

[2.1 NESS as the ground state of the objective function L ^† L]

The Lindbrandian solved by equation (11) above is a non-Hermitian matrix. If there is no dissipation or if the dissipation is parity-time symmetric (see, for example, Reference 5), then the Lindbrandian becomes skew-Hermitian. In this case, diagonalizing the Lindbrandian with a unitary matrix results in purely imaginary eigenvalues. Otherwise, the eigenvalues are neither purely real nor purely imaginary, and the eigenmodes are not orthogonal. In other words, standard numerical methods used to diagonalize Hamiltonians cannot be applied.

Reference 5: Carl M Bender., "Making sense of non-hermitian hamiltonians.", Reports on Progress in Physics,70(6):947, 2007.

In this regard, it is possible to use the Schatten norm ∥dρ^(t)/dt∥ _trace as the objective function (see, for example, Reference 6). However, it is not easy to evaluate the Schatten norm ∥dρ^(t)/dt∥ _trace .

Reference 6: Hendrik Weimer., "Variational principle for steady states of dissipative quantum many-body systems.", Phys. Rev. Lett., 114:040402, Jan 2015.

On the other hand, the product of the Hermitian conjugate L^ ^† of the Lindbrandian and the Lindbrandian L^, as shown in the following equation, is a Hermitian matrix having non-negative eigenvalues.

Therefore, the lowest eigenstate with the eigenvalue λ = 0 corresponds to NESS. That is, NESS |ρ _SS > that satisfies the above formula (11) satisfies the following formula (14).

(14)

For this reason, in this embodiment, methods for searching ground states in isolated quantum systems, such as the Lanczos method or variational approaches (see, for example, Reference 7), can be applied to searching the NESS of open quantum systems.

Reference 7: Jian Cui, J. Ignacio Cirac, and Mari Carmen Ba~nuls., "Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems.", Physical Review Letters, 114(22):1-5,jan 2015.

The advantage of variational approaches such as VQE is that they can show how good the approximation is. Specifically, by using the property that the lower bound of the objective function L^ ^† L^ is zero, we can quantify the accuracy of the optimization without computing the exact solution.

For this reason, in this embodiment, the above equation (14) is solved using a parameterized quantum circuit. dVQE, which is the calculation method of this embodiment, optimizes the objective function using a parameterized quantum circuit. Note that VQE is a technology developed for noisy intermediate-scale quantum (NISQ) devices (see, for example, Reference 8). In VQE, the entire calculation is performed by the calculation of the quantum state by the quantum computer 120 and the parameters of the quantum circuit optimized by the classical computer 110 (see, for example, Reference 9).

Reference 8: John Preskill., "Quantum Computing in the NISQ era and beyond.", (July):1-20, 2018.
Reference 9: Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J.Love, Al an Aspuru-Guzik, and Jeremy L. O'Brien. "A variational eigenvalue solver on a photonic quantum processor." Nature Communications, 5:4213, 2014.

In the hybrid system 100 of this embodiment, the processes performed by the quantum computer 120 and the classical computer 110 are shown below.

(0) The classical computer 110 sets the initial values of the quantum circuit parameters.

As described later, the quantum circuit of this embodiment is expressed by the following equation:

(1) The quantum computer 120 samples from the quantum circuit shown in the above equation, and the expected value <L^ ^† L^> of the objective function is calculated.
(2) The classical computer 110 updates the parameters θ and φ based on the expected value of the sampled objective function using an optimization method such as the Broyden-Fletcher-Goldfarb-Shanno algorithm (hereinafter simply referred to as the “BFGS algorithm”).
(3) For example, the above process (1) and process (2) are repeated until the expected value of the objective function <L^ ^† L^> converges to zero.

The number of terms in the objective function grows rapidly due to all the coupling between the physical space and the auxiliary space. However, we can sample from the qubits efficiently enough to get to this point (see, e.g., Ref. 10).

Reference 10: G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme. "Quantum algorithms for fermionic simulations.", Phys. Rev. A, 64:022319, Jul 2001.

[2.2 Quantum circuit assumptions for dVQE]

Next, we will explain the variational approach to calculating NESS using quantum circuits. As mentioned above, in this embodiment, by expressing the density matrix as a vector, we can proceed with the discussion in parallel with VQE in isolated quantum systems (see, for example, References 9, 11, and 12).

Reference 11: Bela Bauer, DaveWecker, Andrew J. Millis, Matthew B. Hastings, and Matthias Troyer. "Hybrid quantum-classical approach to correlated materials.", Phys. Rev. X, 6:031045, Sep 2016.
Reference 12: Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M.Chow, and Jay M. Gambetta. "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets." Nature, 549:242 EP -, 09 2017.

On the other hand, the density matrix ρ^ of an open quantum system must satisfy the following condition:

Therefore, in this embodiment, a quantum circuit is designed to satisfy the above conditions (I), (II), and (III), and the quantum circuit is used to calculate the steady state of an open quantum system and the physical quantities in the steady state.

In this embodiment, a first quantum circuit is used to determine the steady state of an open quantum system, and a second quantum circuit is used to measure the physical quantity of the measurement target in the steady state. Specifically, the hybrid system 100 of this embodiment determines the steady state of the open quantum system using the first quantum circuit, and then measures the physical quantity of the measurement target in the steady state using the second quantum circuit.

Note that condition (I) is satisfied by applying appropriate normalization to the vector representation. However, when optimizing the parameters of the first quantum circuit, condition (I) does not need to be satisfied. Therefore, when determining the structure of the first quantum circuit, the structure is determined so that the above conditions (II) and (III) are satisfied.

Specifically, the normalization of condition (I) can be taken into consideration when measuring with the second quantum circuit. Since the first quantum circuit is expressed as a vector, when optimizing the parameters of the first quantum circuit, the structure of the first quantum circuit is determined so that the following condition is satisfied.

The above conditions (I) to (III) indicate that the parameterization of eigenvalues and the basis transformation are separated. FIG. 4 shows an example of a quantum circuit of this embodiment. The quantum circuit of FIG. 4 is a quantum circuit for NESS and is the first quantum circuit. D ^∼ (θ) generates a distribution of eigenvalues, and U(φ) transforms the basis. Note that the upper half of the qubit represents the physical space, and the lower half of the qubit represents the auxiliary space.

Any density matrix is a Hermitian matrix and can be diagonalized by a unitary matrix. Therefore, in this embodiment, we construct a quantum circuit represented by the following equations (15) to (16).

(15)


(16)

Here, D^(θ) gives the distribution of eigenvalues. Also, U^(θ) corresponds to the basis transformation. Details of D^(θ) and U^(θ) will be given later.

Assuming that all quantum bits are initialized to 0, the initial state ρ ₀ ̂ is expressed by the following equation.

The initial state can also be expressed by the following equation:

The quantum circuit corresponding to the eigenvalue distribution obtained by the mapping shown in the following equation is represented as D ^∼ , where the eigenvalue distribution D(θ) corresponds to the above equation (16), and will be described in detail later.

In this case, the first quantum circuit of this embodiment is given by the following equation (17).

(17)

The unitary operator U^(φ) for basis transformation depends heavily on the problem setting, but has a large effect on the accuracy of the solution. As will be described later, for simplicity, in this embodiment, a hardware-efficient ansatz (see, for example, References 12, 13, and 14) that is efficient for the quantum computer hardware is applied.

Reference 13: E Farhi, J Goldstone, and S Gutmann. "A quantum approximate optimization algorithm.",arXiv:1411.4028, 2014.
Reference 14: KM Nakanishi, K Mitarai, and K Fujii. "Subspace-search variational quantum eigensolver for excited states.", arXiv:1810.09434, 2018.

[2.2.1 Quantum circuit ^D for eigenvalue distribution (θ)]

A unitary gate D ^≠ (θ) that generates the eigenvalue distribution of the density matrix is described below. The diagonal matrix for a system of n physical qubits is mapped to a vector representation as shown in equations (18)-(19) below.

(18)


(19)

The CNOT gate shown in the above formula (19) generates quantum entanglement between the kth quantum bit and the k+nth quantum bit. For example, if the first quantum bit of the states {|00>, |01>, |10>, |11>} represented by the quantum bits represents the control quantum bit and the second quantum bit represents the quantum bit to be controlled, the CNOT gate is represented by the following formula (20).

(20)

Here, it can be seen that the above equation (19) for the distribution of eigenvalues is determined from unitary gates in the left space. An example of D ^∼ (θ) is shown in Figure 5. The rotating Y gate, which is a component of a quantum circuit, is defined by the following equation (21).

(21)

The positive semidefinite property is guaranteed by the following angle restriction:

The quantum circuit D ^∼ (θ) of the eigenvalue distribution shown in Fig. 5 treats each quantum bit independently, while the quantum circuit D ^∼ (θ) shown in Fig. 6 introduces quantum entanglement. As shown in Fig. 5 and Fig. 6, the quantum bits in the upper half represent the physical space, and the quantum bits in the lower half represent the auxiliary space. Note that these figures show the case of n=3.

[Quantum circuit U(φ) for basis transformation]

In this embodiment, a quantum circuit that is efficient for quantum computer hardware is used for the basis conversion (see, for example, References 12, 13, and 14). An example of U(φ) is shown in FIG. 7. RX and RY in U(φ) shown in FIG. 7 are repeated D ₂ times. RX represents a rotating X gate, and RY represents a rotating Y gate. Note that in U(φ) shown in FIG. 7, quantum bits in physical space are depicted.

[2.3 Measurement of physical quantities of objects]

When optimization by the classical computer 110 is completed, the initial value of the measurement quantum bit is set probabilistically and then a measurement is performed to calculate the expected value of the physical quantity to be measured. Here, we will explain the details of dVQE of this embodiment, which is a calculation method suitable for open quantum systems.

[2.3.1 General measurement scheme for NESS in quantum circuits]

While the expected value <L ^† L> of the vector-represented objective function can be measured efficiently, the measurement of physical quantities cannot be performed with the vector representation. For this reason, the hybrid system 100 of the present embodiment stochastically initializes the input n quantum bits. Specifically, the hybrid system 100 of the present embodiment stochastically initializes the state of the n quantum bits based on a set of eigenvalues {λ _i } of the density matrix.

The measurement of the physical quantity A^ of the measurement target under the density matrix ρ^ is expressed in matrix and vector form as shown in the following equations (22) and (23).

(22)


(23)

Here, the unnormalized vector shown in the following equation corresponds to the vector representation of the density matrix representing the fully mixed state.

Note that the state |Ψ(1)> in the above equation is composed of 2 ⁿ non-zero elements, so the above equation is a measurement in vector representation in which 2 ⁿ elements are extracted from the 2 ^{2 n} elements, making the measurement exponentially inefficient with respect to the size of the system.

To avoid this problem, in this embodiment, the expression of the quantum circuit is converted from a vector expression to a matrix expression before measuring the physical quantity. Furthermore, in this embodiment, after the optimization of parameters θ and φ of D ^∼ (θ) and U(φ) in the quantum circuit is completed, n quantum bits are used. Let q _k be the kth quantum bit that takes 1 or 0, and the n-bit array be expressed as q = (q ₁ , ..., q _n ).

The density matrix ρ^ given by the optimized quantum circuit is expressed by the following formula:

The expected value <A> of the physical quantity A to be measured is newly defined by the following formula:

(24)


(25)

The normalization shown in the following equation allows us to consider λ _q as the probability for setting the initial state to q.

Therefore, the algorithm in this embodiment can be summarized as follows:

(0) Find optimal values of parameters θ and φ by optimizing the first quantum circuit. Note that when optimizing parameters θ and φ, 2n quantum bits are used by the VQE.
(1) Set the initial state |q> of the n qubits corresponding to the left space according to the probability λ _q .
(2) The physical quantity A^ of the measurement object is measured according to the basis-transformed |ψ _q >.
(3) Repeat steps (1) and (2) above until a desired accuracy is achieved.

When a circuit as shown in FIG. 5 is adopted as a quantum circuit corresponding to the eigenvalue distribution D ¹ , the following equation (26) can be used for the probability.

(26)

[Operation of the hybrid system 100 of this embodiment]

Next, the specific operation of the hybrid system 100 of this embodiment will be described. Each device of the hybrid system 100 executes the processes shown in FIG. 8.

First, in step S100, the user terminal 130 transmits information about the calculation target input by the user (hereinafter simply referred to as "calculation target information") to the classical computer 110. The calculation target information is information about the problem to be solved by quantum computing, and includes, for example, information for calculating the Hamiltonian H of the calculation target system, information for calculating the dissipation operator Γ of the calculation target system, and information about the physical quantity of the measurement target.

The physical quantity of the object to be measured is "A" shown in the above formula (24) etc., and examples of the physical quantity of the object to be measured include the magnetization or current of the object. The information for calculating the Hamiltonian H and the information for calculating the dissipation operator Γ differ depending on the object to be calculated. Examples of information for calculating the Hamiltonian H include the index of interacting quantum bits and a function system of interactions. Also, examples of information for calculating the dissipation operator Γ include specifying the relaxation time for the phase relaxation term and the energy relaxation term.

Next, in step S102, the classical computer 110 receives the calculation target information transmitted from the user terminal 130. Then, in step S102, the classical computer 110 calculates the Hamiltonian H, which represents the energy state of the system, according to a known formula, based on the information for calculating the Hamiltonian H from the received calculation target information. This determines the interactions within the system.

In addition, in step S102, the classical computer 110 calculates the dissipation operator Γ of the system according to a known formula based on the information for calculating the dissipation operator Γ of the system from the received calculation target information. This determines the interaction of the system with the external environment.

The Hamiltonian H and the dissipation operator Γ will differ depending on the calculation target.

In step S104, the classical computer 110 calculates the Lindbrandian L according to the above equations (8) to (10) based on the Hamiltonian H calculated in step S102 and the dissipation operator Γ.

In step S106, the classical computer 110 calculates an objective function L ^† L according to the Lindbradian L calculated in step S104 above.

In step S108, the classical computer 110 determines the structure of a first quantum circuit for obtaining a stationary state of an open quantum system as shown in Fig. 4. The structure of the first quantum circuit is a quantum circuit represented by 2n quantum bits when the number of quantum bits to be calculated is n quantum bits as shown in Fig. 4. The first quantum circuit is also a quantum circuit including D ^~ (θ) representing a matrix for generating an eigenvalue distribution and U(φ) representing a matrix for performing a basis transformation. For this reason, the first quantum circuit includes a parameter θ representing a matrix for generating an eigenvalue distribution and a parameter φ of a matrix for performing a basis transformation.

As described above, the classical computer 110 determines the structure of the first quantum circuit so as to satisfy the following conditions (II) and (III).

where ρ^ denotes the density matrix of the mixed state in an open quantum system, ρ^ ^† denotes the Hermitian conjugate of ρ^, and |ψ> denotes a quantum state. Moreover, the above (II) holds for any |ψ>.

In step S110, the classical computer 110 outputs the structure of the first quantum circuit determined in step S108 and the initial values of the parameters θ and φ of the first quantum circuit. Specifically, the classical computer 110 transmits the structure of the first quantum circuit determined in step S108, the initial values of the parameters θ and φ of the first quantum circuit, and the optimization method to the quantum computer 120. An example of the optimization method is the BFGS method.

In step S106, the data transmitted from the classical computer 110 to the quantum computer 120 is a data structure related to the configuration of a quantum circuit used in the quantum computation of the quantum computer 120, and is information for determining the stationary state of an open quantum system, including information representing a matrix for generating a distribution of eigenvalues and information representing a matrix for performing a basis transformation.

This data structure is a data structure used for quantum computation by the quantum computer 120, and is used in the process of generating parameters θ ^' , φ ^' of the first quantum circuit by quantum computation by the quantum computer 120 using VQE based on the first quantum circuit and the initial values of parameters θ and φ of the first quantum circuit.

In step S112, the control device 121 receives the structure of the first quantum circuit, the initial values of the parameters θ and φ of the first quantum circuit, and the optimization method transmitted from the classical computer 110 in step S108. Then, the control device 121 causes the quantum computer 120 to execute quantum computation using VQE according to the structure of the first quantum circuit, the initial values of the parameters θ and φ of the first quantum circuit, and the optimization method.

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 accordance with 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 a quantum circuit according to the initial information, thereby generating parameters θ ^' , φ ^' of a first quantum circuit that is an optimal quantum circuit. The gate operation of each quantum gate included in the quantum circuit 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 parameters θ ^' , φ ^' of the first quantum circuit.

In step S114, the control device 121 transmits the parameters θ ^′ , φ ^′ of the first quantum circuit obtained in step S108 above to the classical computer 110.

In step S116, the classical computer 110 receives the parameters θ ^' , φ ^' of the first quantum circuit transmitted from the control device 121 in step S110. Then, the classical computer 110 determines information about the quantum measurement according to the parameters θ ^' , φ ^' of the first quantum circuit.

Specifically, the classical computer 110 determines information about the quantum measurement, including information representing the physical quantity to be measured, a second quantum circuit that is a quantum circuit for measurement, and an initial value of the quantum bit for measurement.

First, the classical computer 110 sets a second quantum circuit for determining the physical quantity of the measurement target in a steady state. The second quantum circuit is a quantum circuit for measuring the physical quantity, and satisfies the above conditions (I), (II), and (III). Specifically, the classical computer 110 sets the unitary circuit U(φ ^' ) of the first quantum circuit as the second quantum circuit. In an open quantum system, unlike a normal problem setting, the first quantum circuit obtained by optimization cannot be used for measurement. Therefore, in this embodiment, the unitary circuit U(φ ^' ) of the first quantum circuit is set as the second quantum circuit, which is a quantum circuit for measurement.

The parameter φ ^′ of the unitary circuit U(φ ^′ ) is the parameter φ ^′ received in step S114. Therefore, the second quantum circuit is the unitary circuit U(φ ^′ ) of the first quantum circuit in which the parameter φ ^′ generated by the quantum computer 120 is reflected.

Next, the classical computer 110 sets the initial value of the measurement quantum bit according to the parameter θ ^′ of the first quantum circuit output from the quantum computer 120 .

Specifically, when setting the initial value of the measurement quantum bit, the classical computer 110 calculates the probability λ _q (θ ^' ) according to the parameter θ ^' received in step S114, for example, according to the above formula (26). Then, the classical computer 110 sets the initial value of the measurement quantum bit q according to the calculated probability λ _q (θ ^' ). For example, the probability λ _qA (θ ^' ) of the measurement quantum bit _qA , the probability λ _qB (θ ^' ) of the measurement quantum bit _qB , and the probability λ _qC (θ ^' ) of the measurement quantum bit _qC are each calculated, and one of the quantum bits _qA , _qB , and _qC is set as the initial value according to the probability.

Note that the information about the quantum measurement that represents the physical quantity to be measured has already been acquired in step S100 above.

In step S118, the classical computer 110 transmits to the quantum computer 120 the information about the quantum measurement determined in step S116 above, which is information representing the physical quantity to be measured, the second quantum circuit which is the quantum circuit for measurement, and the initial value of the quantum bit for measurement.

In step S120, the control device 121 receives the information representing the physical quantity to be measured, the second quantum circuit which is the quantum circuit for measurement, and the initial value of the quantum bit for measurement, which were transmitted from the classical computer 110 in step S118 above. Then, the control device 121 causes the quantum computer 120 to execute a quantum calculation according to the information representing the physical quantity to be measured, the second quantum circuit which is the quantum circuit for measurement, and the initial value of the quantum bit for measurement.

Specifically, quantum computer 120 generates electromagnetic waves to be irradiated to at least any one of quantum bits in quantum bit group 123 in accordance with the control of control device 121. Then, quantum computer 120 measures information observed by executing the second quantum circuit by irradiating the generated electromagnetic waves to at least any one of quantum bits in quantum bit group 123. This results in a measurement result of physical quantity A received in step S102 above. Then, quantum computer 120 outputs the measurement result of physical quantity A obtained by quantum calculation.

In step S122, the control device 121 transmits the measurement result of physical quantity A obtained in step S120 to the classical computer 110.

In step S124, the classical computer 110 receives the measurement result of physical quantity A transmitted from the control device 121 in step S122.

The processes in steps S116 to S122 are repeated until the results of the quantum computation by the quantum computer 120 converge. For example, the setting of the initial value of the quantum bit for measurement by the classical computer 110 and the execution of the quantum computation according to the initial value of the quantum bit for measurement by the quantum computer 120 are repeated until the fluctuation of the physical quantity A obtained at each repetition of the quantum computation becomes equal to or less than a predetermined threshold. In this case, in each repetition, the average value of the physical quantity A obtained up to that point is calculated, and the processes in steps S116 to S122 are repeated until the fluctuation of the average value becomes equal to or less than a predetermined threshold.

In step S124, the classical computer 110 acquires each of the physical quantities A obtained in each of the repeated processes in steps S116 to S122. The classical computer 110 then performs statistical processing on the physical quantities A obtained in each iteration to calculate the expected value <A> of the physical quantity A shown in the above formula (24) etc.

In step S126, the classical computer 110 transmits the calculation result of the expected value <A> of the physical quantity obtained in step S124 to the user terminal 130.

In step S128, the user terminal 130 receives the calculation result of the expected value <A> of the physical quantity sent from the classical computer 110 in step S124.

As described above, in the hybrid system of this embodiment, the classical computer calculates the Lindebrandian based on the Hamiltonian and the dissipation operator. Then, the classical computer calculates an objective function according to the Lindebrandian and determines the structure of the first quantum circuit so as to satisfy the condition expressed in the following formula (1). Then, the classical computer outputs the objective function, the structure of the first quantum circuit, and the initial values of the parameters of the first quantum circuit. Then, the quantum computer executes quantum calculation using VQE according to the objective function, the structure of the first quantum circuit, and the initial values of the parameters output from the classical computer, generates parameters of the first quantum circuit, and outputs the parameters of the first quantum circuit. Then, the classical computer sets the initial values of the quantum bits for measurement according to the parameters of the first quantum circuit output from the quantum computer. Then, the classical computer outputs information representing the physical quantity to be measured, the second quantum circuit which is a quantum circuit for measurement, and the initial values of the quantum bits for measurement. The quantum computer then executes quantum computation according to the information representing the physical quantity of the measurement target output from the classical computer, the second quantum circuit, and the initial value of the measurement quantum bit, and outputs the measurement result of the quantum computation. The classical computer then outputs the computation result of the physical quantity of the measurement target according to the measurement result output from the quantum computer. This makes it possible to calculate the steady state of the physical quantity of the target in the open quantum system.

In addition, the hybrid system of this embodiment can efficiently obtain the steady state of a target physical quantity in an open quantum system by appropriately dividing up the roles between the classical computer and the quantum computer.

In addition, the hybrid system of this embodiment has a first quantum circuit for determining the steady state of the open quantum system and a second quantum circuit for measuring the physical quantity of the object to be measured in the steady state, which are different quantum circuits, making it possible to obtain the steady state of the open quantum system and the physical quantity of the object to be measured in the steady state.

In addition, in the hybrid system of this embodiment, when optimizing the parameters of the first quantum circuit, the quantum circuit is expressed as a vector representation, and when measuring a physical quantity using the second quantum circuit, the quantum circuit is expressed as a matrix representation. This makes it possible to obtain the steady state of the open quantum system and the physical quantity of the measurement target in the steady state.

Next, we will explain an example. In this example, we performed a numerical simulation using a one-dimensional transverse magnetic field Ising model with dissipation (1dTFIM).

First, we will explain the model for performing dVQE. First, when the above formula (10) is applied to a one-dimensional transverse magnetic field Ising model, the NESS of the one-dimensional transverse magnetic field Ising model is expressed by the following formulas (27) and (28).

(27)


(28)

The state in which the objective function L ^† L is optimized by dVQE can be checked sequentially by the BFGS method (see, for example, Reference 15). The BFGS method is implemented, for example, by the SciPy library (see, for example, Reference 16).

Reference 15: Jorge Nocedal and Stephen J. Wright. "Numerical Optimization.", Springer, New York, NY, USA, second edition, 2006.
Reference 16: Eric Jones, Travis Oliphant, Pearu Peterson, et al., "SciPy: Open source scientic tools for Python", 2001.

First, we calculate the NESS of the one-dimensional dissipative transverse-field Ising model (1dTFIM). The Lindbrandian L^ of the one-dimensional dissipative transverse-field Ising model is given by the following equation (29).

(29)

Note that the NESS of the one-dimensional transverse-field Ising model with dissipation deviates slightly from a simple all-down state due to correlations induced by nearest-neighbor interactions.

FIG. 9 shows the results of this embodiment. FIG. 9(A) shows the density matrix obtained by the dVQE of this embodiment. FIG. 9(B) shows the density matrix obtained by strict diagonalization. Note that the number of quantum bits in the physical system is n=4, and the parameters are g=h=1. Also, σ ^x _i represents the x component of the Pauli matrix acting on the i-th spin, σ ^z _i represents the z component of the Pauli matrix acting on the i-th spin, and σ ^- _i represents the annihilation operator of the angular momentum acting on the i-th spin. Also, the strength of dissipation γ _i is all constant and γ _i =1.

In Figures 9(A) and (B), the left side shows the calculation results of the density matrix of the real part, and the right side shows the calculation results of the density matrix of the imaginary part. As shown in Figure 9, it can be seen that the density matrix obtained by dVQE of this embodiment is in good agreement with the density matrix obtained by strict diagonalization.

Figure 10 shows the behavior of the objective function value as the optimization is repeated. The fidelity F as a vector expression is expressed by the following equation (30).

(30)

Here, |ρ _ED > is a vector expression of the density matrix obtained by exact diagonalization, and |ρ _dVQE > is a vector expression of the density matrix obtained by dVQE.

As shown in Figure 10, as the optimization progresses, the value of the objective function decreases, and the fidelity F increases.

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 classical computer 110 repeatedly sets the initial values of the measurement quantum bits and the quantum computer 120 executes the quantum calculation according to the initial values of the measurement quantum bits until the calculation result of the quantum calculation by the quantum computer 120 converges. However, the present invention is not limited to this. For example, the classical computer 110 may set multiple sets of initial values of the measurement quantum bits at once, and the quantum computer 120 may sequentially execute the quantum calculation according to the multiple initial values of the measurement quantum bits. Alternatively, each of the multiple initial value sets of the measurement quantum bits may be transmitted to multiple quantum computers 120, each of the multiple quantum computers 120 may execute the quantum calculation according to the received initial value, and the classical computer 110 may receive each calculation result from the multiple quantum computers 120 and calculate the expected value.

In the above embodiment, the objective function is <L ^† L>, but the present invention is not limited to this.

In the above embodiment, as an example of the second quantum circuit U(φ), a quantum circuit that is efficient for the hardware of a quantum computer as shown in FIG. 7 is used, but the present invention is not limited to this. For example, other types of quantum circuits, such as a unitary coupled cluster quantum circuit, may be used as the second quantum circuit U(φ).

In the above embodiment, the probability for setting the initial value is calculated using the above formula (29), but this is not limiting, and the probability may be set using other formulas.

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. For example, if the user who operates the user terminal 130 can express the problem as a Hamiltonian, the classical computer 110 may receive the Hamiltonian H as the calculation target information. The user terminal 130 may transmit the calculation target information to the classical computer 110 or a storage medium or storage device accessible to the classical computer 110 via a computer network such as an IP network, or may store the calculation target information in a storage medium or storage device and hand it over to the operator of the classical computer 110, who may then input 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 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 not necessary to transmit quantum computing information from the classical computer 110 to the quantum computer 120, and it is not necessary to transmit measurement results from the quantum computer 120 to the classical computer 110. In this case, it is also possible that the control device 121 of the quantum computer 120 plays the role of the classical computer 110 in the above description.

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 also be taken into consideration. As an example, the statement "in the case of a, do b" does not necessarily mean "in the case of a, always do b", unless expressly stated.

In addition, in the above embodiment, expressions such as "optimize" or "optimized parameters" are used, but 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.

In addition, 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.

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 (14)

A quantum information processing method executed by a hybrid system including a classical computer and a quantum computer, comprising:
The classical computer calculates an objective function in an open quantum system ;
The classical computer determines a structure of a first quantum circuit for obtaining a stationary state of an open quantum system;
The classical computer outputs the objective function, a structure of the first quantum circuit, and initial values of parameters of the first quantum circuit;
The quantum computer executes a quantum computation to approximately calculate a minimum eigenvalue of a Hamiltonian by sequentially updating parameters of the quantum circuit according to the objective function, the structure of the first quantum circuit, and the initial values of the parameters output from the classical computer, to generate parameters of the first quantum circuit , and to output the parameters of the first quantum circuit;
The classical computer sets an initial value of a measurement quantum bit in accordance with the parameter of the first quantum circuit output from the quantum computer;
the classical computer outputs information representing a physical quantity of a measurement object, a second quantum circuit for measuring the physical quantity of the measurement object in a steady state, and an initial value of the measurement quantum bit;
The quantum computer executes a quantum computation according to the information representing the physical quantity of the measurement object output from the classical computer, the second quantum circuit, and the initial value of the measurement quantum bit, and outputs a measurement result of the quantum computation;
The classical computer outputs a calculation result of the physical quantity of the measurement target in response to the measurement result output from the quantum computer.
-Quantum information processing method for open quantum systems including quantum processing. When determining the structure of the first quantum circuit, the classical computer determines the structure of the first quantum circuit so as to satisfy conditions (II) and (III) shown in the following formula (A),
When outputting the second quantum circuit, the classical computer outputs the second quantum circuit that satisfies conditions (I), (II), and (III) shown in the following formula (A).
Quantum information processing method for an open quantum system according to claim 1.



(A)
where ρ^ denotes the density matrix of the mixed state in an open quantum system, ρ^ ^† denotes the Hermitian conjugate of ρ^, and |ψ> denotes a quantum state. Moreover, the above (II) holds for any |ψ>. The matrix representing the first quantum circuit includes a matrix for generating a distribution of eigenvalues and a matrix for performing basis transformation.
3. A quantum information processing method for an open quantum system according to claim 1 or 2. The parameters of the first quantum circuit include a parameter θ that controls a distribution of eigenvalues and a parameter φ of a matrix for performing basis transformation.
A quantum information processing method for an open quantum system according to any one of claims 1 to 3. When setting an initial value of the measurement quantum bit, the classical computer calculates a probability λ _q (θ ′) according to a parameter θ ^′ generated by the quantum computer, and sets an initial value of the measurement quantum bit according to the calculated probability λ _q ( ^{θ ′ )} .
5. A quantum information processing method for an open quantum system according to claim 4. The second quantum circuit is a unitary circuit U(φ ^′ ) of the first quantum circuit, in which a parameter φ ^′ generated by the quantum computer is reflected.
6. A quantum information processing method for an open quantum system according to claim 4 or 5. repeating setting an initial value of the measurement quantum bit by the classical computer and executing the quantum computation by the quantum computer according to the initial value of the measurement quantum bit until a calculation result of the quantum computation by the quantum computer converges;
A quantum information processing method for an open quantum system according to any one of claims 1 to 6. the classical computer and the quantum computer are connected via a computer network;
The classical computer and the quantum computer transmit and receive information via the computer network.
A quantum information processing method for an open quantum system according to any one of claims 1 to 7. Classical computers,
Calculate the objective function in the open quantum system ,
Determining the structure of a first quantum circuit for obtaining a stationary state of an open quantum system;
Outputting a structure of the first quantum circuit and initial values of parameters of the first quantum circuit;
setting an initial value of a quantum bit for measurement according to parameters of the first quantum circuit calculated based on a structure of the first quantum circuit and initial values of the parameters output from a quantum computer, the parameters of the first quantum circuit being generated by a quantum calculation that approximately calculates a minimum eigenvalue of a Hamiltonian by sequentially updating the parameters of the quantum circuit;
outputting information representing a physical quantity of an object to be measured, a second quantum circuit for measuring the physical quantity of the object to be measured in a steady state, and an initial value of the quantum bit for measurement;
outputting a calculation result of the physical quantity of the object to be measured according to a measurement result of a quantum computation according to information representing the physical quantity of the object to be measured output from the quantum computer, a second quantum circuit which is the quantum circuit for measurement, and an initial value of the quantum bit for measurement;
A quantum information processing method for an open quantum system performing the processing. Quantum computers,
Executing a quantum computation to approximately calculate a minimum eigenvalue of a Hamiltonian by sequentially updating parameters of a quantum circuit according to an objective function in an open quantum system , a structure of a first quantum circuit for obtaining a stationary state of the open quantum system , and initial values of parameters of the first quantum circuit, which are output from a classical computer, to generate parameters of the first quantum circuit, and output the parameters of the first quantum circuit;
execute a quantum computation according to information indicating a physical quantity of the measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a steady state, and an initial value of a measurement quantum bit set according to a parameter of the first quantum circuit, and output a measurement result of the quantum computation.
A quantum information processing method for an open quantum system performing the processing. Calculate the objective function in the open quantum system ,
Determining the structure of a first quantum circuit for obtaining a stationary state of an open quantum system;
Outputting a structure of the first quantum circuit and initial values of parameters of the first quantum circuit;
setting an initial value of a quantum bit for measurement according to parameters of the first quantum circuit, the parameters of the first quantum circuit being calculated based on a structure of the first quantum circuit and initial values of the parameters output from a quantum computer, and generated by a quantum calculation that approximately calculates a minimum eigenvalue of a Hamiltonian by sequentially updating the parameters of the quantum circuit;
outputting information representing a physical quantity of an object to be measured, a second quantum circuit for measuring the physical quantity of the object to be measured in a steady state, and an initial value of the quantum bit for measurement;
outputting a calculation result of the physical quantity of the object to be measured according to a measurement result of a quantum computation corresponding to information indicating the physical quantity of the object to be measured output from the quantum computer, a second quantum circuit which is the quantum circuit for measurement, and an initial value of the quantum bit for measurement;
A classical computer performs the processing. Executing a quantum computation to approximately calculate a minimum eigenvalue of a Hamiltonian by sequentially updating parameters of a quantum circuit according to an objective function in an open quantum system , a structure of a first quantum circuit for obtaining a stationary state of the open quantum system , and initial values of parameters of the first quantum circuit, which are output from a classical computer, to generate parameters of the first quantum circuit, and output the parameters of the first quantum circuit;
execute a quantum computation according to information indicating a physical quantity of the measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a steady state, and an initial value of a measurement quantum bit set according to a parameter of the first quantum circuit, and output a measurement result of the quantum computation.
The quantum computer that performs the processing. Calculate the objective function in the open quantum system ,
Determining the structure of a first quantum circuit for obtaining a stationary state of an open quantum system;
Outputting a structure of the first quantum circuit and initial values of parameters of the first quantum circuit;
setting an initial value of a quantum bit for measurement according to parameters of the first quantum circuit calculated based on a structure of the first quantum circuit and initial values of the parameters output from a quantum computer, the parameters of the first quantum circuit being generated by a quantum calculation that approximately calculates a minimum eigenvalue of a Hamiltonian by sequentially updating the parameters of the quantum circuit;
outputting information representing a physical quantity of an object to be measured, a second quantum circuit for measuring the physical quantity of the object to be measured in a steady state, and an initial value of the quantum bit for measurement;
outputting a calculation result of the physical quantity of the object to be measured according to a measurement result of a quantum computation corresponding to information indicating the physical quantity of the object to be measured output from the quantum computer, a second quantum circuit which is the quantum circuit for measurement, and an initial value of the quantum bit for measurement;
A quantum information processing program that allows processing to be performed by a classical computer. Executing a quantum computation to approximately calculate a minimum eigenvalue of a Hamiltonian by sequentially updating parameters of a quantum circuit according to an objective function in an open quantum system , a structure of a first quantum circuit for obtaining a stationary state of the open quantum system , and initial values of parameters of the first quantum circuit, which are output from a classical computer, to generate parameters of the first quantum circuit, and output the parameters of the first quantum circuit;
execute a quantum computation according to information indicating a physical quantity of the measurement object output from the classical computer, a second quantum circuit for measuring the physical quantity of the measurement object in a steady state, and an initial value of a measurement quantum bit set according to a parameter of the first quantum circuit, and output a measurement result of the quantum computation.
A quantum information processing program that causes processing to be performed by a quantum computer.