JP2023113956A - Quantum information processing method for open quantum system, classic computer, quantum computer, quantum information processing program, and data structure - Google Patents

Abstract

To calculate a steady state and a physical quantity in the steady state in an open quantum system.SOLUTION: A classic computer 110 outputs an objective function associated with Lindbladian, a structure of a first quantum circuit, and an initial value of a parameter of the first quantum circuit. A quantum computer 120 executes quantum calculation in accordance with the objective function, the structure of the first quantum circuit, and the initial value of the parameter to generate a parameter. The classic computer 110 sets an initial value of a quantum bit for measurement in accordance with the parameter. The classic computer 110 outputs a physical quantity of a measurement object, a second quantum circuit, and the initial value of the quantum bit for measurement. The quantum computer 120 executes quantum calculation in accordance with the physical quantity of the measurement object, the second quantum circuit, and the initial value of the quantum bit for measurement and outputs a measurement result of the quantum calculation. The classic computer 110 outputs a calculation result of the physical quantity of the measurement object in accordance with the measurement result.SELECTED DRAWING: Figure 1

Description

The disclosed technique relates to a quantum information processing method, a classical computer, a quantum computer, a quantum information processing program, and a data structure for an open quantum system.

Conventionally, variational-quantum-eigensolver (VQE: Variational-Quantum-Eigensolver) (hereinafter simply referred to as "VQE") is known (see, for example, Non-Patent Document 1). VQE approximately computes the smallest eigenvalue of the Hamiltonian by successively updating the parameters of the 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 Non-Patent Document 1 above assumes an isolated quantum system and does not consider an open quantum system.

The disclosed technology has been made in view of the above circumstances, and includes a quantum information processing method for an open quantum system, a classical computer, which can calculate the steady state and the physical quantity in the steady state in the open quantum system, It aims at providing a quantum computer, a quantum information processing program, and a data structure.

To achieve the above object, a 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, A computer calculates a Lindbladian based on the Hamiltonian and a dissipation operator, the classical computer calculates an objective function according to the Lindbladian, and the classical computer calculates a steady state of an open quantum system and 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. , the quantum computer performs 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, which are output from the classical computer; to generate the parameters of the first quantum circuit, output the parameters of the first quantum circuit, and perform the classical computer according to the parameters of the first quantum circuit output from the quantum computer to set an initial value of a quantum bit for measurement, and the classical computer performs information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and the measurement and the initial value of the qubit for measurement, and the quantum computer outputs the information representing the physical quantity to be measured, the second quantum circuit, and the qubit for measurement, which are output from the classical computer. Quantum computation is performed according to the initial values, the measurement result of the quantum computation is output, and the classical computer computes the calculation result of the physical quantity to be measured according to the measurement result output from the quantum computer. A quantum information processing method for an open quantum system, including processing, outputting.

The classical computer according to the second aspect of the present disclosure, when determining the structure of the first quantum circuit, satisfies the conditions (II) and conditions (III) expressed in the following formula (A): 1, and the classical computer, when outputting the second quantum circuit, satisfies conditions (I), (II), and (III) expressed in the following equation (A): is a quantum information processing method for an open quantum system that outputs the second quantum circuit that satisfies

(A)

where ρ̂ represents the density matrix of the mixed state in the open quantum system, ρ̂ ^† represents the Hermitian conjugate of ρ̂̂, and |ψ> represents the quantum state. Also, the above (II) holds for any |ψ>.

A 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 basis transformation.

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

The classical computer of the fifth aspect of the present disclosure calculates the probability λ _q (θ ^' ) according to the parameter θ ^' generated by the quantum computer when setting the initial value of the measurement qubit. and set the initial value of the quantum bit for measurement according to the calculated probability λ _q (θ ^′ ).

The second quantum circuit of the sixth aspect of the present disclosure is the 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 is to set the initial value of the measurement qubit by the classical computer and the measurement qubit by the quantum computer until the calculation result of the quantum computation by the quantum computer converges. Execution of quantum calculation according to the initial value, and repeat.

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 that a classical computer calculates a Lindbladian based on a Hamiltonian and a dissipation operator, calculates an objective function according to the Lindbladian, and calculates a steady state of an open quantum system. determining the structure of the first quantum circuit to be obtained, outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit, and outputting the first quantum circuit from a quantum computer; According to the parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure of the quantum circuit and the initial values of the parameters, the initial value of the quantum bit for measurement setting a value, outputting information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the quantum bit for measurement; A measurement result of quantum computation according to the information output from the computer representing the physical quantity to be measured, the second quantum circuit that is the quantum circuit for measurement, and the initial value of the qubit for measurement A quantum information processing method for an open quantum system that executes processing, outputting a calculation result of the physical quantity to be measured in response.

A tenth aspect of the present disclosure is that the quantum computer outputs an objective function according to the Lindbladian calculated based on the Hamiltonian and the dissipation operator output from the classical computer, and the steady state of the open quantum system. Quantum calculation using a Variational-Quantum-Eigensolver (VQE) is performed according to the structure of the first quantum circuit and the initial values of the parameters of the first quantum circuit, and the parameters of the first quantum circuit and outputs the parameters of the first quantum circuit, the information representing the physical quantity to be measured, which is output from the classical computer, and the second quantum for measuring the physical quantity to be measured in a steady state performing a process of performing quantum computation according to a circuit and an initial value of a quantum bit for measurement set according to parameters of the first quantum circuit, and outputting a measurement result of the quantum computation; A quantum information processing method for open quantum systems.

An eleventh aspect of the present disclosure calculates a Lindbladian based on a Hamiltonian and a dissipation operator, calculates an objective function according to the Lindbladian, and obtains a steady state of an open quantum system. determining the structure of one quantum circuit, outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit, and outputting the structure of the first quantum circuit from a quantum computer; An initial value of a qubit for measurement is set according to the parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure and the initial values of the parameters. , information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement, and output from the quantum computer Further, according to the measurement result of the quantum calculation according to the information representing the physical quantity to be measured, the second quantum circuit that is the measurement quantum circuit, and the initial value of the measurement qubit, It is a classical computer that executes processing and outputs the calculation result of the physical quantity to be measured.

A twelfth aspect of the present disclosure is an objective function according to the Lindbladian calculated based on the Hamiltonian and the dissipation operator output from the classical computer, and the first quantum for obtaining the steady state of the open quantum system performing quantum computation using a Variational-Quantum-Eigensolver (VQE) according to the circuit structure and the initial values of the parameters of the first quantum circuit to generate the parameters of the first quantum circuit; a second quantum circuit for outputting parameters of the first quantum circuit and output from the classical computer and representing a physical quantity to be measured; a second quantum circuit for measuring the physical quantity to be measured in a steady state; A quantum computer that executes a process of executing quantum computation according to an initial value of a measurement quantum bit set according to a parameter of a first quantum circuit and outputting a measurement result of the quantum computation. .

A thirteenth aspect of the present disclosure calculates a Lindbladian based on a Hamiltonian and a dissipation operator, calculates an objective function according to the Lindbladian, and obtains a steady state of an open quantum system. determining the structure of one quantum circuit, outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit, and outputting the structure of the first quantum circuit from a quantum computer; An initial value of a qubit for measurement is set according to the parameters of the first quantum circuit generated by quantum calculation using a VQE (Variational-Quantum-Eigensolver) based on the structure and the initial values of the parameters. , information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement, and output from the quantum computer Further, according to the measurement result of the quantum calculation according to the information representing the physical quantity to be measured, the second quantum circuit that is the measurement quantum circuit, and the initial value of the measurement qubit, It is a quantum information processing program for causing a classical computer to execute processing that outputs calculation results of physical quantities to be measured.

A fourteenth aspect of the present disclosure is an objective function according to the Lindbladian calculated based on the Hamiltonian and the dissipation operator output from the classical computer, and the first quantum for obtaining the steady state of the open quantum system performing quantum computation using a Variational-Quantum-Eigensolver (VQE) according to the circuit structure and the initial values of the parameters of the first quantum circuit to generate the parameters of the first quantum circuit; a second quantum circuit for outputting parameters of the first quantum circuit and output from the classical computer and representing a physical quantity to be measured; a second quantum circuit for measuring the physical quantity to be measured in a steady state; To cause a quantum computer to execute a process of performing quantum computation according to the initial value of the quantum bit for measurement set according to the parameters of the first quantum circuit and outputting the measurement result of the quantum computation is a quantum information processing program.

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

According to the disclosed technique, an effect is obtained in which a steady state and a physical quantity in the steady state can be calculated in an open quantum system.

It is a figure showing an example of a schematic structure of hybrid system 100 of this embodiment. 1 is a schematic block diagram of a computer functioning as classical computer 110, controller 121, and user terminal 130. FIG. It is a figure which shows typically an example of the quantum circuit of this embodiment. It is a figure which shows typically an example of the quantum circuit of this embodiment. FIG. 4 is a diagram schematically showing an example of D ^{1 ˜} (θ) that constitutes the quantum circuit of the present embodiment; FIG. 4 is a diagram schematically showing an example of D ^{1 ˜} (θ) that constitutes the quantum circuit of the present embodiment; It is a figure which shows typically an example of U ((phi)) which comprises the quantum circuit of this embodiment. 4 is a diagram showing an example of calculation processing of the hybrid system 100; FIG. It is a figure which shows the simulation result in an Example. It is a figure which shows the simulation result in an Example.

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

<Hybrid system 100 according to the present embodiment>

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

In the hybrid system 100 of this embodiment, the quantum computer 120 performs 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 calculation results according to quantum calculation to the user terminal 130 . As a result, the hybrid system 100 as a whole performs predetermined calculation processing.

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

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

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

Controller 121 is a classical computer that operates on classical bits and alternatively performs some or all of the processing described herein as performed in 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 the parameters of the quantum circuit. good.

User terminal 130 is a classical computer that operates with classical bits. The user terminal 130 receives information input by the user and executes processing according to the information.

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

The hybrid system 100 of this embodiment computes the steady state of an open quantum system. An outline and prerequisites of the present embodiment will be described below.

[Overview of hybrid system 100 of the present embodiment]

This embodiment proposes VQE (hereinafter simply referred to as "dVQE") in a dissipative-system that is an open quantum system. dVQE uses a quantum circuit represented by a variational parameter (hereinafter simply referred to as a "parameter") to generate a Non-Equilibrium Steady State (NESS) (hereinafter simply referred to as a "steady state") in an open quantum system. (also referred to as "state" or simply "NESS").

In this embodiment, when calculating the steady state in an open quantum system, quantum computation is performed using a quantum circuit to which a predetermined physical requirement is imposed. Specifically, when trying to obtain a steady state of an open quantum system described by n qubits, the quantum circuit of this embodiment is represented by 2n qubits.

FIG. 3 shows an explanatory diagram for explaining the quantum circuit used in this embodiment. As shown in FIG. 3, the quantum circuit of this embodiment consists of n qubits 70 corresponding to the physical space (also referred to as the left space) and the auxiliary space (also referred to as the right space). n qubits 72 corresponding to . Further, as shown in FIG. 3, the quantum circuit of this embodiment includes a portion 74 representing the eigenvalue distribution and a portion 76 representing the basis transformation.

In this embodiment, VQE is first used to optimize the parameters of a quantum circuit represented by 2n quantum bits. This gives the steady state of the open quantum system. After the parameters of the quantum circuit are optimized by VQE, in this embodiment, n qubits out of 2n qubits are excluded. Then, in this embodiment, steady-state physical quantities of an open quantum system described by n quantum bits are obtained. A specific description will be given below.

[1 Representation of Open Quantum Systems by Lindbladian]

An open quantum system is a system that interacts with the external environment. An open quantum system is also a system that follows a non-unitary time evolution. In contrast, solitary 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 Lindbladian is obtained by quantum computation. The Lindbladian has two properties imposed on the operator of the time evolution in open quantum systems. 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 ρ̂(t) is given by the well-known Lindblad equation (see, eg, references 1, 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 Lρ̂(t) of the Lindbladian L on the mixed state ρ̂(t) in the finite-dimensional Hilbert space is expressed by the following equation (1). Note that the "^" attached to the symbol indicates that the symbol is a matrix.

(1)

The space in which Lindbladian acts is given as follows.

A finite-dimensional Hilbert space is expressed by the following equation.

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 super operator D[Γ̂ _k ] in the above formula is applied to the mixed state ρ̂(t), it is represented by the following formula (2). In addition, "†" represents a Hermitian conjugate.

(2)

The dissipation with amplitude γ _k in the second term on the right side of the above equation (1) has the role of destroying the coherence of the pure state, and the space on which the corresponding hyperoperator acts is given as follows: be done.

Note that _Γ̂k in the above equation (2) is the k-th jump operator and determines the details of dissipation.

[1.1 Vector representation of the Lindblad equation]

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

(3)

ρ _SS in equation (3) above represents the density matrix of NESS (see, eg, 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 techniques, in the present embodiment, the mixed state density “matrix” ρ in an open quantum system is mapped to a “vector” |ρ> on a space called “operator space”. be. Note that the space in which the density matrix ρ^ acts is given as follows.

Also, the space of vectors |ρ> is defined by the following equation.

Note that the new representation of the state is called a vector representation and is represented by the following equation (4).

(4)

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

Note that the normalization methods in the two representations are different from each other. The matrix representation is normalized so that the diagonal sum is 1, as shown in the following equation. On the other hand, in the vector representation, |ρ> is normalized so that the L2 norm is one.

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

Note that these two subspaces are distinguished by subscripts in the following equations.

，

Note that the elements of these two subspaces are represented by, for example, the following equations.

，

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

(5)


(6)


(7)

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

(8)


(9)


(10)

The vector representation of the Lindbladian L̂ operates on the space represented by the following equation.

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

(11)

Other eigenstates with non-zero eigenvalues generally satisfy the following equations (12)-(13).

(12)


(13)

where |ρ _i > is the vector representation of the i-th eigenmode in eigenvalues satisfying Also, λ _i is the i-th eigenvalue. In this case |ρ _i > means the damping mode.

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

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

(B1)


(B2)

[2 dVQE calculation method]

This embodiment uses a variational method to obtain the steady state of an open quantum system. 2n qubits are used for the optimization of the quantum circuit of this embodiment. On the other hand, n qubits are used to measure physical quantities in the steady state of an open quantum system. Such a change in the number of qubits corresponds to switching from a vector representation of the density matrix to a matrix representation.

A specific description will be given below.

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

The Lindbradian solved by equation (11) above is a non-Hermitian matrix. If there is no dissipation or if the dissipation is parity-time symmetric (see, eg, Ref. 5), the Lindbladian becomes skewed Hermitian. At this time, the eigenvalues of pure imaginary numbers are obtained by diagonalizing the Lindbradian using a unitary matrix. Otherwise, the eigenvalues are neither purely real nor imaginary, and the eigenmodes are not orthogonal. In other words, standard numerical techniques used for diagonalization of the Hamiltonian cannot be applied.

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

Therefore, Schatten norm ||dρ̂(t)/dt|| _trace may be used as the objective function (see Reference 6, for example). However, evaluation of the Schatten norm ||dρ̂(t)/dt|| _trace is not straightforward.

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 Lindbladian Hermitian conjugate L^ ^† and the Lindbladian L^ as shown in the following equation is a Hermitian matrix having non-negative eigenvalues.

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

(14)

Therefore, in the present embodiment, ground state search methods in isolated quantum systems, such as the Lanczos method or the variational approach (see, for example, Reference 7), can be applied to search for NESS in 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.

An 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 limit of the objective function L̂ ^† L̂ is 0, the accuracy of optimization can be quantified without calculating an exact solution.

Therefore, in this embodiment, the above equation (14) is solved using a parameterized quantum circuit. dVQE, which is the calculation method of the present 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 reference 8, for example). VQE is performed entirely by quantum state calculations by the quantum computer 120 and optimized quantum circuit parameters 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.

The processing executed by the quantum computer 120 and the processing executed by the classical computer 110 in the hybrid system 100 of this embodiment are shown below.

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

As will be described later, the quantum circuit of this embodiment is represented by the following equation.

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

The number of objective function terms increases rapidly due to all the couplings between the physical space and the auxiliary space. In this regard, however, qubits can be sampled sufficiently efficiently (see, for example, reference 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 Hypothetical quantum circuit for dVQE]

Next, we describe a variational approach for computing NESS using quantum circuits. As described above, in the present embodiment, the density matrix is expressed as a vector, so that the discussion can proceed in parallel with VQE in an isolated quantum system (see References 9, 11, and 12, for example).

Reference 11: Bela Bauer, Dave Wecker, 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 ρ̂ of an open quantum system must satisfy the following condition.

Therefore, in this embodiment, a quantum circuit that satisfies the above conditions (I), (II), and (III) is designed, and using the quantum circuit, the steady state of the open quantum system and the steady state Calculate physical quantities.

In this embodiment, a first quantum circuit for finding the steady state of an open quantum system and a second quantum circuit for measuring the physical quantity of the measurement target in the steady state are used. Specifically, the hybrid system 100 of the present embodiment uses the first quantum circuit to determine the steady state of the open quantum system, and then uses the second quantum circuit to measure the physical quantity to be measured in the steady state. do.

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

Specifically, the normalization of condition (I) may be taken into consideration in the measurement by the second quantum circuit. Since the first quantum circuit is represented by a vector, the structure of the first quantum circuit is determined so that the following conditions are satisfied when optimizing the parameters of the first quantum circuit. be.

Conditions (I) to (III) above express that the parameterization of the eigenvalues and the basis transformation are separated. FIG. 4 shows an example of the 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 the distribution of eigenvalues and U(φ) transforms the basis. Note that the top half of the qubit represents the physical space and the bottom half of the qubit represents the auxiliary space.

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

(15)


(16)

where D̂(θ) gives the distribution of the eigenvalues. Also, Û(θ) corresponds to a basis transformation. Details of D̂(θ) and Û(θ) will be described later.

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

Note that the initial state can also be represented by the following equation.

A quantum circuit corresponding to the eigenvalue distribution obtained by the mapping shown in the following equation is represented as D ¹ . Here, the eigenvalue distribution D(θ) corresponds to the above equation (16), and the details will be described later.

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

(17)

The unitary operator for basis transformation, Û(φ), is highly dependent on the problem setting, but has a large impact on the accuracy of the solution. As will be described later, for simplicity, we apply a hardware-efficient ansatz (see, for example, references 12, 13, 14) to the hardware of the quantum computer in this embodiment.

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 forexcited states.", arXiv:1810.09434, 2018.

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

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

(18)


(19)

Here, the CNOT gate shown in equation (19) above produces quantum entanglement between the kth qubit and the k+nth qubit. For example, of the states {|00>, |01>, |10>, |11>} represented by the qubits, the first qubit represents the control qubit, and the second qubit represents the control qubit. When representing a qubit, the CNOT gate is represented by equation (20) below.

(20)

It can now be seen that equation (19) above for the distribution of eigenvalues is determined from the unitary gate in the left space. FIG. 5 shows an example of D ^˜ (θ). A rotating Y gate, which is a component of the quantum circuit, is defined by the following equation (21).

(21)

Note that the property of positive semidefinite value is guaranteed by the angle limitation of the following equation.

The quantum circuit D ^~ (θ) of the eigenvalue distribution shown in FIG. 5 treats each qubit independently, while the quantum circuit D ^~ (θ) shown in FIG. 6 introduces quantum entanglement. there is As shown in FIGS. 5 and 6 above, the upper half qubits represent the physical space and the lower half qubits represent the auxiliary space. These figures show the case of n=3.

[Quantum circuit U(φ) for basis transformation]

In the present embodiment, quantum circuits, which are efficient for quantum computer hardware, are used for basis transformations (see, for example, references 12, 13, and 14). FIG. 7 shows an example of U(φ). RX, RY in U(φ) shown in FIG. 7 are repeated D _twice . RX represents a rotating X gate and RY represents a rotating Y gate. Note that U(φ) shown in FIG. 7 depicts a quantum bit in physical space.

[2.3 Measurement of physical quantity to be measured]

After the optimization by the classical computer 110 is completed, the expected value of the physical quantity to be measured is calculated by stochastically setting the initial value of the measurement qubit and performing the measurement. Here, the details of dVQE of this embodiment, which is a calculation method suitable for open quantum systems, will be described.

[2.3.1 General measurement scheme for NESS in quantum circuits]

While the expected value <L ^† L> of the vector-expressed objective function is efficiently measured, the physical quantity cannot be measured using the vector representation. Therefore, the hybrid system 100 of this embodiment stochastically initializes the input n qubits. Specifically, the hybrid system 100 of this embodiment probabilistically initializes the states of n qubits based on the set of eigenvalues {λ _i } of the density matrix.

The measurement of the physical quantity A ̂ to be measured under the density matrix ρ ̂ is represented by the following equations (22) to (23) using matrix representation and vector representation.

(22)


(23)

where the unnormalized vector shown in the following equation corresponds to the vector representation of the density matrix representing the perfect mixture.

Note that the state |Ψ(1)> shown in the above equation is composed of 2 ⁿ nonzero elements. For this reason, ^{the above equation has the form of picking out 2 n elements out of 2 2n elements} in the measurement in the vector representation, and the measurement becomes exponentially inefficient with respect to the size of the system.

In order to avoid this problem, in this embodiment, the physical quantity is measured after converting the representation of the quantum circuit from the vector representation to the matrix representation. Furthermore, in this embodiment, n quantum bits are used after optimizing the parameters θ and φ of D ¹ (θ) and U(φ) in the quantum circuit. An n-bit array is _expressed as q=(q _{1 ,} .

The density matrix ρ given by the optimized quantum circuit is expressed by the following equation.

Also, the expected value <A> of the physical quantity A to be measured is newly defined by the following equation.

(24)


(25)

The normalization shown in is allowed to consider λ _q as the probability for setting the initial state to q.

Therefore, the algorithm of this embodiment is summarized as follows.

(0) Optimal values of parameters θ and φ are obtained by optimization for the first quantum circuit. Note that 2n quantum bits are used by VQE when optimizing the parameters θ and φ.
(1) Set the initial state |q> of the n qubits corresponding to the left space according to the probability λ _q .
(2) Measure the physical quantity A^ to be measured according to |ψ _q > after the basis conversion.
(3) Repeat (1) and (2) until the desired accuracy is achieved.

If the circuit shown in FIG. 5 is employed as the quantum circuit corresponding to the eigenvalue distribution D ¹ , the probability can be expressed by the following equation (26).

(26)

[Operation of hybrid system 100 of the present embodiment]

Next, specific operations of the hybrid system 100 of this embodiment will be described. In each device of hybrid system 100, each process shown in FIG. 8 is executed.

First, in step S<b>100 , the user terminal 130 transmits to the classical computer 110 information about the calculation target input by the user (hereinafter simply referred to as “calculation target information”). The calculation target information is information about a problem to be solved by quantum computation, 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 on the physical quantity to be measured.

The physical quantity to be measured is "A" shown in the above formula (24) and the like, and an example of the physical quantity to be measured is 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 of calculation. Examples of information for calculating the Hamiltonian H include indices of interacting qubits, interaction functional systems, and the like. An example of information for calculating the dissipation operator Γ is identification of relaxation times for phase relaxation terms and energy relaxation terms.

Next, in step S<b>102 , the classical computer 110 receives calculation target information transmitted from the user terminal 130 . Then, in step S102, the classical computer 110 calculates the Hamiltonian H representing the energy state of the system according to a known calculation formula based on the information for calculating the Hamiltonian H among the received calculation target information. This determines the interactions inside the system.

Further, 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 among the received calculation target information. . This determines the interaction of the system with the external environment.

Note that the Hamiltonian H and the dissipation operator Γ differ depending on the calculation target.

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

At step S106, the classical computer 110 calculates an objective function L ^† L corresponding to the Lindbradian L calculated at step S104.

At step S108, the classical computer 110 determines the structure of the first quantum circuit for finding the steady state of the open quantum system as shown in FIG. The structure of the first quantum circuit is a quantum circuit represented by 2n qubits when the number of qubits to be calculated is n qubits, as shown in FIG. Also, the first quantum circuit is a quantum circuit including D ^∼ (θ) representing a matrix for generating the distribution of eigenvalues and U(φ) representing a matrix for performing basis conversion. Therefore, the first quantum circuit includes a parameter θ representing a matrix for generating the distribution of eigenvalues and a matrix parameter φ for performing 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 ρ̂ represents the density matrix of the mixed state in the open quantum system, ρ̂ ^† represents the Hermitian conjugate of ρ̂̂, and |ψ> represents the quantum state. Also, the above (II) holds for any |ψ>.

At step S110, the classical computer 110 outputs the structure of the first quantum circuit determined at 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. do. The optimization method includes, for example, the BFGS method.

In step S106, the data transmitted from the classical computer 110 to the quantum computer 120 is a data structure relating to the configuration of the quantum circuit used for the quantum computation of the quantum computer 120, and is information for obtaining the steady state of the open quantum system. It contains information representing a matrix for generating a distribution of eigenvalues and information representing a matrix for performing basis transformation.

This data structure is a data structure used for the quantum computation of the quantum computer 120, and the quantum computer 120 using VQE based on the first quantum circuit and the initial values of the parameters θ and φ of the first quantum circuit Quantum computation is used to generate the 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 for irradiating at least one of the quantum bits in the quantum bit group 123 under the control of the control device 121 . Then, the quantum computer 120 irradiates at least one of the quantum bits in the quantum bit group 123 with the generated electromagnetic wave, and executes a quantum circuit according to the initial information, thereby obtaining the optimum quantum circuit. Generate the parameters θ ^' and φ ^' of the quantum circuit of 1. A 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 applied to the quantum bit group 123 by the electromagnetic wave generator 122 . Quantum computer 120 then outputs the parameters θ ^' and φ ^' of the first quantum circuit.

At step S114, the controller 121 transmits the parameters θ ^′ and φ ^′ of the first quantum circuit obtained at step S108 to the classical computer 110. FIG.

At step S116, the classical computer 110 receives the parameters θ ^' , φ ^' of the first quantum circuit transmitted from the control device 121 at step S110. Classical computer 110 then determines information about the quantum measurements according to the parameters θ ^' and φ ^' of the first quantum circuit.

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

First, the classical computer 110 sets up a second quantum circuit for obtaining the physical quantity to be measured in the steady state. The second quantum circuit is a measurement quantum circuit for measuring physical quantities, 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, the first quantum circuit obtained by optimization cannot be used for measurement, unlike the usual problem setting. Therefore, in this embodiment, the unitary circuit U(φ ^′ ) of the first quantum circuits is set as the second quantum circuit for measurement.

Note that 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 values of the quantum bits for measurement according to the parameters θ 1 ^′ of the first quantum circuit output from the quantum computer 120 .

Specifically, when the classical computer 110 sets the initial value of the quantum bit for ^measurement , the probability λ _q ( θ ^′ ). The classical computer 110 then sets the initial value of the measurement qubit q according to the calculated probability λ _q (θ ^′ ). For example, the probability λ _qA (θ ^′ ) of the measurement qubit q _A , the probability λ qB (θ ^′ ) of the measurement qubit q _B , and the probability λ _qC (θ′ ⁾ of the measurement qubit _{q C} ) are calculated, and one of the qubits q _A , q _B , and q _C is set as an initial value according to the probability.

The information representing the physical quantity to be measured, among the information relating to quantum measurement, has already been obtained in step S100.

In step S118, the classical computer 110 retrieves information representing the physical quantity to be measured, which is information about the quantum measurement determined in step S116, a second quantum circuit that is a quantum circuit for measurement, and a quantum circuit for measurement. and the initial values of the bits to the quantum computer 120 .

In step S120, the control device 121 receives the information representing the physical quantity to be measured, the second quantum circuit that is the quantum circuit for measurement, and the quantum bit for measurement that are transmitted from the classical computer 110 in step S118. receive initial values. Then, the control device 121 causes the quantum computer 120 to perform the 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. Let

Specifically, the quantum computer 120 generates electromagnetic waves for irradiating at least one of the quantum bits in the quantum bit group 123 under the control of the control device 121 . Quantum computer 120 measures information observed by executing the second quantum circuit by irradiating at least one quantum bit of quantum bit group 123 with the generated electromagnetic wave. As a result, the measurement result of the physical quantity A received in step S102 is obtained. The quantum computer 120 then outputs the measurement result of the physical quantity A obtained by quantum computation.

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

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

Each process of steps S116 to S122 is repeated until the calculation result of the quantum computation by the quantum computer 120 converges. For example, the initial value of the quantum bit for measurement by the classical computer 110 is set by the classical computer 110 and the initial value of the quantum bit for measurement by the quantum computer 120 is set until the variation of the physical quantity A obtained for each iteration of the quantum calculation becomes equal to or less than a predetermined threshold. Execution of quantum computation according to the value is repeated. In this case, in each iteration, the average value of the physical quantity A obtained so far is calculated, and the processes of steps S116 to S122 are repeated until the variation in the average value becomes equal to or less than a predetermined threshold.

At step S124, the classical computer 110 acquires each of the physical quantities A obtained by each iterative process of steps S116 to S122. Then, the classical computer 110 statistically processes the physical quantity A obtained each time to calculate the expected value <A> of the physical quantity A given by the above equation (24) or the like.

In step S<b>126 , the classical computer 110 transmits the calculation result of the expected value <A> of the physical quantity obtained in step S<b>124 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 transmitted from the classical computer 110 in step S124.

As described above, in the hybrid system of this embodiment, a classical computer calculates the Lindbladian based on the Hamiltonian and the dissipation operator. Then, the classical computer calculates an objective function according to the Lindbladian, and determines the structure of the first quantum circuit so as to satisfy the condition represented by the following equation (1). The classical computer then 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 computation using VQE in accordance with the objective function, the structure of the first quantum circuit, and the initial values of the parameters, which are output from the classical computer, and obtains the parameters of the first quantum circuit and output 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 that is the quantum circuit for measurement, and the initial value of the quantum bit for measurement. Then, the quantum computer executes quantum computation according to the information representing the physical quantity to be measured, the second quantum circuit, and the initial value of the quantum bit for measurement, which are output from the classical computer, and perform the quantum computation. Output the measurement results. Then, the classical computer outputs the calculation result of the physical quantity to be measured according to the measurement result output from the quantum computer. This makes it possible to calculate the steady state of the physical quantity of interest in an open quantum system.

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

Further, in the hybrid system of the present embodiment, the first quantum circuit for obtaining the steady state of the open quantum system and the second quantum circuit for measuring the physical quantity to be measured in the steady state are different quantum circuits. Then, the steady state of the open quantum system and the physical quantity to be measured in the steady state can be obtained.

Further, in the hybrid system of the present embodiment, when optimizing the parameters of the first quantum circuit, the expression of the quantum circuit is expressed as a vector expression, and when the physical quantity is measured by the second quantum circuit, the quantum circuit is expressed as a vector expression. Let the representation of the circuit be a matrix representation. Thereby, the steady state of the open quantum system and the physical quantity to be measured in the steady state can be obtained.

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

First, a model for executing dVQE will be described. First, when the above equation (10) is applied to the one-dimensional transverse magnetic field Ising model, the NESS of the one-dimensional transverse magnetic field Ising model is expressed by the following equations (27) and (28).

(27)


(28)

It should be noted that the state when the objective function L ^† L is optimized by dVQE can be sequentially confirmed by the BFGS method (see reference 15, for example). The BFGS method is for example performed 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 scientific tools for Python", 2001.

First, the NESS of the one-dimensional transverse magnetic field Ising model (1dTFIM) with dissipation is calculated. The dissipative one-dimensional transverse field Ising model Lindbladian L̂ is given by equation (29) below.

(29)

Note that the dissipative one-dimensional transverse Ising model NESS deviates slightly from the simple all-down state due to correlations induced by nearest-neighbour interactions.

FIG. 9 shows the results of this example. FIG. 9A shows the density matrix obtained by dVQE of this embodiment. Also, FIG. 9B shows a 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 i-th spin. represents the annihilation operator of angular momentum acting on Also, the dissipation intensities γ _i are all constant and γ _i =1.

In FIGS. 9A and 9B, the left side represents the calculation result of the density matrix of the real part, and the right side represents the calculation result of the imaginary part of the density matrix. As shown in FIG. 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.

FIG. 10 shows the value of the objective function after repeated optimization. The fidelity F as a vector representation is represented by Equation (30) below.

(30)

where |ρ _ED > is the vector representation of the density matrix obtained by strict diagonalization. Also, |ρ _dVQE > is the vector representation of the density matrix obtained by dVQE.

As shown in FIG. 10, it can be seen that the value of the objective function decreases as the optimization progresses, and the fidelity F increases.

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

For example, in the above embodiments, information may be transmitted and received between the classical computer 110 and the quantum computer 120 in any way. For example, the transmission and reception of quantum circuit parameters and the transmission and reception of measurement results between the classical computer 110 and the quantum computer 120 may be performed sequentially each time a predetermined calculation is completed, or all calculations may be performed. Sending and receiving may occur after completion.

In the above embodiment, until the calculation result of the quantum computation by the quantum computer 120 converges, according to the setting of the initial value of the quantum bit for measurement by the classical computer 110 and the initial value of the quantum bit for measurement by the quantum computer 120 Although the case where the execution of the quantum calculation is repeated has been described as an example, the present invention is not limited to this. For example, the classical computer 110 sets a plurality of initial value sets of the measurement qubits at once, and the quantum computer 120 sequentially performs quantum calculations according to the plurality of initial values of the measurement qubits. You may do so. Alternatively, each of the plurality of initial value sets of the qubits for measurement is transmitted to the plurality of quantum computers 120, and each of the plurality of quantum computers 120 performs quantum calculation according to the received initial values, and the classical computer 110 may receive each calculation result from a plurality of quantum computers 120 and calculate an expected value.

Also, in the above embodiment, the case where the objective function is <L ^† L> has been described as an example, but it is not limited to this.

Further, in the above embodiment, as an example of the second quantum circuit U(φ), the case of using a quantum circuit that is efficient for the hardware of a quantum computer, such as that shown in FIG. 7, has been described as an example. is not limited to For example, other types of quantum circuits such as unitary coupled cluster quantum circuits may be used as the second quantum circuit U(φ).

In the above embodiment, the case where the probability of setting the initial value is obtained by the above equation (29) has been described as an example, but the probability is not limited to this, and the probability can be set using other mathematical expressions. You may do so.

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

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

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

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

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

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

Further, in the specification of the present application, an embodiment in which the program is pre-installed has been described, but it is also possible to store the program in a computer-readable recording medium and provide it.

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 generation 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,
the classical computer computes the Lindbladian based on the Hamiltonian and the dissipation operator;
The classical computer calculates an objective function according to the Lindbladian,
The classical computer determines the structure of a first quantum circuit for determining the steady state of an open quantum system;
the classical computer outputs the objective function, the structure of the first quantum circuit, and initial values of the parameters of the first quantum circuit;
The quantum computer performs quantum calculation using a VQE (Variational-Quantum-Eigensolver) according to the objective function, the structure of the first quantum circuit, and the initial values of the parameters, which are output from the classical computer. Execute, generate parameters for the first quantum circuit, output parameters for the first quantum circuit,
The classical computer sets an initial value of a qubit for measurement according to the parameters of the first quantum circuit output from the quantum computer,
The classical computer outputs information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement,
The quantum computer performs quantum calculation according to the information representing the physical quantity to be measured, the second quantum circuit, and the initial value of the measurement qubit, which are output from the classical computer, Outputting the measurement result of the quantum computation,
The classical computer outputs a calculation result of the physical quantity to be measured according to the measurement result output from the quantum computer;
Quantum information processing methods for open quantum systems involving 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 the conditions (II) and (III) expressed in the following formula (A): decide and
When outputting the second quantum circuit, the classical computer outputs the second quantum circuit that satisfies conditions (I), conditions (II), and conditions (III) expressed in the following formula (A): do,
A quantum information processing method for an open quantum system according to claim 1.







(A)
where ρ̂ represents the density matrix of the mixed state in the open quantum system, ρ̂ ^† represents the Hermitian conjugate of ρ̂̂, and |ψ> represents the quantum state. Also, 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 conversion,
3. A quantum information processing method for an open quantum system according to claim 1 or claim 2. The parameters of the first quantum circuit include a parameter θ for controlling the distribution of eigenvalues and a matrix parameter φ for performing basis conversion.
A quantum information processing method for an open quantum system according to any one of claims 1 to 3. The classical computer calculates the probability λ _q (θ ^' ) according to the parameter θ ^' generated by the quantum computer when setting the initial value of the measurement qubit, and the calculated probability λ _q setting the initial value of the measurement qubit according to (θ ^' );
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 the parameter φ ^' generated by the quantum computer is reflected.
A quantum information processing method for an open quantum system according to claim 4 or 5. Setting the initial value of the measurement qubit by the classical computer and performing quantum computation according to the initial value of the measurement qubit by the quantum computer until the calculation result of the quantum computation by the quantum computer converges Execute and repeat
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. the classical computer
Compute the Lindbladian based on the Hamiltonian and the dissipation operator,
calculating an objective function according to the Lindbladian;
Determining the structure of the first quantum circuit for obtaining the steady state of the open quantum system,
outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit;
parameters of the first quantum circuit generated by quantum calculation using a Variational-Quantum-Eigensolver (VQE) based on the structure of the first quantum circuit and the initial values of the parameters, which are output from a quantum computer; set the initial value of the qubit for the measurement, depending on
Outputting information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement,
Measurement of quantum computation according to the information output from the quantum computer representing the physical quantity to be measured, the second quantum circuit that is the quantum circuit for measurement, and the initial value of the quantum bit for measurement Outputting a calculation result of the physical quantity to be measured according to the result;
Quantum information processing methods for open quantum systems that perform processing. quantum computer
An objective function corresponding to a Lindbradian calculated based on a Hamiltonian and a dissipation operator output from a classical computer, a structure of a first quantum circuit for obtaining a steady state of an open quantum system, and the first performing quantum computation using a Variational-Quantum-Eigensolver (VQE) according to the initial values of the parameters of the quantum circuit of and generating the parameters of the first quantum circuit; and
Information representing the physical quantity to be measured output from the classical computer, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and the parameters of the first quantum circuit. performing quantum calculation according to the initial value of the quantum bit for measurement and outputting the measurement result of the quantum calculation;
Quantum information processing methods for open quantum systems that perform processing. Compute the Lindbladian based on the Hamiltonian and the dissipation operator,
calculating an objective function according to the Lindbladian;
Determining the structure of the first quantum circuit for obtaining the steady state of the open quantum system,
outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit;
parameters of the first quantum circuit generated by quantum calculation using a Variational-Quantum-Eigensolver (VQE) based on the structure of the first quantum circuit and the initial values of the parameters, which are output from a quantum computer; set the initial value of the qubit for the measurement, depending on
Outputting information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement,
Measurement of quantum computation according to the information output from the quantum computer representing the physical quantity to be measured, the second quantum circuit that is the quantum circuit for measurement, and the initial value of the quantum bit for measurement Outputting a calculation result of the physical quantity to be measured according to the result;
A classical computer that performs processing. An objective function corresponding to a Lindbradian calculated based on a Hamiltonian and a dissipation operator output from a classical computer, a structure of a first quantum circuit for obtaining a steady state of an open quantum system, and the first performing quantum computation using a Variational-Quantum-Eigensolver (VQE) according to the initial values of the parameters of the quantum circuit of and generating the parameters of the first quantum circuit; and
Information representing the physical quantity to be measured output from the classical computer, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and the parameters of the first quantum circuit. performing quantum calculation according to the initial value of the quantum bit for measurement and outputting the measurement result of the quantum calculation;
A quantum computer that performs processing. Compute the Lindbladian based on the Hamiltonian and the dissipation operator,
calculating an objective function according to the Lindbladian;
Determining the structure of the first quantum circuit for obtaining the steady state of the open quantum system,
outputting the structure of the first quantum circuit and initial values of the parameters of the first quantum circuit;
parameters of the first quantum circuit generated by quantum calculation using a Variational-Quantum-Eigensolver (VQE) based on the structure of the first quantum circuit and the initial values of the parameters, which are output from a quantum computer; set the initial value of the qubit for the measurement, depending on
Outputting information representing a physical quantity to be measured, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and an initial value of the qubit for measurement,
Measurement of quantum computation according to the information output from the quantum computer representing the physical quantity to be measured, the second quantum circuit that is the quantum circuit for measurement, and the initial value of the quantum bit for measurement Outputting a calculation result of the physical quantity to be measured according to the result;
A quantum information processing program for making a classical computer perform processing. An objective function corresponding to a Lindbradian calculated based on a Hamiltonian and a dissipation operator output from a classical computer, a structure of a first quantum circuit for obtaining a steady state of an open quantum system, and the first performing quantum computation using a Variational-Quantum-Eigensolver (VQE) according to the initial values of the parameters of the quantum circuit of and generating the parameters of the first quantum circuit; and
Information representing the physical quantity to be measured output from the classical computer, a second quantum circuit for measuring the physical quantity to be measured in a steady state, and the parameters of the first quantum circuit. performing quantum calculation according to the initial value of the quantum bit for measurement and outputting the measurement result of the quantum calculation;
A quantum information processing program for making a quantum computer execute processing.