JP7556546B2 - Quantum information processing method, classical computer, quantum computer, hybrid system, and quantum information processing program for determining energy differential or nonadiabatic coupling - Google Patents

Description

The disclosed technology relates to a quantum information processing method, a classical computer, a quantum computer, a hybrid system, and a quantum information processing program for determining energy derivatives or nonadiabatic coupling.

Conventionally, in the field of quantum chemistry, a method called State-Averaged Orbital-Optimization (hereinafter simply referred to as "SA-OO") is known (see, for example, Non-Patent Document 1). By performing SA-OO calculations, it is possible to obtain multiple energy eigenstates of electrons in a molecule. Non-Patent Document 1 discloses a method for performing SA-OO calculations using a quantum computer.

S. Yalouz, B. Senjean, J. Gunther, F. Buda, T. E. O'Brien, and L. Visscher, A state-averaged orbital-optimized hybrid quantum-classical algorithm for a democratic description of ground and excited states, (2021 ), Quantum Science and Technology, Volume 6, Number 2

When SA-OO calculations are performed using a quantum computer, it is possible to obtain energies corresponding to the ground state and excited states. However, in the field of quantum chemistry, it is preferable to obtain not only the energy value itself but also the energy derivative value. Furthermore, non-adiabatic bonds are known as one of the important physical quantities when analyzing the properties of molecules or substances. In the field of quantum chemistry, it is preferable to obtain the value of this non-adiabatic bond as well. In contrast, the above-mentioned Non-Patent Document 1 does not take into account the derivation of energy derivatives and non-adiabatic bonds.

The disclosed technology has been made in consideration of the above circumstances, and aims to provide a quantum information processing method, a classical computer, a quantum computer, a hybrid system, and a quantum information processing program that can obtain the energy differential value or the value of a nonadiabatic coupling even when performing SA-OO calculations using a quantum computer.

In order to achieve the above object, a quantum information processing method according to a 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 Hamiltonian H^(x) having, as a variable, a position parameter x representing a position of an atomic nucleus of a molecule, determines an orbital rotation operator U^ _OO (κ) which is an operator having, as a variable, an orbital parameter κ which is a parameter related to a molecular orbital and which represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having, as a variable, a circuit parameter θ which is a parameter of a quantum circuit, generates a Hamiltonian H^(x, κ) by performing orbital rotation and projection onto an active space for the Hamiltonian H^(x), sets a first objective function having the circuit parameter θ as a variable, and by iterative calculations between the classical computer and the quantum computer, the classical computer calculates the circuit parameter θ ^* of the quantum circuit U^(θ) such that the first objective function is minimized or maximized based on a measurement result of a quantum measurement by the quantum computer, and the classical computer calculates the circuit parameter θ a classical computer calculates the orbital parameter κ ^* so as to minimize or maximize the second objective function based on a measurement result of a quantum measurement by a quantum computer; the quantum computer acquires a measurement result of the quantum measurement by quantum measuring an expectation value including a derivative of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^{* )} with respect to the position parameter ^x based on the circuit parameter θ* and the orbital parameter κ ^* ; and the classical computer calculates a derivative of the energy corresponding to the Hamiltonian H^(x) based on the measurement result of the quantum measurement.

A quantum information processing method according to a second 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 Hamiltonian H^(x) having, as a variable, a position parameter x representing a position of an atomic nucleus of a molecule, determines an orbital rotation operator U^ _OO (κ) which is an operator having, as a variable, an orbital parameter κ which is a parameter related to a molecular orbital and which represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having, as a variable, a circuit parameter θ which is a parameter of a quantum circuit, generates a Hamiltonian H^(x, κ) by performing orbital rotation and projection onto an active space for the Hamiltonian H^(x), sets a first objective function having the circuit parameter θ as a variable, and by iterative calculations between the classical computer and the quantum computer, the classical computer calculates, based on a measurement result of a quantum measurement by the quantum computer, the circuit parameter θ* of the quantum circuit U^(θ) such that the quantum computer minimizes or maximizes the first objective function, and the classical computer calculates a circuit parameter θ ^* of the quantum circuit U^(θ) such that the quantum computer minimizes or maximizes the first objective function, a classical computer sets a second objective function, which is an objective function having ^θ* as a constant and the orbital parameter κ as a variable, based on a measurement result of a quantum measurement by a quantum computer, the quantum computer calculates the orbital parameter κ ^* such that the second objective function is minimized or maximized, the quantum computer performs quantum measurement of expectation values including derivatives of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO (κ ^* ) with respect to the position parameter ^x based on the circuit parameter θ ^* and the orbital parameter κ* to obtain a measurement result of the quantum measurement, and the classical computer calculates a value of the nonadiabatic coupling based on the measurement result of the quantum measurement.

The disclosed technology has the advantage that it is possible to obtain the energy differential value when performing SA-OO calculations using a quantum computer. In addition, the disclosed technology has the advantage that it is possible to obtain the value of nonadiabatic coupling when performing SA-OO calculations using a quantum computer.

1 is a diagram illustrating an example of a schematic configuration of a hybrid system 100 according to an embodiment of the present invention. FIG. 1 is a schematic block diagram of a classical computer 110, a control device 121, and a computer functioning as a user terminal 130. FIG. 2 is a diagram for explaining an algorithm according to the present embodiment. FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment. FIG. 2 is a diagram showing an example of a sequence executed by the hybrid system 100 of the present embodiment.

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

<Hybrid system 100 according to the embodiment>

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

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

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

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

In the example of FIG. 1, the quantum computer 120 includes a control device 121 that communicates with the classical computer 110, an electromagnetic wave generating device 122 that generates electromagnetic waves in response to a request from the control device 121, and a group of quantum bits 123 that receives electromagnetic wave irradiation from the electromagnetic wave generating device 122. The electromagnetic wave generating device 122 and the group of quantum bits 123 of the quantum computer 120 are also QPUs (Quantum Processing Units). Note that in this embodiment, "quantum computer" does not mean that no operations are performed using classical bits, but refers to a computer that includes operations using quantum bits.

The control device 121 is a classical computer that performs calculations using classical bits, and instead performs some or all of the processing described in this specification as being performed by the classical computer 110. For example, the control device 121 may store or determine a quantum circuit in advance, and generate quantum gate information for executing the quantum circuit U^(θ) in the quantum bit group 123 in response to receiving a parameter θ of the quantum circuit U^(θ).

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

The classical computer 110, the control device 121, and the user terminal 130 can be realized, for example, by a computer 50 shown in FIG. 2. The computer 50 includes a central processing unit (CPU) 51, a memory 52 as a temporary storage area, and a non-volatile storage unit 53. The computer 50 also includes an input/output interface (I/F) 54 to which external devices and output devices are connected, and a read/write (R/W) unit 55 that controls the reading and writing of data from and to a recording medium. The computer 50 also includes a network I/F 56 that is connected to a network such as the Internet. The CPU 51, the memory 52, the storage unit 53, the input/output I/F 54, the R/W unit 55, and the network I/F 56 are connected to one another via a bus 57.

The hybrid system 100 of the embodiment calculates the first derivative of the energy, the second derivative of the energy, or the nonadiabatic coupling value. The following describes the prerequisites.

(Explanation of Hamiltonian)
The Hamiltonian, which represents the electronic state of a molecule, depends on the position parameter x, which represents the position coordinates of the atomic nuclei. The position parameter x is a vector. For a method of calculating the Hamiltonian on a classical computer, see, for example, the following references.

Reference 1: S. McArdle, S. Endo, A. Aspuru-Guzik, SC Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
Reference 2: Y. Cao, J. Romero, JP Olson, M. Degroote, PD Johnson, M. Kieferova, ID Kivlichan, T. Menke, B. Peropadre, NPD Sawaya, S. Sim, L. Veis, and A Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chemical Reviews 119, 10856 (2019).

The Hamiltonian is expressed by the following equation (1).

In addition, in a mathematical formula, a character with a "-" above a symbol (e.g., X) may be hereinafter represented as X- ^, etc. In addition, in a mathematical formula, a character with a "^" above a symbol (e.g., X) may be hereinafter represented as X^.

In the above formula (1), E _c (x) is a scalar value that depends on the location parameter x.

is the creation and annihilation operator for the spin σ in the i-th molecular orbital. This creation and annihilation operator satisfies the following anticommutation relation of fermions:

is the anticommutator. δ is the Kronecker delta.

h _ij (x) is called the one-electron integral, and g _ijkl (x) is called the two-electron integral, where i, j, k, and l are indices. If the position parameter x is the coordinate of the nucleus of the molecule, it is defined as follows:

Here, φ _i (r; x) is the wave function of the electron in the i th molecular orbital, r is the position coordinate of the electron, the coordinate of the I th nucleus is R _I and its charge is Z _I.

When the number of molecular orbitals is N, the size of the Hamiltonian expressed by a matrix is O(e ^N ), which makes it difficult for a classical computer to calculate the Hamiltonian of a large molecule.

On the other hand, when the number of molecular orbitals is N, a quantum computer can handle the Hamiltonian with 2N quantum bits. Therefore, quantum computers have an advantage over classical computers when performing calculations related to molecular orbitals using Hamiltonians.

[State-averaged orbital optimization]
(Active space approximation)
When performing calculations related to molecular orbitals using a Hamiltonian, quantum computers have an advantage over classical computers. However, the computational capabilities of quantum computers at present are still low. Therefore, in this embodiment, the approximation of the active space (Hereinafter, this will be referred to as the active space approximation.) The active space approximation is also introduced in quantum chemical calculations using classical computers.

In the active space approximation, orbitals with high energy (also called virtual orbitals) are considered to be unoccupied, while orbitals with low energy (also called occupied or core orbitals) are considered to be doubly occupied. Since electron superposition is only considered for orbitals in the active space, the number of qubits used in the calculation can be reduced. Mathematically, the wave function in the active space approximation is expressed by the following equation:

is the state corresponding to the virtual orbital. |ψ> is the wave function in the active space and represents the state.

is the state corresponding to an occupied orbital.

The operator P, which performs the projection of the Hamiltonian onto the active space, is expressed by the following equation using an arbitrary state |ψ〉 in the active space.

(Orbital optimization and state average)
In order to improve the accuracy of the active space approximation, optimization of molecular orbitals, called "orbital optimization", may be used. Orbital optimization is performed by using the following unitary operator U^ _OO ( κ) is formulated as follows:

Here, κ is a parameter related to a molecular orbital. Hereinafter, κ will be referred to as an orbital parameter. Note that κ _pq represents elements of the orbital parameter κ, which is a matrix. p and q are indexes representing a molecular orbital.

U^ _OO (κ) is an operator for converting the Hamiltonian H^(x) into U _oo ^ ^† (κ)H^(x)U^ _OO (κ), and is an operator representing the rotation of a molecular orbital having the orbital parameter κ as a variable. Hereinafter, U^ _OO (κ) will be referred to as the orbital rotation operator.

Here, the Hamiltonian H^(x, κ) rotated by the orbital rotation operator U^ _OO (κ) and mapped to the active space is defined by the following equation: where P is the projection operator onto the active space.

In orbital optimization in quantum chemical calculations, an orbital parameter κ is calculated such that the energy corresponding to the Hamiltonian H^(x, κ) in the active space is minimized. In this embodiment, optimization of the state average energy of the Hamiltonian H^(x, κ) in the active space is considered. Specifically, in this embodiment, the state average energy is defined as follows:

Here, the following equations are the K eigenenergies of the Hamiltonian H^(x, κ) in the active space:

The K eigenenergies are the K lowest eigenenergies selected from the energies corresponding to the Hamiltonian H^(x, κ) in the active space.

The following equation is also a positive weighting coefficient:

The positive weight coefficients in the above formula satisfy the following formula:

In this embodiment, the lowest K eigenenergies among the energies corresponding to the Hamiltonian H^(x, κ) are selected as the set of eigenenergies {E _S }, but a set of eigenenergies {E _S } other than the lowest K eigenenergies may also be selected.

In the following references, orbital optimization calculations are performed in combination with a Variational Quantum Eigensolver (VQE).

Reference 3: W. Mizukami, K. Mitarai, YO Nakagawa, T. Yamamoto, T. Yan, and Y.-y. Ohnishi, Orbital optimized unitary coupled cluster theory for quantum computer, Phys. Rev. Research 2, 033421 ( 2020).
Reference 4: T. Takeshita, NC Rubin, Z. Jiang, E. Lee, R. Babbush, and JR McClean, Increasing the representation accuracy of quantum simulations of chemistry without extra quantum resources, Phys. Rev. 2020).

VQE is an algorithm for obtaining the ground state of the Hamiltonian H^(x). Note that the following reference also proposes an orbital optimization calculation for state averaging. On the other hand, subspace search VQE (SSVQE) disclosed in the above non-patent document 1 and the following reference calculates the ground state and excited states.

Reference 5: K. M. Nakanishi, K. Mitarai, and K. Fujii, Subspacesearch variational quantum eigensolver for excited states, Phys. Rev. Research 1, 033062 (2019).

Therefore, in this embodiment, three methods are adopted as algorithms for calculating the eigenstates of the Hamiltonian H^(x, κ). Specifically, in this embodiment, the eigenstates of the Hamiltonian H^(x, κ) are calculated using SSVQE, multistate-contracted VQE (MCVQE), and variational quantum deflation (VQD). MCVQE and VQD are disclosed in the following references.

Reference 6: RM Parrish, EG Hohenstein, PL McMahon, and TJ Martinez, Quantum computation of electronic transitions using a variational quantum eigensolver, Phys. Rev. Lett. 122, 230401 (2019).
Reference 7: O. Higgott, D. Wang, and S. Brierley, Variational Quantum Computation of Excited States, Quantum 3, 156 (2019).

In addition, in this embodiment, the methods that combine the orbital optimization calculation at state average energy (SA-OO) with each of the three algorithms mentioned above are hereinafter referred to as SA-OO-SSVQE, SA-OO-MCVQE, and SA-OO-VQD. Note that SA-OO-VQE disclosed in the above non-patent document 1 corresponds to SA-OO-SSVQE in this embodiment.

Figure 3 shows a diagram for explaining the algorithm of this embodiment. As shown in Figure 3, the above three SA-OO algorithms consist of two steps. Specifically, for the Hamiltonian H^(x, κ) in the active space, orbital optimization as shown on the left side of Figure 3 and wave function optimization as shown on the right side of Figure 3 are repeated. These processes are repeated until the state average energy converges.

More specifically, in the first step, optimization of the wave function (or the state of the quantum circuit) is performed to obtain the eigenstates of the Hamiltonian H^(x, κ) in the active space. In the second step, the orbital parameter κ is updated to reduce the state average energy. Note that in the second step, the density matrix of the wave function optimized in the first step is used. The three algorithms are explained in detail below.

(SA-OO-SSVQE)
First, SA-OO-SSVQE will be described. In SA-OO-SSVQE, the following eigenstates that are orthogonal to each other are set.

Note that S is an index for identifying an eigenstate, and there are K eigenstates.

A quantum circuit U^(θ) is set, which is a unitary operator with a circuit parameter θ. In SA-OO-SSVQE, the circuit parameter θ is optimized so that the following loss function is minimized:

Here, w _s ^VQE is a weighting coefficient that satisfies the following equation.

When the optimal circuit parameters θ ^* that minimize the loss function are used, the K eigenstates are expressed by the following equation.

The specific energy is expressed by the following formula:

It should be noted that the weighting factor w _s ^VQE in SSVQE is not related to the weighting factor w _s ^SA of the state average energy.

In this embodiment, when optimizing the wave function in the SA-OO-SSVQE, a circuit parameter θ ^* is calculated such that the loss function E ^SSVQE of the SSVQE shown in the following equation is minimized.

In equation (9), the Hamiltonian H in equation (8) is replaced with the Hamiltonian H^(x, κ) in the active space. In equation (9), the circuit parameter θ is optimized based on the ^ESSVQE with the orbital parameter κ fixed. Then, the eigenstates and eigenenergies of the Hamiltonian H^(x, κ) are obtained based on the quantum circuit U^(θ ^* ) having the optimal circuit parameter θ ^* . Then, the optimization calculation of the orbital parameters described later is performed.

The state average energy E ^SA used in optimizing the orbital parameters is defined by the following equation: The orbital parameters κ ^* are calculated so that the state average energy E ^SA below is minimized.

In the above formula, θ is the circuit parameter θ ^* optimized by the above-mentioned process. Updating of the circuit parameter θ and updating of the trajectory parameter κ are repeated until the state average energy E ^SA converges.

(SA-OO-VQD)
Next, SA-OO-VQD will be described. VQD is an algorithm for obtaining an excited state of a Hamiltonian. First, a quantum circuit for obtaining an eigenstate is set as shown in the following formula.

Here, |ψ〉 is the initial state, and U^(θ) is the quantum circuit. Suppose we have obtained S-1 eigenstates as shown in the following equation.

In this case, the eigenstate of the S-th Hamiltonian represented by the circuit parameter θ ^* such that the value of the loss function of the VQD for obtaining the S-th eigenstate is minimized is expressed by the following equation.

β _T is a positive constant for ensuring the orthogonality of |ψ(θ _S )>. The S-th eigenstate represented by the circuit parameter θ ^* that minimizes the loss function F _S (θ _S ) is expressed by the following equation.

Optimization of the loss function F 2 _S is performed in order starting from S=0, and the K-th loss function F 2 _S is optimized by repeated calculations.

In the SA-OO-VQD of this embodiment, the S-th circuit parameter θ _S ^* for calculating the eigenstate of the Hamiltonian H^(x,κ) in the active space is obtained by executing VQD. Specifically, the circuit parameters θ ₀ ^* ,...,θ _K-1 ^* are obtained by minimizing F _S ^VQD (θ _S ) shown below for S=0,...,K-1.

When obtaining the circuit parameters θ ₀ ^* , . . . , θ _K-1 ^* by minimizing the above equation (12), the value of the trajectory parameter κ is fixed and then the values of the circuit parameters are calculated.

After the circuit parameters θ ₀ , . . . , θ _K−1 are optimized and the eigenstates are calculated using these circuit parameters, the trajectory parameter κ is calculated so that the state average energy E ^SA in the above equation (10) is minimized. Specifically, the update of the circuit parameter θ and the trajectory parameter κ are repeated until the state average energy E ^SA converges.

(SA-OO-MCVQE)
Next, SA-OO-MCVQE will be described. MCVQE consists of two steps. In the first step, the circuit parameter θ ^* is calculated so that E ^MCVQE (θ) in the following equation is minimized. will be done.

In the second step, the Hamiltonian H is diagonalized using a classical computer. Specifically, the Hamiltonian in the subspace S represented by the following equation is diagonalized.

The Hamiltonian in the subspace is expressed as a K × K matrix. The element h _ST of the Hamiltonian is expressed by the following equation.

The element h _ST of the Hamiltonian can be calculated by a quantum computer if the following superposition state is used.

Moreover, the A-th eigenvector v _S ^(A) obtained from the matrix h representing the Hamiltonian satisfies the following formula.

In addition, E _A in the above formula (15) is also an eigenvalue corresponding to the eigenvector v _S ^(A) . The above formula (15) corresponds to the A-th excited state |Ψ _A > of the Hamiltonian H being expressed by the following formula.

In the SA-OO-MCVQE of this embodiment, the following loss function E ^MCVQE (θ) including the Hamiltonian H^(x, κ) in the activity space of the above formula (6) is defined.

In SA-OO-MCVQE, first, the circuit parameter θ ^* is calculated so that the loss function E ^MCVQE (θ) in the above formula (13) is minimized. Then, the matrix representing the Hamiltonian is diagonalized by a classical computer. As a result, the eigenstate of the Hamiltonian H shown in the above formula (16) and the eigenenergy corresponding to the eigenstate are obtained. Then, the trajectory parameter κ is updated so that the state average energy E ^SA shown in the following formula (10A) is minimized. Then, the update of the circuit parameter θ and the trajectory parameter κ are repeated until the state average energy E ^SA converges.

(10A)

(Regarding the differentiation of energy)
By executing SA-OO-SSVQE, SA-OO-VQD, and SA-OO-MCVQE, the eigenenergy corresponding to the molecular position parameter x is obtained. Here, consider the analytical first derivative of the eigenenergy with respect to x _μ , where x _μ is the μ-th element of the vector position parameter x.

In this embodiment, the analytical first derivative of the eigenenergy is calculated based on the Lagrange multiplier method. Note that there is a slight difference between SA-OO-SSVQE, SA-OO-VQD, and SA-OO-MCVQE.

(Step 0)
First, SA-OO-SSVQE, SA-OO-VQD, or SA-OO-MCVQE is executed. This allows obtaining the optimal trajectory parameters κ ^* and optimal circuit parameters θ ^* that minimize various loss functions. When SA-OO-MCVQE is executed, the vector v ^(A) is also obtained.

(Step 1)
First, the Lagrangian

In addition, extreme value conditions are set for parameters other than the location parameter x. Note that θ ^−A and κ ^−A are undetermined Lagrange multipliers.

(Step 2)
Calculate the Lagrange multipliers by solving the following equation:

Where:

is the Hessian matrix of the Lagrangian L _A. Also, f ^A is the first derivative of the A-th energy with respect to the circuit parameter θ. g ^A is the first derivative of the A-th energy with respect to the orbital parameter κ. The specific format will be described later.

(Step 3)
By using the undetermined Lagrange multiplier obtained by calculating the above equation (18), analytical differentiation becomes possible.

Below, we will provide a detailed explanation of each of the SA-OO-SSVQE, SA-OO-VQD, and SA-OO-MCVQE methods.

(Calculation of derivatives by SA-OO-SSVQE)
In SA-OO-SSVQE, the Lagrangian L _A of the A-th eigenstate is defined by the following equation:

Here, θ _i is the i-th element of the circuit parameter θ, which is a vector, and κ _pq is an element of the trajectory parameter κ, which is a matrix.

In addition, θ ^−A _i and κ ^−A _pq are Lagrange multipliers. By imposing an extreme value condition on all parameters of the Lagrangian L _A except for the location parameter x, the following equation is derived.

Here, E _A represented by the following formula is the energy in a specific state A.

The latter two parts of the above formula (20) are satisfied when the circuit parameter θ is the optimal circuit parameter θ _* and the trajectory parameter κ is the optimal trajectory parameter κ _* . Moreover, by replacing the upper two parts of the above formula (20) with the following formula (21) and simplifying it, the following formula (22) is derived.

In the above formula

and

can be calculated by measurement using a quantum computer.

and

By calculating by a quantum computer, the undetermined Lagrange multipliers θ ^−A* and κ ^−A* are determined.

Then, by imposing an extreme value condition based on the Lagrange multipliers θ ^−A* and κ ^−A* , it becomes possible to calculate the analytical derivative of the energy E _A with respect to the location parameter x.

When the extreme value condition is applied to the Lagrangian L _A , the following relationship holds:

Here, E _A ^* (x) is the energy obtained by SA-OO-SSVQE. Therefore, the above relationship is expressed by the following equation.

More precisely, the above formula (23) is expressed by the following formula:

Each term on the right side of the above formula (24) can be calculated by performing quantum measurement using a quantum computer. Therefore, the classical computer can calculate the first derivative of the energy E _A ^* (x) according to the above formula (24) based on the Lagrange undetermined multipliers and the measurement results obtained by the quantum computer.

(Calculation of Differentials by SA-OO-VQD)
Similarly to the above, the Lagrangian L _A ^VQD shown below is set for the energy obtained by SA-OO-VQD.

In the above formula (25), θ _Si is the i-th element of the circuit parameter θ _S, which is a vector. In addition, θ ^−A _Si and κ ^−A _pq are Lagrange's undetermined multipliers.

Note that in SA-OO-VQD, as can be seen from the above formula (25), the Ath eigenstate includes all the circuit parameters θ ₀ ,...,θ _K-1 and their corresponding undetermined multipliers. Here, by imposing an extreme value condition on all parameters of the Lagrangian L _A ^VQD except for the location parameter x, the following formula is derived.

In addition, E _A expressed by the following formula is the energy for each state.

The latter two equations in the above equation (26) are satisfied when the circuit parameter θ is the optimal circuit parameter θ ^* and the trajectory parameter κ is the optimal trajectory parameter κ ^* . Moreover, by replacing the first two parts of the above equation (26) with the following equation (27) and simplifying it, the following equation (28) is derived.

The following conditions are used when deriving the above formula (28):

Each element in the above formula (28) can be calculated by quantum measurement using a quantum computer and calculation using the Lagrange undetermined multiplier method using a classical computer. Therefore, it is possible to calculate the derivative of the energy E _A ^* (x) in the same way as in the case of the above SA-OO-SSVQE. The derivative of the energy E _A ^* (x) in the SA-OO-VQD is expressed by the following formula.

(Calculation of first derivatives using SA-OO-MCVQE)
In SA-OO-MCVQE, the energy E _A is calculated by solving the above formula (15) as described above. When the Hamiltonian H depends on the location parameter x, each element h _ST of the matrix h, the eigenvector v _S ^(A) , and the energy E _A also depend on the location parameter x. Using the above formula (15), the following formula is established.

It should be noted that the above formula (30) is a formula relating to the μ-th element x _μ of the coordinate parameter x, and is derived by using the following relational expression.

Thus, the derivative of the energy E _A is calculated by differentiating the matrix element h _ST . Now consider the following Lagrangian L _ST :

The following equation also holds:

When considering the extreme value conditions for all parameters other than the location parameter x, the following relationship holds:

When the circuit parameter θ is the optimal circuit parameter θ ^* and the trajectory parameter κ is the optimal trajectory parameter κ ^* , the following equation holds:

If we express the above formula as a linear equation, we obtain the following formula (33).

By replacing each element in the above formula (33) with the following formula (34) and simplifying it, the following formula (35) is derived.

Each term on the right side of the above formula (34) can be quantum measured by a quantum computer. Therefore, a classical computer can calculate the first derivative of the matrix element h _ST according to the above formula (35) based on the Lagrange undetermined multiplier and the measurement result quantum measured by the quantum computer. Then, the first derivative of the energy E _A ^* (x) is calculated by substituting dh _ST /dx _μ calculated by the above formula (35) into the above formula (30).

(2nd order differential)
Next, a method for deriving the second derivative will be described.

(Calculation of second derivatives using SA-OO-SSVQE)
In order to calculate the second derivative of the state-specific energy E _A ^* (x) by SA-OO-SSVQE, the extreme value condition in the Lagrangian of the above equation (20) is differentiated with respect to x to obtain the following equation.

Here, in the above formula (36), α is any one of θ, κ, θ ^−A , and κ ^−A , and i is an index. Here, the Lagrangian is written as the following formula.

By combining the above equations, we arrive at the following equation.

By setting the following equation and setting it to satisfy the above conditions, the following equation (38) is derived.

Using the above relational expressions, the following expressions (39) and (40) are derived, where α, β=θ, κ, θ ^−A , κ ^−A .

It is.

The above equations (39), (40), (37), and (38) can be simplified as shown below.

By combining the above equations, we finally arrive at the following equation:

Where:

represents a term in which the index μ and the index ν are swapped.

(Calculation of second derivatives using SA-OO-MCVQE)
Similar to the first derivative, the second derivative is calculated so as to satisfy the extremum condition of the Lagrangian shown in the above formula (31). Similar to the above formula (37), the following formula is derived.

(Non-adiabatic Bond)
In this embodiment, the nonadiabatic coupling is also calculated. The nonadiabatic coupling between eigenstates S and T is defined by the following equation:

Here, |ψ ^* _S (x)> represents the eigenstate S. A specific description will be given below.

(Non-adiabatic coupling calculations using SA-OO-SSVQE)
In SA-OO-SSVQE, the eigenstate is expressed by the following equation:

The nonadiabatic coupling d _ST ^μ is calculated by the following formula:

The derivatives dθ ^* / _dxμ and dκ ^* / _dxμ of the optimal circuit parameter θ ^* and the optimal trajectory parameter κ ^* with respect to the position parameter _xμ are calculated in the above formula (37). Note that the transition amplitude shown in the following formula can be calculated by a quantum computer.

Each term on the right side of the above formula (45) can be quantum measured by a quantum computer, so the classical computer can calculate the nonadiabatic coupling d _ST ^μ according to the above formula (45) based on the Lagrange undetermined multipliers and the measurement results quantum measured by the quantum computer.

(Non-adiabatic coupling calculations using SA-OO-MCVQE)
The eigenstates in SA-OO-MCVQE are expressed by the following equations:

The nonadiabatic coupling in SA-OO-MCVQE is expressed by the following equation:

The first term on the right hand side of the above equation (46) is calculated using the following formula:

In addition, dh _MN /dx _μ is calculated in the above formula (35). The format of the second term on the right side of the above formula (46) is the same as that of the second term on the right side of the above formula (45). Therefore, the second term on the right side of the above formula (46) can also be calculated by using dθ ^* /dx _μ and dκ ^* /dx _μ obtained by solving the above formula (43). Therefore, the nonadiabatic coupling d _ST ^μ can be calculated according to the above formula (46).

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

Next, the specific operation of the hybrid system 100 of the embodiment will be described. In each device of the hybrid system 100, the processes shown in FIG. 4 and FIG. 5 are executed. Note that the following describes an example in which at least one physical quantity of the first derivative of energy, the second derivative of energy, and nonadiabatic coupling is calculated using SA-OO-SSVQE, SA-OO-VQD, and SA-OO-MCVQE.

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

The calculation target information includes, for example, information on the molecular structure, information on the physical quantities to be calculated, and information on the number K of states to be calculated. The physical quantities to be calculated are at least one of the first derivative of energy, the second derivative of energy, and nonadiabatic bonds. An example of the information on the molecular structure is the position parameter x that represents the position of the atomic nucleus of the molecule to be analyzed.

The calculation method information includes, for example, a Hamiltonian transformation method, initial information of the quantum circuit U^(θ), an initial value of the circuit parameter θ, an initial state of ψ, an optimization method of the circuit parameter θ, a weight for calculating the state average energy, and an initial value of the orbital parameter κ. When SA-OO-SSVQE is used, the calculation method information includes information on a weighting coefficient w _s ^VQE for calculating a loss function for SSVQE. When SA-OO-VQD is used, the calculation method information includes information on a weighting coefficient β _T for calculating a loss function for VQD.

Next, in step S102, the classical computer 110 receives the calculation target information and calculation method information transmitted from the user terminal 130. Then, in step S102, the classical computer 110 calculates the Hamiltonian H^(x) having the position parameter x as a variable, based on the position parameter x representing the position of the atomic nucleus of the molecule in the calculation target information and the Hamiltonian transformation method in the calculation method information.

In step S104, the classical computer 110 determines an orbital rotation operator U^ _{OO(κ) which is an operator having an orbital parameter κ, which is a parameter related to a molecular orbital, as a variable and which represents the rotation of the molecular orbital. Specifically, the classical computer 110 determines the orbital rotation operator U^OO (} κ) according to the above formula (5).

In step S106, the classical computer 110 determines the structure of the quantum circuit U^(θ) having a circuit parameter θ, which is a parameter of the quantum circuit, as a variable. Specifically, the classical computer 110 determines the structure of the quantum circuit U^(θ) based on the initial information of the quantum circuit U^(θ) among the calculation target information received in step S102.

In step S108, the classical computer 110 generates a Hamiltonian H^(x, κ) by performing orbital rotation and projection onto the active space for the Hamiltonian H^(x) calculated in step S102. Specifically, the classical computer 110 generates a Hamiltonian H^(x, κ) by performing the operation shown in the above formula (6) for the Hamiltonian H^(x) calculated in step S102.

In step S109, the classical computer 110 sets a first objective function having the circuit parameter θ as a variable. Specifically, the classical computer 110 sets the first objective function, which is a cost function, based on the weight information for calculating the cost function from the calculation method information received in step S102. Note that the first objective function differs depending on the method used.

When SA-OO-SSVQE is used, the classical computer 110 sets the loss function E ^SSVQE (θ) as shown in the above equation (9) as the first objective function based on the Hamiltonian H^(x, κ) generated in step S108 and the weighting coefficient w _s ^VQE in the calculation method information received in step S102.

When SA-OO-VQD is used, the classical computer 110 sets the loss function F _{S VQD (θ S ) as shown in the above formula (12) as the first objective function based on the Hamiltonian H^(x, κ) generated in step S108 and the weighting coefficient w s} ^VQE _{in the} ^calculation method information received in step S102.

When SA-OO-MCVQE is used, the classical computer 110 sets the loss function E ^MCVQE (θ) as shown in the above equation (17) as the first objective function based on the Hamiltonian H^(x, κ) generated in step S108.

In step S110, the classical computer 110 transmits various information necessary for the quantum computation to the quantum computer 120. Specifically, the classical computer 110 transmits to the quantum computer 120 the first objective function set in step S109, the structure of the quantum circuit U^(θ) determined in step S106, and, from the computation method information received in step S102, the initial value of the trajectory parameter κ, the initial value of the circuit parameter θ, the optimization method for the circuit parameter θ, the number K of states to be computed from the computation target information, and information on the physical quantity of the measurement target.

In step S112, the control device 121 receives various information sent from the classical computer 110 in step S110.

In step S114, the control device 121 causes the quantum computer 120 to execute quantum computation according to the various information received in step S112. The quantum computer 120 executes quantum measurement on the physical quantity of the measurement target according to the control by the control device 121. As will be described later, the classical computer 110 obtains a circuit parameter θ ^* that minimizes a first objective function based on the measurement result of the quantum measurement by the quantum computer 120.

Specifically, the quantum computer 120 generates an electromagnetic wave to be irradiated to at least any one of the quantum bits of the quantum bit group 123 in accordance with the control of the control device 121. Then, the quantum computer 120 irradiates the generated electromagnetic wave to at least any one of the quantum bits of the quantum bit group 123, and executes a quantum circuit according to the initial information, thereby generating an optimal circuit parameter θ ^* . The gate operation of each quantum gate included in the quantum circuit is converted into a corresponding electromagnetic wave waveform, and the generated electromagnetic wave is irradiated to the quantum bit group 123 by the electromagnetic wave generating device 122. Then, the quantum computer 120 outputs the circuit parameter θ ^* .

In step S116, the control device 121 transmits the measurement results obtained in step S114 to the classical computer 110.

In step S118, the classical computer 110 receives the measurement result transmitted in step S116 from the control device 121. Then, the classical computer 110 calculates the circuit parameter θ ^* based on the measurement result.

In the above description, the optimal circuit parameter θ ^* is obtained by the process of steps S114 to S118. In the actual process, the optimal circuit parameter θ ^* is obtained by repeatedly performing calculations between the classical computer 110 and the quantum computer 120 according to any one of the methods of SSVQE, VQD, and MCVQE. Specifically, the quantum computer 120 first irradiates the quantum bit group 123 with electromagnetic waves corresponding to a certain circuit parameter θ. Next, the classical computer 110 obtains an expected value E(θ) corresponding to the measurement result of the quantum measurement of the quantum computer 120, and calculates the value of the first objective function. Then, the classical computer 110 updates the circuit parameter θ according to the value of the first objective function so that the value of the first objective function becomes smaller. These calculations are repeated between the classical computer 110 and the quantum computer 120 until the value of the first objective function becomes minimum, and the classical computer 110 obtains the optimal circuit parameter θ ^* .

In step S120, the classical computer 110 sets a second objective function, which is an objective function having the circuit parameter θ ^* stored in the information storage unit 113 in step S118 as a constant and the trajectory parameter κ as a variable. Then, the classical computer 110 calculates the value of the second objective function.

When the SA-OO-SSVQE or SA-OO-VQD method is used, the classical computer 110 sets, as the second objective function, the state average energy E SA (κ) of the above formula (10) determined based on the Hamiltonian H ^{^} (x, κ) generated in step S108, the weighting coefficient w _S ^SA of the calculation method information received in step S102, and the circuit parameter θ ^* stored in the information storage unit 113 in step S118. On the other hand, when the SA-OO-MCVQE method is used, the classical computer 110 sets, as the second objective function, the state average energy E SA (κ) of the above formula (10A) determined based on the Hamiltonian H ^{^} (x, κ) generated in step S108, the weighting coefficient w _S ^SA of the calculation method information received in step S102, and the circuit parameter θ ^* stored in the information storage unit 113 in step S118. Then, the classical computer 110 calculates the value of the second objective function.

In step S122, the classical computer 110 transmits various pieces of information necessary for quantum computation to the quantum computer 120. Specifically, the classical computer 110 transmits various pieces of information, including information about the physical quantities of the measurement target that correspond to the following part in the above equation (10), to the quantum computer 120:

In step S124, the control device 121 receives various information including information about the physical quantity of the measurement target transmitted from the classical computer 110.

In step S125, the control device 121 causes the quantum computer 120 to execute quantum computation according to the various information received in step S124. The quantum computer 120 executes quantum measurement on the physical quantity of the measurement target according to the control by the control device 121. As will be described later, the classical computer 110 obtains the trajectory parameter κ ^* that minimizes the second objective function, based on the measurement result of the quantum measurement by the quantum computer 120.

In step S126, the control device 121 transmits the measurement results obtained in step S124 to the classical computer 110.

In step S128, the classical computer 110 receives the measurement result transmitted from the control device 121 in step S126. Then, the classical computer 110 calculates the trajectory parameter κ ^* based on the measurement result and the weighting coefficient w _s ^SA of the state average energy. The classical computer 110 also calculates the value of the second objective function.

In step S130, the classical computer 110 determines whether the calculation has converged. Specifically, the classical computer 110 determines whether the difference between the value of the second objective function calculated in step S120 and the value of the second objective function calculated in step S128 is equal to or less than a predetermined threshold, and proceeds to step S122 if the difference is equal to or less than the predetermined threshold. On the other hand, if the difference is greater than the predetermined threshold, the process returns to step S108 and repeats the processes of steps S108 to S128.

In step S132, the classical computer 110 transmits the circuit parameters θ ^* and the trajectory parameters κ ^* obtained by the above processes to the quantum computer 120.

In step S134, the quantum computer 120 receives the circuit parameters θ ^* and the trajectory parameters κ ^* transmitted from the classical computer 110 in step S132.

In step S136, the classical computer 110 sets the physical quantity to be calculated based on the calculation target information received in step S102. Specifically, the classical computer 110 sets which physical quantity to calculate: the first derivative of energy, the second derivative of energy, or the nonadiabatic bond, based on the calculation target information received in step S102.

In step S138, the classical computer 110 transmits information about the physical quantity to be measured to the quantum computer 120, depending on the physical quantity set in step S136. The physical quantity to be measured varies depending on the calculation method and the physical quantity to be calculated.

[First derivative of energy]
(In the case of SA-OO-SSVQE)
When calculating the first derivative of energy using SA-OO-SSVQE, the classical computer 110 transmits the physical quantity in the dashed line portion of the following equation (A1) to the quantum computer 120 as the physical quantity to be measured.

(A1)

As described above, A and S in formula (A1) are indexes for identifying eigenstates, E(x) is eigenenergy, μ is an index for identifying an element of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^(θ) is a quantum circuit, ^θ- is an undetermined Lagrange multiplier corresponding to an element of the circuit parameter θ which is a vector, i is an index for identifying an element of the circuit parameter θ which is a vector, ^wVQE is a weighting coefficient for the eigenstate, ^κ- is an undetermined Lagrange multiplier corresponding to a matrix element of the orbital parameter κ which is a matrix, p, q are indexes for identifying matrix elements, and ^wSA is a weighting coefficient for the state average energy. The undetermined Lagrange multiplier is calculated by solving simultaneous equations such as the above formula (18).

(In the case of SA-OO-VQD)
When calculating the first derivative of energy using SA-OO-VQD, the classical computer 110 transmits the physical quantity in the dashed line portion of the following equation (A2) to the quantum computer 120 as the physical quantity to be measured.

(A2)

(In the case of SA-OO-MCVQE)
When calculating the first derivative of energy using SA-OO-MCVQE, the classical computer 110 transmits the physical quantity in the dashed line portion of the following equation (A3) to the quantum computer 120 as the physical quantity to be measured.

(A3)

As mentioned above, T in equation (A3) is an index for identifying an eigenstate, and h is an element of the matrix representing the eigenenergy.

When calculating the first derivative of the energy, in step S142 described below, the quantum computer 120 quantum measures the expectation value including the derivative with respect to the location parameter x of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^* ) as shown in the above equations.

[Second derivative of energy]
(In the case of SA-OO-SSVQE)
When calculating the second derivative of energy using SA-OO-SSVQE, the classical computer 110 transmits the physical quantity of the dashed line portion in the following formula (B1) as the physical quantity to be measured to the quantum computer 120. As described above, E ^SSVQE in formula (B1) is a loss function equivalent to the first objective function when SSVQE is used, E ^SA is the state average energy, μ and ν are indexes for identifying elements of the position parameter x which is a vector, i and j are indexes for identifying vector elements, p, q, m, and n are indexes for identifying matrix elements, and (μ⇔ν) represents a term in which μ and ν of adjacent terms are swapped.

(B1)

Note that formula (B1) also includes a term corresponding to the first derivative of the energy. Therefore, when calculating the second derivative of the energy, the quantum computer 120 performs quantum measurement in step S142 described later while also using the measurement results of the quantum circuit U^(θ ^* ) and the expectation value including the derivative with respect to the location parameter x of the Hamiltonian H^(x, κ ^* ).

[Non-adiabatic bond]
(In the case of SA-OO-SSVQE)
When calculating the value of nonadiabatic coupling using SA-OO-SSVQE, the classical computer 110 transmits the physical quantity in the dashed part of the following equation (C1) to the quantum computer 120 as the physical quantity to be measured.

(C1)

(In the case of SA-OO-MCVQE)
When calculating the value of nonadiabatic coupling using SA-OO-MCVQE, the classical computer 110 transmits the physical quantity in the dashed part of the following equation (C2) to the quantum computer 120 as the physical quantity to be measured.

(C2)

Note that in equation (C2), M and N are indexes for identifying the eigenstates, and v is an eigenvector corresponding to the eigenstate.

When calculating the value of the nonadiabatic coupling, in step S142 described below, the quantum computer 120 quantum measures an expectation value including the derivative of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO (κ ^* ) with respect to the position parameter x, as shown in equation (C1) or (C2).

In step S140, the quantum computer 120 receives information about the physical quantity to be measured that was sent from the classical computer 110 in step S138.

In step S142, the quantum computer 120 acquires a measurement result by performing quantum measurement of the physical quantity of the measurement target based on the information related to the physical quantity of the measurement target received in step S140 and various information included in the calculation method information received in step S120.

In step S144, the quantum computer 120 transmits the quantum measurement result to the classical computer 110.

In step S146, the classical computer 110 receives the measurement object transmitted from the quantum computer 120 in step S144.

In step S148, the classical computer 110 calculates the physical quantity to be calculated based on the measurement results received in step S144.

Specifically, the classical computer 110 calculates the sum of the measurement results of each term on the right side of each of ^the above equations based on the measurement results obtained by the quantum computer 120, the circuit parameter θ ^* , the orbital parameter κ ^* , and the weighting coefficients _wsVQE , _wsSA ^, etc., to calculate at least one or more physical quantities, namely the first derivative of the energy, the second derivative of the energy, and the nonadiabatic coupling value, represented on the left side of each of the above equations.

In step S150, the classical computer 110 transmits the calculation result obtained in step S148 to the user terminal 130.

In step S152, the user terminal 130 receives the calculation results sent from the classical computer 110.

As described above, in the hybrid system of the embodiment, the classical computer calculates a Hamiltonian H^(x) having a position parameter x representing the position of the atomic nucleus of a molecule as a variable, and determines an orbital rotation operator U^ _OO (κ) which is an operator having an orbital parameter κ, which is a parameter related to a molecular orbital, as a variable and which represents the rotation of the molecular orbital. The classical computer also determines a quantum circuit U^(θ) having a circuit parameter θ, which is a parameter of a quantum circuit, as a variable, and performs orbital rotation and projection onto an active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having the circuit parameter θ as a variable. Then, the quantum computer calculates a circuit parameter θ ^* of the quantum circuit U^(θ) that minimizes or maximizes the first objective function. The classical computer sets a second objective function, which is an objective function having the circuit parameter θ ^* as a constant and the orbital parameter κ as a variable. The quantum computer calculates an orbital parameter κ ^* that minimizes or maximizes the second objective function. The quantum computer acquires a measurement result of ^the quantum measurement by performing quantum measurement on the expectation value including the derivative of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^{* )} with respect to the position parameter x based on the circuit parameter θ* and the orbital parameter κ*. Then, the classical computer calculates the derivative of the energy corresponding to the Hamiltonian H^(x) based on the measurement result of the quantum measurement. This makes it possible to obtain the derivative value of the energy when performing the SA-OO calculation using the quantum computer.

In addition, in the hybrid system of the embodiment, the classical computer calculates a Hamiltonian H^(x) having a position parameter x representing the position of the atomic nucleus of the molecule as a variable, and determines an orbital rotation operator U^ _OO (κ) which is an operator having an orbital parameter κ, which is a parameter related to the molecular orbital, as a variable and which represents the rotation of the molecular orbital. In addition, the classical computer determines a quantum circuit U^(θ) having a circuit parameter θ, which is a parameter of a quantum circuit, as a variable, and performs orbital rotation and projection onto the active space for the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having the circuit parameter θ as a variable. Then, the quantum computer calculates a circuit parameter θ ^* of the quantum circuit U^(θ) that minimizes or maximizes the first objective function. The classical computer sets a second objective function, which is an objective function having the circuit parameter θ ^* as a constant and the orbital parameter κ as a variable. The quantum computer calculates the orbital parameter κ ^* that minimizes or maximizes the second objective function. The quantum computer acquires a measurement result of ^the quantum measurement by performing a quantum measurement on an expectation value including a derivative with respect to the position parameter x of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO ( ^{κ *} ) based on the circuit parameter θ* and the orbital parameter κ*. Then, the classical computer calculates a value of the nonadiabatic coupling based on the measurement result of the quantum measurement. In this way, the value of the nonadiabatic coupling can be obtained when performing a SA-OO calculation using the quantum computer.

In addition, by appropriately dividing the roles between classical and quantum computers, it is possible to obtain the values of energy derivatives and nonadiabatic coupling when performing SA-OO calculations.

The technology disclosed herein is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the spirit and scope of the invention.

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

In the above embodiment, the calculation target information is transmitted from the user terminal 130 to the classical computer 110, and the classical computer 110 executes a calculation according to the calculation target information. However, the present invention is not limited to this. The user terminal 130 may transmit the calculation target information to the classical computer 110 or a storage medium or storage device accessible to the classical computer 110 via a computer network such as an IP network, or may store the calculation target information in a storage medium or storage device and hand it over to the operator of the classical computer 110, who then inputs the calculation target information to the classical computer 110 using the storage medium or storage device.

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

In the above embodiment, the loss function is set as the first objective function or the second objective function, and parameters are calculated so that the values of these objective functions are minimized. However, this is not limiting. For example, the inverse of the loss function may be set as the first objective function or the second objective function, and parameters are calculated so that the value of the objective function is maximized.

In the above embodiment, the quantum computer 120 performs quantum computation, but the present invention is not limited to this. For example, quantum computation may be performed by a classical computer that mimics the behavior of a quantum computer.

In addition, in the above embodiment, 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.

Furthermore, each component of the hybrid system of this embodiment does not have to be realized by a single computer or server, but may be realized in a distributed manner across multiple computers connected by a network.

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

For example, when calculating the derivative of energy, one of the multiple classical computers calculates a Hamiltonian H^(x) having a position parameter x representing the position of the atomic nucleus of a molecule as a variable, determines an orbital rotation operator U^ _OO (κ) which is an operator having an orbital parameter κ, which is a parameter related to a molecular orbital, as a variable and which represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having a circuit parameter θ, which is a parameter of a quantum circuit, as a variable, and performs orbital rotation and projection onto the active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having the circuit parameter θ as a variable. Then, by repeated calculations between one classical computer and one quantum computer, one of the multiple classical computers calculates a circuit parameter θ ^* of the quantum circuit U^(θ) that minimizes or maximizes the first objective function based on the measurement result of the quantum measurement by the one quantum computer. One of the classical computers sets a second objective function, which is an objective function having a circuit parameter θ ^* as a constant and an orbital parameter κ as a variable. One of the classical computers calculates an orbital parameter κ ^* that minimizes or maximizes the second objective function based on a measurement result of a quantum measurement by one quantum computer. One of the quantum computers performs quantum measurement on an expectation value including a derivative of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^* ) with respect to the position parameter x based on the circuit parameter θ ^* and the orbital parameter κ ^* to obtain a measurement result of the quantum measurement. Then, a specific classical computer that is one of the classical computers calculates a derivative of the energy corresponding to the Hamiltonian H^(x) based on the measurement result of the quantum measurement.

Also, for example, when calculating the value of a nonadiabatic bond, one of the multiple classical computers calculates a Hamiltonian H^(x) having a position parameter x representing the position of the atomic nucleus of a molecule as a variable, determines an orbital rotation operator U^ _OO (κ) which is an operator having an orbital parameter κ, which is a parameter related to a molecular orbital, as a variable and which represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having a circuit parameter θ, which is a parameter of a quantum circuit, as a variable, and performs orbital rotation and projection onto the active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having the circuit parameter θ as a variable. Then, by repeated calculations between one classical computer and one quantum computer, one of the multiple classical computers calculates a circuit parameter θ ^* of the quantum circuit U^(θ) that minimizes or maximizes the first objective function based on the measurement result of the quantum measurement by one quantum computer. Then, one classical computer of the multiple classical computers sets a second objective function, which is an objective function having the circuit parameter θ ^* as a constant and the orbital parameter κ as a variable. One classical computer of the multiple classical computers calculates the orbital parameter κ ^* that minimizes or maximizes the second objective function based on the measurement result of the quantum measurement by one quantum computer. Then, one quantum computer of the multiple quantum computers performs quantum measurement of expectation values including derivatives of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO (κ ^* ) with respect to the position parameter x based on the circuit parameter θ ^* and the orbital parameter ^κ * to obtain the measurement result of the quantum measurement. Then, a specific classical computer that is one classical computer of the multiple classical computers calculates a value of the nonadiabatic coupling based on the measurement result of the quantum measurement.

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 (18)

A quantum information processing method executed by a hybrid system including a classical computer and a quantum computer, comprising:
a classical computer calculates a Hamiltonian H^(x) having as a variable a position parameter x that represents the position of the atomic nucleus of the molecule, determines an orbital rotation operator U^ _OO (κ) that is an operator having as a variable an orbital parameter κ that is a parameter related to a molecular orbital and that represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having as a variable a circuit parameter θ that is a parameter of a quantum circuit, performs orbital rotation and projection onto an active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having as a variable the circuit parameter θ;
Through repeated calculations between a classical computer and a quantum computer, the classical computer calculates the circuit parameters θ* of the quantum circuit U^( ^θ ) so as to minimize or maximize the first objective function based on the measurement results of the quantum measurement by the quantum computer;
A classical computer sets a second objective function, which is an objective function having the circuit parameter θ ^* as a constant and the trajectory parameter κ as a variable;
A classical computer calculates the trajectory parameter κ ^* so as to minimize or maximize the second objective function based on the measurement result of the quantum measurement by the quantum computer;
a quantum computer performs quantum measurements on expectation values including differentials of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^* ) with respect to the position parameter x based on the circuit parameter θ ^* and the trajectory ^parameter κ*, thereby acquiring a measurement result of the quantum measurement;
A classical computer calculates an energy derivative corresponding to the Hamiltonian H^(x) based on the measurement result of the quantum measurement.
Quantum information processing methods including quantum information processing. When calculating the first derivative of the energy using the subspace-search variational quantum eigensolver (SSVQE),
The quantum computer performs quantum measurement on each term on the right side of the following formula (A1) to output a measurement result of the quantum measurement of each term on the right side:
The classical computer calculates the sum of the measurement results of each term on the right side of the following formula (A1) to calculate a first-order differential value of the energy represented by the left side of the following formula (A1):
2. The quantum information processing method according to claim 1.

(A1)
In addition, A and S are indexes for identifying eigenstates, E(x) is eigenenergy, μ is an index for identifying an element of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^(θ) is the quantum circuit, ^θ- is an undetermined Lagrange multiplier corresponding to an element of the circuit parameter θ which is a vector, i is an index for identifying an element of the circuit parameter θ which is a vector, ^wVQE is a weighting coefficient for the eigenstate, ^κ- is an undetermined Lagrange multiplier corresponding to a matrix element of the orbital parameter κ which is a matrix, p, q are indices for identifying matrix elements, and ^wSA is a weighting coefficient of the state average energy. When calculating the first derivative of the energy using variational quantum deflation (VQD),
The quantum computer performs quantum measurement on each term on the right side of the following formula (A2), and outputs a measurement result of the quantum measurement of each term on the right side:
The classical computer calculates a first-order differential value of the energy represented by the left side of the following formula (A2) by calculating the sum of the measurement results of each term on the right side of the following formula (A2):
2. The quantum information processing method according to claim 1.

(A2)
In addition, A and S are indexes for identifying eigenstates, E(x) is eigenenergy, μ is an index for identifying an element of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^(θ) is the quantum circuit, ^θ- is an undetermined Lagrange multiplier corresponding to an element of the circuit parameter θ which is a vector, i is an index for identifying an element of the circuit parameter θ which is a vector, ^κ- is an undetermined Lagrange multiplier corresponding to a matrix element of the orbital parameter κ which is a matrix, p, q are indices for identifying matrix elements, and ^wSA is a weighting coefficient of the state average energy. When calculating the first derivative of the energy using MCVQE (Multistate-contracted variational quantum eigensolver),
The quantum computer performs quantum measurement on each term on the right side of the following formula (A3), and outputs a measurement result of the quantum measurement of each term on the right side:
The classical computer calculates the sum of the measurement results of each term on the right side of the following formula (A3) to calculate a first-order differential value of the energy represented by the left side of the following formula (A3):
2. The quantum information processing method according to claim 1.

(A3)
In addition, S and T are indices for identifying eigenstates, h is an element of a matrix representing eigenenergy, μ is an index for identifying an element of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^(θ) is the quantum circuit, ^θ- is an undetermined Lagrange multiplier corresponding to an element of the circuit parameter θ which is a vector, ^κ- is an undetermined Lagrange multiplier corresponding to a matrix element of the orbital parameter κ which is a matrix, p, q are indices for identifying matrix elements, and ^wSA is a weighting coefficient of the state average energy. When calculating the second derivative of the energy using the subspace-search variational quantum eigensolver (SSVQE),
The quantum computer performs quantum measurement on each term on the right side of the following formula (B1) to output a measurement result of the quantum measurement of each term on the right side,
The classical computer calculates the sum of the measurement results of each term on the right side of the following formula (B1) to calculate a second derivative of the energy represented by the left side of the following formula (B1):
2. The quantum information processing method according to claim 1.

(B1)
In addition, A is an index for identifying an eigenstate, E(x) is an eigenenergy, E ^SSVQE is a loss function equivalent to the first objective function when SSVQE is used, E ^SA is a state average energy, μ, ν are indexes for identifying elements of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^(θ) is the quantum circuit, ^θ- is an undetermined Lagrange multiplier corresponding to an element of the circuit parameter θ which is a vector, i, j are indexes for identifying vector elements, ^κ- is an undetermined Lagrange multiplier corresponding to a matrix element of the orbital parameter κ which is a matrix, and p, q, m, n are indexes for identifying matrix elements,

represents a term in which the adjacent terms μ and ν are swapped. A quantum information processing method executed by a hybrid system including a classical computer and a quantum computer, comprising:
a classical computer calculates a Hamiltonian H^(x) having as a variable a position parameter x that represents the position of the atomic nucleus of the molecule, determines an orbital rotation operator U^ _OO (κ) that is an operator having as a variable an orbital parameter κ that is a parameter related to a molecular orbital and that represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having as a variable a circuit parameter θ that is a parameter of a quantum circuit, performs orbital rotation and projection onto an active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x,κ), and sets a first objective function having as a variable the circuit parameter θ;
Through repeated calculations between a classical computer and a quantum computer, the classical computer calculates the circuit parameters θ* of the quantum circuit U^(θ ⁾ based on the measurement results of the quantum measurement by the quantum computer such that the quantum computer minimizes or maximizes the first objective function;
A classical computer sets a second objective function, which is an objective function having the circuit parameter θ ^* as a constant and the trajectory parameter κ as a variable;
A classical computer calculates the trajectory parameter κ ^* based on a measurement result of the quantum measurement by the quantum computer so that the quantum computer minimizes or maximizes the second objective function;
a quantum computer performs quantum measurement of expectation values including differentials of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO (κ ^* ) with respect to the position parameter ^x based on the circuit parameter θ ^* and the orbital parameter κ*, thereby acquiring a measurement result of the quantum measurement;
A classical computer calculates a value of the nonadiabatic coupling based on the measurement result of the quantum measurement.
Quantum information processing methods including quantum information processing. When calculating nonadiabatic coupling using the subspace-search variational quantum eigensolver (SSVQE),
The quantum computer performs quantum measurement on each term on the right side of the following formula (C1) to output a measurement result of the quantum measurement of each term on the right side,
The classical computer calculates the sum of the measurement results of each term on the right side of the following formula (C1) to calculate the value of the nonadiabatic bond represented by the left side of the following formula (C1):
The quantum information processing method according to claim 6, further comprising the step of:

(C1)
In addition, S and T are indices for identifying eigenstates, μ is an index for identifying an element of the position parameter x which is a vector, ψ is an eigenstate in the active space, U^ _OO (κ) is the orbit rotation operator, U^(θ) is the quantum circuit, j is an index for identifying an element of the circuit parameter θ which is a vector, κp _,q are matrix elements of the orbital parameter κ which is a matrix, and p, q are indices for identifying matrix elements. When calculating nonadiabatic coupling using MCVQE (Multistate-contracted variational quantum eigensolver),
The quantum computer performs quantum measurement on each term on the right side of the following formula (C2), and outputs a measurement result of the quantum measurement of each term on the right side:
The classical computer calculates the sum of the measurement results of each term on the right side of the following formula (C2) to calculate the value of the nonadiabatic coupling represented by the left side of the following formula (C2):
The quantum information processing method according to claim 6, further comprising the step of:

(C2)
Note that S, T, M, and N are indexes for identifying eigenstates, μ is an index for identifying an element of the position parameter x, which is a vector, ψ is an eigenstate in the active space, v is an eigenvector corresponding to the eigenstate, U^ _OO (κ) is the orbit rotation operator, and U^(θ) is the quantum circuit. 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.
The quantum information processing method according to any one of claims 1 to 8. A quantum information processing method in which a quantum computer executes a part of the quantum information processing method according to any one of claims 1 to 9 that is executed by the quantum computer. A quantum information processing method in which a classical computer executes the part of the quantum information processing method according to any one of claims 1 to 9 that is executed by a classical computer. A quantum computer that performs the process of the quantum computer-executed portion of the quantum information processing method according to any one of claims 1 to 9. A classical computer that executes the part of the quantum information processing method according to any one of claims 1 to 9 that is executed by a classical computer. A hybrid system comprising a classical computer according to claim 13 and a quantum computer according to claim 12. A quantum information program for causing a quantum computer to execute the part of the quantum information processing method according to any one of claims 1 to 9 that is executed by the quantum computer. A quantum information program for causing a classical computer to execute the part of the quantum information processing method according to any one of claims 1 to 9 that is executed by a classical computer. A quantum information processing method executed by a specific classical computer, which is one classical computer in a hybrid system including at least one classical computer and at least one quantum computer, comprising:
one classical computer calculates a Hamiltonian H^(x) having as a variable a position parameter x that represents the position of the atomic nucleus of a molecule, determines an orbital rotation operator U^ _OO (κ) that is an operator having as a variable an orbital parameter κ that is a parameter related to a molecular orbital and that represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) having as a variable a circuit parameter θ that is a parameter of a quantum circuit, performs orbital rotation and projection onto an active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function having as a variable the circuit parameter θ;
By repeated calculations between one classical computer and one quantum computer, one classical computer calculates the circuit parameters θ ^* of the quantum circuit U^(θ) so as to minimize or maximize the first objective function based on the measurement results of the quantum measurement by the quantum computer;
A classical computer sets a second objective function that has the circuit parameter θ ^* as a constant and the trajectory parameter κ as a variable;
A classical computer calculates the trajectory parameters κ ^* based on the measurement results of the quantum measurement by the quantum computer so as to minimize or maximize the second objective function;
a quantum computer performs quantum measurement of an expectation value including a derivative of the quantum circuit U^(θ ^* ) and the Hamiltonian H^(x, κ ^* ) with respect to the position parameter x based on the circuit parameter θ ^* and ^the trajectory parameter κ*, thereby obtaining a measurement result of the quantum measurement;
A specific classical computer calculates an energy derivative corresponding to the Hamiltonian H^(x) based on the measurement result of the quantum measurement.
A quantum information processing method for performing the process. A quantum information processing method executed by a specific classical computer, which is one classical computer in a hybrid system including at least one classical computer and at least one quantum computer, comprising:
one classical computer calculates a Hamiltonian H^(x) having as a variable a position parameter x that represents the position of the atomic nucleus of a molecule, determines an orbital rotation operator U^ _OO (κ) that is an operator having as a variable an orbital parameter κ that is a parameter related to a molecular orbital and that represents the rotation of the molecular orbital, determines a quantum circuit U^(θ) that has as a variable a circuit parameter θ that is a parameter of a quantum circuit, performs orbital rotation and projection onto an active space on the Hamiltonian H^(x) to generate a Hamiltonian H^(x, κ), and sets a first objective function that has as a variable the circuit parameter θ;
Through repeated calculations between one classical computer and one quantum computer, one classical computer calculates the circuit parameters θ* of the quantum circuit U^(θ) based on the measurement results of the ^quantum measurement by the one quantum computer such that the first objective function is minimized or maximized;
A classical computer sets a second objective function that has the circuit parameter θ ^* as a constant and the trajectory parameter κ as a variable;
A classical computer calculates the trajectory parameters κ ^* based on the measurement results of the quantum measurement by the quantum computer so as to minimize or maximize the second objective function;
a quantum computer performs quantum measurement of an expectation value including a derivative of the quantum circuit U^(θ ^* ) and the orbital rotation operator U^ _OO (κ ^* ) with respect to the position parameter ^x based on the circuit parameter θ ^* and the orbital parameter κ*, thereby obtaining a measurement result of the quantum measurement;
A specific classical computer calculates the value of the nonadiabatic coupling based on the measurement result of the quantum measurement.
A quantum information processing method for performing the process.