WO2020100698A1 - Simulation device, computer program, and simulation method - Google Patents

Simulation device, computer program, and simulation method Download PDF

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WO2020100698A1
WO2020100698A1 PCT/JP2019/043571 JP2019043571W WO2020100698A1 WO 2020100698 A1 WO2020100698 A1 WO 2020100698A1 JP 2019043571 W JP2019043571 W JP 2019043571W WO 2020100698 A1 WO2020100698 A1 WO 2020100698A1
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hamiltonian
trotter
spin
probability distribution
variable
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French (fr)
Japanese (ja)
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真之 大関
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国立大学法人京都大学
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N99/00—Subject matter not provided for in other groups of this subclass
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16Z—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS, NOT OTHERWISE PROVIDED FOR
    • G16Z99/00—Subject matter not provided for in other main groups of this subclass

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  • the present invention relates to a simulation device, a computer program, and a simulation method.
  • quantum annealing for the optimization problem.
  • quantum fluctuations that produce superposition are applied to the optimization problem to be solved.
  • a quantum computer capable of introducing quantum fluctuation has been developed, but the number of bits that can be handled is, for example, 2048 qubits, the number of spins is limited to 2048, and only two-body interaction is considered. Therefore, the practical limit is large. Therefore, the simulation on a normal digital computer is the key for large-scale real problems.
  • Patent Document 1 it is possible to perform, on a digital computer, a simulation in which a quantum mechanical effect other than a so-called transverse magnetic field, which has been conventionally impossible to simulate on a digital computer, is set.
  • the quantum Monte Carlo method is disclosed.
  • Patent Document 1 can obtain an accurate solution, it requires a long calculation time for a large-scale optimization problem. Therefore, it is desired that the calculation result can be obtained in a short time even if the accuracy is somewhat sacrificed.
  • the present invention has been made in view of such circumstances, and an object thereof is to provide a simulation device, a computer program, and a simulation method that can obtain an approximate solution of an optimization problem at high speed.
  • a plurality of spins of the Ising model are represented by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and an x component of the Pauli matrix corresponding to the plurality of spins.
  • a simulation device that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with time.
  • a z-component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first computing unit that computes a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable, .
  • the x-component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the product of the replaced spin variables is summed over the trotter direction using an exponential operator
  • a second computing unit that computes a second probability distribution function for the Hamiltonian, a first message that includes a probability distribution function that sums the first probability distribution functions in the trotter direction, and a second message that includes the second probability distribution function.
  • an expected value calculation unit that calculates an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm using.
  • a computer program causes a computer to represent a plurality of spins of the Ising model by z components of a Pauli matrix, an objective Hamiltonian expressing an optimization problem, and a Pauli matrix corresponding to the plurality of spins.
  • the quantum fluctuation is reduced over time to simulate the expected value of the spin configuration of the target Hamiltonian.
  • a computer program wherein a z component of the Pauli matrix is replaced by a spin variable by Suzuki Trotter decomposition, and a first probability distribution function for the target Hamiltonian is calculated using an exponential operator including the replaced spin variable.
  • a plurality of spins of the Ising model are represented by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and an x component of the Pauli matrix corresponding to the plurality of spins.
  • a simulation method that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation over time.
  • the z component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first probability distribution function for the target Hamiltonian is calculated by using an exponential operator including the replaced spin variable, and the Pauli matrix
  • the target Hamiltonian is calculated by a belief propagation algorithm using a first message including a probability distribution function obtained by calculating a distribution function and summing the first probability distribution functions in the trotter direction, and a second message including the second probability distribution function. Calculate the expected value of spin coordination of.
  • an approximate solution of a large-scale optimization problem can be calculated at high speed.
  • FIG. 1 is an explanatory diagram showing an example of the configuration of the simulation apparatus 100 according to the present embodiment.
  • the simulation device 100 includes a control unit 10 that controls the entire device, an input unit 11, a target Hamiltonian calculation unit 12, an initial Hamiltonian calculation unit 13, a density matrix calculation unit 14, a probability distribution function calculation unit 15, an output unit 18, and a belief propagation process. It includes a unit 19, an expected value calculation unit 20, a magnetization variable calculation unit 21, and a storage unit 22.
  • the probability distribution function calculator 15 includes a first calculator 16 and a second calculator 17.
  • the input unit 11 acquires input data and parameters for executing a simulation.
  • the input data is, for example, data expressing (translating) an optimization problem (also referred to as “combination optimization problem”) in the Ising model.
  • the Ising model is a mathematical model that describes the behavior of a magnetic substance.
  • the parameters include, for example, temperature, a quantum mechanical effect due to a transverse magnetic field, a quantum mechanical effect other than the transverse magnetic field, a Trotter number, and the like.
  • the output unit 18 outputs the output data that is the result of the simulation.
  • the output data is the expected value (spin configuration expected value) of each spin variable of the Ising model in which the optimization problem is expressed.
  • Formula (1) expresses the optimization problem with the Ising model and is also called a cost function.
  • ⁇ i is a binary variable (for example, ⁇ 1), and the subscript i represents the place where the spin, which is the degree of freedom of the Ising model, is arranged.
  • ⁇ is a subscript indicating the interaction between spins, and J ⁇ is the strength of the interaction.
  • i ⁇ means “i around ⁇ ”. The meaning of expression (1) will be described with a simple example.
  • FIG. 2 is a schematic diagram showing an example of interaction between spins of the Ising model.
  • spins are represented by ⁇ 1 to ⁇ 8 (spin number 8).
  • the Ising model Hamiltonian (energy) Can be expressed as ⁇ J ⁇ ⁇ 1 ⁇ 2 ⁇ 3 ⁇ 4 ⁇ J ⁇ ′ ⁇ 4 ⁇ 5 ⁇ J ⁇ ′ ′ ⁇ 4 ⁇ 6 ⁇ 7 ⁇ 8 and this cost function can be minimized (or maximized). By doing so, the optimization problem can be solved.
  • the present embodiment is not limited to the two-body interaction (for example, it can be represented by the product of two spins like J ⁇ ⁇ 1 ⁇ 2 ), and three or more bodies are used. Can handle the interaction of. Also, with respect to the spin number N, if the memory capacity of the digital computer is increased, it is possible to handle spin numbers of several thousands or more, and it is possible to solve a large-scale optimization problem.
  • the target Hamiltonian calculator 12 calculates the target Hamiltonian H 0 hat based on the input optimization problem. Specifically, the target Hamiltonian calculation unit 12 calculates the target Hamiltonian H 0 hat represented by Expression (3) using Expression (2) in Expression (1).
  • the hat ( ⁇ ) indicates that it is a matrix.
  • the initial Hamiltonian calculation unit 13 calculates an initial Hamiltonian as indicated by the second term on the right side of Expression (4).
  • the x component of the Pauli matrix shown in (5) is included. Since the x component of the Pauli matrix has a non-diagonal component, spins directed in the z direction can be inverted and quantum fluctuations can be produced.
  • N is the spin number.
  • Expression (6) shows a specific example of the initial Hamiltonian calculated by the initial Hamiltonian operation unit 13.
  • the first term on the right side of Expression (6) includes the first-order term of the average component of the sum of the x components of the Pauli matrix, and can give a quantum mechanical effect due to the transverse magnetic field.
  • the coefficient ⁇ is a parameter that controls the strength of quantum fluctuations. In the quantum annealing, the coefficient ⁇ is controlled so as to decrease with the passage of time.
  • the second term on the right side of the equation (6) includes a quadratic term of the average component of the sum of x components of the Pauli matrix, represents a so-called antiferromagnetic XX interaction, and may give a quantum mechanical effect other than the transverse magnetic field. it can.
  • ⁇ is a predetermined coefficient.
  • the magnetic field function is not limited to equation (6).
  • the magnetic field function may include more than one power of the average component of the sum of the x components of the Pauli matrix.
  • the density matrix calculation unit 14 calculates the density matrix ⁇ hat for the Hamiltonian H hat shown in Expression (4).
  • the density matrix ⁇ hat can be expressed by Expression (7).
  • the probability distribution function followed by microscopic variables is called Gibbs-Boltzmann distribution, but in quantum mechanics, a density matrix replaced with a matrix is used instead of the probability distribution.
  • ⁇ is the reciprocal of the temperature T, as expressed by Expression (8), Z is the normalization constant expressed by Expression (9), and is called a partition function.
  • Tr is a symbol representing the sum (trace) of diagonal elements of the matrix.
  • the density matrix calculation unit 14 calculates the difference between the average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and the magnetization variable m x in the x direction as a variable, and the exponential operator including the initial Hamiltonian.
  • the product is used to calculate the density matrix for the initial Hamiltonian.
  • the density matrix of the initial Hamiltonian on the left side of Expression (11) can be expressed as on the right side of Expression (11) by using the delta function.
  • the magnetization variable m x is a physical quantity indicating how the spins are aligned as a whole in the Ising model.
  • Equation (12) is a delta function formula, and by using the equation (12), the right side of the equation (11) (initial Hamiltonian density matrix) can be expressed as the equation (13). That is, the density matrix calculation unit 14 calculates the density matrix of the initial Hamiltonian represented by the equation (13).
  • the x tilde indicates that it is an operator for the physical quantity x.
  • the density matrix with respect to the initial Hamiltonian is expressed by the equation (13), and the first-order term (that is, the higher-order term higher than the second-order term is not included) of the sum of x components of the Pauli matrix is used. ),
  • quantum mechanical effects antiferromagnetic XX interaction, etc.
  • other than the transverse magnetic field can be replaced with only the quantum mechanical effect due to the transverse magnetic field.
  • Equation (14) shows the density matrix for the Hamiltonian H hat (Summary of the target Hamiltonian H 0 hat and the initial Hamiltonian) when the Suzuki Trotter decomposition is performed.
  • ⁇ indicates the Trotter number.
  • FIG. 3 is a schematic diagram showing an example of Suzuki Trotter decomposition.
  • the horizontal axis represents the spin variable arranged at each site, and represents the so-called real space direction.
  • the vertical axis is the direction introduced by Suzuki Trotter decomposition (trotter direction), and the state variables are arranged on the two-dimensional lattice points. For example, ⁇ i1 , ..., ⁇ ik , ⁇ i (k + 1) , ..., ⁇ i ⁇ are arranged in the trotter direction with respect to the spin variable ⁇ i .
  • the quantum model can be considered to have been transformed into a classical model having a state space with an increased dimension by Suzuki Trotter decomposition.
  • the z component of the Pauli matrix can be transformed as shown in equation (15), and the x component of the Pauli matrix can be transformed as shown in equation (16).
  • the density matrix expressed by the equation (14) can be expressed by the equation (17). That is, the density matrix calculation unit 14 calculates the density matrix represented by Expression (17). It is possible to rewrite the optimization problem by expanding the Ising model in the virtual imaginary time direction (Trotter direction) by using Taylor expansion and algebraic correspondence.
  • ⁇ is an update variable when the iterative process is performed by the belief propagation algorithm described later, and the update variable ⁇ is expressed by Expression (18) as an update function ⁇ tanh ( ⁇ ⁇ m x tilde / ⁇ ) ⁇ , It can be obtained as the function value of the update function.
  • the belief propagation method (BP: Belief Propagation, also called the error propagation method) describes the dependency relationship between multiple random variables and density functions with a graph that connects nodes, and uses the graph structure to quickly calculate the probability distribution. It is inferred, and the global (overall) probability distribution is inferred by performing local message exchange and processing on the graph.
  • FIG. 4 is a schematic diagram showing an example of message processing in the interaction node.
  • FIG. 4 for convenience, four variable nodes and an interaction node ⁇ connected to the four variable nodes are illustrated.
  • the message M ⁇ ⁇ i from the interaction node ⁇ to the variable node i can be expressed by Expression (19).
  • ⁇ / i means “around ⁇ except i”
  • l ⁇ / i indicates variable nodes l1, l2, and l3 in the example of FIG.
  • the density function fu can be expressed by Expression (20), and as can be seen from Expression (1), the cost function of the optimization problem is expressed.
  • FIG. 5 is a schematic diagram showing an example of message processing in the variable node.
  • FIG. 5 for convenience, four interactions and variable nodes i connected to the four interaction nodes are illustrated.
  • the message Mi ⁇ ⁇ from the variable node i to the interaction node ⁇ can be expressed by equation (21).
  • ⁇ i / ⁇ means “around i except ⁇ ”, and ⁇ i / ⁇ indicates interaction nodes ⁇ 1, ⁇ 2, and ⁇ 3 in the example of FIG.
  • the density function fi can be expressed by Expression (22), and the variable node i propagates the sum of the messages M ⁇ ⁇ i to the interaction node ⁇ as the message Mi ⁇ ⁇ without any processing.
  • the value of the message converges, and the variable that minimizes (or maximizes) the cost function can be obtained.
  • FIG. 6 is a schematic diagram showing an example of an extended graph structure used in the belief propagation algorithm.
  • the diagram on the left side of FIG. 6 shows a graph structure in which four variable nodes are connected around one interaction node ⁇ .
  • One of the four variable nodes is represented by ⁇ i .
  • the z component of ⁇ i included in the target Hamiltonian H 0 hat expressed by Expression (3) and the x component of ⁇ i included in the initial Hamiltonian expressed by Expression (6) are defined.
  • the diagram on the right side of FIG. 6 is obtained by expanding the graph structure on the left side in the trotter direction (imaginary time direction) by Suzuki Trotter decomposition.
  • This is a copy of the graph structure in which four variable nodes are connected around one interaction node ⁇ by the Trotter number ⁇ .
  • variable node ⁇ i has the sum of ⁇ ik 's trotter number ⁇ (the z component of ⁇ i is replaced) and ⁇ ik ⁇ i (k + 1) There is a variable of the sum of ⁇ minutes of the trotter number (the x component of ⁇ i is replaced).
  • sigma ik sigma i (k + 1) represents the interaction of the k th variable node sigma i and (k + 1) of th variable node sigma i (k + 1).
  • FIG. 7 is a schematic diagram showing an example of message processing in an interaction node having an expanded graph structure.
  • the message M ⁇ ⁇ i from the interaction node ⁇ to the variable node i can be expressed by Expression (23).
  • equation (23) the density function fu can be expressed by equation (24).
  • the first operation unit 16 replaces the z component of the Pauli matrix with the spin variable ⁇ ik by Suzuki Trotter decomposition, and uses the exponential operator that includes the replaced spin variable ⁇ ik.
  • the density function fu as the first probability distribution function for the Hamiltonian is calculated.
  • the message Mi ⁇ ⁇ from the variable node i to the interaction node ⁇ can be expressed by equation (25).
  • the second operation unit 17 replaces the x component of the Pauli matrix with spin variables ⁇ ik and ⁇ i (k + 1) which are adjacent to each other along the trotter direction by Suzuki Trotter decomposition.
  • the density function fi as the second probability distribution function for the initial Hamiltonian is calculated using an exponential operator including a value obtained by summing the products of spin variables ⁇ ik ⁇ i (k + 1) over the trotter direction.
  • the interaction between ⁇ ik ⁇ i (k + 1) adjacent to each other along the trotter direction is considered.
  • the second calculation unit 17 may perform Suzuki Trotter decomposition on the density matrix calculated by the density matrix calculation unit 14 and represented by Expression (11) to calculate the density function fi as the second probability distribution function. it can.
  • the equation (26) includes the update variable ⁇ .
  • Update variables alpha, formula (18) update function represented by ⁇ tanh ( ⁇ ⁇ m x tilde / tau) ⁇ be a function value of the formula (18) is the m x tilde formula below (28) As shown in, it is a derivative g ′ (m x ) of the magnetic field function g (m x ) having the magnetization variable m x in the x direction as a variable. That is, the second calculation unit 17 uses the exponential operator that further includes the update variable ⁇ based on the update function having the derivative of the magnetic field function having the magnetization variable in the x direction as the variable, and the second probability for the initial Hamiltonian. The density function fi as a distribution function is calculated.
  • the belief propagation processing unit 19 generates the message M ⁇ ⁇ i at the interaction node as the first message represented by Expression (23).
  • the message M ⁇ ⁇ i at the interaction node contains a probability distribution function that sums the density function fu over the trotter direction.
  • the belief propagation processing unit 19 also generates the message Mi ⁇ ⁇ at the variable node as the second message represented by the equation (25).
  • the message Mi ⁇ ⁇ at the variable node contains a density function fi with an exponential operator containing the sum of the product of the spin variables ⁇ ik ⁇ i (k + 1) over the trotter direction.
  • the belief propagation processing unit 19 performs a process of repeating the calculation and propagation of the messages M ⁇ ⁇ i and Mi ⁇ ⁇ .
  • the expected value calculation unit 20 gives an appropriate initial value to each spin of the variable node, repeats the message calculation and propagation processing by the belief propagation processing unit 19, and the values of the messages M ⁇ ⁇ i and Mi ⁇ ⁇ converge. Then, a spin variable (expected value of spin configuration) that minimizes (or maximizes) the cost function for the optimization problem is calculated.
  • the magnetization variable calculation unit 21 sums the power operations with the update function as the base and the product ⁇ ik ⁇ i (k + 1) of the spin variables adjacent to each other along the trotter direction as the power index over the spin number and
  • the magnetization variable m x in the x direction is calculated by summing over the number.
  • the magnetization variable m x can be calculated by the equation (27).
  • tanh ( ⁇ ⁇ m x tilde / ⁇ ) is the update function.
  • the update variable ⁇ can be represented by a function value obtained by the logarithm of the update function.
  • m x tilde can be calculated by Expression (28).
  • the m x tilde can be represented by the derivative of the magnetic field function g (m x ) having the magnetization variable m x as a variable.
  • the magnetization variable m x is calculated by substituting ⁇ ik and ⁇ i (k + 1) (+1 or ⁇ 1) when the message is calculated and propagated into the equation (27).
  • the calculated magnetization variable m x is substituted into equation (28) to calculate the m x tilde.
  • the update variable ⁇ is calculated.
  • the message Mi ⁇ ⁇ and the message M ⁇ ⁇ i are calculated again, and the propagation processing is performed. Note that if the message Mi ⁇ ⁇ can be calculated, the message M ⁇ ⁇ i can be calculated by the equation (23).
  • the storage unit 22 can store input data, processing results obtained during simulation, output data, and the like.
  • FIG. 9 is a flowchart showing an example of the quantum annealing processing procedure by the simulation apparatus 100 of the present embodiment.
  • the control unit 10 acquires data representing the optimization problem with the Ising model (S11) and sets parameters (S12).
  • the parameter is data expressing a magnetic field function that determines the temperature and the quantum fluctuation action, and includes a transverse magnetic field, an antiferromagnetic XX interaction, a term of the third power or more of the sum of x components of the Pauli matrix, and the like.
  • the parameter includes the number of trotter.
  • the control unit 10 calculates the target Hamiltonian (S13) and the initial Hamiltonian (S14).
  • the target Hamiltonian can be calculated by the equation (3), and the initial Hamiltonian can be calculated by the equation (6), for example. It should be noted that the initial Hamiltonian is not limited to that expressed by the equation (6).
  • the initial Hamiltonian can also include terms of the third power or higher of the average component of the sum of the x components of the Pauli matrix.
  • the control unit 10 replaces the average component of the sum of the x components of the Pauli matrix of the initial Hamiltonian with the magnetization variable m x (S15), calculates the first probability distribution function for the target Hamiltonian (S16), and determines the second probability for the initial Hamiltonian.
  • a distribution function is calculated (S17).
  • the first probability distribution function is expressed by Expression (24), and the second probability distribution function is expressed by Expression (26).
  • the control unit 10 calculates the first message M ⁇ ⁇ i (S18).
  • the first message M ⁇ ⁇ i is represented by Expression (23).
  • the control unit 10 calculates the magnetization variable m x (S19).
  • the magnetization variable m x is represented by Expression (27).
  • the control unit 10 calculates the update variable ⁇ (S20).
  • the update variable ⁇ is represented by Expression (18).
  • the control unit 10 calculates the second message Mi ⁇ ⁇ (S21).
  • the second message Mi ⁇ ⁇ is represented by Expression (25).
  • the control unit 10 performs the belief propagation process using the first message and the second message (S22), calculates the expected value of the spin coordination when the value of the message converges (S23), and ends the process.
  • FIG. 10 is an explanatory diagram showing another example of the configuration of the simulation apparatus according to the present embodiment.
  • reference numeral 300 is an ordinary computer.
  • the computer 300 includes a control unit 30, an input unit 40, an output unit 50, an external I / F (interface) unit 60, and the like.
  • the control unit 30 includes a CPU 31, a ROM 32, a RAM 33, an I / F (interface) 34, and the like.
  • the input unit 40 acquires input data for simulation.
  • the output unit 50 outputs output data that is a simulation result.
  • the I / F 34 has an interface function between the control unit 30 and each of the input unit 40, the output unit 50, and the external I / F unit 60.
  • the external I / F unit 60 can read the computer program from a recording medium M (for example, a medium such as a DVD) recording the computer program.
  • a recording medium M for example, a medium such as a DVD
  • the computer program recorded in the recording medium M is not limited to the one recorded in a medium that can be carried around freely, and a computer program transmitted through the Internet or another communication line may be used. Can be included.
  • the computer includes a computer system including one computer equipped with a plurality of processors or a plurality of computers connected via a communication network.
  • the Ising model is extended in the imaginary time direction by Suzuki Trotter decomposition to rewrite the optimization problem, and the belief propagation algorithm is applied to the rewritten optimization problem. Since the extended level belief propagation algorithm is used, an approximate solution of a large-scale optimization problem (combinational optimization problem) can be obtained at high speed on a normal digital computer without going through a quantum annealing machine (quantum computer). be able to.
  • the quantum mechanical effect including the term of the third power or more of the average component of the sum of the x components of the Pauli matrix including the antiferromagnetic XX interaction is included.
  • the solution candidates (first-stage solutions) for the large-scale optimization problem obtained by the simulation apparatus and the simulation method according to the present embodiment can be changed only by the transverse magnetic field. It is also possible to provide a quantum computer capable of handling quantum mechanical effects and perform a two-step solution.
  • the simulation device of this embodiment can be implemented by a personal computer, workstation, server, GPU, FPGA, or the like.
  • a plurality of spins of the Ising model are represented by z components of the Pauli matrix, and the objective Hamiltonian expressing the optimization problem and the x component of the Pauli matrix corresponding to the plurality of spins are included in the quantum device.
  • a simulation device for simulating the expected value of the spin configuration of the target Hamiltonian by reducing quantum fluctuations over time, A z-component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first computing unit that computes a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable;
  • the x component of the matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the exponential operator containing the sum of the products of the replaced spin variables over the trotter direction is used to calculate the first Hamiltonian for the initial Hamiltonian.
  • a second calculation unit that calculates a probability distribution function, a first message that includes a probability distribution function that sums the first probability distribution functions in the trotter direction, and a second message that includes the second probability distribution function are used.
  • An expected value calculation unit that calculates an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm.
  • the computer program according to the present embodiment causes the computer to represent a plurality of spins of the Ising model by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and x components of the Pauli matrix corresponding to the plurality of spins.
  • a computer program that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with the passage of time, for a Hamiltonian that combines the initial Hamiltonian expressing the quantum fluctuation with a coefficient that changes with time.
  • the x-component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the product of the replaced spin variables is summed over the trotter direction using an exponential operator
  • a process of calculating a second probability distribution function for the Hamiltonian, a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction, and a second message including the second probability distribution function are used.
  • a process of calculating an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm are used.
  • a plurality of spins of the Ising model are represented by z components of the Pauli matrix, and the objective Hamiltonian expressing the optimization problem and the x component of the Pauli matrix corresponding to the plurality of spins are included in the quantum.
  • a simulation method for simulating the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with the passage of time, with respect to the Hamiltonian combining the initial Hamiltonian expressing fluctuation and the coefficient with time change The z component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and the first probability distribution function for the target Hamiltonian is calculated using an exponential operator including the replaced spin variable, and the x component of the Pauli matrix is calculated.
  • the second probability distribution function for the initial Hamiltonian is calculated by using an exponential operator including Suzuki spin trotter decomposition to replace adjacent spin variables along the trotter direction, and including the sum of the products of the replaced spin variables over the trotter direction.
  • the spin distribution of the target Hamiltonian is calculated by a belief propagation algorithm using a first message including a probability distribution function that is calculated by summing the first probability distribution function in the trotter direction and a second message including the second probability distribution function. Calculate the expected value of rank.
  • the first calculation unit replaces the z component of the Pauli matrix with a spin variable by Suzuki Trotter decomposition, and calculates the first probability distribution function fu for the target Hamiltonian using an exponential operator that includes the replaced spin variable ⁇ i .
  • the second arithmetic unit replaces the x component of the Pauli matrix with spin variables ⁇ ik and ⁇ i (k + 1) that are adjacent to each other along the trotter direction by Suzuki Trotter decomposition, and the product ⁇ ik ⁇ i (k ) of the replaced spin variables.
  • the second probability distribution function fi with respect to the initial Hamiltonian is calculated using an exponential operator including the sum of ( +1) over the trotter direction.
  • the expected value calculation unit uses the belief propagation algorithm using the first message including the probability distribution function obtained by summing the first probability distribution functions fu in the trotter direction and the second message including the second probability distribution function fi to calculate the target Hamiltonian of the target Hamiltonian. Calculate the expected value of spin coordination.
  • a site where spins are arranged is a variable node i, an interaction between spins is an interaction node ⁇ , and in a belief propagation algorithm on a graph structure connecting the variable node i and the interaction node ⁇ , a variable is generated from the interaction node ⁇ .
  • the message to the node i is the first message M ⁇ ⁇ i
  • the message from the variable node i to the interaction node ⁇ is the second message Mi ⁇ ⁇ .
  • the first message M ⁇ ⁇ i includes a probability distribution function in which the first probability distribution function fu at the interaction node ⁇ is summed over the trotter direction.
  • the second message Mi ⁇ ⁇ includes the second probability distribution function fi at the variable node i.
  • the quantum mechanical effect (quantum fluctuation action) represented by the initial Hamiltonian having the x component of the Pauli matrix can be replaced with the trotter interaction, and the numerical calculation can be executed.
  • the Ising model is extended in the virtual imaginary time direction (trotter direction), and the optimal propagation is performed by using the belief propagation algorithm (extended level belief propagation algorithm) on the extended graph structure.
  • the approximate solution of the optimization problem can be obtained at high speed.
  • the simulation apparatus of the present embodiment includes a delta function having a difference between an average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and a magnetization variable in the x direction as a variable, and an exponential operator including the initial Hamiltonian. And a density matrix calculation unit that calculates a density matrix for the initial Hamiltonian using a product of and, and the second calculation unit performs Suzuki Trotter decomposition on the density matrix to calculate the second probability distribution function. .
  • the density matrix calculation unit uses a product of a delta function whose variable is the difference between the average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and the magnetization variable in the x direction, and an exponential operator including the initial Hamiltonian. Compute the density matrix for the initial Hamiltonian.
  • the density matrix for the initial Hamiltonian can be expressed only by the first-order term (that is, the second-order or higher-order terms are not included) of the sum of the x components of the Pauli matrix, and the quantum mechanical effects other than the transverse magnetic field ( Antiferromagnetic XX interaction, etc.) can be replaced only by the quantum mechanical effect due to the transverse magnetic field.
  • the second calculation unit performs Suzuki Trotter decomposition on the density matrix to calculate the second probability distribution function fi.
  • quantum mechanical effects antiferromagnetic XX interaction, etc.
  • the initial Hamiltonian having the x component of the Pauli matrix can be replaced with only the quantum mechanical effect due to the transverse magnetic field.
  • the second calculation unit uses an exponential operator that further includes an update variable based on an update function having a derivative of a magnetic field function having a magnetization variable in the x direction as a variable.
  • a second probability distribution function for the initial Hamiltonian is calculated.
  • a variable e.g., ⁇ tanh (beta ⁇ g' of the magnetic field function g whose variable is the x direction of magnetization variable m x (m x) ( m x) / ⁇
  • Trotter interactions ⁇ ik ⁇ i (k + 1 ) is changed and change the magnetization variable m x, 'changes the value of (m x), the derivative g' the magnetization variable m x varies derivative g (m x ),
  • the update variable ⁇ changes.
  • the second probability distribution function fi changes, and when the second probability distribution function fi changes, the second message Mi ⁇ ⁇ and the first message M ⁇ ⁇ i change.
  • the second message Mi ⁇ ⁇ and the first message M ⁇ ⁇ i change, the trotter interaction ⁇ ik ⁇ i (k + 1) changes.
  • the expected value of the spin configuration of each spin variable can be calculated.
  • the simulation device of the present embodiment has the update function as a base, and sums over the number of spins the power operation with the exponent of the product of spin variables adjacent to each other along the trotter direction as the power exponent. And a magnetization variable calculation unit that calculates a magnetization variable in the x direction.
  • the magnetization variable calculator calculates the product ⁇ ik ⁇ i (k + 1) of the spin variables adjacent to each other along the trotter direction with the update function (for example, ⁇ tanh ( ⁇ ⁇ m x tilde / ⁇ ) ⁇ as the base).
  • the update function for example, ⁇ tanh ( ⁇ ⁇ m x tilde / ⁇ ) ⁇ as the base.
  • m x tilde g ′ (m x G (m x ) is a magnetic field function, which makes it possible to formulate that the magnetization variable m x changes when the trotter interaction ⁇ ik ⁇ i (k + 1) changes.
  • the magnetic field function includes a power of 2 or more of the average component of the sum of x components of the Pauli matrix.
  • the magnetic field function g (m x ) includes a term that is a power of 2 or more of the average component of the sum of x components of the Pauli matrix.
  • the quantum mechanical effect can be an effect due to antiferromagnetic XX interaction.
  • the extended belief propagation algorithm on the graph structure composed of the variable nodes and the interaction nodes has been described, but the graph structure is not limited to this, and for example, Bayesian network. Also in, the extended belief propagation algorithm can be applied to obtain similar results.

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Abstract

Provided are a simulation device, a computer program, and a simulation method, with which an approximate solution for an optimization problem can be determined rapidly.　The simulation device is equipped with: a first computation unit for substituting a z-component of a Pauli matrix with a spin variable and using an index function operator in which the substituted spin variable is included to compute a first probability distribution function for an intended Hamiltonian; a second computation unit for substituting an x-component of the Pauli matrix with a spin variable that is adjacent in the Trotter direction and using an index function operator that includes a value obtained by totaling a product of the substituted spin variable across the Trotter direction to compute a second probability distribution function for an initial Hamiltonian; and an expected value calculation unit for calculating an expected value of the spin configuration of the intended Hamiltonian by means of a probability propagation algorithm that uses a first message, which includes a probability distribution function in which a first probability distribution function is totaled across the Trotter direction, and a second message, which includes a second probability distribution function.

Description

シミュレーション装置、コンピュータプログラム及びシミュレーション方法Simulation device, computer program, and simulation method
　本発明は、シミュレーション装置、コンピュータプログラム及びシミュレーション方法に関する。 The present invention relates to a simulation device, a computer program, and a simulation method.
　最適化問題に対して、量子アニーリングと呼ばれる汎用的解法が存在する。量子アニーリングでは、解きたい最適化問題に対して、重ね合わせの状態を生み出す量子揺らぎを印加する。さらに、量子揺らぎの導入が可能な量子コンピュータが開発されているが、扱えるビット数が、例えば２０４８量子ビットであり、スピン数は２０４８個までに限られ、また２体相互作用しか考慮されていないので、実用上の制限が大きい。このため、大規模な実問題に対しては、通常のデジタルコンピュータ上でのシミュレーションが鍵となる。 There is a general-purpose solution called quantum annealing for the optimization problem. In quantum annealing, quantum fluctuations that produce superposition are applied to the optimization problem to be solved. Furthermore, a quantum computer capable of introducing quantum fluctuation has been developed, but the number of bits that can be handled is, for example, 2048 qubits, the number of spins is limited to 2048, and only two-body interaction is considered. Therefore, the practical limit is large. Therefore, the simulation on a normal digital computer is the key for large-scale real problems.
　特許文献１には、従来、デジタルコンピュータ上ではシミュレーションすることができなかった、反強磁性ＸＸ相互作用、いわゆる横磁場以外の量子力学的効果が設定されたシミュレーションをデジタルコンピュータ上で実施できる適応的量子モンテカルロ法が開示されている。 In Patent Document 1, it is possible to perform, on a digital computer, a simulation in which a quantum mechanical effect other than a so-called transverse magnetic field, which has been conventionally impossible to simulate on a digital computer, is set. The quantum Monte Carlo method is disclosed.
特開２０１８－０６７２００号公報JP, 2008-067200, A
　しかし、特許文献１の方法は、精度の良い解は得られるものの、大規模な最適化問題では、計算時間が長くなる。このため、精度は多少犠牲にしても短時間で計算結果が得られることが望まれる。 However, although the method of Patent Document 1 can obtain an accurate solution, it requires a long calculation time for a large-scale optimization problem. Therefore, it is desired that the calculation result can be obtained in a short time even if the accuracy is somewhat sacrificed.
　本発明は斯かる事情に鑑みてなされたものであり、最適化問題の近似的解を高速で求めることができるシミュレーション装置、コンピュータプログラム及びシミュレーション方法を提供することを目的とする。 The present invention has been made in view of such circumstances, and an object thereof is to provide a simulation device, a computer program, and a simulation method that can obtain an approximate solution of an optimization problem at high speed.
　本発明の実施の形態に係るシミュレーション装置は、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション装置であって、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する第１演算部と、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する第２演算部と、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する期待値算出部とを備える。 In the simulation device according to the embodiment of the present invention, a plurality of spins of the Ising model are represented by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and an x component of the Pauli matrix corresponding to the plurality of spins. For a Hamiltonian that combines an initial Hamiltonian that expresses a quantum fluctuation with a coefficient that changes with time, a simulation device that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with time. Then, a z-component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first computing unit that computes a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable, , The x-component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the product of the replaced spin variables is summed over the trotter direction using an exponential operator A second computing unit that computes a second probability distribution function for the Hamiltonian, a first message that includes a probability distribution function that sums the first probability distribution functions in the trotter direction, and a second message that includes the second probability distribution function. And an expected value calculation unit that calculates an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm using.
　本発明の実施の形態に係るコンピュータプログラムは、コンピュータに、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートさせるコンピュータプログラムであって、コンピュータに、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する処理と、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する処理と、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する処理とを実行させる。 A computer program according to an embodiment of the present invention causes a computer to represent a plurality of spins of the Ising model by z components of a Pauli matrix, an objective Hamiltonian expressing an optimization problem, and a Pauli matrix corresponding to the plurality of spins. For the Hamiltonian in which the initial Hamiltonian expressing the quantum fluctuation including the x component of and the coefficient with time change are combined, the quantum fluctuation is reduced over time to simulate the expected value of the spin configuration of the target Hamiltonian. A computer program, wherein a z component of the Pauli matrix is replaced by a spin variable by Suzuki Trotter decomposition, and a first probability distribution function for the target Hamiltonian is calculated using an exponential operator including the replaced spin variable. And the x component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and an exponential operator including the sum of products of the replaced spin variables over the trotter direction is used. Calculating a second probability distribution function for the initial Hamiltonian, a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction, and a second message including the second probability distribution function. And a process of calculating an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm using.
　本発明の実施の形態に係るシミュレーション方法は、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション方法であって、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算し、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算し、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する。 In the simulation method according to the embodiment of the present invention, a plurality of spins of the Ising model are represented by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and an x component of the Pauli matrix corresponding to the plurality of spins. For a Hamiltonian that combines an initial Hamiltonian that expresses a quantum fluctuation with a coefficient that changes with time, a simulation method that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation over time Then, the z component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first probability distribution function for the target Hamiltonian is calculated by using an exponential operator including the replaced spin variable, and the Pauli matrix A second probability for the initial Hamiltonian using the exponential operator that replaces the x component with adjacent spin variables along the Trotter direction by Suzuki Trotter decomposition and includes the sum of the products of the replaced spin variables over the Trotter direction. The target Hamiltonian is calculated by a belief propagation algorithm using a first message including a probability distribution function obtained by calculating a distribution function and summing the first probability distribution functions in the trotter direction, and a second message including the second probability distribution function. Calculate the expected value of spin coordination of.
　本発明によれば、大規模な最適化問題の近似的解を高速に算出することができる。 According to the present invention, an approximate solution of a large-scale optimization problem can be calculated at high speed.
本実施の形態のシミュレーション装置の構成の一例を示す説明図である。It is explanatory drawing which shows an example of a structure of the simulation apparatus of this Embodiment. イジング模型のスピン間の相互作用の一例を示す模式図である。It is a schematic diagram which shows an example of the interaction between spins of an Ising model. 鈴木トロッター分解の一例を示す模式図である。It is a schematic diagram which shows an example of Suzuki Trotter decomposition. 相互作用ノードでのメッセージの処理の一例を示す模式図である。It is a schematic diagram which shows an example of the process of the message in an interaction node. 変数ノードでのメッセージの処理の一例を示す模式図である。It is a schematic diagram which shows an example of the process of the message in a variable node. 確率伝播アルゴリズムに用いる拡張されたグラフ構造の一例を示す模式図である。It is a schematic diagram which shows an example of the extended graph structure used for a belief propagation algorithm. 拡張されたグラフ構造の相互作用ノードでのメッセージの処理の一例を示す模式図である。It is a schematic diagram which shows an example of the process of the message in the interaction node of the extended graph structure. 拡張されたグラフ構造の変数ノードでのメッセージの処理の一例を示す模式図である。It is a schematic diagram which shows an example of the process of the message in the variable node of the extended graph structure. 本実施の形態のシミュレーション装置による量子アニーリング処理手順の一例を示すフローチャートである。7 is a flowchart showing an example of a quantum annealing processing procedure by the simulation device of the present embodiment. 本実施の形態のシミュレーション装置の構成の他の例を示す説明図である。It is explanatory drawing which shows the other example of a structure of the simulation apparatus of this Embodiment.
　以下、本発明をその実施の形態を示す図面に基づいて説明する。図１は本実施の形態のシミュレーション装置１００の構成の一例を示す説明図である。シミュレーション装置１００は、装置全体を制御する制御部１０、入力部１１、目的ハミルトニアン演算部１２、初期ハミルトニアン演算部１３、密度行列算出部１４、確率分布関数演算部１５、出力部１８、確率伝播処理部１９、期待値算出部２０、磁化変数算出部２１、及び記憶部２２を備える。確率分布関数演算部１５は、第１演算部１６及び第２演算部１７を備える。 Hereinafter, the present invention will be described based on the drawings showing an embodiment thereof. FIG. 1 is an explanatory diagram showing an example of the configuration of the simulation apparatus 100 according to the present embodiment. The simulation device 100 includes a control unit 10 that controls the entire device, an input unit 11, a target Hamiltonian calculation unit 12, an initial Hamiltonian calculation unit 13, a density matrix calculation unit 14, a probability distribution function calculation unit 15, an output unit 18, and a belief propagation process. It includes a unit 19, an expected value calculation unit 20, a magnetization variable calculation unit 21, and a storage unit 22. The probability distribution function calculator 15 includes a first calculator 16 and a second calculator 17.
　入力部１１は、シミュレーションを実行するための入力データ及びパラメータを取得する。入力データは、例えば、最適化問題（「組み合せ最適化問題」ともいう）をイジング模型に表現（翻訳）したデータである。イジング模型は、磁性体の振る舞いを記述する数理的な模型であり、±１という２つの値をとるスピン変数σi (＝±１)がサイトと称される格子点に配置され、この変数σi は磁気モーメントの向きを示す微視的変数として利用される。パラメータは、例えば、温度、横磁場による量子力学的効果、横磁場以外の量子力学的効果、トロッター数などを含む。 The input unit 11 acquires input data and parameters for executing a simulation. The input data is, for example, data expressing (translating) an optimization problem (also referred to as “combination optimization problem”) in the Ising model. The Ising model is a mathematical model that describes the behavior of a magnetic substance. A spin variable σ i (= ± 1) that takes two values of ± 1 is placed at a lattice point called a site, and this variable σ i is used as a microscopic variable that indicates the direction of the magnetic moment. The parameters include, for example, temperature, a quantum mechanical effect due to a transverse magnetic field, a quantum mechanical effect other than the transverse magnetic field, a Trotter number, and the like.
　出力部１８は、シミュレーションの結果である出力データを出力する。出力データは、最適化問題が表現されたイジング模型の各スピン変数の期待値（スピン配位の期待値）である。 The output unit 18 outputs the output data that is the result of the simulation. The output data is the expected value (spin configuration expected value) of each spin variable of the Ising model in which the optimization problem is expressed.
　以下、目的ハミルトニアンの算出方法について説明する。 The following explains the calculation method of the objective Hamiltonian.
　式（１）は、最適化問題をイジング模型で表現したものであり、コスト関数とも称される。 Formula (1) expresses the optimization problem with the Ising model and is also called a cost function.
Figure JPOXMLDOC01-appb-M000001
Figure JPOXMLDOC01-appb-M000001
　式（１）において、σi は二値変数（例えば、±１）であり、添え字のｉはイジング模型の自由度であるスピンが配置された場所を表す。μはスピン間の相互作用を表す添え字であり、Ｊμは相互作用の強さを表す。また、ｉ∈∂μは、「μの周りのｉ」という意味である。式（１）の意味を簡単な例で説明する。 In Expression (1), σ i is a binary variable (for example, ± 1), and the subscript i represents the place where the spin, which is the degree of freedom of the Ising model, is arranged. μ is a subscript indicating the interaction between spins, and J μ is the strength of the interaction. Further, iε∂μ means “i around μ”. The meaning of expression (1) will be described with a simple example.
　図２はイジング模型のスピン間の相互作用の一例を示す模式図である。図２に示すように、スピンをσ1 からσ8 で表す（スピン数８）。σ1 からσ4 の相互作用をμとし、σ4  とσ5 の相互作用をμ′とし、σ4 及びσ6 からσ8 の相互作用をμ′′とすると、イジング模型のハミルトニアン（エネルギー）は、－Ｊμσ1 σ2 σ3 σ4－Ｊμ′σ4 σ5－Ｊμ′′σ4 σ6σ7σ8で表すことができ、このコスト関数を最小（または最大）にすることにより最適化問題を解くことができる。図２からも分かるように、本実施の形態では、２体相互作用（例えば、Ｊμσ1 σ2のように、２つのスピンの積で表すことができる）に限定されず、３体以上の相互作用を扱うことができる。また、スピン数Ｎもデジタルコンピュータのメモリ容量を増加すれば、数千以上のスピン数を扱うことができ、大規模な最適化問題を解くことができる。 FIG. 2 is a schematic diagram showing an example of interaction between spins of the Ising model. As shown in FIG. 2, spins are represented by σ 1 to σ 8 (spin number 8). If the interaction between σ 1 and σ 4 is μ, the interaction between σ 4 and σ 5 is μ ′, and the interaction between σ 4 and σ 6 and σ 8 is μ ″, then the Ising model Hamiltonian (energy) Can be expressed as −J μ σ 1 σ 2 σ 3 σ 4 −J μ ′ σ 4 σ 5 −J μ ′ ′ σ 4 σ 6 σ 7 σ 8 and this cost function can be minimized (or maximized). By doing so, the optimization problem can be solved. As can be seen from FIG. 2, the present embodiment is not limited to the two-body interaction (for example, it can be represented by the product of two spins like J μ σ 1 σ 2 ), and three or more bodies are used. Can handle the interaction of. Also, with respect to the spin number N, if the memory capacity of the digital computer is increased, it is possible to handle spin numbers of several thousands or more, and it is possible to solve a large-scale optimization problem.
　量子アニーリングでは、最適化問題に対して、重ね合わせの状態を生み出す量子揺らぎを印加する。重ね合わせの状態とは、２つの異なる状態１、２（例えば、スピンσ＝＋１と、σ＝－１）が同時に存在するという量子力学的な状態であり、例えば、繰り返し観測を行った場合に、ある観測時には状態1が観測され、別の観測時には状態２が観測され、状態１、２がそれぞれある確率で存在するという意味である。量子揺らぎを印加するために、以下のような変換処理を行う。すなわち、イジング模型のスピンの二値変数σi （＝±１）を、式（２）で示すパウリ行列のｚ成分に対応付ける。 In quantum annealing, quantum fluctuations that produce superposed states are applied to the optimization problem. The superposition state is a quantum mechanical state in which two different states 1 and 2 (for example, spin σ = + 1 and σ = −1) exist at the same time. For example, when repeated observation is performed, , State 1 is observed during one observation, state 2 is observed during another observation, and states 1 and 2 exist with a certain probability. In order to apply quantum fluctuation, the following conversion process is performed. That is, the binary variable σ i (= ± 1) of the spin of the Ising model is associated with the z component of the Pauli matrix shown in Expression (2).
Figure JPOXMLDOC01-appb-M000002
Figure JPOXMLDOC01-appb-M000002
　目的ハミルトニアン演算部１２は、入力された最適化問題に基づいて、目的ハミルトニアンＨ0 ハットを算出する。具体的には、目的ハミルトニアン演算部１２は、式（１）において、式（２）を用いて、式（３）で表される目的ハミルトニアンＨ0 ハットを算出する。なお、ハット（∧）は行列であることを示す。 The target Hamiltonian calculator 12 calculates the target Hamiltonian H 0 hat based on the input optimization problem. Specifically, the target Hamiltonian calculation unit 12 calculates the target Hamiltonian H 0 hat represented by Expression (3) using Expression (2) in Expression (1). The hat (∧) indicates that it is a matrix.
　初期ハミルトニアン演算部１３は、式（４）の右辺第２項で示すような初期ハミルトニアンを算出する。初期ハミルトニアンは、式（４）の右辺第２項のように、磁場関数ｇで表すことができ、磁場関数ｇの変数には、複数のスピンσi （＝±１）に対応させて、式（５）に示すパウリ行列のｘ成分を含む。パウリ行列のｘ成分は、非対角成分を持っているので、ｚ方向に向いたスピンを反転させることができ、量子揺らぎを生み出すことができる。式（４）において、Ｎはスピン数である。 The initial Hamiltonian calculation unit 13 calculates an initial Hamiltonian as indicated by the second term on the right side of Expression (4). The initial Hamiltonian can be expressed by the magnetic field function g as in the second term on the right side of the equation (4), and the variable of the magnetic field function g is associated with a plurality of spins σ i (= ± 1). The x component of the Pauli matrix shown in (5) is included. Since the x component of the Pauli matrix has a non-diagonal component, spins directed in the z direction can be inverted and quantum fluctuations can be produced. In the equation (4), N is the spin number.
　式（６）は、初期ハミルトニアン演算部１３が算出する、初期ハミルトニアンの具体例を示す。式（６）の右辺第１項は、パウリ行列のｘ成分の和の平均成分の一次項を含み、横磁場による量子力学的効果を与えることができる。係数Γは、量子揺らぎの強さを制御するパラメータである。量子アニーリングにおいて、係数Γは、時間の経過とともに小さくなるように制御する。式（６）の右辺第２項は、パウリ行列のｘ成分の和の平均成分の二次項を含み、いわゆる、反強磁性ＸＸ相互作用を表し、横磁場以外の量子力学的効果を与えることができる。γは所定の係数である。なお、磁場関数は、式（６）に限定されるものではない。例えば、磁場関数は、パウリ行列のｘ成分の和の平均成分の２以上の累乗を含めることもできる。 Expression (6) shows a specific example of the initial Hamiltonian calculated by the initial Hamiltonian operation unit 13. The first term on the right side of Expression (6) includes the first-order term of the average component of the sum of the x components of the Pauli matrix, and can give a quantum mechanical effect due to the transverse magnetic field. The coefficient Γ is a parameter that controls the strength of quantum fluctuations. In the quantum annealing, the coefficient Γ is controlled so as to decrease with the passage of time. The second term on the right side of the equation (6) includes a quadratic term of the average component of the sum of x components of the Pauli matrix, represents a so-called antiferromagnetic XX interaction, and may give a quantum mechanical effect other than the transverse magnetic field. it can. γ is a predetermined coefficient. The magnetic field function is not limited to equation (6). For example, the magnetic field function may include more than one power of the average component of the sum of the x components of the Pauli matrix.
　式（４）において、初期状態（時刻ｔ＝０）では、係数Γを非常に大きな値とし、時間の経過とともに係数Γを小さくし、最終的には０にする。最初は大きな量子揺らぎによって数多くの状態の重ね合わせを実現して状態探査をする。各時刻における瞬間的な基底状態を連続的にたどり、次第にΓが小さくなると、初期ハミルトニアンに比べて目的ハミルトニアンの相対的な重みが大きくなり、最終的には、目的ハミルトニアンの基底状態に到達する。この状態で、最適化問題の解が得られ、スピン配位の期待値を算出することができる。 In the equation (4), in the initial state (time t = 0), the coefficient Γ is set to a very large value, the coefficient Γ is reduced with the passage of time, and finally set to 0. At the beginning, state exploration is performed by realizing superposition of many states by large quantum fluctuations. When the instantaneous ground state at each time is continuously traced and Γ gradually decreases, the relative weight of the target Hamiltonian becomes larger than that of the initial Hamiltonian, and finally reaches the ground state of the target Hamiltonian. In this state, the solution of the optimization problem is obtained, and the expected value of the spin configuration can be calculated.
　密度行列算出部１４は、式（４）で示すハミルトニアンＨハットについて、密度行列ρハットを算出する。密度行列ρハットは、式（７）で表すことができる。微視的変数が従う確率分布関数は、ギブスボルツマン分布と呼ばれるが、量子力学では、確率分布の代わりに行列に置き換えられた密度行列が用いられる。 The density matrix calculation unit 14 calculates the density matrix ρ hat for the Hamiltonian H hat shown in Expression (4). The density matrix ρ hat can be expressed by Expression (7). The probability distribution function followed by microscopic variables is called Gibbs-Boltzmann distribution, but in quantum mechanics, a density matrix replaced with a matrix is used instead of the probability distribution.
Figure JPOXMLDOC01-appb-M000003
Figure JPOXMLDOC01-appb-M000003
　式（７）において、βは、式（８）で表すように、温度Ｔの逆数であり、Ｚは式（９）で表される規格化定数であり、分配関数と呼ばれる。式（９）において、Ｔｒは行列の対角成分の和（トレース）を表す記号である。式（７）のＨハットに式（４）を代入すると式（１０）が得られる。 In Expression (7), β is the reciprocal of the temperature T, as expressed by Expression (8), Z is the normalization constant expressed by Expression (9), and is called a partition function. In Expression (9), Tr is a symbol representing the sum (trace) of diagonal elements of the matrix. By substituting the equation (4) into the H hat of the equation (7), the equation (10) is obtained.
　密度行列算出部１４は、初期ハミルトニアンのパウリ行列のｘ成分の和の平均成分とｘ方向の磁化変数ｍx との差を変数とするデルタ関数δと、初期ハミルトニアンを含む指数関数演算子との積を用いて、初期ハミルトニアンに対する密度行列を算出する。具体的には、式（１１）の左辺の初期ハミルトニアンの密度行列は、デルタ関数を用いることにより、式（１１）の右辺のように表すことができる。磁化変数ｍx は、イジング模型においてスピンが全体としてどれだけ揃っているかを示す物理量である。 The density matrix calculation unit 14 calculates the difference between the average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and the magnetization variable m x in the x direction as a variable, and the exponential operator including the initial Hamiltonian. The product is used to calculate the density matrix for the initial Hamiltonian. Specifically, the density matrix of the initial Hamiltonian on the left side of Expression (11) can be expressed as on the right side of Expression (11) by using the delta function. The magnetization variable m x is a physical quantity indicating how the spins are aligned as a whole in the Ising model.
　式（１２）はデルタ関数の公式であり、式（１２）を用いることにより、式（１１）の右辺（初期ハミルトニアンの密度行列）は、式（１３）のように表すことができる。すなわち、密度行列算出部１４は、式（１３）で表す、初期ハミルトニアンの密度行列を算出する。式（１２）において、なお、ｘチルダは、物理量ｘに対する演算子であることを示す。上述のように、フーリエ積分を利用して、式（１３）で示すように、初期ハミルトニアンに対する密度行列が、パウリ行列のｘ成分の和の一次項（すなわち、二次項以上の高次項を含まない）だけで表現することができ、横磁場以外の量子力学的効果（反強磁性ＸＸ相互作用など）を横磁場による量子力学的効果のみに置き換えることができる。 [Equation (12) is a delta function formula, and by using the equation (12), the right side of the equation (11) (initial Hamiltonian density matrix) can be expressed as the equation (13). That is, the density matrix calculation unit 14 calculates the density matrix of the initial Hamiltonian represented by the equation (13). In Expression (12), the x tilde indicates that it is an operator for the physical quantity x. As described above, using the Fourier integral, the density matrix with respect to the initial Hamiltonian is expressed by the equation (13), and the first-order term (that is, the higher-order term higher than the second-order term is not included) of the sum of x components of the Pauli matrix is used. ), And quantum mechanical effects (antiferromagnetic XX interaction, etc.) other than the transverse magnetic field can be replaced with only the quantum mechanical effect due to the transverse magnetic field.
　式（１４）は、鈴木トロッター分解したときのハミルトニアンＨハット（目的ハミルトニアンＨ0 ハットと初期ハミルトニアンとの和）に対する密度行列を示す。式（１４）において、τはトロッター数を示す。 Equation (14) shows the density matrix for the Hamiltonian H hat (Summary of the target Hamiltonian H 0 hat and the initial Hamiltonian) when the Suzuki Trotter decomposition is performed. In Expression (14), τ indicates the Trotter number.
Figure JPOXMLDOC01-appb-M000004
Figure JPOXMLDOC01-appb-M000004
　図３は鈴木トロッター分解の一例を示す模式図である。図３において、横軸は各サイトに配置されたスピン変数を表し、いわゆる実空間方向を示す。縦軸は鈴木トロッター分解によって導入された方向（トロッター方向）であり、２次元の格子点上に状態変数が配置される。例えば、スピン変数σi に対して、トロッター方向に向かって、σi1、…、σik、σi(k+1)、…、σiτが配置される。このように、鈴木トロッター分解によって量子モデルは、次元が一つ増えた状態空間を持つ古典モデルに変換されたと考えることができる。 FIG. 3 is a schematic diagram showing an example of Suzuki Trotter decomposition. In FIG. 3, the horizontal axis represents the spin variable arranged at each site, and represents the so-called real space direction. The vertical axis is the direction introduced by Suzuki Trotter decomposition (trotter direction), and the state variables are arranged on the two-dimensional lattice points. For example, σ i1 , ..., σ ik , σ i (k + 1) , ..., σ i τ are arranged in the trotter direction with respect to the spin variable σ i . In this way, the quantum model can be considered to have been transformed into a classical model having a state space with an increased dimension by Suzuki Trotter decomposition.
　鈴木トロッター分解により、パウリ行列のｚ成分は、式（１５）のように変換することができ、パウリ行列のｘ成分は、式（１６）のように変換することができる。式（１５）及び式（１６）を用いることにより、式（１４）で表した密度行列は、式（１７）のように表すことができる。すなわち、密度行列算出部１４は、式（１７）で表す密度行列を算出する。テイラー展開、代数的対応関係を用いてイジング模型を仮想的な虚時間方向（トロッター方向）へ拡張して、最適化問題を書き換えることができる。式（１７）において、αは、後述の確率伝播アルゴリズムで繰り返し処理を行う際の更新変数であり、更新変数αは、式（１８）のように更新関数{ tanh（β・ｍxチルダ／τ）}に基づいて、更新関数の関数値として求めることができる。 By Suzuki Trotter decomposition, the z component of the Pauli matrix can be transformed as shown in equation (15), and the x component of the Pauli matrix can be transformed as shown in equation (16). By using the equations (15) and (16), the density matrix expressed by the equation (14) can be expressed by the equation (17). That is, the density matrix calculation unit 14 calculates the density matrix represented by Expression (17). It is possible to rewrite the optimization problem by expanding the Ising model in the virtual imaginary time direction (Trotter direction) by using Taylor expansion and algebraic correspondence. In Expression (17), α is an update variable when the iterative process is performed by the belief propagation algorithm described later, and the update variable α is expressed by Expression (18) as an update function {tanh (β · m x tilde / τ )}, It can be obtained as the function value of the update function.
　次に、確率伝播アルゴリズムについて説明する。確率伝播法（ＢＰ：Belief Propagation、誤差伝播法ともいう）は、複数の確率変数と密度関数との依存関係をノードで接続したグラフで記述し、そのグラフ構造を利用して高速に確率分布を推論するものであり、そのグラフ上での局所的なメッセージの交換及び処理を行うことにより大域的（全体）の確率分布を推論する。 Next, the belief propagation algorithm will be explained. The belief propagation method (BP: Belief Propagation, also called the error propagation method) describes the dependency relationship between multiple random variables and density functions with a graph that connects nodes, and uses the graph structure to quickly calculate the probability distribution. It is inferred, and the global (overall) probability distribution is inferred by performing local message exchange and processing on the graph.
　図４は相互作用ノードでのメッセージの処理の一例を示す模式図である。図４では、便宜上、４つの変数ノードと、４つの変数ノードに繋がる相互作用ノードμとを図示している。相互作用ノードμから変数ノードｉへのメッセージＭμ→ｉは、式（１９）で表すことができる。 FIG. 4 is a schematic diagram showing an example of message processing in the interaction node. In FIG. 4, for convenience, four variable nodes and an interaction node μ connected to the four variable nodes are illustrated. The message Mμ → i from the interaction node μ to the variable node i can be expressed by Expression (19).
Figure JPOXMLDOC01-appb-M000005
Figure JPOXMLDOC01-appb-M000005
　式（１９）において、∂μ／ｉは、「ｉを除くμの周り」という意味であり、ｌ∈∂μ／ｉは、図４の例では、変数ノードｌ１、ｌ２、ｌ３を示す。図４に示すように、メッセージＭμ→ｉは、変数ノードｌ（＝ｌ１、ｌ２、ｌ３）から相互作用ノードμへのメッセージＭｌ→μの和に、相互作用ノードμの密度関数ｆｕを積算して求めることができる。 In Expression (19), ∂μ / i means “around μ except i”, and lε∂μ / i indicates variable nodes l1, l2, and l3 in the example of FIG. As shown in FIG. 4, the message Mμ → i is obtained by multiplying the sum of the message Ml → μ from the variable node l (= l1, l2, l3) to the interaction node μ by the density function fu of the interaction node μ. Can be asked.
　密度関数ｆｕは、式（２０）で表すことができ、式（１）から分かるように、最適化問題のコスト関数が表現されている。 The density function fu can be expressed by Expression (20), and as can be seen from Expression (1), the cost function of the optimization problem is expressed.
　図５は変数ノードでのメッセージの処理の一例を示す模式図である。図５では、便宜上、４つの相互作用と、４つの相互作用ノードに繋がる変数ノードｉとを図示している。変数ノードｉから相互作用ノードμへのメッセージＭｉ→μは、式（２１）で表すことができる。 FIG. 5 is a schematic diagram showing an example of message processing in the variable node. In FIG. 5, for convenience, four interactions and variable nodes i connected to the four interaction nodes are illustrated. The message Mi → μ from the variable node i to the interaction node μ can be expressed by equation (21).
Figure JPOXMLDOC01-appb-M000006
Figure JPOXMLDOC01-appb-M000006
　式（２１）において、∂ｉ／μは、「μを除くｉの周り」という意味であり、ν∈∂ｉ／μは、図５の例では、相互作用ノードν１、ν２、ν３を示す。図５に示すように、メッセージＭｉ→μは、相互作用ノードν（＝ν１、ν２、ν３）から変数ノードｉへのメッセージＭν→ｉの和に、変数ノードｉの密度関数ｆｉを積算して求めることができる。 In Expression (21), ∂i / μ means “around i except μ”, and νε∂i / μ indicates interaction nodes ν1, ν2, and ν3 in the example of FIG. As shown in FIG. 5, the message Mi → μ is obtained by multiplying the sum of the messages Mν → i from the interaction node ν (= ν1, ν2, ν3) to the variable node i by the density function fi of the variable node i. You can ask.
　密度関数ｆｉは、式（２２）で表すことができ、変数ノードｉでは、特に何の処理もせずに、メッセージＭν→ｉの和をメッセージＭｉ→μとして相互作用ノードμへ伝播させる。変数ノードの各変数に適当な初期値を与えて、メッセージの算出及び伝播を繰り返すことにより、メッセージの値が収束し、コスト関数を最小（又は最大）にする変数を求めることができる。 The density function fi can be expressed by Expression (22), and the variable node i propagates the sum of the messages Mν → i to the interaction node μ as the message Mi → μ without any processing. By giving an appropriate initial value to each variable of the variable node and repeating the calculation and propagation of the message, the value of the message converges, and the variable that minimizes (or maximizes) the cost function can be obtained.
　次に、鈴木トロッター分解によって書き換えられた最適化問題に対して確率伝播アルゴリズム（拡張レベルの確率伝播アルゴリズム）を適用する方法について説明する。 Next, we will explain the method of applying the belief propagation algorithm (extended level belief propagation algorithm) to the optimization problem rewritten by Suzuki Trotter decomposition.
　図６は確率伝播アルゴリズムに用いる拡張されたグラフ構造の一例を示す模式図である。図６の左側の図は、１つの相互作用ノードμの周りに４つの変数ノードが接続されたグラフ構造を示す。４つの変数ノードのうち、１つの変数ノードをσi で表す。変数ノードσi には、式（３）で表した、目的ハミルトニアンＨ0ハットに含まれるσiのｚ成分と、式（６）で表した、初期ハミルトニアンに含まれるσiのｘ成分との行列成分が存在する。 FIG. 6 is a schematic diagram showing an example of an extended graph structure used in the belief propagation algorithm. The diagram on the left side of FIG. 6 shows a graph structure in which four variable nodes are connected around one interaction node μ. One of the four variable nodes is represented by σ i . In the variable node σ i , the z component of σ i included in the target Hamiltonian H 0 hat expressed by Expression (3) and the x component of σ i included in the initial Hamiltonian expressed by Expression (6) are defined. There are matrix elements.
　図６の右側の図は、鈴木トロッター分解により、左側のグラフ構造をトロッター方向（虚時間方向）に拡張したものである。１つの相互作用ノードμの周りに４つの変数ノードが接続されたグラフ構造をトロッター数τだけコピーしたものである。ここで、ｋ＝１、…、ｋ、ｋ＋１、…、ｋ＝τ（トロッター数τ）としている。鈴木トロッター分解により拡張されたグラフ構造では、変数ノードσi には、σikのトロッター数τ分の和（σiのｚ成分を置き換えたもの）、及びσikσi(k+1)のトロッター数τ分の和（σiのx成分を置き換えたもの）の変数が存在する。ここで、σikσi(k+1)は、ｋ番目の変数ノードのσi と（ｋ＋１）番目の変数ノードのσi(k+1) との相互作用を表す。 The diagram on the right side of FIG. 6 is obtained by expanding the graph structure on the left side in the trotter direction (imaginary time direction) by Suzuki Trotter decomposition. This is a copy of the graph structure in which four variable nodes are connected around one interaction node μ by the Trotter number τ. Here, k = 1, ..., K, k + 1, ..., K = τ (Trotter number τ). In the graph structure expanded by the Suzuki Trotter decomposition, the variable node σ i has the sum of τ ik 's trotter number τ (the z component of σ i is replaced) and σ ik σ i (k + 1) There is a variable of the sum of τ minutes of the trotter number (the x component of σ i is replaced). Here, sigma ik sigma i (k + 1) represents the interaction of the k th variable node sigma i and (k + 1) of th variable node sigma i (k + 1).
　図７は拡張されたグラフ構造の相互作用ノードでのメッセージの処理の一例を示す模式図である。図７では、便宜上、相互作用ノードμと、変数ノードｉとのグラフ構造を、ｋ＝１、２、３、…τだけ拡張したものを図示している。相互作用ノードμから変数ノードｉへのメッセージＭμ→ｉは、式（２３）で表すことができる。 FIG. 7 is a schematic diagram showing an example of message processing in an interaction node having an expanded graph structure. In FIG. 7, for convenience, the graph structure of the interaction node μ and the variable node i is expanded by k = 1, 2, 3, ... τ. The message Mμ → i from the interaction node μ to the variable node i can be expressed by Expression (23).
Figure JPOXMLDOC01-appb-M000007
Figure JPOXMLDOC01-appb-M000007
　式（２３）において、密度関数ｆｕは、式（２４）で表すことができる。 In equation (23), the density function fu can be expressed by equation (24).
　式（２４）に示すように、第１演算部１６は、パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数σikに置き換え、置き換えたスピン変数σikが含まれる指数関数演算子を用いて目的ハミルトニアンに対する第１確率分布関数としての密度関数ｆｕを演算する。 As shown in Expression (24), the first operation unit 16 replaces the z component of the Pauli matrix with the spin variable σ ik by Suzuki Trotter decomposition, and uses the exponential operator that includes the replaced spin variable σ ik. The density function fu as the first probability distribution function for the Hamiltonian is calculated.
　式（２４）と式（２０）とを対比すると、式（２０）のσi が式（２４）ではσikに置き換わっている。また、図７及び式（２３）に示すように、相互作用ノードμから変数ノードｉへのメッセージＭμ→ｉは、相互作用ノードμへのメッセージＭｌ→μの和と密度関数ｆｕとの積をトロッター数τ分だけ加えたものとなっている。 Comparing Expression (24) and Expression (20), σ i in Expression (20) is replaced with σ ik in Expression (24). Further, as shown in FIG. 7 and Expression (23), the message Mμ → i from the interaction node μ to the variable node i is the product of the sum of the message Ml → μ to the interaction node μ and the density function fu. It has been added by the number of trotter τ.
　図８は拡張されたグラフ構造の変数ノードでのメッセージの処理の一例を示す模式図である。図８では、図７と同様に、便宜上、相互作用ノードμと、変数ノードｉとのグラフ構造を、ｋ＝１、２、３、…τだけ拡張したものを図示している。変数ノードｉから相互作用ノードμへのメッセージＭｉ→μは、式（２５）で表すことができる。 FIG. 8 is a schematic diagram showing an example of message processing in the variable node of the expanded graph structure. Similar to FIG. 7, FIG. 8 illustrates the graph structure of the interaction node μ and the variable node i expanded by k = 1, 2, 3, ... τ for convenience. The message Mi → μ from the variable node i to the interaction node μ can be expressed by equation (25).
Figure JPOXMLDOC01-appb-M000008
Figure JPOXMLDOC01-appb-M000008
　式（２５）において、密度関数ｆｉは、式（２６）で表すことができる。 In Expression (25), the density function fi can be expressed by Expression (26).
　式（２６）に示すように、第２演算部１７は、パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数σik、σi(k+1)に置き換え、置き換えたスピン変数の積σikσi(k+1) をトロッター方向に亘って合計した値を含む指数関数演算子を用いて初期ハミルトニアンに対する第２確率分布関数としての密度関数ｆｉを演算する。図８及び式（２６）に示すように、変数ノードｉでは、トロッター方向に沿って隣り合うσikσi(k+1) の相互作用が考慮されている。 As shown in Expression (26), the second operation unit 17 replaces the x component of the Pauli matrix with spin variables σ ik and σ i (k + 1) which are adjacent to each other along the trotter direction by Suzuki Trotter decomposition. The density function fi as the second probability distribution function for the initial Hamiltonian is calculated using an exponential operator including a value obtained by summing the products of spin variables σ ik σ i (k + 1) over the trotter direction. As shown in FIG. 8 and Expression (26), in the variable node i, the interaction between σ ik σ i (k + 1) adjacent to each other along the trotter direction is considered.
　第２演算部１７は、密度行列算出部１４が算出した、式（１１）で表す、密度行列に対して、鈴木トロッター分解を行って第２確率分布関数としての密度関数ｆｉを演算することができる。 The second calculation unit 17 may perform Suzuki Trotter decomposition on the density matrix calculated by the density matrix calculation unit 14 and represented by Expression (11) to calculate the density function fi as the second probability distribution function. it can.
　また、式（２６）には、更新変数αが含まれている。更新変数αは、式（１８）で表す更新関数{ tanh（β・ｍxチルダ／τ）}の関数値とすることができ、式（１８）のｍxチルダは、後述の式（２８）に示すように、ｘ方向の磁化変数ｍxを変数とする磁場関数ｇ（ｍx）の導関数ｇ′（ｍx）である。すなわち、第２演算部１７は、ｘ方向の磁化変数を変数とする磁場関数の導関数を変数とする更新関数に基づく更新変数αをさらに含む指数関数演算子を用いて初期ハミルトニアンに対する第２確率分布関数としての密度関数ｆｉを演算する。 Further, the equation (26) includes the update variable α. Update variables alpha, formula (18) update function represented by {tanh (β · m x tilde / tau)} be a function value of the formula (18) is the m x tilde formula below (28) As shown in, it is a derivative g ′ (m x ) of the magnetic field function g (m x ) having the magnetization variable m x in the x direction as a variable. That is, the second calculation unit 17 uses the exponential operator that further includes the update variable α based on the update function having the derivative of the magnetic field function having the magnetization variable in the x direction as the variable, and the second probability for the initial Hamiltonian. The density function fi as a distribution function is calculated.
　確率伝播処理部１９は、式（２３）で表す第１メッセージとしての相互作用ノードでのメッセージＭμ→ｉを生成する。相互作用ノードでのメッセージＭμ→ｉは、密度関数ｆｕをトロッター方向に亘って合計した確率分布関数を含む。また、確率伝播処理部１９は、式（２５）で表す第２メッセージとしての変数ノードでのメッセージＭｉ→μを生成する。変数ノードでのメッセージＭｉ→μは、スピン変数の積σikσi(k+1)をトロッター方向に亘って合計した値を含む指数関数演算子を有する密度関数ｆｉを含む。 The belief propagation processing unit 19 generates the message Mμ → i at the interaction node as the first message represented by Expression (23). The message Mμ → i at the interaction node contains a probability distribution function that sums the density function fu over the trotter direction. The belief propagation processing unit 19 also generates the message Mi → μ at the variable node as the second message represented by the equation (25). The message Mi → μ at the variable node contains a density function fi with an exponential operator containing the sum of the product of the spin variables σ ik σ i (k + 1) over the trotter direction.
　確率伝播処理部１９は、メッセージＭμ→ｉ、及びＭｉ→μの算出及び伝播を繰り返す処理を行う。 The belief propagation processing unit 19 performs a process of repeating the calculation and propagation of the messages Mμ → i and Mi → μ.
　期待値算出部２０は、変数ノードの各スピンに適当な初期値を与えて、確率伝播処理部１９によるメッセージの算出及び伝播の処理を繰り返し、メッセージＭμ→ｉ、及びＭｉ→μの値が収束したときの、最適化問題についてのコスト関数を最小（又は最大）にするスピン変数（スピン配位の期待値）を算出する。 The expected value calculation unit 20 gives an appropriate initial value to each spin of the variable node, repeats the message calculation and propagation processing by the belief propagation processing unit 19, and the values of the messages Mμ → i and Mi → μ converge. Then, a spin variable (expected value of spin configuration) that minimizes (or maximizes) the cost function for the optimization problem is calculated.
　次に、確率伝播アルゴリズムでの更新変数αの決定方法について説明する。 Next, the method of determining the update variable α in the belief propagation algorithm will be explained.
　磁化変数算出部２１は、更新関数を底とし、トロッター方向に沿って隣り合うスピン変数の積σikσi(k+1) を冪指数とする冪演算をスピン数に亘って合計するとともにトロッター数に亘って合計してｘ方向の磁化変数ｍxを算出する。磁化変数ｍxは、式（２７）により算出することができる。式（２７）において、tanh（β・ｍxチルダ／τ）が更新関数である。 The magnetization variable calculation unit 21 sums the power operations with the update function as the base and the product σ ik σ i (k + 1) of the spin variables adjacent to each other along the trotter direction as the power index over the spin number and The magnetization variable m x in the x direction is calculated by summing over the number. The magnetization variable m x can be calculated by the equation (27). In Expression (27), tanh (β · m x tilde / τ) is the update function.
Figure JPOXMLDOC01-appb-M000009
Figure JPOXMLDOC01-appb-M000009
　式（１８）により、更新変数αは、更新関数の対数によって得られる関数値で表すことができる。 According to the equation (18), the update variable α can be represented by a function value obtained by the logarithm of the update function.
　式（２７）において、ｍxチルダは、式（２８）により求めることができる。ｍxチルダは、磁化変数ｍxを変数とする磁場関数ｇ（ｍx）の導関数で表すことができる。 In Expression (27), m x tilde can be calculated by Expression (28). The m x tilde can be represented by the derivative of the magnetic field function g (m x ) having the magnetization variable m x as a variable.
　メッセージを算出し伝播させたときのσik、σi(k+1)（＋１、又は－１）を式（２７）に代入して、磁化変数ｍxを算出する。算出した磁化変数ｍxを式（２８）に代入して、ｍxチルダを算出する。算出したｍxチルダを式（１８）に代入して、更新変数αを算出する。算出した更新変数αを式（２６）に代入して、再度、メッセージＭｉ→μとメッセージＭμ→ｉとを算出し伝播処理を行う。なお、メッセージＭｉ→μが算出できると、式（２３）により、メッセージＭμ→ｉを算出できる。伝播処理によって得られたσik、σi(k+1)（＋１、又は－１）を式（２７）に代入して、以下、同様の処理を、メッセージの値が収束するまで繰り返す。メッセージの値が収束すると、スピン配位の期待値の近似的解を得ることができる。 The magnetization variable m x is calculated by substituting σ ik and σ i (k + 1) (+1 or −1) when the message is calculated and propagated into the equation (27). The calculated magnetization variable m x is substituted into equation (28) to calculate the m x tilde. Substituting the calculated m x tilde into equation (18), the update variable α is calculated. Substituting the calculated update variable α into the equation (26), the message Mi → μ and the message Mμ → i are calculated again, and the propagation processing is performed. Note that if the message Mi → μ can be calculated, the message Mμ → i can be calculated by the equation (23). Substituting σ ik and σ i (k + 1) (+1 or −1) obtained by the propagation processing into the equation (27), the same processing is repeated until the message values converge. When the message values converge, an approximate solution of the expected value of spin configuration can be obtained.
　記憶部２２は、入力データ、シミュレーション中に得られた処理結果、出力データなどを記憶することができる。 The storage unit 22 can store input data, processing results obtained during simulation, output data, and the like.
　次に、本実施の形態のシミュレーション装置１００の動作について説明する。図９は本実施の形態のシミュレーション装置１００による量子アニーリング処理手順の一例を示すフローチャートである。以下では、便宜上、処理の主体を制御部１０として説明する。制御部１０は、最適化問題をイジング模型で表現したデータを取得し（Ｓ１１）、パラメータを設定する（Ｓ１２）。パラメータは、温度、量子揺らぎ作用を決定付ける磁場関数を表現するデータであり、横磁場、反強磁性ＸＸ相互作用、パウリ行列のｘ成分の和の３次以上の累乗の項などを含む。また、パラメータには、トロッター数が含まれる。 Next, the operation of the simulation apparatus 100 according to this embodiment will be described. FIG. 9 is a flowchart showing an example of the quantum annealing processing procedure by the simulation apparatus 100 of the present embodiment. Hereinafter, for convenience, the main body of processing will be described as the control unit 10. The control unit 10 acquires data representing the optimization problem with the Ising model (S11) and sets parameters (S12). The parameter is data expressing a magnetic field function that determines the temperature and the quantum fluctuation action, and includes a transverse magnetic field, an antiferromagnetic XX interaction, a term of the third power or more of the sum of x components of the Pauli matrix, and the like. In addition, the parameter includes the number of trotter.
　制御部１０は、目的ハミルトニアンを算出し（Ｓ１３）、初期ハミルトニアンを算出する（Ｓ１４）。目的ハミルトニアンは、式（３）により算出することができ、初期ハミルトニアンは、例えば、式（６）により算出することができる。なお、初期ハミルトニアンは、式（６）で表現されるものに限定されない。初期ハミルトニアンに、パウリ行列のｘ成分の和の平均成分の三次以上の累乗の項を含めることもできる。 The control unit 10 calculates the target Hamiltonian (S13) and the initial Hamiltonian (S14). The target Hamiltonian can be calculated by the equation (3), and the initial Hamiltonian can be calculated by the equation (6), for example. It should be noted that the initial Hamiltonian is not limited to that expressed by the equation (6). The initial Hamiltonian can also include terms of the third power or higher of the average component of the sum of the x components of the Pauli matrix.
　制御部１０は、初期ハミルトニアンのパウリ行列のｘ成分の和の平均成分を磁化変数ｍxで置き換え（Ｓ１５）、目的ハミルトニアンに対する第１確率分布関数を算出し（Ｓ１６）、初期ハミルトニアンに対する第２確率分布関数を算出する（Ｓ１７）。第１確率分布関数は、式（２４）で表され、第２確率分布関数は、式（２６）で表される。 The control unit 10 replaces the average component of the sum of the x components of the Pauli matrix of the initial Hamiltonian with the magnetization variable m x (S15), calculates the first probability distribution function for the target Hamiltonian (S16), and determines the second probability for the initial Hamiltonian. A distribution function is calculated (S17). The first probability distribution function is expressed by Expression (24), and the second probability distribution function is expressed by Expression (26).
　制御部１０は、第１メッセージＭμ→ｉを算出する（Ｓ１８）。第１メッセージＭμ→ｉは、式（２３）で表される。制御部１０は、磁化変数ｍxを算出する（Ｓ１９）。磁化変数ｍxは、式（２７）で表される。制御部１０は、更新変数αを算出する（Ｓ２０）。更新変数αは、式（１８）で表される。 The control unit 10 calculates the first message Mμ → i (S18). The first message Mμ → i is represented by Expression (23). The control unit 10 calculates the magnetization variable m x (S19). The magnetization variable m x is represented by Expression (27). The control unit 10 calculates the update variable α (S20). The update variable α is represented by Expression (18).
　制御部１０は、第２メッセージＭｉ→μを算出する（Ｓ２１）。第２メッセージＭｉ→μは、式（２５）で表される。制御部１０は、第１メッセージ及び第２メッセージを用いて確率伝播処理を行い（Ｓ２２）、メッセージの値が収束したときのスピン配位の期待値を算出し（Ｓ２３）、処理を終了する。 The control unit 10 calculates the second message Mi → μ (S21). The second message Mi → μ is represented by Expression (25). The control unit 10 performs the belief propagation process using the first message and the second message (S22), calculates the expected value of the spin coordination when the value of the message converges (S23), and ends the process.
　図１０は本実施の形態のシミュレーション装置の構成の他の例を示す説明図である。図１０において、符号３００は、通常のコンピュータである。コンピュータ３００は、制御部３０、入力部４０、出力部５０、外部Ｉ／Ｆ（インタフェース）部６０などを備える。制御部３０は、ＣＰＵ３１、ＲＯＭ３２、ＲＡＭ３３、Ｉ／Ｆ（インタフェース）３４などを備える。 FIG. 10 is an explanatory diagram showing another example of the configuration of the simulation apparatus according to the present embodiment. In FIG. 10, reference numeral 300 is an ordinary computer. The computer 300 includes a control unit 30, an input unit 40, an output unit 50, an external I / F (interface) unit 60, and the like. The control unit 30 includes a CPU 31, a ROM 32, a RAM 33, an I / F (interface) 34, and the like.
　入力部４０は、シミュレーションのための入力データを取得する。出力部５０は、シミュレーション結果である出力データを出力する。Ｉ／Ｆ３４は、制御部３０と、入力部４０、出力部５０及び外部Ｉ／Ｆ部６０それぞれとの間のインタフェース機能を有する。 The input unit 40 acquires input data for simulation. The output unit 50 outputs output data that is a simulation result. The I / F 34 has an interface function between the control unit 30 and each of the input unit 40, the output unit 50, and the external I / F unit 60.
　外部Ｉ／Ｆ部６０は、コンピュータプログラムを記録した記録媒体Ｍ（例えば、ＤＶＤなどのメディア）からコンピュータプログラムを読み取ることが可能である。 The external I / F unit 60 can read the computer program from a recording medium M (for example, a medium such as a DVD) recording the computer program.
　なお、図示していないが、記録媒体Ｍに記録されたコンピュータプログラムは、持ち運びが自由なメディアに記録されたものに限定されるものではなく、インターネット又は他の通信回線を通じて伝送されるコンピュータプログラムも含めることができる。また、コンピュータには、複数のプロセッサを搭載した１台のコンピュータ、あるいは、通信ネットワークを介して接続された複数台のコンピュータで構成されるコンピュータシステムも含まれる。 Although not shown, the computer program recorded in the recording medium M is not limited to the one recorded in a medium that can be carried around freely, and a computer program transmitted through the Internet or another communication line may be used. Can be included. Further, the computer includes a computer system including one computer equipped with a plurality of processors or a plurality of computers connected via a communication network.
　本実施の形態のシミュレーション装置及びシミュレーション方法によれば、鈴木トロッター分解による虚時間方向へのイジング模型の拡張を行って最適化問題を書き換え、書き換えられた最適化問題に対して確率伝播アルゴリズムを適用した拡張レベルの確率伝播アルゴリズムを用いるので、量子アニーリングマシン（量子コンピュータ）を介さずに、通常のデジタルコンピュータ上で、大規模な最適化問題（組み合わせ最適化問題）の近似的解を高速に求めることができる。 According to the simulation apparatus and the simulation method of the present embodiment, the Ising model is extended in the imaginary time direction by Suzuki Trotter decomposition to rewrite the optimization problem, and the belief propagation algorithm is applied to the rewritten optimization problem. Since the extended level belief propagation algorithm is used, an approximate solution of a large-scale optimization problem (combinational optimization problem) can be obtained at high speed on a normal digital computer without going through a quantum annealing machine (quantum computer). be able to.
　本実施の形態のシミュレーション装置及びシミュレーション方法によれば、反強磁性ＸＸ相互作用を含む、パウリ行列のｘ成分の和の平均成分の三次以上の累乗の項を含む量子力学的効果（量子揺らぎ作用）を、横磁場のみに置き換えることができるので、本実施の形態のシミュレーション装置及びシミュレーション方法によって得られた大規模な最適化問題の解の候補（第１段階の解）を、横磁場のみによる量子力学的効果を扱うことができる量子コンピュータへ提供して、２段階の解法を行うこともできる。 According to the simulation apparatus and the simulation method of the present embodiment, the quantum mechanical effect (quantum fluctuation action) including the term of the third power or more of the average component of the sum of the x components of the Pauli matrix including the antiferromagnetic XX interaction is included. ) Can be replaced by only the transverse magnetic field, so that the solution candidates (first-stage solutions) for the large-scale optimization problem obtained by the simulation apparatus and the simulation method according to the present embodiment can be changed only by the transverse magnetic field. It is also possible to provide a quantum computer capable of handling quantum mechanical effects and perform a two-step solution.
　本実施の形態のシミュレーション装置は、パーソナルコンピュータ、ワークステーション、サーバ、ＧＰＵ、ＦＰＧＡ等で実装することができる。 The simulation device of this embodiment can be implemented by a personal computer, workstation, server, GPU, FPGA, or the like.
　本実施の形態のシミュレーション装置は、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション装置であって、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する第１演算部と、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する第２演算部と、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する期待値算出部とを備える。 In the simulation device of the present embodiment, a plurality of spins of the Ising model are represented by z components of the Pauli matrix, and the objective Hamiltonian expressing the optimization problem and the x component of the Pauli matrix corresponding to the plurality of spins are included in the quantum device. For a Hamiltonian combining an initial Hamiltonian expressing fluctuations with a coefficient with time change, a simulation device for simulating the expected value of the spin configuration of the target Hamiltonian by reducing quantum fluctuations over time, A z-component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first computing unit that computes a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable; The x component of the matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the exponential operator containing the sum of the products of the replaced spin variables over the trotter direction is used to calculate the first Hamiltonian for the initial Hamiltonian. (2) A second calculation unit that calculates a probability distribution function, a first message that includes a probability distribution function that sums the first probability distribution functions in the trotter direction, and a second message that includes the second probability distribution function are used. An expected value calculation unit that calculates an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm.
　本実施の形態のコンピュータプログラムは、コンピュータに、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートさせるコンピュータプログラムであって、コンピュータに、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する処理と、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する処理と、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する処理とを実行させる。 The computer program according to the present embodiment causes the computer to represent a plurality of spins of the Ising model by z components of the Pauli matrix, an objective Hamiltonian expressing an optimization problem, and x components of the Pauli matrix corresponding to the plurality of spins. A computer program that simulates the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with the passage of time, for a Hamiltonian that combines the initial Hamiltonian expressing the quantum fluctuation with a coefficient that changes with time. Then, a process of causing the computer to replace the z component of the Pauli matrix with a spin variable by Suzuki Trotter decomposition and calculating a first probability distribution function for the target Hamiltonian using an exponential operator that includes the replaced spin variable, , The x-component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the product of the replaced spin variables is summed over the trotter direction using an exponential operator A process of calculating a second probability distribution function for the Hamiltonian, a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction, and a second message including the second probability distribution function are used. And a process of calculating an expected value of the spin configuration of the target Hamiltonian by a belief propagation algorithm.
　本実施の形態のシミュレーション方法は、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション方法であって、前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算し、前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算し、前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する。 In the simulation method of the present embodiment, a plurality of spins of the Ising model are represented by z components of the Pauli matrix, and the objective Hamiltonian expressing the optimization problem and the x component of the Pauli matrix corresponding to the plurality of spins are included in the quantum. A simulation method for simulating the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuation with the passage of time, with respect to the Hamiltonian combining the initial Hamiltonian expressing fluctuation and the coefficient with time change, The z component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and the first probability distribution function for the target Hamiltonian is calculated using an exponential operator including the replaced spin variable, and the x component of the Pauli matrix is calculated. The second probability distribution function for the initial Hamiltonian is calculated by using an exponential operator including Suzuki spin trotter decomposition to replace adjacent spin variables along the trotter direction, and including the sum of the products of the replaced spin variables over the trotter direction. The spin distribution of the target Hamiltonian is calculated by a belief propagation algorithm using a first message including a probability distribution function that is calculated by summing the first probability distribution function in the trotter direction and a second message including the second probability distribution function. Calculate the expected value of rank.
　第１演算部は、パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数σi が含まれる指数関数演算子を用いて目的ハミルトニアンに対する第１確率分布関数ｆｕを演算する。 The first calculation unit replaces the z component of the Pauli matrix with a spin variable by Suzuki Trotter decomposition, and calculates the first probability distribution function fu for the target Hamiltonian using an exponential operator that includes the replaced spin variable σ i .
　第２演算部は、パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数σik、σi(k+1)に置き換え、置き換えたスピン変数の積σikσi(k+1)をトロッター方向に亘って合計した値を含む指数関数演算子を用いて初期ハミルトニアンに対する第２確率分布関数ｆｉを演算する。 The second arithmetic unit replaces the x component of the Pauli matrix with spin variables σ ik and σ i (k + 1) that are adjacent to each other along the trotter direction by Suzuki Trotter decomposition, and the product σ ik σ i (k ) of the replaced spin variables. The second probability distribution function fi with respect to the initial Hamiltonian is calculated using an exponential operator including the sum of ( +1) over the trotter direction.
　期待値算出部は、第１確率分布関数ｆｕをトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び第２確率分布関数ｆｉを含む第２メッセージを用いた確率伝播アルゴリズムにより目的ハミルトニアンのスピン配位の期待値を算出する。 The expected value calculation unit uses the belief propagation algorithm using the first message including the probability distribution function obtained by summing the first probability distribution functions fu in the trotter direction and the second message including the second probability distribution function fi to calculate the target Hamiltonian of the target Hamiltonian. Calculate the expected value of spin coordination.
　スピンが配置されるサイトを変数ノードｉとし、スピン間の相互作用を相互作用ノードμとし、変数ノードｉ及び相互作用ノードμを繋ぐグラフ構造上での確率伝播アルゴリズムにおいて、相互作用ノードμから変数ノードｉへのメッセージを第１メッセージＭμ→ｉとし、変数ノードｉから相互作用ノードμへのメッセージを第２メッセージＭｉ→μとする。 A site where spins are arranged is a variable node i, an interaction between spins is an interaction node μ, and in a belief propagation algorithm on a graph structure connecting the variable node i and the interaction node μ, a variable is generated from the interaction node μ. The message to the node i is the first message Mμ → i, and the message from the variable node i to the interaction node μ is the second message Mi → μ.
　第１メッセージＭμ→ｉは、相互作用ノードμでの第１確率分布関数ｆｕをトロッター方向に亘って合計した確率分布関数を含む。第２メッセージＭｉ→μは、変数ノードｉでの第２確率分布関数ｆｉを含む。 The first message Mμ → i includes a probability distribution function in which the first probability distribution function fu at the interaction node μ is summed over the trotter direction. The second message Mi → μ includes the second probability distribution function fi at the variable node i.
　第２確率分布関数ｆｉにより、パウリ行列のｘ成分を持つ初期ハミルトニアンで表す量子力学的効果（量子揺らぎ作用）を、トロッター間相互作用に置き換えることができ、数値計算を実行することが可能となる。また、鈴木トロッター分解によって、仮想的な虚時間方向（トロッター方向）へのイジング模型の拡張を行い、拡張したグラフ構造上での確率伝播アルゴリズム（拡張レベルの確率伝播アルゴリズム）を用いることにより、最適化問題の近似的解を高速で求めることができる。 With the second probability distribution function fi, the quantum mechanical effect (quantum fluctuation action) represented by the initial Hamiltonian having the x component of the Pauli matrix can be replaced with the trotter interaction, and the numerical calculation can be executed. . In addition, by using Suzuki Trotter decomposition, the Ising model is extended in the virtual imaginary time direction (trotter direction), and the optimal propagation is performed by using the belief propagation algorithm (extended level belief propagation algorithm) on the extended graph structure. The approximate solution of the optimization problem can be obtained at high speed.
　本実施の形態のシミュレーション装置は、前記初期ハミルトニアンの前記パウリ行列のｘ成分の和の平均成分とｘ方向の磁化変数との差を変数とするデルタ関数と、前記初期ハミルトニアンを含む指数関数演算子との積を用いて前記初期ハミルトニアンに対する密度行列を算出する密度行列算出部を備え、前記第２演算部は、前記密度行列に対して鈴木トロッター分解を行って前記第２確率分布関数を演算する。 The simulation apparatus of the present embodiment includes a delta function having a difference between an average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and a magnetization variable in the x direction as a variable, and an exponential operator including the initial Hamiltonian. And a density matrix calculation unit that calculates a density matrix for the initial Hamiltonian using a product of and, and the second calculation unit performs Suzuki Trotter decomposition on the density matrix to calculate the second probability distribution function. .
　密度行列算出部は、初期ハミルトニアンのパウリ行列のｘ成分の和の平均成分とｘ方向の磁化変数との差を変数とするデルタ関数と、初期ハミルトニアンを含む指数関数演算子との積を用いて初期ハミルトニアンに対する密度行列を算出する。 The density matrix calculation unit uses a product of a delta function whose variable is the difference between the average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and the magnetization variable in the x direction, and an exponential operator including the initial Hamiltonian. Compute the density matrix for the initial Hamiltonian.
　これにより、初期ハミルトニアンに対する密度行列が、パウリ行列のｘ成分の和の一次項（すなわち、二次項以上の高次項を含まない）だけで表現することができ、横磁場以外の量子力学的効果（反強磁性ＸＸ相互作用など）を横磁場による量子力学的効果のみに置き換えることができる。 Accordingly, the density matrix for the initial Hamiltonian can be expressed only by the first-order term (that is, the second-order or higher-order terms are not included) of the sum of the x components of the Pauli matrix, and the quantum mechanical effects other than the transverse magnetic field ( Antiferromagnetic XX interaction, etc.) can be replaced only by the quantum mechanical effect due to the transverse magnetic field.
　第２演算部は、密度行列に対して鈴木トロッター分解を行って第２確率分布関数ｆｉを演算する。第２確率分布関数ｆｉにより、パウリ行列のｘ成分を持つ初期ハミルトニアンで表す横磁場以外の量子力学的効果（反強磁性ＸＸ相互作用など）を横磁場による量子力学的効果のみに置き換えることができ、置き換えられた横磁場による量子力学的効果（量子揺らぎ作用）を、トロッター間相互作用に置き換えることができ、数値計算を実行することが可能となる。 The second calculation unit performs Suzuki Trotter decomposition on the density matrix to calculate the second probability distribution function fi. With the second probability distribution function fi, quantum mechanical effects (antiferromagnetic XX interaction, etc.) other than the transverse magnetic field represented by the initial Hamiltonian having the x component of the Pauli matrix can be replaced with only the quantum mechanical effect due to the transverse magnetic field. , It is possible to replace the quantum mechanical effect (quantum fluctuation action) due to the replaced transverse magnetic field with the interaction between trotter, and it is possible to execute the numerical calculation.
　本実施の形態のシミュレーション装置において、前記第２演算部は、ｘ方向の磁化変数を変数とする磁場関数の導関数を変数とする更新関数に基づく更新変数をさらに含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する。 In the simulation device according to the present embodiment, the second calculation unit uses an exponential operator that further includes an update variable based on an update function having a derivative of a magnetic field function having a magnetization variable in the x direction as a variable. A second probability distribution function for the initial Hamiltonian is calculated.
　第２演算部は、ｘ方向の磁化変数ｍxを変数とする磁場関数ｇ（ｍx）の導関数ｇ′（ｍx）を変数とする更新関数（例えば、{ tanh（β・ｇ′（ｍx）／τ）}に基づく更新変数α（例えば、α＝ｌｎ{ tanh（β・ｇ′（ｍx）／τ）}をさらに含む指数関数演算子を用いて初期ハミルトニアンに対する第２確率分布関数を演算する。βは温度Ｔの逆数であり、τはトロッター数である。 The second operation unit, the derivative g '(m x) an update function to a variable (e.g., {tanh (beta · g' of the magnetic field function g whose variable is the x direction of magnetization variable m x (m x) ( m x) / τ)} updated variable alpha (e.g., α = ln {tanh (β · g '(m x) based on / tau)} second probability distribution for the initial Hamiltonian using exponential operator further including Calculate a function, β is the reciprocal of temperature T, and τ is the Trotter number.
　トロッター間相互作用σikσi(k+1)が変わると磁化変数ｍxが変わり、磁化変数ｍxが変わると導関数ｇ′（ｍx）の値が変わり、導関数ｇ′（ｍx）の値が変わると、更新変数αが変わる。更新変数αが変わると、第２確率分布関数ｆｉが変わり、第２確率分布関数ｆｉが変わると、第２メッセージＭｉ→μ及び第１メッセージＭμ→ｉが変わる。第２メッセージＭｉ→μ及び第１メッセージＭμ→ｉが変わると、トロッター間相互作用σikσi(k+1)が変わる。このような更新を繰り返し、メッセージの値が収束すると、各スピン変数のスピン配位の期待値を算出することができる。 Trotter interactions σ ik σ i (k + 1 ) is changed and change the magnetization variable m x, 'changes the value of (m x), the derivative g' the magnetization variable m x varies derivative g (m x ), The update variable α changes. When the update variable α changes, the second probability distribution function fi changes, and when the second probability distribution function fi changes, the second message Mi → μ and the first message Mμ → i change. When the second message Mi → μ and the first message Mμ → i change, the trotter interaction σ ik σ i (k + 1) changes. When such updating is repeated and the value of the message converges, the expected value of the spin configuration of each spin variable can be calculated.
　本実施の形態のシミュレーション装置は、前記更新関数を底とし、トロッター方向に沿って隣り合うスピン変数の積を冪指数とする冪演算をスピン数に亘って合計するとともにトロッター数に亘って合計してｘ方向の磁化変数を算出する磁化変数算出部を備える。 The simulation device of the present embodiment has the update function as a base, and sums over the number of spins the power operation with the exponent of the product of spin variables adjacent to each other along the trotter direction as the power exponent. And a magnetization variable calculation unit that calculates a magnetization variable in the x direction.
　磁化変数算出部は、更新関数（例えば、{ tanh（β・ｍxチルダ／τ）}を底とし、トロッター方向に沿って隣り合うスピン変数の積σikσi(k+1)を冪指数とする冪演算をスピン数（例えば、Ｎ）に亘って合計するとともにトロッター数τに亘って合計してｘ方向の磁化変数ｍxを算出する。ここで、ｍxチルダ＝ｇ′（ｍx）である。ｇ（ｍx）は磁場関数である。これにより、トロッター間相互作用σikσi(k+1)が変わると磁化変数ｍxが変わることを定式化できる。 The magnetization variable calculator calculates the product σ ik σ i (k + 1) of the spin variables adjacent to each other along the trotter direction with the update function (for example, {tanh (β · m x tilde / τ)} as the base). To calculate the magnetization variable m x in the x direction by summing the power operations for the spin number (for example, N) and for the trotter number τ, where m x tilde = g ′ (m x G (m x ) is a magnetic field function, which makes it possible to formulate that the magnetization variable m x changes when the trotter interaction σ ik σ i (k + 1) changes.
　本実施の形態のシミュレーション装置において、前記磁場関数は、前記パウリ行列のｘ成分の和の平均成分の２以上の累乗の項を含む。 In the simulation apparatus according to the present embodiment, the magnetic field function includes a power of 2 or more of the average component of the sum of x components of the Pauli matrix.
　磁場関数ｇ（ｍx）は、パウリ行列のｘ成分の和の平均成分の２以上の累乗の項を含む。２乗の場合には、量子力学的効果は、反強磁性ＸＸ相互作用による効果とすることができる。これにより、横磁場以外の量子力学的効果を考慮したシミュレーションを行うことができ、汎用性の高い最適化問題の近似的解を高速で求めることができる。 The magnetic field function g (m x ) includes a term that is a power of 2 or more of the average component of the sum of x components of the Pauli matrix. In the case of the square, the quantum mechanical effect can be an effect due to antiferromagnetic XX interaction. As a result, a simulation considering quantum mechanical effects other than the transverse magnetic field can be performed, and an approximate solution of a highly versatile optimization problem can be obtained at high speed.
　上述の実施の形態では、変数ノードと相互作用ノードとで構成されるグラフ構造上での拡張された確率伝播アルゴリズムについて説明したが、グラフ構造はこれに限定されるものではなく、例えば、ベイジアンネットワークにおいても拡張された確率伝播アルゴリズムを適用して同様の結果を得ることができる。 In the above embodiment, the extended belief propagation algorithm on the graph structure composed of the variable nodes and the interaction nodes has been described, but the graph structure is not limited to this, and for example, Bayesian network. Also in, the extended belief propagation algorithm can be applied to obtain similar results.
　１０、３０　制御部
　１１、４０　入力部
　１２　目的ハミルトニアン演算部
　１３　初期ハミルトニアン演算部
　１４　密度行列算出部
　１５　確率分布関数演算部
　１６　第１演算部
　１７　第２演算部
　１８、５０　出力部
　１９　確率伝播処理部
　２０　期待値算出部
　２１　磁化変数算出部
　２２　記憶部
　３１　ＣＰＵ
　３２　ＲＯＭ
　３３　ＲＡＭ
　３４　Ｉ／Ｆ
　６０　外部Ｉ／Ｆ部 10, 30 Control unit 11, 40 Input unit 12 Objective Hamiltonian calculation unit 13 Initial Hamiltonian calculation unit 14 Density matrix calculation unit 15 Probability distribution function calculation unit 16 First calculation unit 17 Second calculation unit 18, 50 Output unit 19 Belief propagation processing Part 20 Expected value calculation part 21 Magnetization variable calculation part 22 Storage part 31 CPU
32 ROM
33 RAM
34 I / F
60 External I / F section

Claims (7)

  1. 　イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション装置であって、
    　前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する第１演算部と、
    　前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する第２演算部と、
    　前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する期待値算出部と
    　を備えるシミュレーション装置。     A plurality of spins of the Ising model are represented by z components of the Pauli matrix, and an objective Hamiltonian that expresses the optimization problem and an initial Hamiltonian that expresses a quantum fluctuation including the x component of the Pauli matrix corresponding to the plurality of spins are expressed in time. For a Hamiltonian combined with a coefficient with a change, a simulation apparatus for simulating the expected value of the spin configuration of the target Hamiltonian by reducing the quantum fluctuations over time,
    A z-component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first computing unit that computes a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable;
    The x component of the Pauli matrix is replaced by spin variables adjacent to each other along the trotter direction by Suzuki Trotter decomposition, and the initial Hamiltonian is calculated using an exponential operator including the sum of products of the replaced spin variables over the trotter direction. A second calculation unit that calculates a second probability distribution function for
    Expectation of spin configuration of the target Hamiltonian by a belief propagation algorithm using a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction and a second message including the second probability distribution function. A simulation device comprising: an expected value calculation unit that calculates a value.
  2. 　前記初期ハミルトニアンの前記パウリ行列のｘ成分の和の平均成分とｘ方向の磁化変数との差を変数とするデルタ関数と、前記初期ハミルトニアンを含む指数関数演算子との積を用いて前記初期ハミルトニアンに対する密度行列を算出する密度行列算出部を備え、
    　前記第２演算部は、
    　前記密度行列に対して鈴木トロッター分解を行って前記第２確率分布関数を演算する請求項１に記載のシミュレーション装置。     The initial Hamiltonian is obtained by using a product of a delta function having a difference between an average component of the sum of x components of the Pauli matrix of the initial Hamiltonian and a magnetization variable in the x direction and an exponential function operator including the initial Hamiltonian. A density matrix calculation unit that calculates a density matrix for
    The second arithmetic unit is
    The simulation device according to claim 1, wherein the second probability distribution function is calculated by performing Suzuki Trotter decomposition on the density matrix.
  3. 　前記第２演算部は、
    　ｘ方向の磁化変数を変数とする磁場関数の導関数を変数とする更新関数に基づく更新変数をさらに含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する請求項１又は請求項２に記載のシミュレーション装置。     The second arithmetic unit is
    The second probability distribution function for the initial Hamiltonian is calculated using an exponential operator that further includes an update variable based on an update function having a derivative of a magnetic field function having a magnetization variable in the x direction as a variable. The simulation device according to claim 2.
  4. 　前記更新関数を底とし、トロッター方向に沿って隣り合うスピン変数の積を冪指数とする冪演算をスピン数に亘って合計するとともにトロッター数に亘って合計してｘ方向の磁化変数を算出する磁化変数算出部を備える請求項３に記載のシミュレーション装置。 The update function is the base, and the power operations with the exponent of the product of spin variables adjacent to each other along the trotter direction are summed over the spin number, and the magnetization variables in the x direction are calculated over the trotter number. The simulation device according to claim 3, further comprising a magnetization variable calculation unit.
  5. 　前記磁場関数は、前記パウリ行列のｘ成分の和の平均成分の２以上の累乗の項を含む請求項３又は請求項４に記載のシミュレーション装置。 The simulation device according to claim 3 or 4, wherein the magnetic field function includes a term that is a power of 2 or more of an average component of a sum of x components of the Pauli matrix.
  6. 　コンピュータに、イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートさせるコンピュータプログラムであって、
    　コンピュータに、
    　前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算する処理と、
    　前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算する処理と、
    　前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出する処理と
    　を実行させるコンピュータプログラム。     In the computer, a plurality of spins of the Ising model are represented by z components of the Pauli matrix, and an objective Hamiltonian for expressing an optimization problem and an initial Hamiltonian for expressing quantum fluctuations including x components of the Pauli matrix corresponding to the plurality of spins. A computer program for simulating the expected value of the spin configuration of the target Hamiltonian by reducing quantum fluctuations over time, for a Hamiltonian combining and with a coefficient that changes with time,
    On the computer,
    A process of replacing the z component of the Pauli matrix with a spin variable by Suzuki Trotter decomposition, and calculating a first probability distribution function for the target Hamiltonian using an exponential operator including the replaced spin variable;
    The x component of the Pauli matrix is replaced by spin variables adjacent to each other along the trotter direction by Suzuki Trotter decomposition, and the initial Hamiltonian is calculated using an exponential operator including the sum of products of the replaced spin variables over the trotter direction. Processing a second probability distribution function for
    Expectation of spin configuration of the target Hamiltonian by a belief propagation algorithm using a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction and a second message including the second probability distribution function. A computer program that executes the process of calculating a value.
  7. 　イジング模型の複数のスピンがパウリ行列のｚ成分で表され、最適化問題を表現する目的ハミルトニアンと、前記複数のスピンに対応するパウリ行列のｘ成分を含み量子揺らぎを表現する初期ハミルトニアンとを時間変化を伴う係数で組み合わせたハミルトニアンに対して、時間経過とともに量子揺らぎを小さくして前記目的ハミルトニアンのスピン配位の期待値をシミュレートするシミュレーション方法であって、
    　前記パウリ行列のｚ成分を鈴木トロッター分解によってスピン変数に置き換え、置き換えたスピン変数が含まれる指数関数演算子を用いて前記目的ハミルトニアンに対する第１確率分布関数を演算し、
    　前記パウリ行列のｘ成分を鈴木トロッター分解によってトロッター方向に沿って隣り合うスピン変数に置き換え、置き換えたスピン変数の積をトロッター方向に亘って合計した値を含む指数関数演算子を用いて前記初期ハミルトニアンに対する第２確率分布関数を演算し、
    　前記第１確率分布関数をトロッター方向に亘って合計した確率分布関数を含む第１メッセージ及び前記第２確率分布関数を含む第２メッセージを用いた確率伝播アルゴリズムにより前記目的ハミルトニアンのスピン配位の期待値を算出するシミュレーション方法。
          A plurality of spins of the Ising model are represented by z components of the Pauli matrix, and an objective Hamiltonian that expresses the optimization problem and an initial Hamiltonian that expresses quantum fluctuations including the x component of the Pauli matrix corresponding to the plurality of spins are expressed in time. For a Hamiltonian combined with coefficients with changes, a simulation method for simulating the expected value of the spin configuration of the target Hamiltonian by reducing quantum fluctuations over time,
    The z component of the Pauli matrix is replaced with a spin variable by Suzuki Trotter decomposition, and a first probability distribution function for the target Hamiltonian is calculated using an exponential operator that includes the replaced spin variable,
    The x component of the Pauli matrix is replaced by adjacent spin variables along the trotter direction by Suzuki Trotter decomposition, and the initial Hamiltonian is calculated using an exponential operator including the sum of products of the replaced spin variables over the trotter direction. Compute the second probability distribution function for
    Expectation of spin configuration of the target Hamiltonian by a belief propagation algorithm using a first message including a probability distribution function obtained by summing the first probability distribution functions in the trotter direction and a second message including the second probability distribution function. Simulation method to calculate the value.
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CN112784472A (en) * 2021-01-27 2021-05-11 电子科技大学 Simulation method for simulating quantum condition principal equation in quantum transport process by using cyclic neural network
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Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
BALDASSI, CARLO ET AL.: "Efficiency of quantum versus classical annealing in non-convex learning problems", ARXIV:1706.08470V3, 16 October 2017 (2017-10-16), pages 1 - 31, XP081401505, Retrieved from the Internet <URL:https://arxiv.org/pdf/1706.08470v3.pdf>> [retrieved on 20200110] *

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112784472A (en) * 2021-01-27 2021-05-11 电子科技大学 Simulation method for simulating quantum condition principal equation in quantum transport process by using cyclic neural network
CN112784472B (en) * 2021-01-27 2023-03-24 电子科技大学 Simulation method for simulating quantum condition principal equation in quantum transport process by using cyclic neural network
WO2024038694A1 (en) * 2022-08-16 2024-02-22 国立大学法人東北大学 Optimization device, optimization method, and program

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