WO2020100698A1 - Dispositif de simulation, programme informatique et procédé de simulation - Google Patents

Dispositif de simulation, programme informatique et procédé de simulation 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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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

L'invention concerne un dispositif de simulation, un programme informatique et un procédé de simulation, permettant de déterminer rapidement une solution approximative à un problème d'optimisation.　Le dispositif de simulation est équipé : d'une première unité de calcul permettant de substituer une composante z d'une matrice de Pauli par une variable de spin et d'utiliser un opérateur de fonction d'indice incluant la variable de spin substituée pour calculer une première fonction de distribution de probabilité pour un hamiltonien prévu ; d'une seconde unité de calcul permettant de substituer une composante x de la matrice de Pauli par une variable de spin qui est adjacente dans la direction de Trotter et d'utiliser un opérateur de fonction d'indice comprenant une valeur obtenue en totalisant un produit de la variable de spin substituée dans la direction de Trotter pour calculer une seconde fonction de distribution de probabilité pour un hamiltonien initial ; et d'une unité de calcul de valeur attendue permettant de calculer une valeur attendue de la configuration de spin de l'hamiltonien prévu au moyen d'un algorithme de propagation de probabilité qui utilise un premier message, qui comprend une fonction de distribution de probabilité dans laquelle une première fonction de distribution de probabilité est totalisée dans la direction de Trotter, et un second message, qui comprend une seconde fonction de distribution de probabilité.
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CN112784472A (zh) * 2021-01-27 2021-05-11 电子科技大学 循环神经网络模拟量子输运过程中的量子条件主方程的模拟方法
WO2024038694A1 (fr) * 2022-08-16 2024-02-22 国立大学法人東北大学 Dispositif d'optimisation, procédé d'optimisation, et programme

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JP2021171810A (ja) 2020-04-30 2021-11-01 株式会社神戸製鋼所 溶接情報の学習モデル生成方法、学習モデル、プログラム及び溶接システム

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CN112784472A (zh) * 2021-01-27 2021-05-11 电子科技大学 循环神经网络模拟量子输运过程中的量子条件主方程的模拟方法
CN112784472B (zh) * 2021-01-27 2023-03-24 电子科技大学 循环神经网络模拟量子输运过程中的量子条件主方程的模拟方法
WO2024038694A1 (fr) * 2022-08-16 2024-02-22 国立大学法人東北大学 Dispositif d'optimisation, procédé d'optimisation, et programme

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