JP2022177458A - Information processing device, method for processing information, and program - Google Patents

Abstract

To provide an information processing device that can obtain an excellent solution promptly.SOLUTION: An annealing control unit 110 controls an annealing step and also controls a parameter of a temperature, a parameter of a self action, and update try rates of spins in the Ising model. An instantaneous magnetic field calculation unit 120 calculates instantaneous magnetic fields of spins in parallel. An update probability calculation unit 130 calculates update probabilities of spins in parallel on the basis of the instantaneous magnetic fields. A spin state determination unit 150 determines states of spins in parallel on the basis of the update probability and the update try rates.SELECTED DRAWING: Figure 9

Description

The present invention relates to an information processing device, an information processing method, and a program.

Annealing algorithms are known for solving the problem of searching for the ground state of the Ising model, for example quantum annealing or simulated annealing. For example, Patent Literature 1 discloses an information processing device capable of parallel updating the states of all spins of an Ising model by introducing self-action parameters.

Japanese Patent Application Laid-Open No. 2021-43508

In the technique disclosed in Patent Document 1, by updating the states of all spins of the Ising model in parallel, it is possible to realize faster processing than simulated annealing that sequentially updates the states of spins. In addition, in the technique according to Patent Document 1, when the value of the self-action parameter is decreased, the number of spins to be updated tends to increase, so it is possible to further speed up the processing. However, if the value of the self-action parameter is reduced, the state in the Ising model may not converge normally. Therefore, with the technique according to Patent Document 1, it is difficult to normally converge the state of the Ising model while realizing high-speed processing. Therefore, it is difficult for the technique disclosed in Patent Literature 1 to obtain a high-quality solution at high speed.

An object of the present invention is to provide an information processing apparatus, an information processing method, and a program capable of obtaining a high-quality solution at high speed.

An information processing apparatus according to the present invention is an information processing apparatus that obtains a solution using an Ising model, and controls an annealing step, a temperature parameter, a self-action parameter, and a plurality of spins in the Ising model. An annealing control unit that controls the update trial rate of the spins, an instantaneous magnetic field calculation unit that calculates the instantaneous magnetic fields of the plurality of spins in parallel, and an update probability of the plurality of spins based on the instantaneous magnetic fields is calculated in parallel. An update probability calculation unit and a spin state determination unit that determines states of the plurality of spins in parallel based on the update probability and the update trial rate.

Further, an information processing method according to the present invention is an information processing method for obtaining a solution using an Ising model. and controlling the update trial rate of the spins, calculating the instantaneous magnetic fields of the plurality of spins in parallel, calculating the update probabilities of the plurality of spins in parallel based on the instantaneous magnetic fields, and calculating the update probability and the update The states of the plurality of spins are determined in parallel based on the trial rate.

Further, a program according to the present invention is a program that implements an information processing method for obtaining a solution using an Ising model, and controls an annealing step, a temperature parameter, a self-action parameter, and the Ising model. A process of controlling the update trial rate of a plurality of spins in the above, a process of calculating the instantaneous magnetic fields of the plurality of spins in parallel, and a process of calculating the update probabilities of the plurality of spins in parallel based on the instantaneous magnetic fields. and determining the states of the plurality of spins in parallel based on the update probability and the update trial rate.

ADVANTAGE OF THE INVENTION According to this invention, the information processing apparatus, information processing method, and program which can obtain a good solution at high speed can be provided.

FIG. 10 is a diagram showing a spin system according to a comparative example; FIG. 9 is a diagram schematically showing updating of spin states according to a comparative example; 9 is a flowchart showing annealing processing according to a comparative example; FIG. 10 is a diagram illustrating simulation results when an annealing algorithm is executed by a technique according to a comparative example; FIG. 10 is a diagram illustrating simulation results when an annealing algorithm is executed by a technique according to a comparative example; FIG. 10 is a diagram illustrating simulation results when an annealing algorithm is executed by a technique according to a comparative example; FIG. 4 is a diagram for explaining an update trial rate according to the embodiment; FIG. FIG. 4 is a diagram for explaining an update trial rate according to the embodiment; FIG. 1 is a diagram showing a configuration of an information processing apparatus according to a first embodiment; FIG. 4 is a diagram illustrating a function of update trial rate according to the first embodiment; FIG. 4 is a flow chart showing an information processing method executed by the information processing apparatus according to the first embodiment; 4 is a diagram for explaining processing of a spin state determination unit according to the first embodiment; FIG. FIG. 7 is a diagram illustrating simulation results when an annealing algorithm is executed by the technique according to the first embodiment; FIG. 7 is a diagram illustrating simulation results when an annealing algorithm is executed by the technique according to the first embodiment; FIG. 7 is a diagram illustrating simulation results when an annealing algorithm is executed by the technique according to the first embodiment; FIG. 7 is a diagram for explaining another algorithm that can be implemented by the information processing apparatus according to the first embodiment; FIG.

(Overview of Embodiment)
Prior to describing the embodiments, an outline of the embodiments according to the present invention will be described. Although embodiments of the present invention will be described below, the following embodiments do not limit the invention according to the claims. Also, not all combinations of features described in the embodiments are essential for the solution of the invention.

・・・（１） <About Simulated Annealing>
First, a combinatorial optimization problem by general simulated annealing (SA) will be described. The combinatorial optimization problem is equivalent to finding σ that minimizes the energy H shown in Equation 1 below. In other words, the combinatorial optimization problem is equivalent to the problem of finding the minimum energy state (ground state) of the Ising model whose energy H is given by Equation (1). Equation 1 shows the energy function (Hamiltonian) in simulated annealing.

... (1)

・・・（２） Here, σ on the left side of Equation 1 is a vector (combination of σ _i ; spin array) defined by Equation 2 below. Also, σ _i (and σ _j ) are components of σ.

... (2)

Here, σ _i is a binary parameter (state parameter) indicating the state (Ising spin) of spin (node; lattice point) i in the Ising model. As shown in Equation 2, σ _i takes on values of either −1 or +1. In the following description, σ _i may be used to mean spin i for convenience. Also, N is the number of spins. J _ij in Equation 1 is an interaction coefficient (coupling coefficient) that indicates the interaction (coupling strength) between spin i (σ _i ) and spin j (σ _j ). The interaction coefficient J _ij indicates, for example, the distance between spin i and spin j, or the weight between spin i and spin j. Also, h _i is an external magnetic field coefficient indicating an external magnetic field for spin i(σ _i ). The interaction coefficient J _ij and the external magnetic field coefficient h _i are given in advance according to the problem to be solved by the annealing algorithm. That is, once the combinatorial optimization problem to be solved is determined, the interaction coefficient J _ij and the external magnetic field coefficient h _i can be uniquely determined.

・・・（３） Also, in simulated annealing, the combinatorial optimization problem corresponds to searching for σ that minimizes the energy H in Eq. The spin state in the ground state is represented by Equation 3 below. σ _g denotes the spin state combination that takes the minimum energy state.

... (3)

Here, in general, σ that minimizes H cannot be solved analytically. Therefore, the search is done by probabilistically sampling the solution. For example, a simulated annealing algorithm based on the Monte Carlo method is performed. Then, as the annealing step progresses, the number of updated spins (the number of spins updated in each annealing step; the number of flips) decreases, and the simulated annealing algorithm is operated so that the energy H shown in Equation 1 decreases. executed. As the ground state is approached, the number of updated spins converges to 0, and the energy H converges to the lowest energy. That is, as the execution of the annealing algorithm progresses, the state in the Ising model converges.

Here, in simulated annealing, in a fully connected (complete graph) Ising model with interactions between all spins, updating (flipping) the state of each spin affects the state of all other spins. . Therefore, the states of the spins cannot be updated in parallel, and the states of each spin are updated sequentially. Therefore, simulated annealing may slow down the processing speed (convergence speed).

On the other hand, in the technique according to Patent Document 1 (hereinafter referred to as a "comparative example"), by introducing the self-action parameter q, it is possible to update the states of all spins of the Ising model in parallel. . In a comparative example, an annealing algorithm is performed using a technique called stochastic cellular automata annealing (SCA). A technique according to a comparative example will be described below.

<Comparative example>
FIG. 1 is a diagram showing a spin system according to a comparative example. FIG. 1 shows an example in which the number of spins N=5. In FIG. 1, each spin is indicated by a circle, and the direction of the arrow within the circle indicates the state of the corresponding spin. Here, an upward arrow indicates that the corresponding spin state is "+1", and a downward arrow indicates that the corresponding spin state is "-1". Therefore, σ ₁ =−1, σ ₂ =+1, σ ₃ =−1, σ ₄ =+1, σ ₅ =+1.

Also, τ _i corresponds to the next state of σ _i . In other words, the comparative example introduces a replica spin τ _i that is a copy of each spin σ _i . Here, in FIG. 1, each spin σ _i is coupled with all replica spins τ, but is not coupled with other spins σ _j . Similarly, each replica spin τ _i is uncoupled with other replica spins τ _j . This eliminates the σ-σ interaction (dependence) that prevents parallel update of the spin state. Therefore, in the comparative example, it is possible to update the spin states in parallel. In addition, in FIG. 1, the thick arrow pointing from σ _i to τ _i indicates self-action.

・・・（４） Here, the energy function (Hamiltonian) according to the comparative example is represented by Equation 4 below. In other words, the energy function shown in Eq. 1 is rewritten as Eq.

... (4)

Here, in Equation 4, τ on the left side is a vector (combination of τ _i ; spin arrangement) corresponding to Equation 2 above. Also, J _ij denotes the interaction between the spins σ _i and τ _j . Also, q is a self-action coefficient (parameter) indicating the self-action between the spin σ _i and the spin τ _i . The self-action coefficient q has a function of suppressing spin update.

Also, the third term on the right side functions as a penalty term that increases the energy H by +2q when the spin state is updated. That is, if the state of spin σ _i changes after the spin update process (that is, σ _i ≠τ _i ), the third term on the right side of Equation 4 becomes +2q. This indicates that a change in one spin state increases the energy of the system by 2q. On the other hand, if the state of spin σ _i does not change (that is, σ _i =τ _i ), the third term on the right side of Equation 4 is zero. Also, at this time, the first and second terms on the right side of Equation 4 are substantially the same as the first and second terms on the right side of Equation 1, respectively. Therefore, Equation 4 is equivalent to Equation 1 at this time. Here, when the self-action coefficient q is large, the third term tends to function as a penalty term, so that the spin is less likely to be reversed. That is, when the self-action coefficient q is large, the number of update spins decreases. On the other hand, the smaller the self-action coefficient q, the easier the spin reversal. That is, when the self-action coefficient q is small, the number of updated spins increases.

As described above, the number of updated spins decreases as the ground state is approached. Therefore, in the ground state, σ _i =τ _i (and σ _j =τ _j ). Therefore, in theory, Equation 4 can be equivalent to Equation 1 in the ground state. Therefore, under appropriate conditions, even in simulated annealing, the ground state (minimum energy state).

FIG. 2 is a diagram schematically showing updating of the spin state according to the comparative example. As mentioned above, each spin σ _i is not coupled with other spins σ _j . Similarly, each spin τ _i is uncoupled with other spins τ _j . This eliminates the σ-σ interaction that hinders the parallel update of the spin state, so that the spin state can be updated in parallel in the comparative example, as shown in FIG. 2 . That is, in one Monte Carlo step, the state of σ ₁ is updated to the state of τ ₁ , the state of σ ₂ is updated to the state of τ ₂ , the state of σ ₃ is updated to the state of τ ₃ , and the state of σ _{4 is updated} . from the state of τ to the state of τ ₄ can be performed in parallel. Note that τ ₁ to τ ₄ become σ ₁ to σ ₄ in the next Monte Carlo step.

FIG. 3 is a flowchart showing annealing processing according to a comparative example. First, data is initialized (process S101). Specifically, the annealing step t is set to the initial value "0". Also, the maximum step (t _max ) of the annealing step t is set, and how many times the spin update is repeated. Also, the temperature T is set to an initial value _T0 . Alternatively, the inverse temperature β (=1/T) is set to the initial value _β0 .

Also, an initial state is randomly set for each of the N spins (nodes) σ in the Ising model. Although the spin value may be binary, it is assumed here to be +1 and -1. Also, an external magnetic field coefficient h _i is determined for each node σ _i and an interaction coefficient J _ij between each node (between σ _i and σ _j ) is determined. Then, the self-action coefficient q is set to the initial value _q0 . These data are determined based on the model and set in the annealing machine.

・・・（５） Next, the spin state is updated (processes S102 to S104). First, in process S102, an instantaneous magnetic field (local field) is calculated. The instantaneous magnetic field is calculated by Equation 5 below.

... (5)

Here, for the sake of convenience, the left side of Equation 5 is expressed as " ^~ h _i " in this specification. The instantaneous magnetic field ^~ h _i is computed in parallel for each spin. where σ _i ^(t) denotes the spin state of spin i at step t.

・・・（６） Then, in process S103, update probabilities P _i are calculated in parallel for each spin σ _i (i=1 to N). Specifically, as shown in Equation 6 below, a sigmoid function is used to calculate the update probability P _i .

... (6)

In step S104, whether or not each spin σ _i (i=1 to N) can be updated is determined based on the spin update probability P _i , and updated in parallel. If the spin state is updated, then σ _i ←−σ _i . This gives the spin state σ of each spin at that annealing step t. For example, a uniform random number rand of 0 or more and less than 1 is generated for each spin, the random number rand is compared with the update probability P _i , and if P _i >rand, the state σ _i of the spin i is updated.

In process S105, the annealing parameters are updated. Specifically, the annealing step is advanced by 1 (t←t+1). Also, the temperature T or the inverse temperature β is updated ((T←T(t)) or (β←β(t))). Also, the self-action coefficient q is updated (q←q(t)). Here, T(t), β(t) and q(t) are functions predefined by the user.

The settings of T(t), β(t), and q(t) can be arbitrarily selected by the user. In general, as the annealing step progresses, the temperature T gradually decreases and the inverse temperature β, which is the reciprocal of T, gradually increases. In principle, q(t) has the same value in the ground state for the i-th spin σ i of the upper spin group and the i-th spin _{τ i of} the lower spin group shown in FIG. should be set as follows. That is, in the final step of annealing, a function is chosen such that q(t) is sufficiently large. In general, at the last annealing step, q(t) takes its maximum value.

In operation S106, it is determined whether the annealing step t is less than _tmax . If t<t _max (YES in S106), the processes of S102 to S105 are repeated. On the other hand, when the annealing step t reaches t _max (NO in S106), the spin array σ _i (i=1 to N) is read out and output as an optimized spin array (process S107).

<Problem of Comparative Example>
Here, problems of the comparative example will be described. As mentioned above, a large self-action coefficient q reduces the number of updated spins and thus reduces the convergence speed. On the other hand, when the self-action coefficient q is small, the number of update spins increases and thus the convergence speed increases. Therefore, in order to quickly obtain a good solution (a spin state in which the energy in the ground state is low), it is necessary to reduce the self-action coefficient q. However, as described below, if the self-action coefficient q is small, the state (updated spin number and energy) in the Ising model may not converge normally.

4 to 6 are diagrams exemplifying simulation results when the annealing algorithm is executed by the method according to the comparative example. 4 to 6 show an example in which the number of spins N=2000. 4 to 6, the horizontal axis indicates the number of Monte Carlo steps, that is, the annealing step t. Also, the vertical axis on the left side indicates the energy H. FIG. The vertical axis on the right side indicates the number of update spins. These are the same for other simulation results to be described later.

In FIGS. 4 to 6, the curve indicated by arrow A indicates the energy H of the system (SA) indicated by Equation (1). That is, the curve indicated by the arrow A is the energy H obtained by substituting the spin state σ obtained by executing the annealing algorithm by the method according to the comparative example into Equation 1 for each annealing step. Show the trajectory.

A curve indicated by an arrow B indicates the energy H of the system (SCA) indicated by Equation 4. That is, the curve indicated by arrow B is the energy H obtained by substituting the spin states σ and τ obtained by executing the annealing algorithm by the method according to the comparative example into Equation 4 for each annealing step. shows the trajectory of each

Also, the curve indicated by arrow C indicates the trajectory of the updated spin number for each annealing step. The number of updated spins is the number of spins whose spin state changed (+1→−1 or −1→+1) before and after each annealing step. In addition, in the lower tables of FIGS. 4 to 6, H _mean is the average value of the energy of the system shown in Equation 1 in the final state of the simulation when the experiment of the annealing algorithm is executed multiple times (here, 128 times). is. Also, H _min is the minimum value of the energy of the system represented by Equation 1 in the final state of the simulation in that case. Also, H _max is the maximum value of the energy of the system represented by Equation 1 in the final state of the simulation in that case. These are the same for other simulation results to be described later. Note that ideally, the final state of the simulation corresponds to the ground state.

Here, when the annealing algorithm according to the comparative example is executed, a combination of spin states σ (σ _i (i=1 to N)) is obtained for each annealing step (S104 in FIG. 3). Then, by substituting each obtained spin state σ _i (i=1 to N) into Equation 1, the energy H of the system shown in Equation 1 can be obtained. Similarly, by substituting each obtained spin state σ _i (i=1 to N) and τ _i (spin state next to σ _i ) corresponding to the spin state σ _i into Equation 4, Equation 4 The energy H of the system shown by is obtained.

Also, FIG. 4 shows the simulation results for q=10.538. Also, FIG. 5 shows the simulation results for q=5.296. Also, FIG. 6 shows the simulation results for q=3.513. In the examples of FIGS. 4 and 5, the energy of the system represented by Equation 1, the energy of the system represented by Equation 4, and the number of updated spins converge normally. Therefore, in this case, a state close to the ground state can be appropriately obtained. Also, from the values of the energy H in the lower tables of FIGS. 4 and 5, it can be seen that the smaller the value of q, the lower the energy H in the final state of the simulation (a state close to the ground state).

On the other hand, in the example of FIG. 6, as indicated by arrow B, the energy of the system indicated by formula 4 apparently converges, but as indicated by arrows A and C, the energy of the system indicated by formula 1 and The number of update spins has not converged. Therefore, in the example of FIG. 6, it cannot be said that the state in the Ising model converges normally.

That is, the update probability p _i expressed by Equation 6 is configured such that the energy function H shown by Equation 4 decreases as the annealing step progresses. Then, when q is large as in the examples of FIGS. 4 and 5, the influence of q is large, so the spin state is updated so as to reduce the third term on the right side of Equation 4. FIG. On the other hand, when q is small as in the example of FIG. 6, the influence of the first and second terms on the right side of Equation 4 can be greater than the influence of the third term on the right side. Therefore, the spin state is updated so that the influence of the third term on the right side of Equation 4 is canceled by the first and second terms on the right side. In this case, the energy in Equation 4 may be low if σ _i ≠τ _i all the time. That is, in a state (vibrational state) in which all spins repeatedly update their spin states as indicated by arrow C in FIG. In this case, as indicated by arrow C in FIG. 6, the number of update spins is the maximum (2000). Also, as indicated by arrow A in FIG. 6, the energy in Equation 1 diverges into a higher energy state. Therefore, in the technique according to the comparative example, if the self-action coefficient q is small, the state in the Ising model may not converge normally. Therefore, it is difficult for the technique of the comparative example to obtain a high-quality solution at high speed.

On the other hand, in the present embodiment, an update trial rate ε is introduced into the annealing algorithm. The update attempt rate ε indicates the ratio of the number of spins whose spin states are attempted to be updated (spins that may be updated) to the total number of spins N. In other words, the update trial rate ε is the ratio of the number of spins for which it is determined whether or not to update the spin state (spins subject to update determination) to the total number of spins. Here, the number of update determination target spins corresponds to the maximum number of update spins. It should be noted that the spin state of the update determination target spin is not necessarily updated, but is updated according to the update probability. By introducing the update trial rate ε, the number of spins updated in one annealing step is suppressed. Therefore, it is possible to suppress the phenomenon that all spins are repeatedly updated as shown in FIG.

7 and 8 are diagrams for explaining the update trial rate ε according to this embodiment. FIG. 7 shows a spin system according to a comparative example, and FIG. 8 shows a spin system according to this embodiment. 7 and 8 show an example in which the number of spins N=5, as in FIG. Here, in FIGS. 7 and 8, the spins surrounded by a rectangle with a dashed line are update determination target spins. As shown in FIG. 7, in the comparative example, all spins are update determination target spins at all times.

On the other hand, as shown in FIG. 8, according to the present embodiment, there are spins that are not update determination target spins depending on the update attempt rate ε. For example, when ε=0.6, 60% of all spins are update determination target spins. In the example of FIG. 8, spins σ ₁ , σ ₃ , and σ ₄ are the target spins for update determination in the annealing step t _n . Therefore, σ ₁ , σ ₃ , σ ₄ may be updated depending on the corresponding update probabilities. On the other hand, σ ₂ and σ ₅ which are not update determination target spins are not updated even if the corresponding update probability is high.

At the annealing step tn ₊₁ , spins σ ₂ , σ ₄ , and σ ₅ are the target spins for update determination. Therefore, σ ₂ , σ ₄ , σ ₅ may be updated depending on the corresponding update probabilities. On the other hand, σ ₁ and σ ₃ , which are not update determination target spins, are not updated even if the update probability is high. In this way, the update determination target spin can be changed for each annealing step. It should be noted that the ratio of the number of update determination target spins to the total number of spins does not need to strictly match the update trial rate ε. That is, even if ε=0.6, the ratio of the number of spins subject to update determination to the total number of spins does not need to be strictly 60%. In other words, if the number of spins subject to update determination is N _f , then ε≈N _f /N.

(Embodiment 1)
Hereinafter, embodiments will be described with reference to the drawings. For clarity of explanation, the following descriptions and drawings are omitted and simplified as appropriate. Moreover, in each drawing, the same elements are denoted by the same reference numerals, and redundant description is omitted as necessary. A first embodiment described below corresponds to an improved annealing algorithm (SCA) according to the comparative example described above.

<Information processing device>
FIG. 9 is a diagram showing the configuration of the information processing apparatus 100 according to the first embodiment. The information processing apparatus 100 has a control unit 102, a storage unit 104, a communication unit 106, and an interface unit 108 (IF: Interface) as main hardware components. The control unit 102, storage unit 104, communication unit 106 and interface unit 108 are interconnected via a data bus or the like.

The control unit 102 is a processor such as a CPU (Central Processing Unit), for example. The control unit 102 has a function of executing control processing, arithmetic processing, and the like. The storage unit 104 is, for example, a storage device such as memory or hard disk. The storage unit 104 is, for example, a ROM (Read Only Memory) or a RAM (Random Access Memory). The storage unit 104 has a function of storing a control program, an arithmetic program, and the like executed by the control unit 102 . The storage unit 104 also has a function of temporarily storing processing data and the like. Storage unit 104 may include a database.

The communication unit 106 performs processing necessary for communicating with other devices via a wired or wireless network. Communication unit 106 may include communication ports, routers, firewalls, and the like. The interface unit 108 is, for example, a user interface (UI). The interface unit 108 has an input device such as a keyboard, touch panel, or mouse, and an output device such as a display or speaker. The interface unit 108 receives data input operations by the user and outputs information to the user.

Further, the information processing apparatus 100 includes, as components, an annealing control unit 110, an instantaneous magnetic field calculation unit 120, an update probability calculation unit 130, a random number generation unit 140, a spin state determination unit 150, and a spin array output unit 160. and Annealing control section 110 also includes step control section 112 , temperature control section 114 , self-action control section 116 , and update trial rate control section 118 .

Each component can be realized by executing a program under the control of the control unit 102, for example. More specifically, each component can be implemented by control unit 102 executing a program stored in storage unit 104 . Further, each component may be realized by recording a necessary program in an arbitrary non-volatile recording medium and installing the program as necessary. Moreover, each component may be implemented by any combination of hardware, firmware, and software, without being limited to being implemented by program software. Also, each component may be implemented using a user-programmable integrated circuit such as an FPGA (field-programmable gate array) or a microcomputer. In this case, this integrated circuit may be used to implement a program composed of the above components.

An annealing control unit 110 is configured to control the annealing step. The annealing control unit 110 is also configured to update parameters in the annealing algorithm. Specifically, the annealing control unit 110 is configured to control a temperature parameter, a self-action parameter, and an update trial rate.

The step controller 112 is configured to control the annealing step t in the annealing algorithm. Specifically, the step controller 112 updates the annealing step t as the annealing step progresses. In other words, the step control unit 112 increments the annealing step t by one each time the processing for each spin is completed in a certain annealing step. The step controller 112 updates the annealing step t up to the maximum step t _max .

The temperature control unit 114 is configured to control the temperature T or the inverse temperature β, which is a temperature parameter. Specifically, the temperature controller 114 updates the temperature T or the inverse temperature β for each annealing step. Here, similarly to the comparative example described above, the temperature T is a function of the annealing step t. Specifically, the temperature T(t) is a function that decreases as the annealing step t increases. Also, the inverse temperature β is a function of the annealing step t. Specifically, the inverse temperature β(t) is a function that increases as the annealing step t increases. Temperature T(t) and inverse temperature β(t) are functions predetermined by the user.

The self-action control unit 116 is configured to control the self-action coefficient q. Specifically, the self-action control unit 116 updates the self-action coefficient q for each annealing step. Here, the self-action coefficient q may be a function of the annealing step t. The self-action coefficient q(t) may be a function that increases as the annealing step t increases. Also, the self-action coefficient q(t) may be a constant value (that is, q(t)=q ₀ ). The self-action coefficient q(t) is a function predetermined by the user.

The update trial rate control unit 118 is configured to control the update trial rate ε. Specifically, update trial rate control section 118 updates update trial rate ε for each annealing step. Here, the update trial rate ε may be a function of the annealing step t. The update attempt rate ε(t) may be a function that decreases as the annealing step t increases. In other words, the update trial rate control unit 118 may control the update trial rate ε so that it decreases as the annealing step elapses. Also, the update trial rate ε(t) may be a constant value (that is, ε(t)=ε ₀ ). The update attempt rate ε(t) is a function predetermined by the user.

FIG. 10 is a diagram exemplifying a function of the update trial rate ε according to the first embodiment; The update trial rate ε(t) illustrated in FIG. 10 is a linear function that takes an initial value ε _init at t=1 and a final value ε _fin at t=t _max . Here, the update trial rate ε(t) is maximized at the initial value ε _init and minimized at the final value ε _fin . Therefore, the update trial rate ε(t) gradually decreases as the annealing step t elapses. Note that the update trial rate ε(t) does not have to be a linear function. Also, the update trial rate ε(t) does not have to be a continuous function. The update trial rate ε(t) may be a function that decreases stepwise as the annealing step t elapses. In other words, the update trial rate ε(t) need not necessarily decrease each time the annealing step t increases by one.

The instantaneous magnetic field calculator 120 is configured to calculate the instantaneous magnetic fields of multiple spins in parallel. Specifically, the instantaneous magnetic field calculator 120 uses Equation 5 above to calculate the instantaneous magnetic field ^~ h _i for each spin i=1 to N in parallel.

The update probability calculator 130 is configured to calculate the update probabilities of a plurality of spins in parallel based on the instantaneous magnetic field calculated by the instantaneous magnetic field calculator 120 . Specifically, the update probability calculator 130 calculates in parallel the update probabilities P _i for each spin i=1 to N using Equation 6 above, for example.

The random number generator 140 is configured to generate random numbers. Specifically, the random number generation unit 140 generates rand, which is a uniform random number equal to or greater than 0 and less than 1. The random number generator 140 generates a random number rand for each of the plurality of spins in each annealing step. Therefore, the random number rand can be different for each spin. More specifically, random number generator 140 generates rand _(P) and rand _(ε) . rand _(P) (first random number) is compared with the update probability P _i in the spin state determination unit 150 . rand _(ε) (second random number) is compared with the update trial rate ε in spin state determination section 150 .

The spin state determination unit 150 is configured to determine states of a plurality of spins in parallel based on the update probability and the update trial rate. Specifically, when the update probability P _i is greater than the random number rand _(P) and the update trial rate ε is greater than the random number rand _(ε) for each of the plurality of spins, the spin state determination unit 150 Update the status of the spin. Details will be described later.

The spin array output unit 160 is configured to output the spin array at the annealing step t=t _max . _{It is expected that the spin alignment σ_1̂(t max ) ,} .

<Information processing method>
FIG. 11 is a flowchart showing an information processing method executed by the information processing apparatus 100 according to the first embodiment. First, the annealing control unit 110 initializes data (process S202). Specifically, the step controller 112 sets the annealing step t to the initial value "1". The step control unit 112 also sets the maximum step (t _max ) of the annealing step t, and sets how many times the spin update is repeated. Also, the temperature control unit 114 sets the temperature T to the initial temperature _T1 . Alternatively, the temperature control unit 114 sets the inverse temperature β (=1/T) to the initial inverse temperature β ₁ (=1/T ₁ ).

・・・（７） Also, the annealing control unit 110 randomly sets the initial states σ_1̂(0), . Although the spin value may be binary, it is assumed here to be +1 and -1.

... (7)

Also, the annealing control unit 110 determines the external magnetic field coefficient h _i of each node σ _i for all i (i=1, . . . , N). Also, the annealing control unit 110, for all i (i=1, . . . , N) and j (j=1, . . . , N), between each node (between σ _i −σ _j ) Define the interaction coefficients J _ij of .

Further, the self-action control unit 116 sets the self-action coefficient q to the initial value _q1 . Further, the update trial rate control unit 118 sets the update trial rate ε to the initial value _ε1 . For example, the initial value ε ₁ corresponds to the initial value ε _init in FIG.

Next, the information processing apparatus 100 executes the annealing algorithm (process S204 to process S212). First, the instantaneous magnetic field calculator 120 calculates an instantaneous magnetic field (process S204). Specifically, the instantaneous magnetic field calculator 120 calculates in parallel the instantaneous magnetic fields ^˜h _i for all spins i=1 to N using Equation 5 above, as described above.

Next, the update probability calculator 130 calculates the spin update probability based on the instantaneous magnetic field calculated by the instantaneous magnetic field calculator 120 (process S206). Specifically, update probability calculation section 130 calculates update probabilities P _i for all spins i=1 to N in parallel using Equation 6 above.

Next, the spin state determination unit 150 determines the next state of each spin (process S208). Specifically, for each spin i=1 to N, the spin state determination unit ₁₅₀ determines the spin state σ Determine _i ^(t) . Note that σ _i ^(t) corresponds to the following spin states. That is, if t=1, σ _i ^(t) (ie, σ _i ⁽¹⁾ ) corresponds to the next state after the initial state σ _i ⁽⁰⁾ . Specific processing of the spin state determination unit 150 will be described later.

Next, the step controller 112 determines whether or not t=t _max (process S210). If not t=t _max , that is, if the annealing step t has not reached the maximum step (NO in S210), the annealing control unit 110 updates the annealing parameters (process S212). Specifically, the step controller 112 advances the annealing step by 1 (t←t+1). Also, the temperature control unit 114 updates the temperature T (T←T(t)). Alternatively, the temperature control unit 114 updates the inverse temperature β (β←β(t)). Further, the self-action control unit 116 updates the self-action coefficient q (q←q(t)). Furthermore, the update trial rate control unit 118 updates the update trial rate ε (ε←ε(t)). Here, T(t), β(t), q(t), and ε(t) are functions predefined by the user. Then, the processing of S204 to S212 is repeated. This allows the execution of the annealing algorithm to continue and the annealing steps to proceed.

・・・（８） In the process of S204, the instantaneous magnetic field calculator 120 may calculate the instantaneous magnetic field only for spins whose spin states have been updated. In other words, the instantaneous magnetic field calculator 120 may update the instantaneous magnetic field for all spins j (j=1 to N) for the updated spin i according to Equation 8 below. In this case, the updated spin state σ _i or the spin identifier i may be stored in a list.

... (8)

In the case of calculating the instantaneous magnetic field using Equation 5 regardless of whether or not the spin state has been updated, and in the case of updating the instantaneous magnetic field only for the spins whose spin state has been updated as described above, Note that the instantaneous magnetic field for each spin is the same. That is, calculating the instantaneous magnetic field using Equation 5 regardless of whether the spin state has been updated results in updating the instantaneous magnetic field for spins whose spin state has been updated. On the other hand, for the spins whose spin state has not been updated, the same instantaneous magnetic field as in the previous annealing step is calculated. However, by updating the instantaneous magnetic field only for spins whose spin states have been updated, unnecessary calculations are suppressed, so the load on computational resources can be suppressed.

On the other hand, if t=t _max , that is, if the annealing step t reaches the maximum step (YES in S210), the spin array output unit 160 outputs the spin array at the annealing step t=t _max (process S214). . Specifically, the spin array output unit 160 reads out the spin arrays σ _i (i=1 to N) at the annealing step t=t _max to obtain the optimized spin arrays σ_1̂(t _max ), . , σ_N̂(t _max ).

FIG. 12 is a diagram for explaining the processing (S208) of the spin state determination unit 150 according to the first embodiment. FIG. 12 shows an example in which the number of spins N=5. As described above, for the annealing step (t−1), the instantaneous magnetic field ( ^˜h _i ) of each spin is calculated (S204) and the update probability P _i of each spin is calculated (S206). Then, in S208, the random number generator 140 generates rand _(P) and rand _(ε) for each spin. Here, as described above, 0≦rand _(P) <1 and 0≦rand _(ε) <1.

The spin state determination unit 150 makes a determination as the determination A based on the update probability P _i . Specifically, for all spins i ₍ i=1, _{. (P)} is determined. For each spin, the spin state determination unit 150 sets the determination result to “1” if the determination result is true, and sets the determination result to “0” if the determination result is false. In the example of FIG. 12, the determination results of determination A for spins σ ₁ , σ ₂ , σ ₃ , σ ₄ , and σ ₅ are “1”, “1”, “0”, “1”, “0 ”.

Further, spin state determining section 150 makes determination B based on update trial rate ε. Specifically, the spin state determination unit 150 uses rand _(ε) independently generated for each spin for all spins i (i=1, . . . , N), Determine whether ε>rand _(ε) . Note that a random number rand _(ε) is generated for each spin i, and judgments are made independently of each other, so the result of judgment B may differ for each spin. For each spin, the spin state determination unit 150 sets the determination result to “1” if the determination result is true, and sets the determination result to “0” if the determination result is false. In the example of FIG. 12, the determination results of determination B for spins σ ₁ , σ ₂ , σ ₃ , σ ₄ , and σ ₅ are “1”, “0”, “1”, “1”, “0 ”.

・・・（９） Then, the spin state determination unit 150 updates the spin state of each spin i (i=1, . judge. That is, for each spin i (i=1, . Determine update.

... (9)

In the example of FIG. 12, the spin update determinations for spins σ ₁ , σ ₂ , σ ₃ , σ ₄ , and σ ₅ are "1", "0", "0", "1", and "0", respectively. be. Here, it is assumed that the spin state is updated when the determination is "1". As a result, the spin state determining unit 150 determines the spin states σ ₁ ^(t) , σ ₂ ^(t) , σ ₃ ^(t) , σ ₄ ^(t) , and σ ₅ ^(t) of each spin for the annealing step t as decide. In the example of FIG. 12, σ ₁ ^(t) =−σ ₁ ^(t−1) , σ ₂ ^(t) =σ ₂ ^(t−1) , σ ₃ ^(t) =σ ₃ ^(t−1) , σ ₄ ^(t) = -σ ₄ ^(t-1) and σ ₅ ^(t) = σ ₅ ^(t-1) . If the spin state σ _i ^(t−1) is “σ _i ”, the spin state σ ₁ ^(t) corresponds to “τ _i ”.

Here, in the example of FIG. 12, the spin state of spin σ ₂ is not updated even though its update probability P ₂ is greater than the random number rand _(P) . On the other hand, in the comparative example, the spin states of all spins whose update probability is greater than the random number rand are updated.

<Simulation result>
13 to 15 are diagrams illustrating simulation results when the annealing algorithm is executed by the technique according to the first embodiment. The curve indicated by the arrow A is the energy H obtained by substituting the spin state σ obtained by executing the annealing algorithm by the method according to the first embodiment into Equation 1 for each annealing step. Show the trajectory. In addition, the curve indicated by arrow B is the energy H obtained by substituting the spin states σ and τ obtained by executing the annealing algorithm by the method according to Embodiment 1 into Equation 4 for each annealing step. Traces for each annealing step are shown. The curve indicated by arrow C shows the trajectory of the updated spin number for each annealing step.

FIG. 13 shows simulation results for q=3.513, similar to the example shown in FIG. Also, in FIG. 13, the update trial rate ε changes from 1.0 to 0.5 as the annealing step progresses. That is, ε _init =1.0 and ε _fin =0.5. Here, as described above, in the example of FIG. 6, the state in the Ising model does not converge normally. On the other hand, in FIG. 13, the energy (arrow A) and the updated spin number (arrow C) of the system shown in Equation 1 converge. That is, unlike the example in FIG. 6, the state in the Ising model converges normally in the example in FIG.

Also, FIG. 14 shows the simulation results when q=2.0. Also, in FIG. 14, the update trial rate ε changes from 0.95 to 0.3 as the annealing step progresses. That is, ε _init =0.95 and ε _fin =0.3. Even in this case, the energy (arrow A) and the updated spin number (arrow C) of the system shown in Equation 1 converge. That is, as shown in FIG. 14, even when the self-action coefficient q is smaller than in the example of FIG. 13, the state in the Ising model converges normally.

Also, FIG. 15 shows the simulation results when q=1.5. Also, in FIG. 15, the update trial rate ε changes from 0.95 to 0.3 as the annealing step progresses. That is, ε _init =0.95 and ε _fin =0.3. Even in this case, the energy (arrow A) and the updated spin number (arrow C) of the system shown in Equation 1 converge. That is, as shown in FIG. 15, even when the self-action coefficient q is smaller than in the example of FIG. 14, the state in the Ising model converges normally.

Thus, in Embodiment 1, even when the self-action coefficient q is small, a state close to the ground state (minimum energy state) can be appropriately obtained. That is, in Embodiment 1, the maximum number of spins to be updated is suppressed by introducing the update trial rate ε. As a result, only some of all spins are updated in any annealing step. Therefore, it is possible to suppress the occurrence of a phenomenon (vibration state) in which all the spins are repeatedly updated when the self-action coefficient q is small, as in the comparative example illustrated in FIG. As a result, as shown in FIGS. 13 to 15, a state close to the normal ground state is obtained. Therefore, the information processing apparatus 100 according to the first embodiment can obtain high-quality solutions at high speed.

Also, from the values of the energy H in the lower tables of FIGS. 13 to 15, it can be seen that the smaller the value of q, the lower the energy H in the final state (state close to the ground state). That is, in Embodiment 1, it is possible to appropriately obtain a good solution. Therefore, in the first embodiment, by performing spin update in parallel, it is possible to obtain a high-quality solution while performing high-speed processing.

Further, in the information processing apparatus 100 according to the first embodiment, for each of a plurality of spins, the update probability P _i is larger than rand _(P) (first random number), and the update trial rate ε is rand _(ε) (first random number). 2), the state of the spin is updated. This makes it possible to easily implement an algorithm capable of suppressing the occurrence of the phenomenon (oscillating state) in which all spins repeatedly update their spin states, as described above.

Further, the information processing apparatus 100 according to the first embodiment controls the update trial rate ε so as to decrease as the annealing step elapses. This can more reliably make it difficult for the spin state to be updated as the annealing step elapses. Here, as described above, in the annealing algorithm, the number of updated spins decreases as the ground state is approached. Therefore, by controlling the update trial rate ε as described above, it is possible to more reliably obtain a state close to the ground state. Therefore, it is possible to obtain a good solution.

(Modification of Embodiment 1)
FIG. 16 is a diagram for explaining another algorithm that can be implemented by the information processing apparatus 100 according to the first embodiment. As described above, the information processing apparatus 100 according to the first embodiment is configured to implement an algorithm obtained by improving the SCA algorithm according to the comparative example. However, the information processing apparatus 100 according to the first embodiment can also be applied to algorithms other than SCA. For example, as described below, the information processing apparatus 100 according to the first embodiment can also be applied to momentum annealing (MA). That is, the algorithm according to the information processing apparatus 100 according to the first embodiment is not limited to SCA. The information processing apparatus 100 according to the first embodiment is applicable to any algorithm that can update spin states of a plurality of spins in parallel.

Similar to SCA, momentum annealing can update the states of all spins of the Ising model in parallel. As indicated by an arrow M1 in FIG. 16, the Ising model in momentum annealing is a fully coupled (perfect graph) model having interactions between all spins. The example of FIG. 16 shows an example of a model consisting of spins σ _i (i=1 to 5) with the number of spins N=5.

This model is converted into a graph that is divided into left and right halves as indicated by an arrow M2. In this transformed model 300, σ _i ^R on the right corresponds to the next spin state of σ _i ^L on the left. That is, in model 300, σ _i ^L and σ _i ^R are paired. In momentum annealing, the configuration of this transformed model 300 is predetermined.

Here, as indicated by an arrow M3, the spin state update process is shown in chronological order. From t=0 to t=t _max , the spin state σ_î(t) of each spin i is updated. Here, in the model 300, when the spin state σ_i^(0) at t=0 corresponds to the spin state _σiL , the spin state σ_i^(1) at t=1 ^corresponds to the spin state σ_i^ corresponds to the spin state σ _i ^R paired with the spin state σ _i ^L corresponding to (0). Then, in the model 300, the spin state σ_î(2) at t=2 corresponds to the spin state next to the spin state σ_î(0) at the position of ^the spin state _σiL . Also, in the model 300, the spin state σ_î(3) at t=3 corresponds to the spin state next to the spin state σ_î(1) at the position of ^the spin state _σiR . As a result, the spin state σ_î(2) and the spin state σ_î(3) are paired. Likewise, in the model 300, when the spin state σ_i^(t _max ) at t=t _max corresponds to the spin state σ _i ^R , the spin state σ_i^ at t=(t _max −1) (t _max −1) corresponds to the spin state σ _i ^L. As a result, the spin state σ_î(t _max −1) and the spin state σ_î(t _max ) are paired.

Here, in the model 300 related to momentum annealing, the spin state of spin i corresponding to the position of σ _i ^L is based on the spin state one before the position of σ _i ^L. Similarly, in model 300, the spin state of spin i corresponding to a σ _i ^R position is based on the spin state that precedes this σ _i ^R position. In other words, the spin state of each spin is updated based on the spin state in the previous annealing step in the update process shown in chronological order (indicated by arrow M3). For example, the update of spin state σ_î(2) is based on spin state σ_î(0).

・・・（１０） Here, when the information processing apparatus 100 according to the first embodiment described above is applied to the momentum annealing algorithm, the flowchart shown in FIG. 11 may be modified as described below. That is, in S202, the annealing control unit 110 sets the initial states of the N spins not only to those shown in Equation 7, but also to the initial states σ_1^(-1 ), . . . , σ_N^(-1) are set randomly.

(10)

・・・（１１） Further, in S206, the update probability calculation unit 130 calculates the update probabilities P _i for all spins i=1 to N in parallel using the following equation 11 based on the instantaneous magnetic field calculated in the process S204. calculate.

(11)

・・・（１２） In S208, the spin state determination unit 150 performs an AND operation on the determination results of determination A and determination B described using FIG. 12 for each spin i (i=1, . . . , N). , the spin state update is determined by Equation 12 below.

(12)

As described above, the information processing apparatus 100 according to the first embodiment is applicable not only to SCA but also to momentum annealing. That is, the information processing apparatus 100 according to the first embodiment is not limited to SCA. The information processing apparatus 100 according to the first embodiment is applicable to any algorithm that can update spin states of a plurality of spins in parallel.

(Modification)
It should be noted that the present invention is not limited to the above embodiments, and can be modified as appropriate without departing from the scope of the invention. For example, in the flowcharts described above, the order of each process (step) can be changed as appropriate. Also, one or more of a plurality of processes (steps) may be omitted.

For example, in the flowchart shown in FIG. 11, the process of S212 may be executed after the process S208 instead of the process of S210. In this case, a process substantially similar to S106 in the flowchart of FIG. 3 may be executed after the process of S212. Also, in the process of S210, it may be determined whether or not "t≧t _max ".

Further, in the first embodiment described above, the spin state determination unit 150 determines that, for each of a plurality of spins, when the update probability is greater than the random number rand _(P) and the update trial rate is greater than the random number rand _(ε) , the state of the spin is updated. However, it is not limited to such a configuration. The spin state of spin i may be updated when P _i ×ε>rand. In other words, spin state determination section 150 may determine to update the spin state of spin i when the product of update probability P _i and update trial rate ε is greater than a random number. In other words, the determination based on the update probability P _i and the determination based on the update trial rate ε need not be performed independently of each other.

In the above examples, the program includes instructions (or software code) that, when read into the computer, cause the computer to perform one or more of the functions described in the embodiments. The program may be stored in a non-transitory computer-readable medium or tangible storage medium. By way of example, and not limitation, computer readable media or tangible storage media may include random-access memory (RAM), read-only memory (ROM), flash memory, solid-state drives (SSD) or other memory technology, CDs -ROM, digital versatile disk (DVD), Blu-ray disc or other optical disc storage, magnetic cassette, magnetic tape, magnetic disc storage or other magnetic storage device. The program may be transmitted on a transitory computer-readable medium or communication medium. By way of example, and not limitation, transitory computer readable media or communication media include electrical, optical, acoustic, or other forms of propagated signals.

100 information processing device 110 annealing control unit 112 step control unit 114 temperature control unit 116 self-action control unit 118 update trial rate control unit 120 instantaneous magnetic field calculation unit 130 update probability calculation unit 140 random number generation unit 150 spin state determination unit 160 spin array output part 300 model

Claims (5)

An information processing device for obtaining a solution using an Ising model,
an annealing control unit that controls the annealing step and also controls the temperature parameter, the self-action parameter, and the update trial rate of a plurality of spins in the Ising model;
an instantaneous magnetic field calculation unit that calculates the instantaneous magnetic fields of the plurality of spins in parallel;
an update probability calculator that calculates in parallel the update probabilities of the plurality of spins based on the instantaneous magnetic field;
a spin state determination unit that determines states of the plurality of spins in parallel based on the update probability and the update trial rate;
Information processing device having The annealing control unit controls the update trial rate so that the update trial rate decreases as the annealing step elapses.
The information processing device according to claim 1 . a random number generator that generates a first random number and a second random number for each of the plurality of spins;
further having
The spin state determination unit updates the state of each of the plurality of spins when the update probability is greater than the first random number and the update trial rate is greater than the second random number. ,
The information processing apparatus according to claim 1 or 2. An information processing method for obtaining a solution using an Ising model,
controlling the annealing step, controlling the temperature parameter, the self-action parameter, and the update trial rate of multiple spins in the Ising model;
calculating in parallel the instantaneous magnetic fields of the plurality of spins;
calculating in parallel update probabilities of the plurality of spins based on the instantaneous magnetic field;
determining states of the plurality of spins in parallel based on the update probability and the update trial rate;
Information processing methods. A program for realizing an information processing method for obtaining a solution using an Ising model,
A process of controlling the annealing step and controlling the temperature parameter, the self-action parameter, and the update trial rate of a plurality of spins in the Ising model;
a process of calculating in parallel the instantaneous magnetic fields of the plurality of spins;
a process of calculating in parallel update probabilities of the plurality of spins based on the instantaneous magnetic field;
A process of determining states of the plurality of spins in parallel based on the update probability and the update trial rate;
A program that makes a computer run

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