Optical ising machine based on cholesky decomposition

Abstract

The application discloses an optical Ising machine based on Cholesky decomposition. The application encodes the interaction matrix of the Ising model into an optical modulation system after Cholesky decomposition, inputs an optical signal carrying the spin group state information of the Ising model, and outputs an optical intensity signal encoded with the Hamiltonian information of the Ising model, so that the high-speed calculation of an arbitrary Ising model Hamiltonian is realized. The application has the advantages of large scale, wide application range, high speed and the like.

Description

Technical Field

[0001] This invention relates to the field of optical information processing, and more particularly to an optical Ising machine based on Cholesky decomposition. Background Technology

[0002] The Ising model is a complex system composed of multiple spins, each of which can only be in one of two states: up or down. The Hamiltonian of the Ising model can be written as... Where N is the number of spins in the Ising model, J jk For spin σ j With spin σ k The interaction strength between them, σ i =±1 represent the two orientations of the spin, h i For other parameters.

[0003] In recent years, various Ising machines based on principles such as optical parametric oscillation and FPGA feedback, quantum adiabatic computing, and spatial optical modulation have emerged. Solving for the ground state of the Ising model involves calculating the minimum value of the Hamiltonian of the Ising model. Solving for the ground state of the Ising model is mathematically consistent with solving the optimal solution of combinatorial optimization problems. Combinatorial optimization problems, such as graph partitioning, graph coloring, Hamiltonian cycle problems, and the peddler's problem, have significant value in fields such as transportation planning, drug design, financial strategy, and circuit design.

[0004] Optical computing boasts advantages such as ultra-fast speed, large scale, and high parallelism, and currently there are three main types of optical Ising machines. The first type, based on optical parametric oscillations and FPGA feedback, modulates the spin onto the pulse phase in a resonant cavity, using the FPGA to calculate the feedback signal to determine the next sampling round. Since the FPGA can only modulate a single pulse at a time, the iteration time increases linearly with the problem size. Existing Ising machines based on spatial phase modulation utilize the Fourier properties of the lens system to achieve spin coupling and Hamiltonian calculation. However, they cannot represent or calculate the Hamiltonian of arbitrary Ising models; they can only calculate the Hamiltonian of Ising models with Mattis-type interactions.

[0005] Cholesky decomposition is a matrix factorization method that decomposes a symmetric matrix into multiple sets of column vectors and their transposes. Cholesky decomposition can decompose a symmetric interaction matrix into multiple sets of fully paired and fully coupled Mattis-type interaction matrices, thereby enabling the computation of the Hamiltonian of any Ising model. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of existing technologies by proposing a method for implementing an optical Ising machine based on Cholesky decomposition and an optical Ising machine itself, which has advantages such as simple structure, large scale, and strong applicability.

[0007] This invention reorders the spin numbers, reducing the number of matrix elements after Cholesky decomposition, and then performs Cholesky decomposition on the interaction matrix. For Ising models with few spin connections or local clusters, the space required by this invention is far less than that of decomposition methods such as eigenvalue decomposition. Even in the case of full connectivity, only half the space required by eigenvalue decomposition methods is needed. In reality, combinatorial optimization problems often do not require fully coupled interactions, but only local interactions with a finite number of connections. For Ising models with a connection number proportional to the number of spins and many clusters, the space required by this invention is proportional to the number of spins, not the square of the number of spins.

[0008] The technical solution adopted by this invention to solve the problem is as follows:

[0009] According to a first aspect of this specification, a method for implementing an optical Ising machine based on Cholesky decomposition is provided, comprising:

[0010] Interaction matrix encoding based on Cholesky decomposition: After reordering the interaction matrix of the Ising model, Cholesky decomposes it into multiple Mattis-type interaction matrices, and the decomposition results are encoded in the optical modulation system;

[0011] The optical modulation system calculates the Hamiltonian of the Ising model: an optical Fourier transform is performed using a lens system to make the light fields carrying different spin and interaction information interfere and superimpose. The interference signal contains the Hamiltonian information of the Ising model encoded by the optical modulation system, thereby calculating the Hamiltonian of the Ising model.

[0012] The spin configuration of the Ising model is updated cyclically: a new spin configuration is generated, and whether to retain the newly generated spin configuration is determined according to the Hamiltonian. The spin configuration to be retained is then fed back into the time or spatial phase of the optical modulation system until a spin configuration that meets the requirements is found.

[0013] Furthermore, the reordering methods include, but are not limited to, the Cuthill-Mckee algorithm, the minimum degree algorithm, and the nested partitioning permutation algorithm, which can change the spin index and interchange the rows and columns of the interaction matrix, thereby reducing the number of matrix elements after Cholesky decomposition.

[0014] Furthermore, for the input Ising model interaction matrix J and magnetic field parameters {h} i Cholesky decomposition decomposes the Hamiltonian H of the Ising model into multiple Mattis-type interacting Ising model Hamiltonians H. i The sum is as follows:

[0015] First, using the Cholesky decomposition method, the input interaction matrix J is decomposed into the product of matrix L and its transpose, J = LL. T At this time, the matrix element J jk For the j-th spin σ j With the k-th spin σ k The interaction strength between them, where N is the number of spins in the Ising model. It is the element in the j-th row and i-th column of matrix L. This represents the interaction strength between the j-th spin and the k-th spin in the i-th group of Mattis-type interactions obtained from the Cholesky decomposition.

[0016] Next, define Where i = 1, 2, ..., N, such that h i These are the input magnetic field parameters.

[0017] Ultimately, the Hamiltonian H of the Ising model is decomposed into multiple Mattis-type interacting Ising model Hamiltonians H i sum:

[0018]

[0019] in σ N+1 =1, H0 is based on the input interaction matrix J and magnetic field parameters {h i} Determined parameters.

[0020] Furthermore, in the process of Hamiltonian optical calculation, in order to enable the light fields to interfere and superimpose, coherent light sources, including but not limited to lasers, are required.

[0021] Furthermore, in the Hamiltonian optical calculation process, spin and interaction information is temporally or spatially modulated using an optical modulator, thereby encoding it onto the incident beam.

[0022] Furthermore, the formula for calculating the Hamiltonian is: Where A is the normalization coefficient, I i It refers to the intensity of the emitted light beam at a specific location or at a specific wavelength.

[0023] Furthermore, the initial spin configuration is a randomly generated ±1 sequence.

[0024] Furthermore, the feedback method includes, but is not limited to, simulated annealing. The simulated annealing method specifically involves changing a portion of the spin configuration based on the original Hamiltonian. If the new Hamiltonian H... t The Hamiltonian H is greater than that of the previous iteration. t-1 According to probability The decision is made on whether to retain the new Hamiltonian, where ΔH = H. t -H t-1 k is the Boltzmann constant, and T is the annealing temperature; if the new Hamiltonian H t Less than or equal to the Hamiltonian H of the previous iteration t-1 The new spin configuration is retained. Alternatively, it can be done only in the new Hamiltonian H. t Less than or equal to the Hamiltonian H of the previous iteration t-1 At that time, the new spin configuration is retained, thus obtaining the local minimum of the Hamiltonian of the Ising model.

[0025] According to a second aspect of this specification, an optical Ising machine based on Cholesky decomposition is provided, including an interaction preprocessing module, an optical calculation module, and a feedback control module;

[0026] The interaction preprocessing module is used to reorder the input Ising model interaction matrix and then decompose it into multiple Mattis-type interaction matrices by Cholesky, and encode the decomposition results and output them to the optical computing module.

[0027] The optical computing module includes an optical modulation system and a signal detection device; the optical modulation system includes a spin phase modulation device and an interaction optical modulation device.

[0028] The spin phase modulation device enables the beam to carry spin configuration information and inputs it into the interaction optical modulation device;

[0029] The interaction optical modulation device enables the light beam to carry interaction information and uses a lens system to achieve optical Fourier transform, so that the light fields carrying different spins and interaction information are superimposed and the interference signal is output to the signal detection device.

[0030] The signal detection device is used to detect interference signals, which contain Ising model Hamiltonian information encoded by the optical modulation system, thereby calculating the Ising model Hamiltonian.

[0031] The feedback control module is used to cyclically update the spin configuration, generate a new set of spin configurations, obtain the Hamiltonian information of the Ising model from the signal detection device, determine whether to retain the newly generated spin configuration based on the Hamiltonian, and encode the spin configuration to be retained to the spin phase modulation device for time or space phase modulation until a spin configuration that meets the requirements is found.

[0032] The beneficial effects of this invention are as follows: This invention employs the Cholesky decomposition method, which can greatly reduce the space requirement for encoding the Ising model in the optical Ising machine, thereby expanding the scale of problems that the optical Ising machine can handle. It features large scale, wide applicability, and high speed. Attached Figure Description

[0033] Figure 1 This is a schematic diagram of an optical Ising machine based on Cholesky decomposition provided in an embodiment of the present invention;

[0034] Figure 2 A schematic diagram of the interaction preprocessing module provided in an embodiment of the present invention;

[0035] Figure 3 This is a schematic diagram of the Cholesky decomposition of the Ising model provided in an embodiment of the present invention;

[0036] Figure 4 The optical path diagram of the Ising machine based on Cholesky decomposition provided in this embodiment of the invention;

[0037] Figure 5 The optical path diagram of the Ising machine based on Cholesky decomposition and gauge transformation is provided for embodiments of the present invention. Detailed Implementation

[0038] The embodiments of the present invention are described in detail below. These embodiments are implemented based on the technical solution of the present invention, and provide detailed implementation methods and specific operation processes. However, the scope of protection of the present invention is not limited to the following embodiments.

[0039] This invention provides a method for implementing an optical Ising machine based on Cholesky decomposition, comprising:

[0040] S1, Interaction matrix encoding based on Cholesky decomposition: After reordering the interaction matrix of the Ising model, Cholesky decomposes it into multiple Mattis-type interaction matrices, and encodes the decomposition results in the optical modulation system;

[0041] S2, Calculation of the Hamiltonian of the Ising model by optical modulation system: Optical Fourier transform is performed using a lens system to make the light fields carrying different spin and interaction information superimposed by interference. The interference signal contains the Hamiltonian information of the Ising model encoded by the optical modulation system, thereby calculating the Hamiltonian of the Ising model.

[0042] S3, Cyclic update of the spin configuration of the Ising model: generate a new spin configuration, determine whether to retain the newly generated spin configuration based on the Hamiltonian, and feed back the spin configuration to be retained to the time or space phase of the optical modulation system until a spin configuration that meets the requirements is found.

[0043] In one embodiment, for the input Ising model interaction matrix J and magnetic field parameters {h} iCholesky decomposition decomposes the Hamiltonian H of the Ising model into multiple Mattis-type interacting Ising model Hamiltonians H. i sum:

[0044]

[0045] Where N is the number of spins in the Ising model, J jk For the j-th spin σ j With the k-th spin σ k The interaction strength between them, σ i =±1 represent the two orientations of the i-th spin, respectively, h i These are the input magnetic field parameters. This represents the interaction strength between the j-th spin and the k-th spin in the i-th group of Mattis-type interactions obtained from the Cholesky decomposition; H0 is a parameter determined based on the input interaction matrix and magnetic field parameters, σ N+1 =1.

[0046] In one embodiment, the feedback method in S3 adopts simulated annealing, specifically: based on the original Hamiltonian, some spin configurations are changed, if the new Hamiltonian H... t The Hamiltonian H is greater than that of the previous iteration. t-1 According to probability The decision is made on whether to retain the new Hamiltonian, where ΔH = H. t -H t-1 k is the Boltzmann constant, and T is the annealing temperature; if the new Hamiltonian H t Less than or equal to the Hamiltonian H of the previous iteration t-1 The new spin configuration is retained. Furthermore, it is also possible to retain the new Hamiltonian H only. t Less than or equal to the Hamiltonian H of the previous iteration t-1 At that time, the new spin configuration is retained, thus obtaining the local minimum of the Hamiltonian of the Ising model.

[0047] In another embodiment, such as Figure 1 As shown, an optical Ising machine based on Cholesky decomposition is provided, including: an interaction preprocessing module, an optical calculation module, and a feedback control module.

[0048] like Figure 2 , 3 As shown, the interaction preprocessing module includes:

[0049] The reordering of the interaction matrix in the Ising model, specifically, includes, but is not limited to, the Cuthill-Mckee algorithm, the minimum degree algorithm, and the nested partitioning permutation algorithm, which can change the spin index and interchange the rows and columns of the interaction matrix, thereby reducing the number of matrix elements after Cholesky decomposition.

[0050] Cholesky decomposition: Performing Cholesky decomposition on the rearranged Ising model interaction matrix, such as... Figure 3 As shown, any Ising model is transformed into an Ising model with multiple Mattis-type interactions.

[0051] Encoding: Calculating the optical modulation required for the experiment, in one embodiment, such as... Figure 4 As shown, encoding and... on the spatial light modulator SLM1 Amplitude modulation that is proportional to the amplitude.

[0052] Furthermore, the optical computing module includes an optical modulation system and a signal detection device; the optical modulation system includes a spin phase modulation device and an interaction optical modulation device.

[0053] Spin phase modulation devices encode spin information on a light beam and can be used as standalone spin phase modulation devices or integrated with gauge transform and interaction optical modulation devices. In one embodiment, such as Figure 4 As shown, the spin phase modulation device consists of a laser source, a lens group, an aperture, a reflective spatial light modulator, and a semi-transparent mirror. The laser source C emits a collimated beam, which is expanded by the lens group L1 and L2, then its profile is limited by a rectangular aperture. After passing through a polarizer P, it is converted into linearly polarized light with the same polarization response direction as the spatial light modulator, and imaged onto the spatial light modulator SLM1. For an upward spin, σ = +1, and the corresponding modulation phase on the spatial light modulator is 0; for a downward spin, σ = -1, and the corresponding modulation phase on the spatial light modulator is π. After passing through the spin phase modulation device, the beam becomes a rectangular profile beam carrying spin phase information.

[0054] In addition, such as Figure 5 As shown, by adopting the canonical transformation method, the spin phase modulation device and the interaction optical modulation device are integrated on the same spatial light modulator, which can make the Ising machine more compact and reduce errors.

[0055] An interaction-based optical modulation device receives an input beam carrying spin information, encodes the interaction matrix information, and achieves the interaction between different spins through the superposition of light interference. In such a device... Figure 4In the illustrated embodiment, the beam encoded with spin phase information is imaged onto the spatial light modulator SLM2 via lenses L3 and L4. SLM2 encodes amplitude phase modulation containing interaction matrix information on the beam, and lens L5 performs an optical Fourier transform on the spatial light field, causing light carrying different spin and interaction information to interfere in the detector plane, thereby achieving the summation of terms in the Hamiltonian of the Ising model.

[0056] Furthermore, the signal detection device includes a detector for receiving optical signals and a signal processing program. The detector receives optical signals containing information about the Ising model Hamiltonian, and the signal processing program samples the signal at a specific time or spatial location, outputting the experimentally measured Hamiltonian to the feedback control module for further processing. The detector includes, but is not limited to, a camera.

[0057] Furthermore, the feedback control module is used to control the spin phase modulation device with electrical signals based on the Hamiltonian information provided by the optical signal. Specifically, the spin configuration is updated cyclically to generate a new set of spin configurations. The Ising model Hamiltonian information is obtained from the signal detection device, and the Hamiltonian determines whether to retain the newly generated spin configuration. The spin configuration to be retained is then fed back and encoded and output to the spin phase modulation device for time or spatial phase modulation until a satisfactory spin configuration is found. The implementation of the feedback control module includes, but is not limited to, a computer and a field-programmable gate array (FPGA).

[0058] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for implementing an optical Ising machine based on Cholesky decomposition, characterized in that, include: Interaction matrix encoding based on Cholesky decomposition: After reordering the Ising model interaction matrix, Cholesky decomposes it into multiple Mattis-type interaction matrices, and the decomposition results are encoded in the optical modulation system; specifically, for the input Ising model interaction matrix... With magnetic field parameters Cholesky decomposition transforms the Hamiltonian of the Ising model. Decomposed into multiple Mattis-type interacting Ising model Hamiltonians sum: in For the spin number of the Ising model, For the first Each spin With the k-th spin The strength of their interaction These represent the two orientations of the i-th spin. These are the input magnetic field parameters. This represents the interaction strength between the j-th spin and the k-th spin in the i-th group of Mattis-type interactions obtained from the Cholesky decomposition; , The parameters are determined based on the input interaction matrix and magnetic field parameters. ; The optical modulation system calculates the Hamiltonian of the Ising model: an optical Fourier transform is performed using a lens system to make the light fields carrying different spin and interaction information interfere and superimpose. The interference signal contains the Hamiltonian information of the Ising model encoded by the optical modulation system, thereby calculating the Hamiltonian of the Ising model. The spin configuration of the Ising model is updated cyclically: a new spin configuration is generated, and whether to retain the newly generated spin configuration is determined according to the Hamiltonian. The spin configuration to be retained is then fed back into the time or spatial phase of the optical modulation system until a spin configuration that meets the requirements is found.

2. The method for implementing the optical Ising machine based on Cholesky decomposition according to claim 1, characterized in that, The reordering methods include the Cuthill-Mckee algorithm, the minimum degree algorithm, and the nested partitioning permutation algorithm, which can change the spin index and interchange the rows and columns of the interaction matrix, thereby reducing the number of matrix elements after Cholesky decomposition.

3. The method for implementing the optical Ising machine based on Cholesky decomposition according to claim 1, characterized in that, In the process of Hamiltonian optical calculation, coherent light sources are required to enable the light fields to interfere and superimpose.

4. The method for implementing the optical Ising machine based on Cholesky decomposition according to claim 1, characterized in that, In Hamiltonian optical calculations, spin and interaction information is temporally or spatially modulated using an optical modulator, thereby encoding it onto the incident beam.

5. The method for implementing the optical Ising machine based on Cholesky decomposition according to claim 1, characterized in that, The specific method for feedback is as follows: in the new Hamiltonian Less than or equal to the Hamiltonian of the previous iteration At that time, the new spin configuration is retained, thus obtaining the local minimum of the Hamiltonian of the Ising model.

6. The method for implementing the optical Ising machine based on Cholesky decomposition according to claim 1, characterized in that, The feedback uses a simulated annealing method, specifically: based on the original Hamiltonian, some spin configurations are changed, and if the new Hamiltonian... Greater than the Hamiltonian of the previous iteration According to probability The decision on whether to retain the new Hamiltonian, among which , Boltzmann's constant, The annealing temperature; if the new Hamiltonian... Less than or equal to the Hamiltonian of the previous iteration The new spin configuration is retained.

7. An optical Ising machine based on Cholesky decomposition, characterized in that, It includes an interaction preprocessing module, an optical computing module, and a feedback control module; The interaction preprocessing module is used to reorder the input Ising model interaction matrix and perform Cholesky decomposition into multiple Mattis-type interaction matrices, and then encode and output the decomposition results to the optical computing module; specifically, for the input Ising model interaction matrix... With magnetic field parameters Cholesky decomposition transforms the Hamiltonian of the Ising model. Decomposed into multiple Mattis-type interacting Ising model Hamiltonians sum: in For the spin number of the Ising model, For the first Each spin With the k-th spin The strength of their interaction These represent the two orientations of the i-th spin. These are the input magnetic field parameters. This represents the interaction strength between the j-th spin and the k-th spin in the i-th group of Mattis-type interactions obtained from the Cholesky decomposition; , The parameters are determined based on the input interaction matrix and magnetic field parameters. ; The optical computing module includes an optical modulation system and a signal detection device; the optical modulation system includes a spin phase modulation device and an interaction optical modulation device. The spin phase modulation device enables the beam to carry spin configuration information and inputs it into the interaction optical modulation device; The interaction optical modulation device enables the light beam to carry interaction information and uses a lens system to achieve optical Fourier transform, so that the light fields carrying different spins and interaction information are superimposed and the interference signal is output to the signal detection device. The signal detection device is used to detect interference signals, which contain Ising model Hamiltonian information encoded by the optical modulation system, thereby calculating the Ising model Hamiltonian. The feedback control module is used to cyclically update the spin configuration, generate a new set of spin configurations, obtain the Hamiltonian information of the Ising model from the signal detection device, determine whether to retain the newly generated spin configuration based on the Hamiltonian, and output the spin configuration to be retained as a feedback code to the spin phase modulation device for time or space phase modulation until a spin configuration that meets the requirements is found.

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