Methods for implementing the Ising model

Abstract

This invention relates to a method for realizing the Ising model. A Kerr nonlinear optical parametric oscillator is driven by an external pump to obtain a Hamiltonian based on the Kerr nonlinear optical parametric oscillator. The Hamiltonian based on the Kerr nonlinear optical parametric oscillator is transformed into an evolutionary model with generalized coordinates and generalized momentum. The generalized coordinates and generalized momentum are iteratively evolved, and the values ​​of predetermined parameters in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator are obtained from their iterative evolution. Based on the Ising Hamiltonian and the values ​​of the predetermined parameters, the evolution result of the evolutionary model corresponds to the ground state result of the Ising Hamiltonian.

Description

Technical Field

[0001] This invention mainly relates to the field of Ising model technology, and more specifically, to a method for implementing the Ising model. Background Technology

[0002] Combinatorial optimization problems arise in various social and industrial contexts, and their rapid resolution can improve efficiency. However, these problems are notoriously difficult due to combinatorial explosion; the number of candidate solutions grows exponentially with the problem size. Therefore, new computational methods for combinatorial optimization are desired. Hardware solutions for these problems have recently been developed through various approaches. In particular, the Ising machine, designed to find the ground state of the Ising spin model, has recently attracted significant attention because many combinatorial optimization problems can be mapped to the Ising problem, such as very large-scale integrated circuit design, drug design, and financial portfolio management. Notable examples include quantum annealing furnaces and quantum annealing-based superconducting circuits, coherent Ising machines, and interferometric networks based on coherent laser pulses.

[0003] However, the Ising machine is complex in construction and operates under demanding conditions, such as quantum annealing furnaces based on superconducting circuits, typically requiring ultra-low operating temperatures and the inability to achieve full interconnection between qubits. The feedback control portion of the coherent Ising machine can be implemented using digital circuits, such as field-programmable gate arrays (FPGAs). The large number of computations and their corresponding speeds in this type of feedback control, based on classical computer principles, limit the machine's solution efficiency. Furthermore, the clock synchronization rate with the coupler and the contrast of light interference also significantly affect the accuracy of the solution. Therefore, the industry has begun to focus on building software-based solution algorithms by reproducing the principles of solving such hardware models.

[0004] The Ising spin dynamics model was originally a model describing the magnetism of crystals. The Ising machine, also known as an annealing furnace, is a specialized computer capable of solving specific classes of combinatorial optimization problems very quickly and efficiently. Such a specialized computational architecture can provide orders of magnitude speedup when solving combinatorial optimization problems. Given an initial input configuration, the Ising machine transforms that initial input configuration into an output configuration that minimizes the total energy of the system. Typically, the input configuration is a two-dimensional matrix with two spins (e.g., +1 and -1, 0 or 1, or spin-up and spin-down). It is widely used in condensed matter physics, materials science, magnetic theory, statistical physics, and mechanics to describe and study phenomena such as the properties of uniaxial anisotropic magnetic systems, the properties of public transportation control systems or glassy materials, the activity properties of protein molecules, and phase transitions in binary solutions.

[0005] Classical computers can solve some combinatorial optimization problems of relatively low complexity, but for more difficult problems such as the maximum cut problem, their time complexity is often unacceptable. Emerging quantum computers, based on the natural evolution of hardware computing methods, possess powerful parallel computing capabilities, enabling them to quickly complete calculations that classical computers cannot. The Ising model, initially designed to describe various physical phenomena, has evolved to solve combinatorial optimization and nondeterministic polynomial problems. By mapping these problems to the Ising Hamiltonian, the optimal solution can be found by solving the Hamiltonian's ground state.

[0006] Optimization is the process of finding the best or near-best solution from a set of many possible solutions. Combinatorial optimization is a subset of optimization, involving finding the best or near-best solution from a finite set of objects. Examples of combinatorial optimization problems include task scheduling, transportation route planning, portfolio optimization, and the like. The main concern is that most combinatorial optimization problems are very difficult to solve in terms of the impractical time required to obtain an exact optimal solution, such as years, several years, or even longer. Therefore, many approximation techniques have been designed to obtain near-optimal solutions in practical time units such as seconds, minutes, hours, or days. Examples of such approximation techniques are simulated annealing or evolutionary optimization.

[0007] By leveraging traditional resources, conventional computers and various processors can be used to construct Ising models to solve or attempt to solve the Ising problem, thus addressing the Ising problem and other combined problems based on conventional hardware and software resources. Summary of the Invention

[0008] This application discloses a method for implementing the Ising model, including:

[0009] The Hamiltonian based on the Kerr nonlinear optical parametric oscillator is obtained by using an external pump to drive the Kerr nonlinear optical parametric oscillator.

[0010] The Hamiltonian based on the Kerr nonlinear optical parametric oscillator condition is transformed into an evolutionary model with generalized coordinates and generalized momentum;

[0011] The generalized coordinates and the generalized momentum are iteratively evolved, and the values ​​of predetermined parameters in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator are obtained according to their iterative evolution.

[0012] Based on the Ising Hamiltonian and the values ​​of the predetermined parameters, the evolution result of the evolution model corresponds to the ground state result of the Ising Hamiltonian.

[0013] The above method is characterized by:

[0014] The Hamiltonian H based on the Kerr nonlinear optical parametric oscillator condition is expressed as:

[0015]

[0016] Where K is the nonlinear Kerr coefficient, and the detuning frequency Δ is... i ε0 represents the difference between the resonant frequency of the i-th resonant oscillator in N resonant oscillators and half of the pump frequency, where ε0 is the coupling strength, and J0 is the coupling strength. ij Corresponding to the coupling term of the Ising Hamiltonian;

[0017] as well as and a i These are the generation operator and annihilation operator for the i-th harmonic oscillator, respectively, and P is the pump rate.

[0018] The above method is characterized by:

[0019] The evolutionary model H(x, p) with generalized coordinates x and generalized momentum p is expressed as:

[0020]

[0021] in and a i The relationship with the generalized coordinate x and the generalized momentum p satisfies:

[0022]

[0023] The above method is characterized by:

[0024] The iterative process of the dynamic evolution of the generalized coordinate x and the generalized momentum p includes:

[0025]

[0026]

[0027] Where f represents the Gaussian white noise term, t is time, and n represents the transformation factor.

[0028] The above method is characterized by:

[0029] Set the initial values ​​of the generalized coordinates and generalized momentum to zero.

[0030] The above method is characterized by:

[0031] While the generalized coordinates and the generalized momentum undergo multiple iterative evolutions, the ground state results obtained in the previous iteration are compared with the ground state results obtained in the subsequent iteration. If the ground state results obtained in the previous iteration and the subsequent iteration are the same, then the ground state result at this time is considered to be the correct ground state result.

[0032] The above method is characterized by:

[0033] While the generalized coordinates and generalized momentum undergo multiple iterative evolutions, the ground state results obtained in the previous iteration are compared with the ground state results obtained in the subsequent iteration. If the ground state results obtained in the previous iteration and the subsequent iteration are different, the values ​​of the predetermined parameters are readjusted and iterative evolution continues until the ground state results are the same.

[0034] The above method is characterized by:

[0035] The Ising Hamiltonian is expressed as:

[0036]

[0037] Where σ i Let be the Pauli operator corresponding to the i-th harmonic oscillator, whose ground state result is represented by ±1 of the Ising spin state.

[0038] The above method is characterized by:

[0039] The predetermined parameters include at least the detuning frequency Δ. i and transformation factor n.

[0040] The above method is characterized by:

[0041] If the ground state results obtained in the previous iteration and the next iteration are the same, then the value of the predetermined parameter at this time is defined as the final value after iterative optimization.

[0042] This application utilizes existing software and hardware resources to solve the Ising problem and implement the Ising model in multi-spin fully connected systems. It solves the Ising problem without using any complex Ising machines or demanding conditions such as quantum annealing furnaces based on superconducting circuits, and without requiring ultra-low operating temperatures. Compared to existing implementations of the Ising model, this algorithm offers faster computation and a lower error rate, avoiding complex hardware operations. Attached Figure Description

[0043] To make the above-mentioned objectives, features and advantages more apparent and understandable, the specific embodiments are explained in detail below with reference to the accompanying drawings. After reading the following detailed description and referring to the following drawings, the features and advantages of this application will become obvious.

[0044] Figure 1 This is a schematic diagram of the general architecture of a Kerr nonlinear optical parametric oscillator driven by an external pump.

[0045] Figure 2 This is a flowchart illustrating the method of implementing the Ising model through evolutionary iteration using an evolutionary model.

[0046] Figure 3 This is a step in the implementation process of the Ising model that incorporates predetermined parameters for optimization.

[0047] Figure 4 This is an example of continuously optimizing predetermined parameters by comparing ground state results.

[0048] Figure 5 The final parameters are obtained by comparing the ground state results and ensuring consistency of the ground state results. Detailed Implementation

[0049] The present invention will be clearly and completely described below with reference to various embodiments. The described embodiments are only embodiments used for illustrative purposes and not all embodiments. Based on these embodiments, solutions obtained by those skilled in the art without creative effort are within the protection scope of the present invention.

[0050] See Figure 1 With the continuous development and improvement of nonlinear optics, the pumping methods of optical parametric oscillators (OPOs) are also constantly changing. On the one hand, the properties of nonlinear crystals are gradually developing towards higher destruction thresholds, wider transparency regions, lower absorption losses, and larger sizes; on the other hand, the pump source itself is also constantly evolving, with characteristics such as high peak power, narrower linewidth, and higher average power appearing in pumps. Kerr nonlinearity OPOs are a class of optical parametric oscillators. Improving the conversion efficiency of OPOs, reducing the pump threshold of parametric oscillations, and controlling the linewidth of parametric light have become important research areas in this field. Its main objectives are twofold: First, an optical parametric oscillator (OPO) with high conversion efficiency can generate parametric light with good beam quality and other indicators. Second, if the pump threshold is successfully reduced, the stringent requirements on the pump source, nonlinear crystal, and cavity diaphragm can be relaxed. Third, the widespread application of parametric light has led to more stringent parametric light quality indicators, such as narrower linewidth, which has also promoted research on various operating parameters of the optical parametric oscillator.

[0051] See Figure 1Regarding Kerr nonlinear optical parametric oscillators (KERR OPO): The Hamiltonian based on the Kerr nonlinear optical parametric oscillator can be obtained by externally pumping the PU (Polymer). The most typical external pump is a laser pump source. In recent years, the rapidly developing optical superlattice materials have presented various novel quasi-phase-matched optical parametric oscillation theories. Because optical superlattice materials possess many advantages not found in intrinsic crystals, they offer broader application prospects for optical parametric oscillators. Currently, optical parametric oscillators operate in various modes, including pulsed, continuous, and mode-locked operation, and also have multimode and single-mode operating modes. Pump PUs or pump sources are now widely used in solid-state, gas, dye, and even excimer laser types. Optical coupling devices (CPs) can couple the pump beam to the optical transmission medium, such as optical fiber, used in the Kerr nonlinear optical parametric oscillator.

[0052] See Figure 1 Regarding Optical Parametric Oscillators (OPOs): Two beams of light with different frequencies incident on a nonlinear crystal will generate polarized traveling waves with different frequencies. If the speed of the polarized traveling waves in the crystal is the same as or approximately the speed of free propagation of electromagnetic waves, cumulative growth will occur. These two incident beams can be called the "pump light" and the "signal light," and the accompanying third beam can be called the "idle light." Under appropriate conditions, the idle light and the pump light can be mixed to generate a polarized traveling wave with the signal frequency, whose phase substantially strengthens the signal light. This process continues as both the signal light and the idle light are strengthened, while the pump light attenuates with increasing propagation distance in the crystal; this process is parametric amplification, and when it occurs in a resonant cavity, it is called parametric oscillation. In certain optical wavelengths, optical parametric oscillation is the only, or the most effective, way to obtain tunable coherent light.

[0053] See Figure 1Regarding Optical Parametric Oscillators (OPOs): As a wide-tunable coherent light source, they overcome the limitations of output wavelengths in solid-state and gas lasers, generating ultraviolet to far-infrared lasers. When a high-frequency, high-intensity laser beam and a low-frequency, low-intensity beam simultaneously pass through the same nonlinear medium, the signal wave is amplified, and a third light wave (as mentioned earlier) is generated. The frequency of the idler wave is exactly equal to the frequency of the pump wave. This nonlinear optical phenomenon, known as optical parametric amplification, is a crucial optical characteristic: if a nonlinear medium is placed within an optical resonant cavity, and the pump wave, signal wave, and idler wave repeatedly pass through this medium, laser oscillation can occur within the resonant cavity once the gain from optical parametric amplification exceeds their losses within the cavity. Simultaneously, in practical optical parametric oscillators, a specific intensity of signal wave input from the outside is not required, as it can also be generated spontaneously within the nonlinear medium.

[0054] See Figure 1 The main characteristics of optical parametric oscillators (OPOs) include: wide tuning range, highly compact laser design, and the ability to achieve high power and narrow linewidth output. One condition for an OPO is the threshold pump power condition: after the pump power reaches a certain value, the gain of the signal light wave is equal to or greater than its optical loss within the cavity. If the pump light intensity exceeds the threshold, then the energy of the pump light is mainly converted into coherent signal light or idle light wave output. A second condition is the energy conservation condition: during parametric amplification, for every photon increase in the signal light wave and idle light wave, the pump light wave loses one photon; therefore, all three must satisfy the energy conservation condition. A third condition is that the OPO should achieve corresponding phase matching.

[0055] See Figure 1Due to the characteristic frequency of the working material, a particular laser can typically only output laser light at a fixed wavelength, which greatly limits its applications. Expanding the laser wavelength coverage generally utilizes tunable lasers and the nonlinear effects of nonlinear optical crystals, a problem of common concern in the physics, optics, materials science, and technology communities. In recent years, many technologically advanced solid-state tunable lasers have emerged, especially all-solid-state tunable lasers, which represent the development direction of this field. An optical parametric laser can be roughly understood as a device that uses nonlinear optical crystals to convert the frequency of laser light, thereby generating tunable laser light. It, along with laser mixing techniques such as frequency doubling, summing, or difference mixing, belongs to coherent optical parametric processes. Optical parametric oscillators (OPOs) are widely used in new materials, biology, resonance spectroscopy, chemistry, ranging, radar, and other fields due to their wide tuning range, compact structure, ease of use, high power, wide wavelength coverage, and high energy conversion efficiency.

[0056] See Figure 1 The various technical solutions and features of optical parametric lasers (OPOs) described in the context or disclosed in conventional technology also apply to this application. Therefore, this application will not elaborate further on optical parametric lasers.

[0057] See Figure 2 This application discloses a method for implementing the Ising model, or in other words, a bifurcation algorithm for solving the Ising model based on the principle of an optical parametric oscillator (OPO). In optional embodiments, the method includes: Step 10 mainly uses an external pump to drive the Kerr nonlinear optical parametric oscillator (KERR OPO) to obtain the Hamiltonian under the conditions of the Kerr nonlinear optical parametric oscillator; Step 11 mainly transforms the Hamiltonian under the conditions of the Kerr nonlinear optical parametric oscillator into an evolutionary model with generalized coordinates and generalized momentum; Step 12 mainly iteratively evolves the generalized coordinates and generalized momentum, for example, deriving the values ​​of certain predetermined parameters in the Hamiltonian based on the iterative evolution of the generalized coordinates and generalized momentum; Step 13 mainly extracts the ground state result based on the Ising Hamiltonian and the values ​​of the predetermined parameters, for example, making the evolutionary result of the evolutionary model correspond to the ground state result of the Ising Hamiltonian. The above is the main workflow.

[0058] See Figure 3 This application discloses another method for implementing the Ising model, or in other words, this application discloses another bifurcation algorithm for solving the Ising model based on the principle of optical parametric oscillators. Figure 2However, it additionally includes: Process 14 mainly compares the ground state results obtained from multiple iterations. If the ground state results obtained from multiple iterations of generalized coordinates and generalized momentum are consistent or identical, i.e., the comparison result is YES, then process 15 is executed, i.e., the ground state result is output and is now correct. If the ground state results obtained from multiple iterations of generalized coordinates and generalized momentum are inconsistent or different, i.e., the comparison result is NO, then process 12 is executed, i.e., the generalized coordinates and generalized momentum are continued to undergo multiple subsequent iterations. Note that process 12 also includes a parameter optimization step, and the parameter optimization process requires readjusting the values ​​of predetermined parameters in the Hamiltonian. If the values ​​of some predetermined parameters in the Hamiltonian are not the optimized final values, then the ground state results are unlikely to be consistent.

[0059] See Figure 3 In addition, it can be combined with Figure 2 As an example, the Hamiltonian H of the Kerr nonlinear optical parametric oscillator (KERR) under the rotating representation can be expressed as follows.

[0060]

[0061] See Figure 3 In addition, it can be combined with Figure 2 An example. In the Hamiltonian H: K is the nonlinear Kerr coefficient; Δ i It is the detuning frequency and represents the difference between the resonant frequency of the i-th harmonic oscillator and half of the pump frequency; ε0 is the coupling strength, and the Hamiltonian also includes the Ising Hamiltonian H. Ising Coupling term J ij etc.

[0062]

[0063] See Figure 3 In addition, it can be combined with Figure 2 An example. Where P (uppercase) is the pump rate. When the pump rate gradually and slowly increases from zero to a large value, the energy state with the lowest threshold (ground state) first appears in the optical parametric oscillator or the Kerr nonlinear optical parametric oscillator, suppressing high-energy states through mode competition. Furthermore, it is worth noting that the state energy bifurcation of each harmonic oscillator approximates the following equation:

[0064] The state energy of a harmonic oscillator related to pump rate, etc., is expressed as:

[0065] See Figure 3 In addition, it can be combined with Figure 2 Examples. Where σ iLet be the Pauli operator corresponding to the i-th harmonic oscillator, and let its eigenenergy in its own representation be ±1. Similarly, where σ... j The Pauli operator corresponding to the j-th harmonic oscillator has an eigenenergy of ±1 in its own representation. Therefore, the final evolution result of the Kerr nonlinear parametric oscillator (KERR OPO) corresponds to the ground state solution of the Ising Hamiltonian.

[0066] See Figure 3 In addition, it can be combined with Figure 2 Examples. Among them and a i These are the generation and annihilation operators of the i-th harmonic oscillator, respectively, and can also be transformed into the form of generalized coordinate x and generalized momentum p (lowercase).

[0067] The production and annihilation operators can be represented as follows:

[0068] See Figure 3 In addition, it can be combined with Figure 2 An example. The Hamiltonian of the Kerr nonlinear parametric oscillator (KERR OPO) can be expressed as an evolutionary model H(x, p) using the generalized coordinates x and the generalized momentum p.

[0069]

[0070] According to Hamilton's canonical equations, generalized coordinates and generalized momentum can satisfy the following:

[0071]

[0072] See Figure 3 In addition, it can be combined with Figure 2 An example. The evolution of generalized coordinates and generalized momentum can be transformed into a classical system of differential equations, note that the pumping rate is in uppercase while the generalized momentum is in lowercase:

[0073]

[0074] See Figure 3 In addition, it can be combined with Figure 2 An example. The dynamic evolution of generalized coordinates and generalized momentum allows for iteration using an explicit Sinusoidal method, note that the pumping rate is in uppercase while the generalized momentum is in lowercase:

[0075]

[0076]

[0077] See Figure 3 In addition, it can be combined with Figure 2Examples of this approach include: In an optional embodiment, where the initial values ​​of generalized coordinates and generalized momentum are both zero, a Gaussian white noise term f is added during iteration to ensure bifurcation evolution, thus avoiding a situation where specific initial values ​​of generalized coordinates and momentum lead to a unique bifurcation result. This inherently introduces uncertainty into the bifurcation of generalized coordinates and momentum. The Gaussian white noise term can have a given value in optional embodiments or be an adjustable parameter in other embodiments. The pump rate P is a time-varying parameter that can slowly increase from zero. When the pump rate exceeds the ground state threshold, if the classical system cannot overcome the gap between the ground state and the excited state, it will tend to evolve towards a ground state-stable bifurcation. The above mainly introduces the evolutionary iteration process of the evolutionary model. The initial values ​​of generalized coordinates and generalized momentum, i.e., their initial values, are crucial for the evolution when associated with the Gaussian white noise term.

[0078] See Figure 3 The pumping characteristics of an optical parametric oscillator (OPO) are as follows: when the pump exceeds a threshold, the conversion efficiency increases with increasing pump intensity; however, after reaching a maximum value, the conversion efficiency decreases with further increases in pump intensity, indicating the existence of an optimal pump value. Furthermore, the optimal pump value varies depending on the output transmittance; that is, the optimal coupling transmittance differs for different pump intensities. This application is based on a bifurcation algorithm for solving the Ising model based on the Kerr nonlinear optical parametric oscillator principle, and therefore the algorithm naturally adapts to these characteristics.

[0079] See Figure 3This application discloses a bifurcation algorithm for solving the Ising model based on the oscillator principle. The algorithm mainly includes the following steps: Based on the Hamiltonian of the Kerr oscillator under external pumping, the evolution equations of the generalized coordinates and generalized momentum in the corresponding classical system are given; the bifurcation evolution of the generalized coordinates and momentum is iterated using feedforward; based on the calculation results of the generalized coordinates, the parameters inside the algorithm are automatically optimized. The final optimized results, such as the ground state or parameters, are transformed into the solution results of the corresponding Ising spins. This application can solve the Ising problem for multi-spin fully connected systems. Compared with the existing Ising machine model, the software solution algorithm based on traditional hardware resources has a faster calculation speed, lower error rate, avoids complex hardware operations, and avoids the stringent operating conditions required by the Ising machine model. This application is applicable to solving combinatorial optimization problems that can be transformed into Ising Hamiltonian solutions. In an optional embodiment, Hamiltonian canonical equations are used to transform the Hamiltonian of the Kerr nonlinear optical parametric oscillator into a model within the realm of classical theoretical physics, resulting in differential forms of generalized coordinates and generalized momentum. The evolution of the bifurcation of generalized coordinates and momentum can be iteratively evolved using an explicit Sinusoidal method. Based on the iterative results of the generalized coordinates, a self-looping approach is used to optimize the internal predetermined parameters to output accurate result parameters. The optimization process combines gradient descent and iterative updates.

[0080] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: the generalized coordinates will bifurcate when the pump rate P reaches a specific value, and the generalized momentum will oscillate slightly around the origin in any case.

[0081] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: due to and The existence of some nonlinear terms, such as J, in different J ij The input may result in an overflow of the final value of the evolution.

[0082] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: to avoid numerical overflow of the output results in a large number of spin systems, in an optional embodiment, the inter-spin coupling strength J in the Ising problem can be adjusted during parameter initialization. ij Perform an identity multiplication transformation on matrix J using smaller numbers.

[0083] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: even for the reflection of the inter-rotational coupling strength J in the Ising problem... ijThe identity multiplication transformation of matrix J does not affect the final result of the ground state solution. The final evolution result of the generalized coordinates is mapped to the value of the Ising spin state ±1.

[0084] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: for the solution of the Ising Hamiltonian with a degenerate ground state, there are two specific cases of the solution, each with a probability of 50%.

[0085] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: the detuning quantity Δ i In other words, the detuning frequency reflects the system's own resonant mode, and its value affects the bifurcation position of the generalized coordinates.

[0086] See Figure 3 Regarding the bifurcation evolution of generalized coordinates and momentum, in the iterative evolution: for solving the Ising problem of multiple self-selected systems, considering that the energy interval between the ground state and excited state is too close, the generalized coordinates will bifurcate directly towards the excited state when the pump rate reaches a certain level, bypassing the ground state and obtaining incorrect solutions. Optimizing the predetermined internal parameters using a self-looping approach is equivalent to providing a method to resolve these incorrect solutions.

[0087] See Figure 3 The parameters are optimized using iterative results of generalized coordinates and a self-looping approach. In this process, based on the bifurcation and evolution of the generalized coordinates corresponding to the exact solution, in optional embodiments, the bifurcation point of the generalized coordinates or the final evolution result can be set within a suitable range (e.g., set within the range...). (Nearby). Therefore, the solution obtained at this point is more accurate and less prone to errors. Thus, choosing the generalized coordinate bifurcation point within this interval results in a high accuracy of the ground-state solution.

[0088] See Figure 3 The parameters are optimized using iterative results from generalized coordinates and a self-looping approach. This utilizes the idea of ​​gradient descent and the detuning amount Δ. i That is, the magnitude of the detuning frequency is positively correlated with the bifurcation position of the generalized coordinate system.

[0089] See Figure 3 The parameters are optimized using iterative results from generalized coordinates and a self-looping approach: This includes the parameter J, which reflects the inter-spin coupling strength in the Ising problem. ij The matrix J undergoes identity multiplication transformation with a transformation factor of n, the magnitude of which is positively correlated with the final evolution result of the generalized coordinates.

[0090] See Figure 3 The parameters are optimized using iterative results from generalized coordinates and a self-looping approach. This optimization considers the differences in evolution results across multiple iterations, and addresses the detuning factor Δ. iIterative optimization is performed using the transformation factor n.

[0091] See Figure 3 The parameters are optimized by iterative results of generalized coordinates and other methods and by a self-looping approach: the ground state results with consistency are output after multiple iterations.

[0092] See Figure 1 In an optional embodiment, this application discloses an alternative algorithm or method for solving the Ising model based on the principle of a Kerr nonlinear optical parametric oscillator, comprising the following steps: Step 1: Based on the Hamiltonian of the Kerr nonlinear optical parametric oscillator under pump drive, transform it into a corresponding evolution equation or evolution model with generalized coordinates and generalized momentum, such as the evolution equation in a classical system; Step 2: Perform bifurcation evolution of the generalized coordinates and generalized momentum and use feedforward iteration; Step 3: Based on the calculation results of the generalized coordinates, automatically optimize the predetermined parameters; Step 4: Transform the final optimized ground state result into the solution result of the corresponding Ising spin. Note that the methods for implementing the Ising model described above are also applicable to this alternative embodiment. This alternative example and Figure 3 Slightly different.

[0093] See Figure 1 In an optional embodiment, this application discloses an alternative algorithm or method for solving the Ising model based on the principle of Kerr nonlinear optical parametric oscillators, comprising the following steps: Step 1, converting the Hamiltonian of the Kerr nonlinear optical parametric oscillator under pump drive into a corresponding evolutionary model with generalized coordinates and generalized momentum; Step 2, setting the coupling term J of the Ising Hamiltonian involved in the evolutionary model. ij Step 3: Set the values ​​of some predetermined parameters in the Hamiltonian evolution model; Step 4: Iterate the dynamic evolution of generalized coordinates and generalized momentum using the explicit Euler method; Step 5: Calculate the ground state results of each harmonic oscillator based on the solution of the explicit Euler method; Step 6: After multiple evolutions of generalized coordinates and generalized momentum, compare the ground state results of each harmonic oscillator before and after iteration to determine if the ground state results before and after iteration are the same. If the ground state results obtained in the previous iteration and the next iteration are different, readjust the values ​​of the predetermined parameters (re-execute Step 3) and continue iterating until the ground state results of each harmonic oscillator are the same before and after iteration; In Step 5, if the ground state results obtained in the previous iteration and the next iteration are the same, directly output the ground state result of each harmonic oscillator. Note that the methods for implementing the Ising model described above are also applicable to this alternative embodiment. This alternative example and Figure 3 There are slight differences.

[0094] See Figure 3In an optional embodiment, the Hamiltonian based on the Kerr nonlinear optical parametric oscillator is first transformed into an evolutionary model with generalized coordinates and generalized momentum. In this model, the generalized coordinates and momentum undergo iterative evolution. Based on this iterative evolution, the values ​​of predetermined parameters in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator are obtained; for example, the predetermined parameters include at least the detuning frequency Δ. i The transformation factor n and predetermined parameters can also include more types, such as Hamiltonian functions and / or other parameters in the evolution model. Based on the Ising Hamiltonian and the values ​​of predetermined parameters, for example, combining the evolution model of the Hamiltonian under Kerr nonlinear optical parametric oscillator conditions with the traditional Ising Hamiltonian expression, the evolution result of the evolution model corresponds to the ground state result of the Ising Hamiltonian. In an alternative example, the evolution model and the Ising Hamiltonian share a common Hamiltonian equation relationship, thereby achieving the ground state solution. In an alternative example, if the values ​​of the predetermined parameters in the evolution model are optimized and finally determined, the evolution result of the evolution model is equivalent to solving for each ground state result, so the evolution result of the evolution model corresponds to the ground state result of the Ising Hamiltonian. The optimization process of the predetermined parameters will be further described below.

[0095] See Figure 3 In an optional embodiment, a method for implementing the Ising model has been described: Step 14 mainly compares the ground state results obtained from multiple iterations. If the ground state results obtained from multiple iterations of generalized coordinates and generalized momentum are consistent or identical, i.e., the comparison result is YES, then step 15 is executed, i.e., the ground state result is output and is now correct. If the ground state results obtained from multiple iterations of generalized coordinates and generalized momentum are inconsistent or different, i.e., the comparison result is NO, then step 12 is executed, i.e., subsequent iterations of evolution are performed on the generalized coordinates and generalized momentum. Before performing subsequent iterations of evolution on the generalized coordinates and generalized momentum, predetermined parameters in the evolution model (such as the detuning frequency Δ) need to be readjusted. i The process of adjusting the values ​​of predetermined parameters in the evolution model (such as transformation factor n, etc.) so that the ground state result of each harmonic oscillator obtained in the previous iteration is similar to the ground state result obtained in the next iteration is the process of parameter optimization.

[0096] See Figure 4 Regarding parameter optimization: First, assume that the iteration number Q is a positive integer. While the generalized coordinates and generalized momentum undergo multiple iterations, compare the ground state results obtained in any previous iteration with the ground state results obtained in any subsequent iteration. If the ground state results obtained in the previous iteration and the subsequent iteration are different, readjust the values ​​of the predetermined parameters and continue iterating the generalized coordinates and generalized momentum until the ground state results tend to be the same.

[0097] See Figure 4 Regarding parameter optimization: the generalized coordinates and generalized momentum yielded the evolutionary results of the evolutionary model during the Qth iteration, namely the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian. i σ j etc.

[0098] See Figure 4 Regarding parameter optimization: the generalized coordinates and generalized momentum yield the evolution results of the evolutionary model, i.e., the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian, during the Q+1 iteration. i σ j Etc. According to the rules designed in this application, the ground state result obtained in any previous iteration (e.g., the Qth iteration) of the generalized coordinates and generalized momentum needs to be compared with the ground state result obtained in the next iteration (e.g., the Q+1th iteration). If the ground state results obtained in the previous iteration and the next iteration are different, such as the ground state result obtained in the Qth iteration being different from the ground state result obtained in the Q+1th iteration, the predetermined parameters (e.g., the detuning frequency Δ) need to be readjusted. i After obtaining the values ​​of the transformation factor n, the generalized coordinates and generalized momentum are iteratively evolved. If the process is re-executed... Figure 3 The iterative evolution process 12.

[0099] See Figure 4The solution of this application can be implemented by a program written on a conventional computer, or the implementation scheme of this application can be implemented by hardware. For example, a computer program stored in the computer stores a program for implementing the Ising model, and the process executed by the processor of the computer includes: iteratively evolving an evolutionary model with generalized coordinates and generalized momentum transformed from the Hamiltonian, obtaining the values ​​of predetermined parameters in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator, and making the evolution result of the evolutionary model correspond to the ground state result of the Ising Hamiltonian based on the values ​​of the predetermined parameters. Then the computer can compare the ground state result obtained in the Qth iteration with the ground state result obtained in the Q+1th iteration. In this example, the comparison process COMP is essentially performed by the computer. If various hardware resources such as digital signal processors, microcontrollers, single-chip microcomputers, or high-reduction instruction set computers are used instead of conventional computers, these hardware resources can also enable the implementation of the Ising model. Taking a digital signal processor (DSP) as an example, an iterative evolution module is provided to perform the task of iteratively evolving the generalized coordinates and generalized momentum. A data comparison module is provided to compare the ground state results of the generalized coordinates and generalized momentum obtained in the previous iteration with the ground state results obtained in the subsequent iteration. The comparison result of the data comparison module also triggers the iterative evolution module to readjust the values ​​of predetermined parameters: if the ground state results obtained in the previous and subsequent iterations are different, the comparison result of the data comparison module notifies the iterative evolution module to readjust the values ​​of the predetermined parameters and continue iterative evolution; if the ground state results obtained in the previous and subsequent iterations are the same, the value of the predetermined parameter in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator can be obtained, and the ground state result at this time is considered to be the correct ground state result, and the value of the predetermined parameter at this time is defined as the final value after iterative optimization. The DSP needs to provide a large number of storage units to store the ground state results obtained in each iteration. The ground state obtained by the iterative evolution module according to the final value after iterative optimization is naturally the correct ground state result. The iterative evolution module, based on the Ising Hamiltonian and predetermined parameter values ​​(e.g., the final values), ensures that the evolutionary result of the evolutionary model corresponds to the ground state result of the Ising Hamiltonian. The measures in this example are for illustrative purposes only and do not constitute a limitation. For example, the digital signal processor can be replaced with a microcontroller, single-chip microcomputer, or advanced reduced instruction set computer (RISC) hardware.

[0100] See Figure 4 Regarding parameter optimization: the generalized coordinates and generalized momentum yield the evolution results of the evolutionary model, i.e., the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian, during the Q+2th iteration. i σ jEtc. According to the design rules of this application, the ground state result obtained by the generalized coordinates and generalized momentum in any previous iteration (Q+1th iteration) must be compared with the ground state result obtained by the generalized coordinates and generalized momentum in a later iteration (e.g., Q+2th iteration). If the ground state results obtained by the previous iteration and the later iteration are different, such as the ground state result obtained by the Q+1th iteration being different from the ground state result obtained by the Q+2th iteration, the predetermined parameters (e.g., the detuning frequency Δ) need to be readjusted. i After obtaining the values ​​of the transformation factor n, the generalized coordinates and generalized momentum are iteratively evolved. If the process is re-executed... Figure 3 The iterative evolution process 12.

[0101] See Figure 5 Regarding parameter optimization: it is also assumed that the iteration number Q is a positive integer. While the generalized coordinates and generalized momentum undergo multiple iterative evolutions, the ground-state results obtained in any previous iteration are compared with those obtained in any subsequent iteration. If the ground-state results obtained in the previous and subsequent iterations are the same, then the current ground-state result is considered correct. Furthermore, it is defined that the values ​​of the predetermined parameters at this point are the final values ​​after iterative optimization, and the values ​​of the predetermined parameters do not require any further readjustment.

[0102] See Figure 5 Regarding parameter optimization: the generalized coordinates and generalized momentum yielded the evolutionary results of the evolutionary model during the Qth iteration, namely the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian. i σ j etc.

[0103] See Figure 5 Regarding parameter optimization: the generalized coordinates and generalized momentum yield the evolution results of the evolutionary model, i.e., the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian, during the Q+1 iteration. i σ j Wait. According to the design rules of this application, the ground state result obtained by the generalized coordinates and generalized momentum in any previous iteration (e.g., the Qth iteration) needs to be compared with the ground state result obtained by the generalized coordinates and generalized momentum in the next iteration (e.g., the Q+1th iteration). If the ground state results obtained by the previous iteration and the next iteration are the same, such as the ground state result obtained by the Qth iteration and the ground state result obtained by the Q+1th iteration, there is no need to adjust the predetermined parameters (e.g., the detuning frequency Δ). iThe values ​​of the predetermined parameters (such as the transformation factor n) are defined as follows. The values ​​of the predetermined parameters at this point are defined as the final values ​​after iterative optimization. The final parameters shown in the figure indicate that if the ground state results obtained in the previous and subsequent iterations are the same, then the values ​​of the predetermined parameters are the final optimized values. Thus, based on the multiple iterative evolution process of the generalized coordinates and generalized momentum, the values ​​of the predetermined parameters in the Hamiltonian of the Kerr nonlinear optical parametric oscillator are obtained. The ground state result at this point is the correct ground state result.

[0104] See Figure 5 Regarding parameter optimization: the generalized coordinates and generalized momentum yield the evolution results of the evolutionary model, i.e., the ground state results σ1, σ2, ..., σ of the Ising Hamiltonian, during the Q+2th iteration. i σ j Wait. According to the rules designed in this application, the ground state result obtained by the generalized coordinates and generalized momentum in any previous iteration (Q+1th iteration) needs to be compared with the ground state result obtained by the generalized coordinates and generalized momentum in a later iteration (e.g., Q+2th iteration). If the ground state results obtained by the previous iteration and the later iteration are the same, such as the ground state result obtained by the Q+1th iteration being the same as the ground state result obtained by the Q+2th iteration, there is no need to adjust the predetermined parameters (e.g., the detuning frequency Δ). i The values ​​of the generalized parameters (such as the transformation factor n) are defined as follows. The values ​​of the predetermined parameters at this point are defined as the final values ​​or final parameters after iterative optimization. In this specific scheme, "comparing the ground-state results of the generalized coordinates and generalized momentum obtained in the previous iteration with the ground-state results obtained in the subsequent iteration, and if the ground-state results obtained in the previous and subsequent iterations are different, then readjusting the values ​​of the predetermined parameters and continuing the iterative evolution of the generalized coordinates and generalized momentum" is of great significance. For solving the Ising problem of multiple self-selected systems, because the energy interval between the ground state and excited state is too close, the generalized coordinates will bifurcate directly towards the excited state when the pump rate reaches a certain level, resulting in incorrect solutions. When the generalized coordinates tend to bifurcate directly towards the excited state, one of the functions of the aforementioned specific scheme is to change the bifurcation direction of the generalized coordinates by readjusting the values ​​of the predetermined parameters, thus avoiding the generalized coordinates bifurcerating directly towards the excited state and preventing errors in the ground-state results evolved by the evolution model.

[0105] See Figure 5 Using traditional computers and various processors to construct the Ising model can solve or partially solve the Ising problem, thus addressing the Ising problem and other combinatorial problems based on traditional hardware and software resources. This approach requires no complex Ising machine and is not limited by stringent conditions such as quantum annealing furnaces based on superconducting circuits.

[0106] The foregoing description and accompanying drawings have provided typical embodiments of specific structures for specific implementations. The above application content presents existing preferred embodiments, but these are not intended to be limiting. Various changes and modifications will undoubtedly be apparent to those skilled in the art after reading the foregoing description. Therefore, the appended claims should be considered as covering all changes and modifications that encompass the true intent and scope of the invention. Any and all equivalent scopes and contents within the scope of the claims should be considered to still fall within the intent and scope of the invention.

Claims

1. A method for implementing the Ising model, characterized in that, include: The Hamiltonian based on the Kerr nonlinear optical parametric oscillator is obtained by using an external pump to drive the Kerr nonlinear optical parametric oscillator. The Hamiltonian based on the Kerr nonlinear optical parametric oscillator condition is transformed into an evolutionary model with generalized coordinates and generalized momentum; The generalized coordinates and the generalized momentum are iteratively evolved, and the values ​​of predetermined parameters in the Hamiltonian based on the Kerr nonlinear optical parametric oscillator are obtained according to their iterative evolution. Based on the Ising Hamiltonian and the values ​​of the predetermined parameters, the evolution result of the evolution model corresponds to the ground state result of the Ising Hamiltonian; The Hamiltonian H based on the Kerr nonlinear optical parametric oscillator condition is expressed as: Where K is the nonlinear Kerr coefficient, and the detuning frequency Δ is... i ε0 represents the difference between the resonant frequency of the i-th resonant oscillator in N resonant oscillators and half of the pump frequency, where ε0 is the coupling strength, and J0 is the coupling strength. ij Corresponding to the coupling term of the Ising Hamiltonian; as well as and a i These are the generation operator and annihilation operator of the i-th harmonic oscillator, respectively, and P is the pump rate; The evolutionary model H(x, p) with generalized coordinates x and generalized momentum p is expressed as: in and a i The relationship with the generalized coordinate x and the generalized momentum p satisfies: The iterative process of the dynamic evolution of the generalized coordinate x and the generalized momentum p includes: Where f represents the Gaussian white noise term, t is time, and n represents the transformation factor.

2. The method according to claim 1, characterized in that: Set the initial values ​​of the generalized coordinates and generalized momentum to zero.

3. The method according to claim 1, characterized in that: While the generalized coordinates and the generalized momentum undergo multiple iterative evolutions, the ground state results obtained in the previous iteration are compared with the ground state results obtained in the subsequent iteration. If the ground state results obtained in the previous iteration and the subsequent iteration are the same, then the ground state result at this time is considered to be the correct ground state result.

4. The method according to claim 1, characterized in that: While the generalized coordinates and generalized momentum undergo multiple iterative evolutions, the ground state results obtained in the previous iteration are compared with the ground state results obtained in the subsequent iteration. If the ground state results obtained in the previous iteration and the subsequent iteration are different, the values ​​of the predetermined parameters are readjusted and iterative evolution continues until the ground state results are the same.

5. The method according to claim 1, characterized in that: The Ising Hamiltonian is expressed as: Where σ i Let be the Pauli operator corresponding to the i-th harmonic oscillator, whose ground state result is represented by ±1 of the Ising spin state.

6. The method according to claim 1, characterized in that: The predetermined parameters include at least the detuning frequency Δ. i and transformation factor n.

7. The method according to claim 3, characterized in that: If the ground state results obtained in the previous iteration and the next iteration are the same, then the value of the predetermined parameter at this time is defined as the final value after iterative optimization.

Images