Coherent Ising Machine with Optical Error Correction for Optimization Solution Generator System and Method

Abstract

The coherent Ising machine can include a pump pulse generator configured to generate an optical signal pulse, an optical error correction circuit configured to generate an optical error pulse, and a main ring cavity configured to store the optical signal pulse and the optical error pulse, where the optical error pulse causes the coherent Ising machine to not converge to a local minimum of the Ising solution but to continue searching for nearby states.

Description

[Technical Field]

[0001] (cross reference) This application claims priority to U.S. Provisional Patent Application No. 63 / 234,127, filed August 17, 2021, which is incorporated herein by reference in its entirety.

[0002] (appendix) Appendices A-F (8 pages) disclose 1) simulation parameter optimization, 2) G-set simulation results, 3) parameter selection rationale, 4) CIM-SFC results and discussion for G-set, 5) similarities and differences between CIM and SBM algorithms, and 6) optical implementation of CIM-SFC, all of which are part of the specification and incorporated herein by reference. Appendix G discloses references.

[0003] The present disclosure relates to systems and methods for solving combinatorial optimization problems using quantum-inspired optimization processes. [Background technology]

[0004] Combinatorial optimization problems are ubiquitous in modern science, engineering, medicine, and business. These problems are often NP-hard, and therefore run times on classical digital computers are expected to scale exponentially. A typical example of an NP-hard combinatorial optimization problem is the non-planar Ising model. Special-purpose quantum hardware devices have been developed to find solutions to Ising problems more efficiently than standard heuristic approaches. One example is the quantum annealing (QA) device, which utilizes the adiabatic progression of a pure state vector through a time-dependent Hamiltonian. Another example is the coherent Ising machine (CIM), which utilizes the quantum-to-classical transition of a mixed density-of-states operator in a network of quantum oscillators. Performance comparisons between QA and CIM for various Ising models, including complete graphs, dense graphs, and sparse graphs, have been reported. Furthermore, theoretical performance comparisons have been performed between ideal gate-model quantum computers implementing the Global Search algorithm or the adiabatic quantum computing algorithm and the CIM. Although CIMs with all-to-all coupling between spins can be effective, the use of external FPGA circuits as well as analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) generates a large amount of energy dissipation, posing a potential bottleneck for high-speed operation. Summary of the Invention

[0005] It has been recognized that the standard linear combination approach of CIM suffers from amplitude non-uniformity between the constituent quantum oscillators. Due to this amplitude non-uniformity, the Ising Hamiltonian is incorrectly mapped to the network loss, leading to operational failure, especially in frustrated spin systems. To solve this problem, an error correction feedback scheme has been developed, which allows CIM to compete in solution accuracy with state-of-the-art heuristics such as breakout local search (BLS).

[0006] A quasi-classical model for error-correcting feedback. There are several mutual coupling and error correction feedback schemes for CIM. To simplify this discussion, a quasi-classical deterministic view is used. The quasi-classical model is an approximate theory of a hypothetical machine as follows: At initial time t = 0, each signal pulse field is prepared in a vacuum state, as shown in Figure 1A. The pump fields p and p i When the beam splitter BS (shown in Figure 1C) is switched on at t ≥ 0, it extracts the beam from the open port. e In this ODL-CIM, the vacuum field incident on the ion beam is compressed / decompressed by a phase-sensitive amplifier (PSA) 102, as shown in FIG. 1C.

[0007]

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[0008] A truncated-Wigner stochastic differential equation (W-SDE) for such a quantum-optical CIM with a compressed reservoir has been derived and studied. This particular CIM achieves the maximum quantum correlation between the OPO pulse fields along the in-phase component and thus the maximum success probability. This is because the quantum correlation between the OPO pulse fields is formed by the mutual coupling of the vacuum fluctuations of the OPO pulse fields without injecting uncorrelated fresh reservoir noise in such a system. As an approximate theory for the above W-SDE, the following quasi-classical model is considered in the limit of a large deamplification factor (G>>1). A more realistic fully quantum description of the CIM with optical error correction circuitry (without reservoir engineering) is discussed below.

[0009] To overcome the problem of amplitude non-uniformity in CIM, the addition of an auxiliary variable for error detection and correction has been proposed. This system is being investigated as a modified version of the measurement feedback CIM. The spin variable (signal pulse amplitude) x i and the auxiliary variable (error pulse amplitude) e i is the following deterministic equation:

[0010]

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[0011] The CIM-CAC equation can be modified as follows: z i =e i Σ j ξJ ij x j (3)

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[0012] A very different set of equations, z i =Σ j ξJ ij x j (6)

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[0014] Another significant difference between CIM-SFC (Equations (6), (7), and (8)) compared to CIM-CAC and CIM-CFC (Equations (1) to (5)) is the auxiliary variable e iHowever, in CIM-CAC and CIM-CFC, e i is intended to be a strictly positive number that varies exponentially and adjusts for the mutual coupling terms. In CIM-SFC, e i is a variable that stores sign information, and the mutual coupling signal z i The error signal e i is basically z i can be thought of as a low-pass filter for the term k(e i -.z i ) is z i (In other words, k(e i -.z i ) only z i (This records the sudden change in .) A way to understand the similarities and differences between CIM-SFC, CIM-CAC, and CIM-CFC is to look at the fixed points. In CIM-CAC and CIM-CFC, the fixed points are x i =λ1σ i (9)

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[0016] The major difference between the fixed points of these two types of systems is that in CIM-CAC and CIM-CFC, the error signal

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[0017] [Figure 1A] 1A and 1B are diagrams illustrating a vacuum state and a squeezed vacuum state. [Figure 1B] FIG. 1 illustrates a coherent state and a compressed coherent state. [Figure 1C] FIG. 1 illustrates a coherent Ising machine. [Figure 2A] FIG. 1 illustrates the operating power of an active photonic device in a 100 GHz CIM. [Figure 2B] FIG. 1 illustrates the energy to solution in three subsystems in CIM. [Figure 2C] FIG. 1 illustrates an example of an OPO amplitude trajectory in a CIM. [Figure 2D] FIG. 10 illustrates an example of an OPO amplitude trajectory in another CIM. [Figure 2E] FIG. 10 illustrates an example of an OPO amplitude trajectory in yet another CIM. [Figure 3] FIG. 1 illustrates the mutual coupling fields and injection feedback fields of four different feedback systems. [Figure 4] FIG. 10 illustrates signal pulse amplitude correlation at different travel times. [Figure 5] FIG. 1 illustrates the trajectory of a signal pulse with fixed and adjusted system parameters. [Figure 6A] FIG. 1 illustrates an optical error correction circuit for a novel CIM. [Figure 6B] FIG. 1D illustrates a pump pulse factory that provides pulses to the main cavity shown in FIG. 1C. [Figure 6C] 6A illustrates a novel CIM with optical feedback, including a main cavity as shown in FIG. 1C, an optical error correction circuit in FIG. 6A, and a pump pulse factory as shown in FIG. 6B. [Figure 7] FIG. 1 illustrates the success probability versus saturation parameter for CIM-SFC and CIM-CFC. [Figure 8] FIG. 1 illustrates the energy cost to solution in Joules for CIM-SFC and CIM-CFC. [Figure 9] FIG. 10 illustrates the estimated energy cost to solution of optical and GPU implementations of CIM-CAC versus problem size. [Figure 10] A diagram illustrating TTS for CIM-CAC, CIM-CFC, and CIM-SFC. [Figure 11] FIG. 1 illustrates TTS versus question size for CIM-SFC and NMFA. [Figure 12] FIG. 10 illustrates the required number of matrix-vector multiplications for CIM-CAC, CIM-CFC, and CIM-SFC versus dSBM. [Figure 13] A diagram illustrating TTS for CIM-CAC, CIM-CFC, CIM-SFC, and dSBM. [Figure 14] FIG. 1 illustrates a number of pending instances for CIM-CAC and dSBM. [Figure 15] FIG. 10 is a diagram illustrating the success probability of CIM-CAC. [Figure 16] FIG. 1 illustrates the performance of CIM-CAC for problem sizes with different parameters. [Figure 16-1] 10 is a table illustrating the success probability and TTS of CIM-CAC, CIM-CFC, and dSBM for G-set graphs. [Figure 17] FIG. 1 illustrates an optical implementation of CIM-SFC. DETAILED DESCRIPTION OF THE INVENTION

[0018] The present disclosure relates to and will be described in the present context as a coherent Ising machine with optical feedback correction (CIM) that includes a combination of a main cavity, optical error correction circuitry, and a pulsed pump factory that can be used to solve combinatorial optimization problems. However, it will be understood that the system and method can be implemented using other variations of the elements disclosed below within the scope of the present disclosure.

[0019] The novel CIM architecture disclosed herein has optically implemented error correction. In this architecture, the computationally intensive matrix-vector multiplication (MVM) and nonlinear feedback functions are implemented by phase-sensitive (degenerate) optical parametric amplifiers, which are essentially the same devices used for the main-cavity optical parametric oscillator (OPO). This novel CIM architecture has the potential to be monolithically implemented in future photonic integrated circuits using a thin-film LiNbO3 platform. A network of open dissipative quantum oscillators with optical error correction circuits is not only promising as a future hardware platform, but also as a quantum-inspired algorithm due to its simple and efficient theoretical explanation. To numerically simulate the time progression of an N-qubit quantum system, 2 NComplex amplitudes can be used. However, various phase-space techniques in quantum optics have been developed for quantum oscillator networks. A complete description of a network of quantum oscillators is now possible through a set of N (or 2N) stochastic differential equations (SDEs) based on the positive P, truncated-Wigner, or truncated Fusimi representation of the master equation. These SDEs can be used as heuristic algorithms on modern digital platforms. To fully describe a network of low-Q quantum oscillators, a discrete map approach using the Gaussian quantum model is available, which is also computationally efficient. Similarly, a network of dissipationless quantum oscillators with adiabatic Hamiltonian tuning can be described by a set of N deterministic equations and is also used as a heuristic algorithm on modern digital platforms. This heuristic algorithm is also called a simulated bifurcation machine (SBM), and a variant of it is disclosed below. While it is interesting to compare these two quantum-inspired algorithms, the version of SBM discussed below (dSBM) is not necessarily a true unitary system because dissipation is artificially added using inelastic walls to improve the algorithm's performance. Both algorithms have matrix-vector multiplications (MVMs) as a computational bottleneck when simulated on a digital computer, so the number of MVMs is a good metric for performance comparison. As described below, both types of systems can have similar performance in most cases, except for graph types where the number of edges per node varies significantly, where SBM consistently struggles.

[0020] Numerical simulation of CIM-CAC, CIM-CFC, and CIM-SFC One known CIM architecture employs simple linear feedback without any error detection / correction mechanism, and the feedback term in equation (1) is simply Σ j ξJ ij x jIn this case, as illustrated in Figure 2C, the inhomogeneity of the OPO amplitudes prevents the Ising Hamiltonian from being properly mapped to the network losses, especially for frustrated spin systems. Such CIMs often fail to find the ground state of the Ising Hamiltonian and instead find the coupling (Jacobian) matrix [J ij ]. This undesired behavior is caused by the formation of inhomogeneous amplitudes in the system. This technical problem is solved by solving the equation tanh(Σ j ξJ ij x j This can be partially resolved by introducing a nonlinear filter function for the feedback pulse, such as ( ). The CIM system can then achieve uniform OPO amplitudes at least well above the threshold and satisfy the appropriate mapping condition toward the end of the system's trajectory, as illustrated in Figure 2D. However, such nonlinear filtering alone may not be powerful enough to prevent the machine from becoming trapped in numerous local minima. As the problem size N increases, the number of local minima is expected to increase exponentially for NP-hard Ising problems, making a system that is too easily trapped ineffective.

[0021] To destabilize the attractor caused by the local minimum and keep the machine searching for the true ground state, an error detection / correction variable, represented by equation (2) or equation (5), can be introduced. When a local minimum becomes unstable, the ground state inevitably becomes unstable as well. Although this is undesirable, the CIM machine can reach many local minima and then discover which one has the lowest energy. Alternatively, system parameters can be adjusted so that the system is more likely to remain in the ground state toward the end of the trajectory, as discussed below.

[0022] By adding another N degrees of freedom, the machine can reach a local minimum, but then escape the minimum and continue exploring nearby states. This is not possible with traditional CIM algorithms. As shown in Figure 2E, the trajectory of a CIM with an error correction variable never reaches equilibrium and continues to explore many states. Conversely, with an error correction variable (e i ) often rapidly converge to a fixed point corresponding to a high-energy excited state of the Ising Hamiltonian. The above problem can be solved using error correction schemes, including CIM-CFC and CIM-SFC. Although CIM-CFC and CIM-SFC are described by very different equations, the two systems were originally conceived with similar concepts. To understand why CIM-CFC and CIM-SFC are similar, consider the "mutual coupling signal" M i (t)=Σ j ξJ ij x j (t) and the "injection feedback signal" I i It is useful to consider these systems by introducing (t). These signals are, for both CIM-CFC and CIM-SFC, M i (t)=Σ j ξJ ij x j (t) (15)

[0023]

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[0024] Although CIM-CFC and CIM-SFC are based on the same principles, the dynamics of the two systems can differ from each other. In particular, CIM-CFC (and CIM-CAC) are almost always characterized by chaotic dynamics, as the trajectories are highly sensitive to initial conditions. In the case of CIM-SFC, the trajectories often quickly become stable periodic orbits unless the parameters are dynamically adjusted.

[0025] To demonstrate this difference, Figure 4 illustrates the correlation of pulse amplitudes between two initial conditions that are very close to each other. The initial condition for pulse amplitude #1 (plotted on the x-axis) is chosen from a zero-mean Gaussian distribution with a standard deviation of 0.25, while the other initial condition for the same pulse #2 (plotted on the y-axis) is equal to #1 plus a small amount of noise (standard deviation of 0.01). Figure 4 illustrates the correlation of all 100 pulse amplitudes between two initial conditions for the Sherrington-Kirkpatrick (SK) spin glass instance f0 with problem size N = 100. In CIM-SFC (first row), the correlation is maintained even after 4000 time steps (round trips). This means that the two initial conditions follow nearly identical trajectories. However, in CIM-CFC (second row), after about 100 time steps, the correlation is maintained even after 4000 time steps (round trips). This means that the two initial conditions follow nearly identical trajectories. i The variables became uncorrelated, qualitatively demonstrating that CIM-CFC is highly sensitive to initial conditions, whereas CIM-SFC is not.

[0026] This pattern tends to hold when different parameters and initial conditions are used. However, while CIM-SFC remains correlated for the most part, in some regions of system parameters and initial conditions, the two trajectories diverge. This means that although CIM-SFC is less sensitive to initial conditions than CIM-CFC, chaotic dynamics are still likely to occur during the exploration, especially when parameters are being tuned.

[0027] Another way to qualitatively understand the difference in dynamics is simply by looking at the trajectories. Figure 5 illustrates example trajectories for both systems with fixed system parameters and with linearly adjustable system parameters (for the SK problem above, 100 x i (Ten of the variables are illustrated). When the parameters are fixed, the differences between the two systems are clear. CIM-SFC quickly traps in a stable periodic attractor, while CIM-CFC continues to explore in an unpredictable manner. This requires that parameters be slowly adjusted in CIM-SFC for the system to find a ground state. CIM-CFC and CIM-CAC are able to find a ground state with fixed parameters. However, tuning the system parameters significantly improves the performance of CIM-CFC and CIM-CAC (see Appendix C for details).

[0028] In the bottom left panel of Figure 5, the parameters c and p of the CIM-SFC increase linearly from low to high values (p ranges from -1 to +1, and c ranges from 1 to 3). As shown, as the parameters change, the system may jump from one attractor to another and eventually reach a fixed point / minimum. In CIM-SFC, by linearly increasing the parameters c and p from low to high values, the nonlinear term tanh(cz i ) is the nonlinear coupling term from the "soft spin" mode, which has a continuous range of values between -1 and 1, as expressed by tanh(cz i) primarily takes values of +1 or -1. This transition is believed to be important for the proper functioning of CIM-SFC.

[0029] For most fixed parameters, CIM-SFC rapidly approaches a periodic or fixed-point attractor as in Figure 5, but as mentioned above, for some specific values of c and p, CIM-SFC may feature chaotic dynamics similar to CIM-CFC. It has been shown that chaotic dynamics can be observed when efficiently solving difficult optimization problems using deterministic systems and in simulated bifurcation machines.

[0030] Implementing CIM with optical error correction circuits. 6A and 6B, in conjunction with FIG. 1C, illustrate the physical setup of a CIM-CAC and a CIM-CFC with optical error correction circuitry. The overall architecture is illustrated in FIG. 6C. In this novel architecture, the main ring cavity (illustrated in FIG. 1C) emits a sine wave with normalized amplitude x i and a signal pulse with normalized amplitude e i where i=1, 2,..., N. The signal pulses are stored in the vacuum state |O>1|O>2...|O> N and is amplified (or de-amplified) along the X coordinate by a positive (or negative) pump speed p.

[0031] The error pulse is a coherent state |α>1|α>2...|α> N and is amplified (or de-amplified) along the X coordinate by the pump speed p', as explained below. The squared amplitude of the error pulse is small compared to the saturation level of the main cavity OPO.

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[0033] The extraction beam splitter (BSe as shown in FIG. 1C) extracts partial waves of the signal and error pulses which are amplified by a noise-free phase-sensitive amplifier (PSA0 as shown in FIG. 6A). The PSA0 amplifies the signal and error pulses to classical levels without introducing additional noise. The extracted amplitudes are

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[0035] A small portion of the PSA0 output is used to measure the amplitude of the extracted signal and error pulses.

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[0036] For example, a signal pulse

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[0038] Then the output of the fan-in circuit is

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[0040] One advantage of this optical implementation of the CIM-CAC and CIM-CFC is that only one type of active device is used: a noiseless phase-sensitive (degenerate optical parametric) amplifier; all other elements are passive. This potentially enables on-chip monolithic integration of the CIM system and also enables low energy dissipation in the computational unit discussed below. A similar optical implementation of the CIM-SFC is illustrated in Appendix F.

[0041] Figure 6(b) shows the second harmonic generation (SHG) pulses from the main cavity PSA, post-amplifier PSA0, delay line amplifiers PSA1, PSA2, ... PSA N and exit amplifier PSA e The pump pulse factory shown in Figure 1 provides a pulse train with a repetition rate of 100 GHz and a wavelength of 1.56 μm. The pulse train is then fed to the pump amplifiers PSA before being split into many branches. p Matrix vector multiplication

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[0044] Another detail that needs to be considered when considering an optical implementation is the calculation of the Ising energy. In the digital simulations used to generate the results in this disclosure, the Ising energy is calculated for each time step (round trip) and the minimum energy obtained is used as the result of the calculation. This is because in an optical implementation, the system

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[0046] This is because in a CIM with optical error correction, the system makes multiple trips to get the calculation results,

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[0047] Quantum noise analysis and energy to solution Because CIM discloses the use of analog optical devices, it is important to use quantum models based on optical implementations to study to what extent noise from the physical system (in this case, quantum noise from the pump source and external reservoir) degrades performance.

[0048] In the optical implementation proposed for CIM-CAC, the real signal pulse amplitude μ i (unit of photon amplitude) is the following truncated-Wigner SDE [22;23]

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[0052] Real error pulse amplitude ν j (unit of photon amplitude) is

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[0054] The error pulse is a coherent state |γ>1|γ>2...|γ> for some positive real number 1 / g>>γ>0. N The absence of a gain saturation term in equation (18) means that the error pulse is always pumped below the threshold. Nevertheless, the error pulse exhibits an exponentially varying amplitude. The parameter β governs the time constant of the error correction dynamics, and α is the square of the target amplitude. This feedback model is based on the exponentially varying error pulse amplitude e i =gν j By the squared signal pulse amplitude

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[0056] CIM-CFC is also realized by the experimental setup illustrated in Fig. 6. In this case, the truncated-Wigner SDE in terms of error pulse amplitude is still given by equation (18) or equation (21), but the pump speed p' i teeth,

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[0057] Finally, CIM-SFC can also be realized by the experimental setup illustrated in Appendix F and Figure 17. In this case, equations (20) and (21) become:

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[0058] Comparing the quasi-classical nonlinear dynamical models of CIM-CAC, CIM-CFC, and CIM-SFC expressed by equations (1) to (8) with the quantum nonlinear dynamical models (truncated-Wigner SDE) equations (20) to (26), the main differences are the vacuum noise and the pump noise term gn i and gm i Another important difference is that in the quantum model

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[0060] It is important to analyze the impact of quantum noise on the performance of the CIM. As shown in equations (20) through (26), the relative magnitude of quantum noise in the signal and error pulses is governed by the saturation parameter g. As g increases, the normalized pulse amplitude (x i ,e i ) and the normalized quantum noise amplitude (gn i and gm i ) decreases. Therefore, increasing g is expected to degrade CIM performance. However, increasing g reduces the OPO threshold pump power (see Figure C1 in

[31] ), suggesting that increasing g can potentially reduce the OPO energy cost to solution.

[0061] As shown in Figure 7, the success probability P s vs. saturation parameter g 2 is plotted. Extraction beam splitter R B The reflectance of R B = 0.1. The success probability P s is g 2 ≦10 -4 As long as 2 However, g 2 is 10 -3 Above this, the probability of success drops off rapidly due to the decreasing signal-to-quantum noise ratio as discussed above.

[0062] In Figure 8, the energy cost for the solution of the Ising problem (SK model) for N = 100 and N = 800 is shown using only the pump power to the main cavity PSA.

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[0065] When the energy costs in the optical error correction circuitry and the pump pulse factory (described in FIG. 6) are included, the energy costs increase by an order of magnitude as illustrated in FIG. 9, where the pump pulse energy is the sum of the PSA1, PSA2, ..., PSA N and PSAe It is assumed that the pump energy dissipated in the optical error correction circuit is 100 fJ / pulse for small signal amplification (about 10 dB) at 100 fJ / pulse and 1 pJ / pulse for large signal amplification (about 50 dB) at 100 fJ / pulse. These values correspond to experimental values for a thin-film LiNO3 ridge waveguide DOPO at a pump wavelength of 780 nm and a pump pulse duration of 100 femtoseconds

[15] . The pump energy dissipated in the optical error correction circuit is correction =[(N+1)×10 -13 The energy consumption in the pump pulse factory is estimated as follows: It consists of three parts: the 100 GHz soliton frequency comb generator, the EOM adjuster, and the phase-sensitive amplifier (Fig. 6(b)). The 100 GHz soliton frequency comb generator requires an input power of about 100 mW.

[16] The 100 GHz EOM adjusters require an electrical input power of about 400 mW each.

[17] PSA p The energy cost per pulse of the PSA is about 1 pJ, while the energy cost of N PSAs in the storage ring cavity is about 100 fJ. N ) is the first round trip time 10 -11 Note that the pump pulse factory only needs to operate for N seconds. The operating power of the active devices at 100 GHz CIM is summarized in Figure 2A. The energy cost in the pump pulse factory is E factory =[1.3×10 -11 N(MVM) + 4 × 10 -12 N 2 +(10 -12 +10 -13 N)(MVM)N](J). Figure 2B summarizes the energy costs of the three parts of CIM. In Figure 9, the energy to solution is illustrated when the CIM-CFC algorithm is implemented on a GPU. A detailed description of this approach is given below. Optical implementation of the error correction circuitry and pump pulse factory as described in Figure 6, although technically challenging, can reduce the energy cost by an order of magnitude compared to state-of-the-art GPUs.

[0066] Scaling Performance of CIM-inspired Heuristic Algorithms - CIM-CAC, CIM-CFC, and CIM-SFC To test whether three classical nonlinear dynamical models, Equation (1), Equation (2), Equation (3), Equation (4), Equation (5), Equation (6), Equation (7), and Equation (8), are good Ising solvers, they can be numerically integrated on a digital platform. Additionally, to ensure numerical stability, the ranges of several variables are restricted, details of which can be found in Appendix A. The relevant performance metric is the time to solution, TTS (the number of integration time steps required to achieve a 99% success rate). In particular, we compare how the median TTS scales as a function of problem size for randomly generated Sherrington-Kirkpatrick (SK) spin glass instances (couplings are randomly chosen between +1 and -1). The median TTS is calculated based on a set of 100 randomly generated instances for each problem size, and 3200 trajectories per instance are used to evaluate the TTS.

[0067] In Figure 10, the median TTS of the three CIM-inspired algorithms (CIM-CAC, CIM-CFC, and CIM-SFC) is plotted against problem size. The shaded region represents the 25th to 75th percentile.

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[0070] CIM-inspired heuristic algorithm - Comparison with Noisy Mean Field Annealing (NMFA) To demonstrate the importance of the auxiliary variable (error pulse) in CIM-SFC, its performance was compared with another CIM-inspired algorithm known as Noisy Mean Field Annealing (NMFA)

[26] . NMFA also applies a hyperbolic tangent function to the mutual coupling terms. However, NMFA does not have an auxiliary variable and relies on (artificial) quantum noise to escape local minima. In Figure 11, the scaling from NMFA to CIM-SFC is compared for various values of the feedback parameter k. The parameter k controls the strength of the destabilizing force caused by the auxiliary variable, so the effect of the term k(z i -e i ) can be measured. For k = 0, CIM-SFC is almost identical to NMFA. The fact that the performance of CIM-SFC for k = 0 is slightly worse indicates that the influence of noise contained in NMFA may be small and may contribute to the destabilization of the local minimum (this can also be observed in Figure 7). The case of k = 0.15 is illustrated as an intermediate case, and k = 0.2 is the optimal value for k (obtained experimentally) in CIM-SFC. The error correction feedback term k(z i -e iAdding auxiliary variables is effective in improving both the scaling and spread of TTS in SK instances. This suggests that the "correlated artificial noise" provided by auxiliary variables is more effective in finding better solutions than the "random quantum noise" from the reservoir.

[0071] CIM-inspired heuristic algorithms - Comparison with discrete simulated branching machines (dSBMs) The performance of the CIM-inspired algorithm can also be compared to another heuristic Ising solver known as the discrete simulated branching machine (dSBM). [27;29;28] Like CIM, dSBM utilizes analog spin and continuous dynamics to solve combinatorial optimization problems. The authors of

[29] appear to claim that dSBM is algorithmically superior to CIM-CAC by comparing the number of matrix-vector multiplications (MVMs) required to the solution. Although the authors of

[29] discussed real-time TTS of implementations on many problem sets, they only used the median TTS (in MVM units) of SK instances for two problem sizes when claiming the algorithm's superiority.

[0072] To evaluate these methods, a more detailed comparison of the three algorithms (CIM-CAC, CIM-CFC, and CIM-SFC) against dSBM can be performed by using the MVM to solution (or equivalently, the integration time step to solution) as a performance metric. This is a good comparison because all of these algorithms involve matrix-vector multiplication as a computational bottleneck when implemented on a digital platform. As discussed above, the calculation of the Ising energy can be left until the end of the trajectory in most cases, so only the MVM involved in the calculation of the mutual coupling terms is considered when calculating the MVM to solution.

[0073] The problem instance set used for comparison may include: 1) a set of 100 randomly generated 800-spin SK instances. This instance set includes fully connected instances with weights of +1, -1; 2) a G-set instance (available at https: / / web.stanford.edu / yyye / yyye / Gset / ) that has been used as a benchmark for Max-Cut performance. For this comparison, 50 instances with problem sizes between 800 and 2000 are used. These instances have varying edge densities and include either weights of +1, 0, or weights of +1, 0, -1; and 3) another set of 1000 randomly generated 800-spin and 1200-spin SK instances used to evaluate worst-case performance.

[0074] The dSBM algorithm was also implemented on the GPU to compare its performance on the 800-spin SK instance. The parameters of dSBM were chosen based on the parameters in

[29] and can be found in Appendix D.

[0075] Figure 12 illustrates the performance of three algorithms (CIM-CAC, CIM-CFC, and CIM-SFC) compared instance-by-instance with dSBM on the 800-spin SK instance set. The ground-state energy used to evaluate the MVM to a solution is the lowest energy found by the four algorithms. Because all four algorithms found the same lowest energy, these are likely to be the true ground-state energies. The parameters for all four systems can be found in Appendix A. Comparing performance on the 800-spin instance in Figure 12, all four systems show remarkably similar performance when the parameters are optimized. It is important to note that with the parameters used in Figure 12, CIM-SFC failed to find the ground state in one instance. However, if different parameters were used, CIM-SFC would find the ground state of this particular instance as well. This means that CIM-SFC can achieve high performance but is very sensitive to parameter selection.

[0076] The median TTS (in MVM units) for CIM-CFC, CIM-SFC, and dSBM were nearly identical, approximately 2 × 10 5 Furthermore, the spread in TTS for these three algorithms is quite similar. While CIM-CAC has a slightly worse (less than 2x worse) median TTS, it is notable that the instances where CIM-CAC performs better than dSBM tend to be the more difficult instances. This may indicate that, of the four algorithms, CIM-CAC may have slightly better worst-case performance. This pattern is also seen in the G set.

[0077] Overall, all four algorithms perform similarly on the fully connected instance set, making it impossible to draw conclusions about which particular algorithm is most effective for this problem type. In addition to similar median TTS and spread, there is a high level of correlation in TTS between all four systems. This may indicate that instance difficulty is a universal property of all Ising heuristics, or that there is something fundamentally common to the four algorithms discussed. Appendix E contains a detailed discussion of the similarities and differences between these four systems.

[0078] CIM-SFC performs well on fully connected problem instances, but struggles on many G-set instances. Appendix D contains partial reasons for this deficiency, but the full reasons are not understood. See Appendix D for CIM-SFC results on the G-set.

[0079] All three algorithms (CIM-CAC, CIM-CFC, and dSBM) perform fairly well on the G set, but of the three, CIM-CAC appears to be the more consistently effective algorithm. While CIM-CAC and dSBM were able to find the best known cut value in 47 / 50 instances, CIM-CFC found the best known cut value in 45 / 50 instances. It is noteworthy that the simulation time

[29] used to calculate the TTS for dSBM was much longer than that used in this comparison. Given the same simulation time, dSBM would likely have only been able to solve 45 / 50 instances. As demonstrated in Figure 13, CIM-CAC and CIM-CFC are faster (in MVM units) than dSBM in most instances. More importantly, among the instances where dSBM is faster, dSBM is never significantly faster than CIM-CAC, except for G37, where CIM-CAC failed to find the best known cut value. On the other hand, we found that CIM-CAC was more than an order of magnitude faster than dSBM, and we found 13 / 50 instances where CIM-CAC was the more reliable algorithm when considering many problem types.

[0080] The differences between CIM-CAC and CIM-CFC are subtle. This is expected, as the dynamics of the two systems are very similar. The performance of the two algorithms on the G set is nearly identical in most cases, but in some more difficult instances, CIM-CFC fails to find the best known cut value or has a significantly longer TSS. This may indicate that CIM-CAC is fundamentally a more promising algorithm or that precise parameter selection for CIM-CFC is required.

[0081] As noted in Figure 12, the worst-case performance of CIM-CAC may be slightly better than that of dSBM. To further evaluate this, new sets of 1000 SK instances with 800 and 1200 spins were created. In Figure 14, the number of solved instances is plotted as a function of the number of MVMs required to achieve a 99% success probability. In both cases, dSBM can solve easier instances with fewer MVMs, but for the most difficult instances, CIM-CAC is faster. This can be seen by observing the intersection of the two curves in Figure 14.

[0082] In nearly all cases, the best Ising energy found was the same for both solvers when a similar number of MVMs were used (see Appendix A for parameters). However, in two instances in the N = 1200 instance set, the Ising energy found by CAC was not found by dSBM. This was true even when 50,000 dSBM trajectories were used in these instances.

[0083] Based on our results, it appears that dSBM may struggle significantly on some of the more difficult SK instances. However, this may be the result of suboptimal parameter selection for dSBM. The parameters used (see Appendix A) were manually optimized to have good median TTS, but may not be optimal if we want to solve the most difficult instances. However, for CIM-CAC, the optimal parameters for median TTS appear to perform well even on the most difficult instances.

[0084] Worst-case performance is crucial to ensure that the Ising solver finds the true ground state of a given problem. To this end, at least for randomly generated SK instances, CIM-CAC may be a fundamentally better algorithm. For other CIM modifications, this may not be the case. In particular, for CIM-SFC, the worst-case performance (as illustrated in Figure 10) is significantly worse than dSBM and CIM-CAC.

[0085] The proposed CIM-inspired algorithm disclosed herein has proven to be a fast and accurate Ising solver, even when implemented on current digital platforms. This performance is very similar to other existing analog system-based algorithms, such as dSBM. This reiterates the question posed in

[11] : whether simulation of analog spins on digital computers can outperform purely discrete heuristic algorithms.

[0086] The above description has been made with reference to specific embodiments for purposes of explanation. However, the above exemplary discussion is not intended to be exhaustive or to limit the disclosure to the precise form disclosed. Many modifications and variations are possible in light of the above teachings. The embodiments have been chosen and described to best explain the principles of the disclosure and its practical application, and thereby enable others skilled in the art to utilize the disclosure and various embodiments to the fullest extent, with various modifications as suited to the particular use contemplated.

[0087] The systems and methods disclosed herein may be implemented via one or more components, systems, servers, devices, or other subcomponents, or may be distributed among such elements. When implemented as a system, such a system may include and / or contain components such as software modules, a general-purpose CPU, RAM, etc., found, among other things, in a general-purpose computer. In implementations where the innovations reside on a server, such a server may include or contain components such as a CPU, RAM, etc., found in a general-purpose computer.

[0088] Additionally, the systems and methods herein may be achieved through implementation using disparate or entirely different software, hardware, and / or firmware components beyond those described above. With respect to such other components (e.g., software, processing components, etc.) and / or computer-readable media associated with or embodying the invention, for example, aspects of the innovations herein may be implemented in conjunction with numerous general-purpose or special-purpose computing systems or configurations. Various exemplary computing systems, environments, and / or configurations that may be suitable for use with the innovations herein may include, but are not limited to, software and / or other components embodied within or in personal computers, servers or server computing devices such as routing / connection components, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set-top boxes, consumer electronic devices, network PCs, other existing computer platforms, distributed computing environments that include one or more of the above systems or devices, and the like.

[0089] In some instances, aspects of the systems and methods may be achieved or performed by logic and / or logic instructions, including, for example, program modules, executed in conjunction with such components or circuitry. Generally, program modules may include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular instructions herein. The present invention may also be realized in the context of a distributed software, computer, or circuit configuration where circuitry is connected via communication buses, circuitry, or links. In a distributed configuration, control / instruction may originate from both local and remote computer storage media, including memory storage devices.

[0090] The software, circuitry, and components herein may include and / or utilize one or more types of computer-readable media. Computer-readable media can be any available medium that resides on, is associated with, or is accessible by such circuits and / or computing components. By way of example and not limitation, computer-readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules, or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVDs) or other optical media, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to store the desired information and that can be accessed by a computing component. Communication media may comprise computer-readable instructions, data structures, program modules, and / or other components. Additionally, communication media may include wired media such as a wired network or direct-wired connection, although as used herein such type of media does not include transitory media. Combinations of any of the above are also included within the scope of computer-readable media.

[0091] In this description, terms such as component, module, device, and the like may refer to any type of logical or functional software element, circuit, block, and / or process that may be implemented in various ways. For example, the functionality of various circuits and / or blocks may be combined into any number of other modules. Each module may be implemented as a software program stored in tangible memory (e.g., random access memory, read-only memory, CD-ROM memory, hard disk drive, etc.) that is read by a central processing unit to perform the functions of the innovations herein. Alternatively, a module may comprise programming instructions transmitted via a transmission carrier wave to a general-purpose computer or processing / graphics hardware. A module may also be implemented as hardware logic circuitry that performs the functions encompassed by the innovations herein. Finally, a module may be implemented using special purpose instructions (SIMD instructions), field programmable logic arrays, or a combination thereof, that provide a desired level of performance and cost.

[0092] As disclosed herein, features consistent with the present disclosure may be implemented via computer hardware, software, and / or firmware. For example, the systems and methods disclosed herein may be embodied in various forms, including, for example, a data processor such as a computer, including a database, digital electronic circuitry, firmware, software, or combinations thereof. Moreover, while some of the disclosed implementations describe specific hardware components, systems and methods consistent with the innovations herein may be implemented in any combination of hardware, software, and / or firmware. Furthermore, the features and other aspects and principles described herein may be implemented in a variety of environments. Such environments and related applications may include general-purpose computers or computing platforms that may be specially constructed to execute various routines, processes, and / or operations in accordance with the invention, or may be selectively activated or reconfigured by code to provide the required functionality. The processes disclosed herein are not inherently related to any particular computer, network, architecture, environment, or other apparatus, but may be implemented by any suitable combination of hardware, software, and / or firmware. For example, various general-purpose machines may be used with programs written in accordance with the teachings of the invention, or it may be more convenient to construct a specialized apparatus or system to perform the required methods and techniques.

[0093] Aspects of the methods and systems described herein, such as logic, may be implemented as programmed functions in any of a variety of circuit configurations, including field programmable gate arrays ("FPGAs"), programmable array logic ("PAL") devices, electrically programmable logic and memory devices, and programmable logic devices ("PLDs") such as application-specific integrated circuits, as well as standard cell-based devices. Some other possibilities for implementing aspects include memory devices, microcontrollers with memory (such as EEPROMs), embedded microprocessors, firmware, software, etc. Additionally, aspects may be embodied in microprocessors with software-based circuit emulation, discrete logic (sequential and combinatorial), custom devices, fuzzy (neural) logic, quantum devices, and hybrids of any of the above device types. The underlying device technology may be provided in a variety of component types, including, for example, metal-oxide-semiconductor field-effect transistor ("MOSFET") technologies such as complementary metal-oxide-semiconductor ("CMOS"), bipolar technologies such as emitter-coupled logic ("ECL"), polymer technologies (such as silicon-conjugated polymer and metal-conjugated polymer-metal structures), and mixed analog and digital technologies.

[0094] It should also be noted that the various logic and / or functionality disclosed herein, in terms of their behavior, register transfers, logic components, and / or other characteristics, may be enabled using any number of combinations of hardware, firmware, and / or data and / or instructions embodied in various machine-readable or computer-readable media. Computer-readable media on which such formatted data and / or instructions may be embodied include, but are not limited to, various forms of non-volatile storage media (e.g., optical, magnetic, or semiconductor storage media), but again, do not include volatile media. Throughout the description, terms such as "comprises," "comprising," and the like, should be construed in an inclusive sense, i.e., "including, but not limited to," rather than an exclusive or exhaustive sense, unless the context clearly requires otherwise. Words using the singular or plural number also include the plural or singular number, respectively. Additionally, the terms "herein," "herein," "upon," "below," and words of similar import refer to this application as a whole, and not to any particular portions thereof. When the word "or" is used in reference to a list of two or more items, the word covers all of the following interpretations of the word: any item in the list, every item in the list, and any combination of items in the list.

[0095] While certain presently preferred implementations of the invention have been specifically described herein, it will be apparent to those skilled in the art to which this invention pertains that variations and modifications of the various implementations shown and described herein can be made without departing from the spirit and scope of the invention. Accordingly, it is intended that the invention be limited only to the extent required by applicable rules of law.

[0096] While the foregoing has been made with reference to particular embodiments of the present disclosure, those skilled in the art will recognize that changes can be made in the embodiments without departing from the principles and spirit of the disclosure, the scope of the invention as defined by the appended claims.

[0097] Appendix A: Optimizing Simulation Parameters Here we summarize the simulation parameters used in our numerical experiments. The parameters were empirically optimized and therefore do not necessarily reflect the true optimum values.

[0098] Parameters in Figure 5

[0099] [Table 1] [Table 2] [Table 3] [Table 4]

[0100] Parameters used in Figures 10, 11, and 12 CIM-CAC In our simulation, x i The variables are in the range

number

[0101] [Table 5]

[0102] CIM-CFC In our simulation, x i The variables are limited to the range [-1.5, 1.5], and e i is limited to the range [0.01,∞]. The parameter p is the first T r The value is linearly adjusted from the start to the end during the time step, and an additional T p The initial value x is kept at its final value for the time step. i has a standard deviation of 0.1 and e i = 1. To evaluate TTS, 3200 trajectories are calculated for each instance. The actual parameters used for the simulation are listed below.

[0103] [Table 6] CIM-SFC This system is numerically more stable, so x i variables and e i No constraints on the variables are required. The parameters p, c, and β are adjusted linearly from the starting value to the ending value during the simulation. The initial value x i has a standard deviation of 0.1 and e i = 0. To evaluate TTS, 3200 trajectories are calculated for each instance. The actual parameters used for the simulation are listed below.

[0104] [Table 7]

[0105] In addition to the above parameters, the normalization coefficient ξ of the mutual coupling term is

number

[31] .

[0106] This choice is crucial for successful performance in CIM-SFC, but not in CIM-CAC and CIM-CFC. Additionally, it is important to note in Figures 10 and 11 that we use the same number of time steps for all problem sizes. Because the optimal number of time steps is likely to be smaller for smaller problem sizes, the scaling of TTS may be slightly worse than the reported scaling when the number of time steps is optimized separately for each problem size. However, we do not expect the difference to be significant. See Appendix C for the scaling of TTS for CIM-CAC when different parameters are chosen.

[0107] dSBM In Figure 12, dSBM is implemented as described in

[29] . The parameters used are: [Table 8] is.

[0108] Parameters used in Figure 14 The parameters for N=800 are the same as those for Figure 12. The parameters for N=1200 are illustrated below. The number of trajectories used for N=1200 was 3200 for most instances, but to accurately evaluate the success probability, 10,000 to 50,000 trajectories were calculated for the 10 most difficult instances for both algorithms. Also, due to the difficulty of the N=1200 case, we are less certain whether the true ground state was found.

[0109] CIM-CAC [Table 9]

[0110] dSBM [Table 10]

[0111] Numerical Integration In all cases (except for dSBM), Euler steps are used for the integration. As explained above, to ensure numerical stability, we use x i We constrain the range of the variables. This is not necessary for performance, but we can increase the integration time step by a factor of two or three without compromising the success probability. In Figure 15, we show the success probability of CIM-CAC versus the time step for both the constrained and unconstrained systems.

[0112] The results in Section 5 on CIM-CFC do not use this numerical restriction; the CIM in Section 5 is intended for a physical machine and a time step of 0.2 is used.

[0113] Appendix B: Simulation results for the G set The dSBM results in Figure 13 are taken directly from the GPU implementation of dSBM in

[29] . The units of TTS in Table 3 are time steps to solution, or equivalently, MVM to solution. In our simulations, depending on the difficulty of the instance, either 3200, 10000, or 32000 trajectories were generated to evaluate TTS. The bold numbers are the best TTSs among the three algorithms.

[0114] CIM-CAC parameters for G set The variables are bounded as described in Appendix A, and the initial conditions are set similarly. The following parameters are the same for all G-set instances: [Table 11]

[0115] Meanwhile, the parameters p, ΔT, and number of time steps used in each phase are selected depending on the instance type as follows: [Table 12]

[0116] CIM-CFC parameters for G sets The variables are bounded as described in Appendix A, and the initial conditions are set similarly. The following parameters are the same for all G-set instances: [Table 13]

[0117] Meanwhile, the parameters p, ΔT, and number of time steps used in each phase are selected depending on the instance type as follows: [Table 14]

[0118] Appendix C: Parameter Selection Rationale The parameters are mostly selected numerically, but the selection of p, α, and β can be understood as follows: It has been observed that the average residual energy reached by CIM-CAC during the search process can be roughly estimated by the following equation:

[11]

[0119]

number

[0120] Here, K is a constant that depends only on the problem type and size. This formula essentially predicts the effective sampling temperature of the system (although the distribution may not be an exact Boltzmann distribution). Based on this underlying principle, we gradually decrease the "system temperature" to create an annealing effect. This is the motivation for increasing p and α. The different choices of the range of p in different G-set instances reflect the widely different values for the constant K, depending on the structure of the max-cut problem. In a more general setting, the value of K can be predicted based on the problem type, and the ranges of p and α can be chosen accordingly. Although not verified, a similar formula is likely to apply to CIM-CFC, and therefore the parameters for CIM-CFC are also selected using the same method.

[0121] Optimal parameters with respect to problem size (CIM-CAC) In Figure 16, we see a difference in scaling when the parameters are fixed (red shading) compared to when they are linearly adjusted (blue shading). In addition, the optimal annealing time (i.e., the optimal adjustment speed) varies with problem size, as larger problem sizes require longer annealing times to achieve good results. This pattern was also used when selecting the parameters for the G set.

[0122] Figure 16 was created using the Gaussian quantum model for CIM-CAC in MFB-CIM (see

[31] ), but g 2 =10 -4 Since a time step of 0.01 is used, the difference in performance between this model and the noise-free model discussed in this paper is insignificant. Note also that a time step of 0.01 is used in Figure 16. This is why the TTS in Figure 16 is an order of magnitude longer than the results discussed in this paper, where a time step of 0.125 was used.

[0123] Appendix D: CIM-SFC Results and Discussion for Set G Based on our current understanding of CIM-SFC, |cz i |>>0 and tanh(cz i )≒sign(cz i ), then the term tanh(cz i ) but CZ i ≒0 and tanh(cz i )≒cz i It is crucial to transition from the "soft spin" mode, where z = 0, to the "discrete spin" mode, which is the mode where z = 0 for a randomly chosen spin configuration. i On average, about

[0124]

number

[0125] In some G-set instances, especially planar graph instances, the degree of some nodes is much larger, so no matter what normalization factor ξ we use, czi will be too large in some cases and too small in others. This may be one reason why CIM-SFC struggles in many G-set instances, especially planar graphs. This may also be why dSBM struggles in planar graphs, since it relies on the same normalization factor to achieve good results. CIM-CAC and CIM-CFC are j J ij σ j Since it automatically compensates for different values of , it does not need this normalization factor, which may be why it works so well on planar graphs.

[0126] On the other hand, for toroidal graphs, for these graphs, Σ j J ij σ j can only take on five different values, and vice versa. This may mean that the transition from "soft spin" to "discrete spin" is very quick for CIM-SFC, so we need to carefully tune the parameters to get good results on these graphs.

[0127] This observation about analog / discrete transitions may partially explain the poor results on the G set, but it is not a sufficient explanation. For example, CIM-SFC struggles on some random graphs (e.g., G9) that do not have the above properties because each node has similar connectivity.

[0128] Below are the results of CIM-SFC on the G set and the parameters used (not all instances were tested).

[0129] CIM-SFC results for G set [Table 15]

[0130] For instances G14-G21 (800-node planar graphs) and G51-G54 (1000-node planar graphs), CIM-SFC exhibits very small, non-zero or zero, success probabilities. The 2000-node instance has not yet been tested.

[0131] CIM-SFC parameters in the G set Common Parameters p|-1.0→1.0

[0132] Parameters selected by question type [Table 16]

[0133] The parameters of CIM-SFC are chosen experimentally, and our understanding of how the parameters affect performance and dynamics is limited. We hope that once this system is studied more thoroughly, we will find a more systematic way to select parameters that will ensure good performance across many different problem types.

[0134] Appendix E: Similarities and Differences Between CIM and SBM Algorithms Using continuous analog dynamics to solve discrete optimization problems is a somewhat new concept, and it is interesting to compare these different approaches. [13,11,29] In this appendix, we briefly discuss the similarities and differences between the three CIM-inspired algorithms and the SBM algorithm.

[0135] All four systems discussed in Section 6, CIM-CAC, CIM-CAC, CIM-SFC, and dSBM, were originally motivated by the same basic principles [10, 27]: function

[0136]

number

[0137] The original CIM algorithm uses gradient descent to find a local minimum of H while H is transformed by increasing p. This system has two major drawbacks

[10] : 1. The minimum value is stable. 2. Due to the non-uniformity of the amplitude, the mapping of the Ising problem to the cost function is incorrect.

[0138] All four algorithms discussed in Section 6 can be considered as modifications of the original CIM algorithm aimed at overcoming these two drawbacks. [11,27,28,29] In all three algorithms, the first drawback is overcome by adding an additional degree of freedom to the system, so that there are now 2N analog variables for N spins, rather than just N. In SBM, this is the position vector x i and the velocity / momentum vector y i However, we have modified the CIM algorithm to include the auxiliary variable e i Add.

[0139] To address the second drawback, the creators of dSBM added discretization and "inelastic walls," whereas CIM-CFC and CIM-SFC do not require this discretization. All three algorithms use different mechanisms to ensure that the system has fixed points only at local minima of the Ising Hamiltonian (during the end of the trajectory), which is often not the case with the original CIM algorithm. Because these systems are fundamentally very similar, it is not surprising that the systems can achieve similar performance to the digital algorithms.

[0140] We also note that to achieve good performance, dSBM requires the use of discretization and inelastic walls, which make the system discontinuous. This is highly favorable for implementations on digital platforms that favor discrete processes, but unfavorable for implementing these algorithms on analog physical platforms. On the other hand, in CIM-CAC, CIM-CFC, and CIM-SFC, the system evolves continuously, making them more suitable for analog implementations such as the optical CIM architecture proposed in this paper.

[0141] One interesting difference between CIM and the original bifurcation machine in

[27] , named aSBM in

[29] , is that aSBM is a completely single, dissipative system. Thus, unlike dissipative CIM or other Ising heuristics (such as simulated annealing or breakout local search

[14] ), which rely on some form of dissipative relaxation, aSBM relies on adiabatic evolution of the computation (similar to quantum annealing). However, in

[29] , the new SBM algorithm deviates from this concept of adiabatic evolution by adding inelastic walls, making the new bifurcation machine a dissipative system that loses information over time. It would be interesting to understand whether dissipation is actually necessary for the system to achieve the high performance of the algorithms discussed in this paper. For example, aSBM could be modified in a different way to eliminate the amplitude nonuniformity problem while maintaining adiabaticity. Whether this is possible is beyond the scope of this paper.

[0142] Appendix F: Optical Implementation of CIM-SFC Figure 17 illustrates an optical implementation of a CIM-SFC that is similar to the optical implementations of the CIM-CAC and CIM-CFC illustrated in Figure 6. Feedback Signal

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Claims

1. A coherent Ising machine, a pump pulse generator configured to generate optical signal pulses; an optical error correction circuit configured to generate optical error pulses; a main ring cavity configured to store the optical signal pulse and the optical error pulse, wherein the optical error pulse causes the coherent Ising machine not to converge to a local minimum of an Ising solution but continues to search for nearby states, and the squared amplitude of the optical error pulse is smaller than a saturation level of an optical parametric oscillator in the main ring cavity.

2. 2. The coherent Ising machine of claim 1, wherein the main ring cavity further comprises a phase-sensitive amplifier configured to compress the optical signal pulses such that the optical signal pulses are stored as compressed optical signal pulses.

3. 2. The coherent Ising machine of claim 1, wherein the main ring cavity further comprises a phase-sensitive amplifier configured to compress the optical error pulse such that the optical error pulse is stored as a compressed optical error pulse.

4. 10. The coherent Ising machine of claim 1, wherein the optical signal pulses are configured to be manipulated in both a linear amplifier / de-amplifier scheme and a nonlinear oscillator scheme.

5. 10. The coherent Ising machine of claim 1, wherein the optical error pulse is configured to be manipulated in a linear amplifier / de-amplifier fashion.

6. A coherent Ising machine, a pump pulse generator configured to generate optical signal pulses; an optical error correction circuit configured to generate optical error pulses; a main ring cavity configured to store the optical signal pulse and the optical error pulse, wherein the optical error pulse causes the coherent Ising machine not to converge to a local minimum of an Ising solution but continues to search for nearby states, and the pump pulse generator, the main ring cavity, and the optical error correction circuit are monolithically integrated on a single chip.

7. The coherent Ising machine of claim 1 , wherein the pump pulse generator is configured to generate the optical signal pulses as parametric down-conversion pulses.

8. 2. The coherent Ising machine of claim 1, wherein the optical error correction circuitry comprises an optical homodyne detector configured to measure the amplitude of the optical signal pulses and the optical error pulses.

9. A coherent Ising machine, a pump pulse generator configured to generate optical signal pulses; an optical error correction circuit configured to generate optical error pulses; a main ring cavity configured to store the optical signal pulse and the optical error pulse, wherein the optical error pulse causes the coherent Ising machine not to converge to a local minimum of an Ising solution but continues to search for nearby states, and the optical error correction circuitry comprises fan-out and fan-in circuits configured to generate the optical error pulse having a predetermined amplitude.

10. 1. A method for solving an Ising problem, comprising: generating an optical signal pulse by a pump pulse generator of the coherent Ising machine; generating an optical error pulse by an optical error correction circuit of the coherent Ising machine; storing the optical signal pulse and the optical error pulse in a main ring cavity of the coherent Ising machine, wherein the optical error pulse causes the coherent Ising machine not to converge to a local minimum of an Ising solution but continues to search for nearby states, and the squared amplitude of the optical error pulse is small compared to a saturation level of an optical parametric oscillator in the main ring cavity.

11. 11. The method of claim 10, further comprising compressing the optical signal pulse with a phase-sensitive amplifier of the coherent Ising machine such that the main ring cavity stores the optical signal pulse as a compressed optical signal pulse.

12. 11. The method of claim 10, further comprising compressing the optical error pulse with a phase-sensitive amplifier of the coherent Ising machine such that the main ring cavity stores the optical error pulse as a compressed optical error pulse.

13. The method of claim 10 , wherein the optical signal pulses are operated in both a linear amplifier / de-amplifier mode and a nonlinear oscillator mode.

14. 11. The method of claim 10, wherein the optical error pulse is manipulated in a linear amplifier / de-amplifier fashion.

15. A method for solving an Ising problem, comprising: generating an optical signal pulse by a pump pulse generator of the coherent Ising machine; generating an optical error pulse by an optical error correction circuit of the coherent Ising machine; storing the optical signal pulse and the optical error pulse by a main ring cavity of the coherent Ising machine, wherein the optical error pulse causes the coherent Ising machine not to converge to a local minimum of an Ising solution but continues to search for nearby states, and wherein the pump pulse generator, the main ring cavity, and the optical error correction circuit are monolithically integrated on a single chip.

16. The method of claim 10 , further comprising generating the optical signal pulses as parametric down-conversion pulses by the pump pulse generator of the coherent Ising machine.

17. 11. The method of claim 10, further comprising measuring the amplitude of the optical signal pulse and the optical error pulse with an optical homodyne detector of the optical error correction circuit.

18. A method for solving an Ising problem, comprising: generating an optical signal pulse by a pump pulse generator of the coherent Ising machine; generating an optical error pulse by an optical error correction circuit of the coherent Ising machine; storing the optical signal pulse and the optical error pulse by a main ring cavity of the coherent Ising machine, wherein the optical error pulse causes the coherent Ising machine to not converge to a local minimum of an Ising solution and to continue searching for nearby states; generating the optical error pulses having a predetermined amplitude by fan-out and fan-in circuits of the optical error correction circuit.

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