Novel oscillator ising machine coupling for high-quality optimization

Abstract

A spin coupling unit which enables high-quality optimization in Ising machines is provided. In embodiments, a spin coupling unit includes a first sampling circuit and a first current source operatively connected to the first sampling circuit. The first current source receives as a first input a first waveform from a first oscillator, receives as a second input a second waveform from a second oscillator, and provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform. The first current source receives as an input the first state value from the first sampling circuit, provides a first current to the first oscillator based on the first state value and a programmable first conductance parameter. Oscillating Ising machines including a plurality of oscillators and spin coupling units are also disclosed.

Description

Attorney Docket # 00495-0023 (B2024-136) NOVEL OSCILLATOR ISING MACHINE COUPLING FOR HIGH-QUALITY OPTIMIZATION STATEMENT OF GOVERNMENT SUPORT

[0001] The invention was made with government support under Grant Number FA8650-23-C-7311 awarded by the DOD Advanced Research Projects Agency. The government has certain rights in the invention. REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of U.S. Provisional Application No.63 / 639,281,filed April 26, 2024, the entire contents of which are hereby incorporated by reference. FIELD OF THE INVENTION

[0003] The present invention generally relates to coupling circuits for use in oscillatorIsing machines. BACKGROUND OF THE INVENTION

[0004] Ising machines are hardware solvers for the Ising problem, a general mathematicalformulation involving an energy-like quantity (the Ising Hamiltonian, a quadratic function of binary problem variables called spins). They have been a focus of research in recent years (e.g.,

[0032] ,[1],

[0026] ,

[0029] ,

[0012] ,

[0028] ,[4]), on account of their ability to solve combinatorial optimization (CO) problems using novel analog mechanisms. Virtually all CO problems can be mapped into Ising form

[0017] , making them amenable to solution using Ising machines, which offer the promise of speed, energy efficiency and miniaturisability.

[0005] The model underlying an Ising machine is similar to a weighted graph, andcomprises, a collection of nodes / vertices and branches / edges between some pairs of nodes, with each branch having a real-number weight. Each node (termed a “spin” in this context) is allowed to take two values, either 1 or -1. Associated with this graph is an expression, the Ising 1 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) Hamiltonian, which multiplies the weight of each branch by the values of the two spins it connects to, and sums over all branches, as shown in Equation 1: ^= −1 2^    ^^^^^^^, where ^^ ∈ {−1, +1} (Equation 1)where H is the ithspin, ^^is the value assignedto the jthspin, and Jijis the coupling weight between the ithspin and the jthspin. The Ising Hamiltonian can be understood to be an “energy” associated with a given configuration of the spins, although in many situations they have no connection with energy in physics. The “Ising problem” is to find spin configurations with the minimum possible energy.

[0006] Oscillator Ising Machines (OIMs) are networks of coupled nonlinear oscillators thatsolve the NP-hard Ising problem. In a scheme proposed in 2016, each of the N spins of an Ising problem is implemented by an oscillator, and the information needed to find a solution of the Ising problem is encoded in the phase of each oscillator. This purely classical scheme had a significant advantage over prior Ising machines: OIMs can potentially be implemented entirely on chip in Complementary Metal-Oxide-Semiconductor (CMOS) device and related technologies, with all the attendant benefits of Integrated Circuit (IC) integration, including small physical size, low power consumption, scalability to many spins and mass production with potentially low cost. Moreover, oscillator Ising machines, a scheme based on the dynamics of suitably-designed networks of coupled oscillators, have shown some promise for high- quality optimization albeit with implementation difficulties to date [28,31].

[0007] So far, OIMs with somewhat idealized phase-domain functions (Fc(·), describedbelow) have shown excellent optimization characteristics, in both simulation [30,31,24] and a custom digital IC emulator

[0023] . High-quality analog OIM implementations can offer important 2 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) practical advantages over simulation and digital emulation, such as significantly lower energy- to-solution.

[0008] Despite these possible advantages, high-quality analog OIM implementations havefaced difficulty in implementation. Indeed, integrated circuit realizations of OIMs that employ real analog oscillators to deliver high-quality optimizations on practically-important problems have not been publicly demonstrated yet.

[0009] For example, as shown in FIG. 7A, in some circumstances, the oscillators in anOIM are coupled using resistors and / or rely on resistive couplings. More precisely, independent of the actual hardware implementation (which can use, e.g., active elements instead of resistors), conventional oscillators couple to one another by injecting a signal proportional to its voltage waveform. A key problem with resistive coupling is that an OIM’s phase-domain functions (Fc(·)) essentially mirrors the oscillator’s phase-sensitivity function (PPV) which can make it difficult to tailor the shape of the phase-domain function to improve the OIM’s optimization performance. As a result, practical analog OIM hardware do not match results from simulation / emulations (which use tailored, idealized, Fc(·) functions). It is also difficult to alter oscillator designs to yield a desired Fc(·) shape with resistive coupling.

[0010] What is needed is a coupling scheme that overcomes these and other issues in theprior art by allowing for a high-quality optimization of oscillator Ising machines. SUMMARY OF THE INVENTION

[0011] In view of the above, it is an object of the present disclosure to overcome these andother issues present in the prior art by providing a spin coupling unit as well as an oscillator Ising machine incorporating a plurality of spin coupling units which provide flexible, programmable phase-domain functions. 3 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0012] In embodiments, a spin coupling unit for use with an Ising machine includes: (a) afirst sampling circuit operatively connected to a first oscillator and a second oscillator that: (i) receives as a first input a first waveform from the first oscillator; (ii) receives as a second input a second waveform from the second oscillator; and (iii) provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform; and (b) a first current source operatively connected to the first sampling circuit that: (i) receives as an input the first state value from the first sampling circuit; and (ii) provides a first current to the first oscillator based on the first state value and a programmable first conductance parameter.

[0013] In embodiments, the first sampling circuit is a flip-flop.

[0014] In embodiments, the programmable first conductance parameter is obtained from aregister.

[0015] In embodiments, the programmable first conductance parameter is based on a firstcoupling weight of the first oscillator and the second oscillator.

[0016] In embodiments, an Ising machine includes: (a) a first oscillator that generates afirst waveform; (b) a second oscillator that generates a second waveform; (c) a plurality of spin coupling units, including a first spin coupling unit and a second spin coupling unit, wherein the first spin coupling unit includes: (i) a first sampling circuit operatively connected to the first oscillator and the second oscillator that: (a) receives as a first input the first waveform from the first oscillator; (b) receives as a second input the second waveform from the second oscillator; (c) provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform; and (ii) a first current source operatively connected to the first sampling circuit that: (a) receives as an input the first state value from the first sampling circuit; and (b) provides a first current to the first oscillator based on the first state value and a programmable first conductance parameter; and wherein the second spin coupling unit includes: (i) a second sampling circuit operatively connected to the first oscillator and the 4 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) second oscillator that: (a) receives as a third input the second waveform from the second oscillator; (b) receives as a fourth input the first waveform from the first oscillator; (c) provides as an output a second state value based on sampling of the first waveform at a transition of the second waveform; and (ii) a second current source operatively connected to the second sampling circuit that: (a) obtains the second state value from the second sampling circuit; and (b) provides a second current to the second oscillator based on the second state value and a programmable second conductance parameter, wherein the first oscillator and the second oscillator are interdependent.

[0017] In embodiments, the Ising machine further includes: (d) a third oscillator thatgenerates a third waveform; and wherein the plurality of spin coupling units further includes a third spin coupling unit and a fourth spin coupling unit.

[0018] In embodiments, the third spin coupling unit includes: (i) a third sampling circuitoperatively connected to the first oscillator and the third oscillator that: (a) receives as a fifth input the first waveform from the first oscillator; (b) receives as a sixth input the third waveform from the third oscillator; (c) provides as an output a third state value based on sampling of the third waveform at a transition of the first waveform; and (ii) a third current source operatively connected to the third sampling circuit that: (c) receives as an input the third state value from the third sampling circuit; and (d) provides a third current to the first oscillator based on the third state value and a programmable third conductance parameter; and wherein the fourth spin coupling unit includes: (i) a fourth sampling circuit operatively connected to the first oscillator and the third oscillator that: (a) receives as a seventh input the third waveform from the third oscillator; (b) receives as an eighth input the first waveform from the first oscillator; (c) provides as an output a fourth state value based on sampling of the first waveform at a transition of the third waveform; and (ii) a fourth current source operatively connected to the fourth sampling circuit that: (a) obtains the fourth state value from the fourth sampling circuit; and 5 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (b) provides a fourth current to the third oscillator based on the fourth state value and a programmable fourth conductance parameter, wherein the first oscillator and the third oscillator are interdependent.

[0019] In embodiments, the second oscillator and the third oscillator are independent.

[0020] In embodiments, the plurality of spin coupling units further includes a fifth spincoupling unit and a sixth coupling unit, wherein the fifth spin coupling unit includes: (i) a fifth sampling circuit operatively connected to the second oscillator and the third oscillator that: (a) receives as a ninth input the second waveform from the second oscillator; (b) receives as a tenth input the third waveform from the third oscillator; (c) provides as an output a fifth state value based on sampling of the third waveform at a transition of the second waveform; and (ii) a fifth current source operatively connected to the fifth sampling circuit that: (a) receives as an input the fifth state value from the fifth sampling circuit; and (b) provides a fifth current to the second oscillator based on the fifth state value and a programmable fifth conductance parameter; and wherein the sixth coupling unit includes: (i) a sixth sampling circuit operatively connected to the second oscillator and the third oscillator that: (a) receives as an eleventh input the third waveform from the third oscillator; (b) receives as a twelfth input the first waveform from the second oscillator; (c) provides as an output a sixth state value based on sampling of the second waveform at a transition of the third waveform; and (ii) a sixth current source operatively connected to the sixth sampling circuit that: (c) obtains the sixth state value from the sixth sampling circuit; and (d) provides a sixth current to the third oscillator based on the sixth state value and a programmable sixth conductance parameter, wherein the second oscillator and the third oscillator are interdependent.

[0021] In embodiments, the first sampling circuit is a flip-flop.

[0022] In embodiments, the second sampling circuit is a flip-flop.

[0023] In embodiments, the third sampling circuit is a flip-flop.6 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0024] In embodiments, the fourth sampling circuit is a flip-flop.

[0025] In embodiments, the fifth sampling circuit is a flip-flop.

[0026] In embodiments the sixth sampling circuit is a flip-flop.

[0027] In embodiments, the programmable first conductance parameter is obtained from aregister.

[0028] In embodiments, the programmable second conductance parameter is obtained froma register.

[0029] In embodiments the programmable third conductance parameter is obtained from aregister.

[0030] In embodiments, the programmable fourth conductance parameter is obtained froma register.

[0031] In embodiments, the programmable fifth conductance parameter is obtained from aregister.

[0032] In embodiments, the programmable sixth conductance parameter is obtained froma register.

[0033] In embodiments the programmable first conductance parameter is based on acoupling weight of the first oscillator and the second oscillator.

[0034] In embodiments, the programmable second conductance parameter is based on acoupling weight of the first oscillator and the second oscillator.

[0035] In embodiments the programmable third conductance parameter is based on acoupling weight of the first oscillator and the third oscillator.

[0036] In embodiments, the programmable fourth conductance parameter is based on acoupling weight of the first oscillator and the third oscillator.

[0037] In embodiments, the programmable fifth conductance parameter is based on acoupling weight of the second oscillator and the third oscillator. 7 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0038] In embodiments, the programmable sixth conductance parameter is based on acoupling weight of the second oscillator and the third oscillator. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] The above and related objects, features and advantages of the present disclosurewill be more fully understood by reference to the following detailed description of the preferred, albeit illustrative, embodiments of the present invention when taken in conjunction with the accompany figures, wherein:

[0040] FIG. 1A provides an ideal sharp plot for idealized oscillators to be used inaccordance with embodiments of the present invention;

[0041] FIG. 1B provides example waveforms of idealized oscillators to be used inaccordance with embodiments of the present invention;

[0042] FIG. 2 depicts a circuit design in accordance with embodiments of the presentinvention;

[0043] FIG.3 depicts sampling couplers coupling oscillators in accordance with exemplaryembodiments of the present invention;

[0044] FIG. 4 is an example of a nominal waveform of an unperturbed oscillator ascompared with a nominal waveform of a perturbed oscillator over a period of time;

[0045] FIGs. 5A-5C depict an Ising machine comprising N oscillators and using samplingcouplers in accordance with exemplary embodiments of the present invention;

[0046] FIG. 6A is a plot of Symbol Error Rates (SERs) against Signal to Noise Ratio (SNR)of various detectors;

[0047] FIG.6B is a plot of Symbol Error Rates (SERs) against Signal to Noise Ratio (SNR)for a resistive coupler OIM, a sampling coupler OIM, the idealized OIM and other heuristics; 8 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0048] FIG.7A provides an example of a resistive coupling via a resistor between oscillatorspresent in the prior art;

[0049] FIG.7B provides an example of active “one-way resistor” coupling utilizing two one-way buffered gm units in accordance with embodiments of the present invention; and

[0050] FIG. 7C provides a schematic of multiplicative coupling in accordance withembodiments of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0051] The present invention generally relates to coupling circuits for use in oscillator Isingmachines.

[0052] Prior to the present invention, it was anticipated that resistive coupling circuitswould provide sufficient OIM performance (see, e.g., FIG. 6A, discussed below). However, surprisingly, such coupling circuits did not provide a SER to SNR ratio which matches simulations (see FIG. 6B, discussed below). The present invention overcomes these technical challenges with the coupling circuits disclosed by exemplary embodiments discussed herein which simulations indicate provide improved OIM optimization performance, particularly in relation to standard analog oscillator designs, and are easy to implement in practice. Unlike resistive coupling, the present invention provides a coupling signal injected into a target oscillator by a source oscillator which depends not only on the source oscillator’s waveform, but also on the target oscillator’s waveform, in a multiplicative manner. In other words, in embodiments of the present invention, multiplicative feedback from the target oscillator modifies the injection from the source oscillator. Because multiplication captures phase differences, the effect in the phase domain is an injection that is dependent on the phase difference between the target and source oscillators in contrast to resistive coupling, where the injection depends only on the source oscillator’s phase. 9 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) Overview of OIMs and OIM Models

[0053] An OIM’s overall model (the generalized Kuramoto model, explained below) isobtained by combining the resistive coupling equation with the oscillators’ nonlinear phase- sensitivity function (the PPV, see below). This combination results in the generalized- Kuramoto function Fc(·), the precise nature of which crucially determines how well the OIM solves optimization problems.

[0054] The present disclosure provides a circuit which is highly effective in enablingexcellent OIM optimization performance using standard analog oscillator designs, which do not perform well with resistive coupling.

[0055] A simple model of the phase dynamics of the oscillator network [29,8,14] is theKuramoto model, see Equation 2: ^ 1 ^ Δ^^(^) = ^^ ^    ^^^sin ^Δ^^(^) − Δ^^(^)^ (Equation 2)where N isKc is a constant, ∆ϕi(t) is the phase change of an ithoscillator due to the influences of the other oscillators via coupling, ∆ϕj(t) is the phase change of a jthoscillator due to the influences of the other oscillators via coupling, and ^^is the unaffected (or “base” / original) waveform. Equation 2 describes the phase change for each oscillator in the system, resulting in a coupled system of nonlinear differential equations.

[0056] This system is (as far as is known) impossible to solve analytically. However, a keyresult that underpins OIM has been established analytically

[0029] . The result is that Equation 3 represents a Lyapunov function E()

[0018] for the system (Equation 2): ^({Δ^^}) ≜ −^^∑^  ∑^  ^^^cos ^Δ^^ − Δ^^^ (Equation 3)4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) where ({Δ^^}) represents the phases of all the oscillators in the system (e.g., five), such that ^({Δ^^}) denotes that the Lyapunov function depends on the phases of all the oscillators (e.g., five oscillators).

[0057] The utility of a Lyapunov function is that it is non-increasing (it always decreases,or remains constant) with time if the phases ∆ϕi(t) obey (Equation 2). In other words, the Lyapunov function always decreases (eventually settling to a constant) as the oscillator system’s dynamics evolve. The minima of the Lyapunov function correspond to stable equilibrium points (DC solutions) of Equation 2, i.e., the coupled oscillator system’s phases settle to values that minimize (locally) the Lyapunov function. The Lyapunov function (3) looks very similar to the Ising Hamiltonian (1) if the coupling weights are the same as (or a scaled version of) the Ising problem’s Jij. The main difference is that the Ising Hamiltonian contains sisjterms, while the Lyapunov function has a cos(∆ϕi− ∆ϕj) term. But in embodiments where ∆ϕiand ∆ϕjis restricted to either 0 or π, with 0 defined as a spin value of 1 and π of −1, would equal sisj, and the oscillator network’s Lyapunov function would simply become a scaled version of the Ising Hamiltonian. In other words, in such embodiments, the oscillator network’s dynamics innately solve the Ising problem, at least to the extent that it would find a local minimum of the Ising Hamiltonian, so long as each oscillator’s phase is restricted to be either 0 or π.

[0058] In general, there is no guarantee that the oscillator’s phases will settle to either 0 orπ; indeed, phases tend settle to steady-state values that range continuously over [0, 2π]. In such cases, the oscillator network’s dynamics do not lead it to solutions that correspond to mimima of the Ising Hamiltonian. It is in this context that “binarizing” the oscillator’s phases using subharmonic injection locking (“SHIL”) becomes important, since it restricts each oscillator’s phase to 0 or π. Modifying the oscillator network with SHIL injection changes the Kuramoto equations to Equation 4: 11 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) ^ 1 ^ Δ^ ^(^) = ^^ ^    ^ sin ^Δ^ (^) − Δ^ (^)^ + ^^sin ^2Δ^ (^)^ (Equation 4)0^^ ^^^ ^ ^ ^^ ^ ^ ^^^Δ^^^^ ≜ −^^ ^    ^    ^^^cos ^Δ^^ − Δ^^^− ^^ ^    cos ^2Δ^^^ (Equation 5)the strength of the SYNC input for inducing SHIL. Since SHIL forces ∆ϕi to either 0 or π, this additional term represents simply an addition of a constant offset to the Ising Hamiltonian, leaving the minima unchanged. Thus, the coupled oscillator network with the addition of a 2nd harmonic SYNC input to induce SHIL naturally settles to phase solutions that locally minimize the Ising Hamiltonian. This result is currently the best-known theoretical basis for oscillator Ising machines.

[0060] The above result only guarantees that the oscillator network will find local minima.In embodiments, to get the network out of local minima and guide it towards the global minimum, additional steps are needed. An effective way to get the network out of local minima is to relax or remove the SYNC signal (which binarizes each oscillator’s phases through SHIL) and restore it again several times. Reducing SYNC allows the oscillators to drift away from 0 / π to continuous values; as SYNC is ramped up again, the system tends to find its way to minima that are lower than previous ones. Adding a moderate amount of noise to the system helps with the energy minimization process.

[0061] In embodiments, the sinusoidal functions in the above equations, originallyproposed by Kuramoto

[0016] , do not suffice for practical oscillators, which require the more general form: 12 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) ^ 1 ^ Δ^ ^(^) = ^^ ^    ^0^^ ^^^^^ ^Δ^^(^) − Δ^^(^)^ +^^^^ ^2Δ^^(^)^ (Equation 6)in (Equation 4) isthat sin(·) functions in Equation 4 are replaced by Fc(·) and Fs(·) in Equation 6. In embodiments, these functions can be extracted from, e.g., the detailed circuit description of an oscillator provided to circuit simulators for low-level electronic simulation. Extracting Fc() and Fs() from low-level circuit differential equations involves first finding an abstraction known as the PPV phase-domain model [8] for the oscillator, as shown in Equation 7: 1 ^ Δ^(^) = ^^⃗ ( ) ^⃗^ ^^^^0^+ Δ^ ^ ^ ⋅ ^ (^) (Equation 7)0where ^^⃗ (^) representsin response to which theoscillator’s phase changes by ∆ϕ(t), the quantity ^(⃗^), called the PPV, represents a “nonlinear sensitivity” of the oscillator’s phase response to input perturbations. It can be extracted from the detailed differential equations of any oscillator using numerical techniques [9].

[0063] In embodiments, once the result of Equation 7 is available, an averaging process [2]is used to extract Fc() and Fs(), with the coupling assumed to be resistive (i.e., the coupled signal into a target oscillator is proportional to the waveform of a source oscillator). Using these in Equation 6, OIM systems with many spins can be simulated quickly to assess Lyapunov / Hamiltonian minimization performance. Simulating a coupled system of PPV equations for the OIM network, as done to generate some of the results presented in this disclosure, provides more accurate results than Equation 6, though it requires somewhat greater computational effort. References [30,31] provide further information about OIMs and their underlying mathematics. Phase Changes in Oscillators 13 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0064] The output waveform of an unperturbed self-sustaining oscillator is a ^-periodicsignal at frequency ^0 = 1 ^, where ^ is the time period. Denote by ^(^) a 1-periodic signal that captures the shape of the waveform (but not its period ^ or frequency ^0, yet), then the oscillator's unperturbed waveform ^^(^), which is the same waveform squeezed / stretched along the time axis to have period ^, is given by Equation 8: ^(^ ^^) = ^ ^^^ = ^^^0^^ (Equation 8)

[0065] The argument of ^the phase of the oscillator. In general, the phase is denoted by ^(^); the nominal phase ^0^ of the unperturbed oscillator can be denoted by ^0(^).

[0066] When an oscillator is disturbed / perturbed by an external input or injection, e.g., dueto coupling with other oscillators, then its phase changes. The deviation of the phase from its nominal value ^0(^) will be denoted by Δ^(^), i.e., as shown in Equation 9: ^(^) = ^ ^^0^ + Δ^(^)^ (Equation 9)

[0067] What Δ^(^) will bethe oscillator is complicated, thoughit can be solvedusing, e.g., the PPV equation or a simplified version, the Adlerized equation, as described herein. The rate at which Δ^(^) is changes (increases or decreases) at a given moment, in effect, increases orinstantaneous frequency at that moment. In fact, the instantaneous frequency ^(^) is defined to be the time-derivative of ^(^), i.e., as given by Equation 10:^(^) ≜^ (^) = ^^ +(^) (Equation 10)where ^^is the nominal14 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0068] Note that if Δ^(^) is constant with time, then the instantaneous frequency remainsconstant, at the nominal frequency ^0 for all time; this is because a constant Δ^ simply shifts the nominal oscillator waveform forward or backward some fixed amount in time.

[0069] FIG. 4 is an example of a nominal waveform of an unperturbed oscillator(waveform 402) as compared with a nominal waveform of a perturbed oscillator (waveform 404) over a period of time.

[0070] If Δ^(^) changes uniformly with time, i.e., Δ^(^) = Δ^ ⋅ ^, then the instantaneousfrequency becomes ^0+ Δ^, ie, goes up or down (depending on the sign of Δ^ ) a constantamount from the nominal frequency ^0. If the frequency increases (Δ^ > 0), then theoscillator's nominal waveform (waveform 402) simply gets squeezed along the time axis,uniformly for all cycles. Conversely, if the frequency decreases (Δ^ < 0), then the oscillator'snominal waveform (waveform 402) gets stretched along the time axis, uniformly for all cycles. As illustrated, waveform 402 exhibits minimal stretching.

[0071] The actual behavior of Δ^(^) is usually not so simple—it can be a smoothlychanging waveform, itself. Roughly though, over any small interval of time, it will be either increasing or decreasing. If it is decreasing, the instantaneous frequency is lower than ^0, i.e., the nominal waveform stretches over that interval, as depicted by the firstof the waveform 404 in FIG.4. If, over some next interval, Δ^(^) starts increasing instead, then the nominal waveform squeezes over that period-this is depicted by the second cycle waveform 404. Over some next interval, if Δ^(^) stays more or less constant, then the nominal waveform remains the same (neither squeezes nor stretches, though there may be a constant time shift), as illustrated by the third cycle of waveform 404.

[0072] In embodiments, the Δ^^(^) of each coupled oscillator in a network will change insuch a way that they all settle to the exact same frequency, with only constant (nonchanging) 15 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) phase shifts between them. This is called injection locking, used in embodiments of the present invention. The Sampling Coupler

[0073] FIG. 1A provides an ideal ‘sharp’ plot for idealized oscillators. As depicted in FIG.1A, the value of Fc(∆ϕ1 − ∆ϕ2) is -1 between 0 and .5 (see, e.g., section 100), and 1 between .5 and 1 (see, e.g., section 102).

[0074] FIG. 1B provides example waveforms of oscillator of idealized oscillators. Inparticular, FIG.1B provides example waveforms of OSC1(corresponding to oscillator 200 of FIG.2) (e.g., waveform 110) and OSC2(corresponding to oscillator 202) (e.g., waveform 120and waveform 130). In embodiments, as shown in FIG.1B, OSC1 satisfies the formula ^^(^) =square(Δ^^ + ^(^), and OSC2 satisfies the formula ^^(^) = square(Δ^^ + ^(^). Waveform120the case where 0 < ∆ϕ1 − ∆ϕ2 < 0.5, and waveformto the case where 0.5 < ∆ϕ1 − ∆ϕ2 < 1. For ease of simplicity, the disclosure refers to 1 as the period of Fc(·) instead of 2π. As depicted in the graph, in accordance with embodiments of the present invention, waveforms 120 and 130 are shown as being sampled at the rising edges (e.g., rising edges 112A, 112B, 112C) of OSC1’s waveform (waveform 110). The samples (shown as bullets) directly provide the values of an ideal Fc(∆ϕ1− ∆ϕ2). The waveforms 120 and 130 in FIG.1 can be said to be unaffected, or unperturbed, as they are not impacted by the waveform 110 / oscillation of oscillator 200.

[0075] FIG. 2 depicts a circuit design in accordance with embodiments of the presentinvention. As shown in FIG. 1, in embodiments, flip-flop DFF1,2 (flip-flop 210) samples thewaveform of OSC2(oscillator 202) at the rising edges of OSC1(e.g., the output of oscillator200), resulting in Fc(∆ϕ1− ∆ϕ2). Similarly, in embodiments, samples of DFF2,1 (flip-flop 212)evaluate Fc(∆ϕ2−∆ϕ1). In embodiments, the flip-flops are a D flip-flop, also known as a data flip-flop. In embodiments, the D flip-flop receives the value of the D-input (e.g., signal x1(t) 16 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) and signal x2(t) at a defined portion of the clock cycle (as shown in FIG. 1, for example, a transition: the rising edge of the clock)). In embodiments, the output (Q) becomes the received value. While the term flip-flop is used, in embodiments, other circuits, such as edge-triggered dynamic D storage elements and d latches, to name a few, are used.

[0076] The precise nature of the 2π-periodic, typically non-sinusoidal, functions Fc(·) andFs(·) strongly influences hypothetical Hamiltonian minimization performance of the OIM. In contrast, the functions that emerge from practical oscillator designs conventionally do not tend to lead to high-quality minimization performance. Moreover, it is very difficult (indeed essentially impossible) to alter an oscillator design to achieve a desired Fc(·) or Fs(·) shape.

[0077] In embodiments, sampling couplers of the present invention use a special case ofmultiplicative coupling that overcomes the issues present in the prior art and results in a near- ideal square wave-shape for Fc(·) or Fs(·). See, e.g., FIGs.1A and FIG.2. FIG.1A depicts the square-wave shape of Fc(·). While the present disclosure largely discusses Fc(·), the principles and techniques discussed herein are applicable to Fs(·). Because of this shape, note that any term Fc(∆ϕi − ∆ϕj) in Equation 6 takes only 2 values, ±1. If ∆ϕi − ∆ϕj is restricted to a single period [−0.5, 0.5], Fc(∆ϕi − ∆ϕj) is −1 if ∆ϕi > ∆ϕj, and +1 if ∆ϕi < ∆ϕj. In other words, in embodiments, the value of the term depends only on whether one phase is ahead of, or behind, the other. The square-wave shape for Fc(·) is desirable because, empirically, it leads to very good optimization performance (e.g., see FIG. 6A); other shapes, e.g., those that emerge naturally for ring and other oscillators may lead to significant performance degradation.

[0078] ∆ϕi and ∆ϕj are the phases of oscillatory waveforms. If these are square as well, asshown FIG. 1B, then simply looking at (or sampling) the value of one waveform at the transition edge of the other suffices to determine if the phase of one is ahead of, or behind, the other. This is called early-late sampling; in embodiments, it achieves evaluating Fc(∆ϕi−∆ϕj). 17 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) A block-level circuit implementing this using D flip-flops is shown in FIG. 2; the outputs Q produce Fc(∆ϕi − ∆ϕj).

[0079] FIG.3 depicts sampling couplers coupling oscillators in accordance with exemplaryembodiments of the present invention. Sampling couplers are also referred to throughout as “spin coupling unit[s]”. In embodiments, the circuit depicted in FIG. 3 is implemented in an oscillating Ising machine. In particular, FIG.3 depicts a two oscillator (i.e., oscillator 200 and oscillator 202) OIM with sampling coupler 210 and sampling coupler 230.

[0080] In embodiments, as depicted in FIG. 3, a spin coupling unit for use with an Isingmachine comprises a first sampling circuit (e.g., flip-flop 210) operatively connected to a first oscillator (e.g., oscillator 200) and a second oscillator (e.g., oscillator 202) and a first current source (e.g., current source 240) operatively connected to the first sampling circuit. The outputs Q of the circuit produce Fc(∆ϕi− ∆ϕj). While not depicted, in embodiments, the sampling circuit is connected to either oscillator via a switch.

[0081] In embodiments, the first sampling circuit is a flip-flop.

[0082] In embodiments, the first sampling circuit receives as a first input (e.g., at the clock)a first waveform (e.g., square wave x1(t), the output of oscillator 200) from the first oscillator (e.g., oscillator 200). In embodiments, the first sampling circuit receives as a second input (e.g., at the input) a second waveform (e.g., square wave x2(t), the output of oscillator 202) from the second oscillator (e.g., oscillator 202). In embodiments, the first sampling circuit provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform. For example, in embodiments, output Q of flip-flop 210 is set to +1 or -1 depending on the waveform from oscillator 202.

[0083] In embodiments, the first current source (e.g., current source 240) is operativelyconnected to the first sampling circuit. In embodiments, the first current source receives as an input the first state value (e.g., output Q of flip-flop 210) from the first sampling circuit (e.g., 18 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) flip-flop 210). In embodiments, the first current source provides a first current (in FIG. 3, bin,1(t)) to the first oscillator based on the first state value and a first conductance parameter (e.g., J1,2). In embodiments, the first conductance parameter is programmable. In embodiments, the first conductance parameter is not programmable. In embodiments, the first conductance parameter (which may be a programmable first conductance parameter) is obtained from a register. In embodiments, the first conductance parameter (which may be a programmable first conductance parameter) is based on a first coupling weight of the first oscillator and the second oscillator.

[0084] In embodiments, OSC1 (oscillator 200) couples with OSC2 (oscillator 202) asfollows: the lower flip-flop 212 samples the output of oscillator 200 (e.g., square wave x1(t)) at the transition edge of the output of oscillator 202 (e.g., the transition of output x2(t)), the resulting sample value of +1 or -1 is then weighted by the Ising coupling weight (J2,1) (also referred to herein as the conductance parameter) and injected into OSC2(oscillator 202). In embodiments, the upper flip-flop 210 couples OSC2to OSC1in a similar manner (but the clock and input D of flip-flop 210 is reversed from the clock and input D of flip-flop 212). In embodiments, the same coupling weight applies to both couplings, which ensures that the Ising coupling model is valid. Accordingly, in embodiments, the Ising coupling weight J1,2 is the same as J2,1. FIGs.5A-5C depict an N oscillator Ising machine using sampling couplers couplingoscillators in accordance with exemplary embodiments of the present invention. As opposed to FIG. 3, which demonstrates the coupling of two oscillators, FIGs. 5A-5C depict a schematicwherein N oscillators are coupled to one another via a number coupling units, where N can be,essentially, any value (e.g., 5, 10, 15, 30, 100, 200, to give a few examples). As depicted in FIGs.5A-5C, N is at least 4. 19 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0086] In embodiments, the sampling couplers depicted in FIGs. 5A-5C operate similar tothe sampling couplers depicted in FIG. 3. In embodiments, each of the N oscillators may optionally be coupled or not coupled with one another. For example, while oscillator 200 is depicted as being coupled with oscillator 202, in embodiments, the oscillators may selectively be coupled with one another (e.g., a switch may disconnect the sampling couplers from the oscillators). In embodiments, for a given ithoscillator, there is no sampling coupler that couples the oscillator to itself.

[0087] In embodiments, as shown in FIGs. 5A-5C, an Ising machine includes a firstoscillator (e.g., oscillator 200) that generates a first waveform (e.g., waveform x1(t)), a second oscillator (e.g., oscillator 202) that generates a second waveform (e.g., waveform x2(t)), and a plurality of spin coupling units (for example, as described with respect to FIG. 3). In embodiments, the first oscillator and the second oscillator, when coupled, are interdependent. In embodiments, the first spin coupling unit includes a first sampling circuit (e.g., flip-flop 210) and a first current source (e.g., current source 240). In embodiments, the second spin coupling unit includes a second sampling circuit (e.g., flip-flop 212) and a second current source (e.g., current source 242).

[0088] In embodiments, the first sampling circuit is operatively connected to the firstoscillator and the second oscillator. In embodiments, the first sampling circuit receives as a first input (e.g., at the clock) a first waveform (e.g., square wave x1(t), the output of oscillator 200) from the first oscillator (e.g., oscillator 200). In embodiments, the first sampling circuit receives as a second input (e.g., at input D) a second waveform (e.g., square wave x2(t), the output of oscillator 202) from the second oscillator (e.g., oscillator 202). In embodiments, the first sampling circuit provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform. For example, in embodiments, output Q of flip- flop 210 is set to +1 or -1 depending on the waveform from oscillator 202. 20 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0089] In embodiments, the first current source (e.g., current source 240) is operativelyconnected to the first sampling circuit. In embodiments, the first current source receives as an input the first state value (e.g., output Q of flip-flop 210) from the first sampling circuit (e.g., flip-flop 210). In embodiments, the first current source provides a first current (e.g., a respective portion of bin,1(t)) to the first oscillator based on the first state value and a first conductance parameter (e.g., J1,2). In embodiments, the first conductance parameter is programmable. In embodiments, the first conductance parameter is not programmable. In embodiments, the first conductance parameter (which may be a programmable first conductance parameter) is obtained from a register. In embodiments, the first conductance parameter (which may be a programmable first conductance parameter) is based on a first coupling weight of the first oscillator and the second oscillator.

[0090] In embodiments, the second sampling circuit (e.g., flip-flop 212) is operativelyconnected to the first oscillator and the second oscillator. In embodiments, the second sampling circuit receives as a third input (e.g., at the clock) the second waveform (e.g., square wave x2(t), the output of oscillator 202) from the second oscillator (e.g., oscillator 202). In embodiments, the second sampling circuit receives as a fourth input (e.g., at input D) the first waveform (e.g., square wave x1(t), the output of oscillator 200) from the first oscillator (e.g., oscillator 200). In embodiments, the second sampling circuit provides as an output a second state value based on sampling of the first waveform at a transition of the second waveform. For example, output Q of flip-flop 212 may be set to +1 or -1 depending on the waveform from oscillator 200.

[0091] In embodiments, the second current source (e.g., current source 242) is operativelyconnected to the second sampling circuit. In embodiments, the second current source receives as an input the second state value (e.g., output Q of flip-flop 212) from the second sampling circuit (e.g., flip-flop 212). In embodiments, the second current source provides a second current (e.g., a respective portion of bin,2(t)) to the second oscillator based on the second state 21 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) value and a second conductance parameter (e.g., J2,1). In embodiments, the second conductance parameter is programmable. In embodiments, the second conductance parameter is not programmable. In embodiments, the second conductance parameter (which may be a programmable second conductance parameter) is obtained from a register. In embodiments, the second conductance parameter (which may be a programmable second conductance parameter) is based on a second coupling weight of the first oscillator and the second oscillator. In embodiments, the first oscillator and the second oscillator are interdependent.

[0092] As shown in FIGs. 5A-5C, in embodiments, an Ising machine includes a thirdoscillator (e.g., OSC3, labelled as oscillator 204) that generates a third waveform (e.g., x3(t)). In embodiments, the plurality of spin coupling units further includes a third spin coupling unit and a fourth spin coupling unit. In embodiments, the third spin coupling unit includes a third sampling circuit (e.g., flip-flop 220) operatively connected to the first oscillator (e.g., oscillator 200) and the third oscillator (e.g., oscillator 204). In embodiments, the third coupling unit includes a third current source (e.g., current source 250) operatively connected to the third sampling circuit. In embodiments, the fourth spin coupling unit includes a fourth sampling circuit (e.g., flip-flop 214) operatively connected to the first oscillator and the third oscillator. In embodiments, the fourth spin coupling unit includes a fourth current source (e.g., current source 244) operatively connected to the fourth sampling circuit.

[0093] In embodiments, the third spin coupling units and fourth spin coupling units operateto couple the third oscillator with another oscillator, for example, oscillator 200, and vice versa.

[0094] In embodiments, the third sampling circuit (e.g., sampling circuit 220) isoperatively connected to the first oscillator and the third oscillator. In embodiments, the third sampling circuit receives as a fifth input (e.g., at the clock) the first waveform (e.g., square wave x1(t), the output of oscillator 200) from the first oscillator (e.g., oscillator 200). In embodiments, the third sampling circuit receives as a sixth input (e.g., at input D) a third 22 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) waveform (e.g., square wave x3(t), the output of oscillator 204) from the third oscillator (e.g., oscillator 204). In embodiments, the third sampling circuit provides as an output a third state value based on sampling of the third waveform at a transition of the first waveform. For example, in embodiments, output Q of flip-flop 220 is set to +1 or -1 depending on the waveform from oscillator 204.

[0095] In embodiments, the third current source (e.g., current source 250) is operativelyconnected to the third sampling circuit. In embodiments, the third current source receives as an input the third state value (e.g., output Q of flip-flop 220) from the third sampling circuit (e.g., flip-flop 220). In embodiments, the third current source provides a third current (e.g., a respective portion of bin,1(t)) to the third oscillator based on the third state value and a third conductance parameter (e.g., J1,3). In embodiments, the third conductance parameter is programmable. In embodiments, the third conductance parameter is not programmable. In embodiments, the third conductance parameter (which may be a programmable third conductance parameter) is obtained from a register. In embodiments, the third conductance parameter (which may be a programmable third conductance parameter) is based on a third coupling weight of the first oscillator and the third oscillator.

[0096] In embodiments, the fourth sampling circuit (e.g., flip-flop 214) is operativelyconnected to the first oscillator and the third oscillator. In embodiments, the fourth sampling circuit receives as a seventh input (e.g., at the clock) the third waveform (e.g., square wave x3(t), the output of oscillator 204) from the third oscillator (e.g., oscillator 204). In embodiments, the third sampling circuit receives as an eighth input (e.g., at input D) the first waveform (e.g., square wave x1(t), the output of oscillator 200) from the first oscillator (e.g., oscillator 200). In embodiments, the fourth sampling circuit provides as an output a fourth state value based on sampling of the first waveform at a transition of the third waveform. For 23 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) example, output Q of flip-flop 214 may be set to +1 or -1 depending on the waveform from oscillator 200.

[0097] In embodiments, the fourth current source (e.g., current source 244) is operativelyconnected to the fourth sampling circuit. In embodiments, the fourth current source receives as an input the fourth state value (e.g., output Q of flip-flop 214) from the fourth sampling circuit (e.g., flip-flop 214). In embodiments, the fourth current source provides a fourth current (e.g., a respective portion of bin,3(t)) to the third oscillator based on the fourth state value and a fourth conductance parameter (e.g., J3,1). In embodiments, the fourth conductance parameter is programmable. In embodiments, the fourth conductance parameter is not programmable. In embodiments, the fourth conductance parameter (which may be a programmable fourth conductance parameter) is obtained from a register. In embodiments, the fourth conductance parameter (which may be a programmable fourth conductance parameter) is based on a fourth coupling weight of the first oscillator and the third oscillator. In embodiments, the first oscillator and the third oscillator are interdependent.

[0098] In embodiments, for example, where the second and third oscillators are notcoupled, the second oscillator and the third oscillator are independent (e.g., the spin of one is not directly related to the other, even if they are indirectly related).

[0099] In embodiments, the plurality of spin coupling units further includes a fifth spincoupling unit and a sixth spin coupling unit. In embodiments, the fifth spin coupling unit includes a fifth sampling circuit (e.g., sampling circuit 222) operatively connected to the second oscillator (e.g., oscillator 202) and the third oscillator (e.g., oscillator 204). In embodiments, the fifth spin coupling unit includes a fifth current source (e.g., current source 252) operatively connected to the fifth sampling circuit. In embodiments, the sixth spin coupling unit includes a sixth sampling circuit (e.g., sampling circuit 224) operatively connected to the second 24 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) oscillator and the third oscillator. In embodiments, the sixth spin coupling unit includes a sixth current source (e.g., current source 254).

[0100] In embodiments, the fifth spin and sixth spin coupling units operate to couple thethird oscillator with another oscillator, for example, oscillator 202.

[0101] In embodiments, the fifth sampling circuit (e.g., sampling circuit 222) is operativelyconnected to the second oscillator and the third oscillator. In embodiments, the fifth sampling circuit receives as an input (e.g., at the clock) the second waveform (e.g., square wave x2(t), the output of oscillator 202) from the second oscillator (e.g., oscillator 202). In embodiments, the fifth sampling circuit receives as a ninth input (e.g., at input D) the third waveform (e.g., square wave x3(t), the output of oscillator 204) from the third oscillator (e.g., oscillator 204). In embodiments, the fifth sampling circuit provides as an output a fifth state value based on sampling of the third waveform at a transition of the second waveform. For example, in embodiments, output Q of flip-flop 222 is set to +1 or -1 depending on the waveform from oscillator 204.

[0102] In embodiments, the fifth current source (e.g., current source 252) is operativelyconnected to the fifth sampling circuit. In embodiments, the fifth current source receives as an input the fifth state value (e.g., output Q of flip-flop 222) from the fifth sampling circuit (e.g., flip-flop 222). In embodiments, the fifth current source provides a fifth current (e.g., a respective portion of bin,2(t)) to the fifth oscillator based on the fifth state value and a fifth conductance parameter (e.g., J2,3). In embodiments, the fifth conductance parameter is programmable. In embodiments, the fifth conductance parameter is not programmable. In embodiments, the fifth conductance parameter (which may be a programmable fifth conductance parameter) is obtained from a register. In embodiments, the fifth conductance parameter (which may be a programmable fifth conductance parameter) is based on a fifth coupling weight of the second oscillator and the third oscillator. 25 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0103] In embodiments, the sixth sampling circuit (e.g., flip-flop 224) is operativelyconnected to the second oscillator and the third oscillator. In embodiments, the sixth sampling circuit receives as an eleventh input (e.g., at the clock) the third waveform (e.g., square wave x3(t), the output of oscillator 204) from the third oscillator (e.g., oscillator 204). In embodiments, the sixth sampling circuit receives as a twelfth input (e.g., at input D) the second waveform (e.g., square wave x2(t), the output of oscillator 202) from the second oscillator (e.g., oscillator 202). In embodiments, the sixth sampling circuit provides as an output a sixth state value based on sampling of the second waveform at a transition of the third waveform. For example, output Q of flip-flop 224 may be set to +1 or -1 depending on the waveform from oscillator 202.

[0104] In embodiments, the sixth current source (e.g., current source 254) is operativelyconnected to the sixth sampling circuit. In embodiments, the sixth current source receives as an input the sixth state value (e.g., output Q of flip-flop 224) from the sixth sampling circuit (e.g., flip-flop 224). In embodiments, the sixth current source provides a sixth current (e.g., a respective portion of bin,3(t)) to the third oscillator based on the sixth state value and a sixth conductance parameter (e.g., J3,2). In embodiments, the sixth conductance parameter is programmable. In embodiments, the sixth conductance parameter is not programmable. In embodiments, the sixth conductance parameter (which may be a programmable sixth conductance parameter) is obtained from a register. In embodiments, the sixth conductance parameter (which may be a programmable sixth conductance parameter) is based on a sixth coupling weight of the second oscillator and the third oscillator. In embodiments, the second oscillator and the third oscillator are interdependent.

[0105] Still referring to FIGs. 5A-5C, in embodiments, the currents used as input to theoscillators of the Ising machine are the sums of the currents provided by the current source. For example, in FIGs.5A-5C (assuming only a first, second and third oscillator, all coupled to 26 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) one another in accordance with embodiments of the present invention), bin,1(t) is the sum of the first current provided by the first current source 240 and the third current provided by the third current source 250, bin,2(t) is the sum of the second current provided by the second current source 242 and the fifth current provided by the fifth current source 252, and bin,3(t) is the sum of the fourth current provided by the fourth current source 244 and the sixth current provided by the sixth current source 254. In embodiments, oscillators may be selectively coupled and uncoupled by respectively connected and disconnecting a respective current source. For example, disconnecting oscillator 204 from current source 244, and disconnecting oscillator 200 from current source 250, will uncouple the oscillators. In embodiments, one or more switches are used to selectively couple and uncouples a given pair of oscillators.

[0106] In general, still referring to FIGs. 5A-5C, in embodiments, coupling from a sourceoscillator i to a target oscillator j is effected by sampling the source oscillator’s square-wave signal at the transition of the target oscillator’s square-wave output (e.g., on a rising waveform); then, weighting the ±1 sampled value by the Ising coupling weight Ji,j; and finally, injecting this value into the target oscillator. This can be extended to N coupled oscillators, depicted in FIGs.5A-5C.

[0107] Still referring to FIGs.5A-5C, in embodiments, the total input current ^^^,^(^) at the^thoscillator’s input node ^^,^^is given by Equation 11: ^=^ ^^ ^ ^ (Equation 11)

[0108] It can be easily shown that the Adlerized differential equation system of the ^oscillator OIM in FIGs.5A-5C takes the form (derivation omitted for brevity): 27 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) ^=^ ∀^, Δ^^̇(^) = ^0 ⋅^^ ^    ^^,^ ⋅ Fc ^Δ^^(^) − Δ^^(^)^ (Equation 12)

[0109] OIM system (see, e.g.,FIG.3), OSC1’s (oscillator 200) PPV equation takes the forms in Equations 13a and 13b: ^˙1 (^) = ^⃗ ^^0^^ + ^1(^)^^ ⋅ ^^⃗1(^), and (Equation 13a)^1(^) ≈ square ^^0^+ ^1(^)^ (Equation 13b)where it is assumed that that ^^(^) is the 'phase shift' of ^^^^, square (⋅) is a 1-periodic square wave, ^⃗(⋅) is the 1 -periodic vector of PPVs of ^^^^, and ^^⃗^(^) is the vector of inputs applied to ^^^^. Note that the only existing input is applied to the node ^^,^^(t) (which is distinguished from a resistive coupler, in that it is not the same as the output ^^(t) of ^^^^).

[0110] Thus, the above equation can be simplified to Equation 14a and 14b:^1̇(^) = ^^^ ^^0^^ + ^1(^)^^ ⋅ ^^^,1(^), and (Equation 14a)(Equation 14b)where ^^^(⋅)perturbation current of the ‘input’ node of ^^^^(e.g., ^^, inin FIG.3).

[0111] Similarly, the second oscillator is represented by Equations 15a and 15b, assumingfor present purposes that the second oscillator has identical attributes to the first oscillator and thus have identical PPVs at the input nodes: ^^̇(^) = ^^^ ^^^^^ + ^^(^)^^ ⋅ ^^^,^(^), and (Equation 15a)^^(^) ≈ square ^^^^ + ^^(^)^ (Equation 15b)

[0112] Variables are introducedderivation. ^(^),^^(^), and ^2(^) aredefined in Equation 16: ^(^) ≜ ^^^^^(^) ≜ ^^(^ + ^^(^))28 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) and ^^(^) ≜ ^^(^ + ^^(^))(Equation 16) where ^(^) is the ‘total phase’ of an unperturbed oscillator, and ^^(^) and ^^(^) are the (perturbed) ‘total phases’ of the two oscillators respectively.

[0113] ‘Relative phase’s’ of the two oscillators (i.e., (Δ^^(^) and Δ^^(^))are defined asby Equations 17a and 17b: Δ^^(^) ≜ ^^(^) − ^(^) = ^^(^ + ^^(^)) − ^^^ = ^^^^(^) (Equation 17a)Δ^2(^) ≜ ^^(^) − ^(^) = ^0(^+ ^^(^)) − ^0^ = ^0^^(^) (Equation 17b)

[0114] Substituting the above definitions into Equations 14a, 14b, 15a and 15b, oneobtains: Δ^1̇(^) = ^0⋅ ^^^ ^Δ^1(^) + ^(^)^ ⋅ ^^^,1(^) (Equation 18a)^1(^) ≈ square ^^^1(^) +^(^)^ (Equation 18b)Δ^2̇(^) = ^0⋅ ^^^ ^Δ^2(^) + ^(^)^ ⋅ ^^^,2(^) (Equation 18c)^2(^) ≈ square ^^^2(^) +^(^)^ (Equation 18d)

[0115] Turning to finding an expression for ^^^,^(^), the output of the flip-flop ^^^^,^ (flip-flop 210) is obtained, followed by the current through ^^^^,^,^(current source 242). This derivation begins with the assumption that the relative phases Δ^^(^) and Δ^^(^) vary ‘slowly’ with respect to the time period of the oscillator, i.e., that Δ^^(^) and Δ^^(^) are approximately constant over one time period of the oscillator. This assumption may not always hold true in embodiments of the present invention (as a wave form may vary over a period, particularly where an oscillator is densely coupled), but applying it, the output voltage of ^^^^,^is equalto Fc^Δ^1(^) − Δ^2(^)^. Additionally, it is assumed that the output Q of ^^^^,^ is available29 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) instantaneously as soon as its clock undergoes a rising edge transition. Note that this constraint is assumed merely to simplify the below derivation, and can be relaxed easily.

[0116] Immediately after the output of ^^^^,^ is available, i.e., immediately after the risingedge of ^^(^), it is assumed that the controlled current source ^^^^,^,^injects exactly one periodof a current waveform ^(^) scaled by ^^,^. F^(Δ^^(^) − Δ^^(^)). Here, ^(⋅) mustsatisfy the following properties: it must be 1-periodic, and the average of ^(^) multiplied by the ‘input’ node's PPV is positive. In other words, Equation 19 applies: ^=1 ^^ > 0, where ^^ ≜ ^    ^^^(^) ⋅ ^(^) ⋅ ^^ (Equation 19)

[0117] 200 (OSC1) is given byEquation 20: ^^^,1(^) = ^1,2 ⋅ Fc ^Δ^1(^) − Δ^2(^)^ ⋅^ ^Δ^1(^) + ^(^)^ (Equation 20)

[0118] is phase shifted by Δ^^(^)since current injected by ^^^^,1,2is always closely aligned with the rising edge of ^^^^'swaveform (i.e., ^1(^)) which is approximately equal to square ^Δ^1(^) + ^(^)^ as stated in(Equation 18).

[0119] Substituting ^^^,1(^) in (Equation 18), one obtains Equation 21:Δ^1̇(^) = ^0⋅ ^^^ + ^(^)^ ⋅ ^1,2 ⋅ Fc ^Δ^1(^) − Δ^2(^)^ ⋅ ^ ^Δ^1(^) + ^(^)^

[0120] Since it was assumed that Δ^^(^) and Δ^^(^) vary ‘slowly’ over one period of theoscillator, the above equation can be approximated as 30 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) Δ^^̇(^) =^^^ ⋅⋅ − ⋅ ^ ⋅ ^^where ^= ^^ ⋅ ^^ ⋅ ^^,^ ⋅ F^(Δ^^(^) − Δ^^(^)), (Equation 22)where Equation 19 and the fact that ^^^(⋅) and ^(⋅) are 1-periodic are used.

[0121] The process can be repeated for OSC2, giving the following equation pair:Δ^^̇(^) = ^^ ⋅ ^^ ⋅ ^^,^ ⋅ F^(Δ^^(^) − Δ^^(^)) (Equation 23a)Δ^^̇(^) = ^^ ⋅ ^^ ⋅ ^^,^ ⋅ F^(Δ^^(^) − Δ^^(^)) (Equation 23b)

[0122] The key result is that this scheme depicted in FIGs. 5A-5C using the samplingcoupler is also captured by the generalized Kuramoto model (Equation 6), with an Fc(·) function that is identical to the square wave one in FIGs. 1A, 1B. Crucially, the square-wave ^ shape of Fc(·) essentially does not depend on the oscillator’s PPV ^⃗ in Equation 7. This feature is in stark contrast to the resistive / proportional coupling scenario, where the shape of Fc(·) is entirely determined by that of ^(⃗·) leading to sub-optimal Hamiltonian minimization performance. Results (Multi-User – Multi-Input Multiple-Output Detection)

[0123] Simulations in MU MIMO detection serves to demonstrate the promise of thepresent invention.

[0124] One of the key features of 5G / 6G technologies is the extensive use of Multi- InputMulti-Output (MIMO) techniques to increase the bandwidth available in the system [7]. 31 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) Consider a MIMO network with Nttransmitting antennae and Nrreceiving antennae, assuming that Nr ≥ Nt, with all the transmitters transmitting electromagnetic signals simultaneously on the same spectrum [7,20]. For simplicity, it is also assumed that the transmitters use Binary Phase Shift Keying (BPSK)

[0020] ; i.e., each transmitted symbol carries one bit of information. The transmitted signals interfere with each other at each receiving antenna, and are corrupted by Additive White Gaussian Noise (AWGN); the vector of received signals (^⃗) can be modelled as given in Equation 24: ^⃗ = ^^^⃗+ ^⃗ (Equation 24)where ^^is the channel matrix, ^⃗ is the vector of transmitted symbols (which are either +1 or -1 due to BPSK modulation), and ^⃗ is a vector of AWGN. The receiver's task is to ‘decode’ the symbols sent by the transmitters (i.e., find ^⃗) given the vector of received signals (^⃗). The most optimal solution is the vector ^∗⃗ which minimizes the norm∥^⃗−^^^∥⃗. The above is a brief description of the Maximum-Likelihood Multi-Input Multi-Output (MU-MIMO) detection problem.

[0125] Assuming that the system uses BPSK modulation, it can be shown that a MU-MIMO detection problem with Nttransmitters can be mapped to an Ising problem with Nt+1 spins

[0015] ; if Quadrature Phase Shift Keying (QPSK)

[0022] is utilized instead, the equivalent Ising problem will have 2Nt+ 1 spins

[0015] .

[0126] To evaluate the sampling coupler, a dataset of 550,000 MU-MIMO detectionproblems was used. The dataset is divided into 11 chunks (of 50,000 problems each) where the Signal to Noise Ratio (SNR) is constant. For a given SNR, there are 1,000 different channels; for each channel there are 50 MU-MIMO detection problems. Note that the channels for the dataset were generated as described in reference

[0011] . It assumes that closely spaced users’ 32 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) channels are more correlated than far away users; this assumption is considered to be more realistic than conventional independent identically distributed Rayleigh fading models

[0020] .

[0127] Consider a MU-MIMO detection problem with Nt transmitting antennae and Nrreceiving antennae. As stated, the above problem can be mapped to an Ising problem with Nt+1 spins. The most optimal solution can be found by brute force, where the minimum Hamiltonian is found by evaluating it at every possible combination of the Nt + 1 spins. However, this approach becomes infeasible as Ntgrows since the number of possible combinations grows exponentially with Nt. Note that there exist faster alternatives to brute force known as Sphere Decoders [19,10,13] that also find the optimal solution.

[0128] In practice, heuristics such as Zero Forcing (ZF) and Linear Minimum Mean SquareError (LMMSE) Estimators are used instead

[0020] . Such algorithms are faster than optimal decoders such as brute force or Sphere Decoders, which allow MIMO techniques to be employed in high bandwidth systems such as 5G / 6G [7]. However, the downside is that the above heuristics are less accurate than the optimal decoder; bit errors in MU-MIMO detection may reduce the effective bandwidth available to the system [7].

[0129] As reported in

[0025] , OIMs perform near-optimally on practically relevant MU-MIMO detection problems.

[0130] FIG. 6A (reproduced from

[0025] with permission) is a plot of Symbol Error Rates(SERs) against Signal to Noise Ratio (SNR) of various detectors. ZF (610) stands for a zero forcing detection technique, LMMSE stands for Linear Minimum Mean Square Estimator (620), Sphere (630) stands for a sphere decoder, ‘Gen-K FE’ (640) stands for Gen-K Forward Euler [5], ‘Gen-K event’ (650) is the event based solver reported in

[0025] , and ‘OIM emulator’ (660) is the digital emulator from

[0023] . ‘Gen-K FE’ can be considered as the idealized OIM with sharp-Fc(·) that is relevant for the discussion in this subsection. ‘Gen-K’ FE denotes the 33 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) idealized OIM with sharp FFc(·).As shown, OIMs are indistinguishable from the Sphere Decoder. Moreover, since oscillators in integrated circuits can operate in the GHz range, high detection throughputs from OIMs can be expected.

[0131] FIG. 6B is a plot of Symbol Error Rates (SERs) against Signal to Noise Ratio (SNR)for a resistive coupler OIM (line 652), a sampling coupler OIM (line 662), the idealized OIM (line 642) and other heuristics. As shown, the sampling coupler OIM matches the SER of the Sphere decoder and outperforms the resistive coupler OIM.

[0132] FIGs. 7A-7C provides schematic representations of different types of coupling unitsthat can be used with OIMs.

[0133] FIG. 7A provides an example of a resistive coupling via a resistor 706 betweenoscillators 702 and 704 present in the prior art. One downside of the such coupling is that it may distort actual coupling values and provide a less-than-ideal phase response.

[0134] FIG. 7B provides an example of active “one-way resistor” coupling. Such couplingmay include two one-way buffered gm units 712 and 714 coupling oscillators 704 and 702. While this eliminates source resistance problems, the oscillator phase response may still be less-than- ideal.

[0135] FIG. 7C provides a schematic of multiplicative coupling in accordance withembodiments of the present invention. In embodiments, by multiplying the injection by a target oscillator’s signal (i.e., at multipliers 722 and 724), the issues present in other types of sampling couplers are overcome. As discussed with respect to FIG. 6A and FIG. 6B, simulations of multiplicative coupling provides improved phase response.

[0136] While the present invention and validation of the sampling couple constitutes a keyadvance for enabling genuinely analog IC realizations of high-performance OIMs, it will be understood that the techniques and principles of the present invention may be implemented for other types of Ising machines that do not use oscillators. 34 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136)

[0137] Now that embodiments of the present invention have been shown and described indetail, various modifications and improvements thereon can become readily apparent to those skilled in the art. Accordingly, the exemplary embodiments of the present invention, as set forth above, are intended to be illustrative, not limiting. The spirit and scope of the present invention is to be construed broadly. 35 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) References 1 A. Marandi, Z. Wang, K. Takata, R. L. Byer and Y. Yamamoto: Network of time- multiplexed optical parametric oscillators as a coherent Ising machine. Nature Pho- tonics 8(12), 937–942 (2014). 2 A. Neogy and J. Roychowdhury: Analysis and Design of Sub-harmonically Injection Locked Oscillators. In: Proc. IEEE DATE (Mar 2012). 3 Adler, R.: A study of locking phenomena in oscillators. Proceedings of the I.R.E. and Waves and Electrons 34, 351–357 (June 1946). 4 Afoakwa, R., Zhang, Y., Vengalam, U.K.R., Ignjatovic, Z., Huang, M.: BRIM: Bistable Resistively-Coupled Ising Machine. In: 2021 IEEE International Sympo- sium on High-Performance Computer Architecture (HPCA). pp.749–760. IEEE (2021). 5 Atkinson, K.E.: An Introduction to Numerical Analysis. John Wiley & Sons, New York, second edn. (1989). 6 Bhansali, P., Roychowdhury, J.: Gen-Adler: The generalized Adler’s equation for injection locking analysis in oscillators. In: Proc. IEEE ASP-DAC. pp. 522–227 (January 2009). 7 Bjornson, E., Hoydis, J., Sanguinetti, L.: Massive MIMO networks: Spectral, en- ergy, and hardware efficiency. Foundations and Trends® in Signal Processing 11(3- 4), 154– 655 (2017). 8 Demir, A., Mehrotra, A., Roychowdhury, J.: Phase Noise in Oscillators: a Unifying Theory and Numerical Methods for Characterization. IEEE Trans. Ckts. Syst.–I: Regular Papers 47(5), 655–674 (May 2000). 9 Demir, A., Roychowdhury, J.: A Reliable and Efficient Procedure for Oscillator PPV Computation, with Phase Noise Macromodelling Applications. IEEE Trans. CAD pp. 188–197 (February 2003). 36 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) 10 Fincke, U., Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Mathematics of Computation 44(170), 463– 471 (1985). 11 Goutay, M., Aoudia, F.A., Hoydis, J.: Deep Hypernetwork-based MIMO Detec- tion. In: 21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC). pp.1–5. IEEE (2020). 12 Hamerly, R., et al.: Experimental investigation of performance differences be- tween coherent Ising machines and a quantum annealer. Science Advances 5(5) (2019). https: / / doi.org / 10.1126 / sciadv.aau0823,https: / / advances.sciencemag.org / content / 5 / 5 / e aau0823 13 Hassibi, B., Vikalo, H.: On the expected complexity of sphere decoding. In: Conf. Record of Thirty-Fifth Asilomar Conf. on Signals, Systems and Computers. vol.2, pp.1051–1055. IEEE (2001). 14 J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort and R. Spigler: The Kuramoto Model: A Simple Paradigm for Synchronization Phenomena. Reviews of Modern Physics 77(1), 137 (2005). 15 Kim, M., Venturelli, D., Jamieson, K.: Leveraging quantum annealing for large MIMO processing in centralized radio access networks. In: Proceedings of the ACM Special Interest Group on Data Communication. pp.241–255. ACM (2019). 16 Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics. pp. 420– 422. Springer (1975). 17 17. Lucas, A.: Ising formulations of many NP problems. Frontiers in Physics 2, 5 (2014). 18 Lyapunov, A.M.: The General Problem of the Stability of Motion. International Journal 37 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) of Control 55(3), 531–534 (1992). 19 Pohst, M.: On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications. SIGSAM Bull.15(1), 37–44 (FEB 1981). 20 Proakis, J.G., Salehi, M.: Digital Communications. McGraw Hill, Boston, fifth edn. (2008). 21 Roychowdhury, J., Wabnig, J., Srinath, K.P.: Performance of Oscillator Ising Ma- chines on Realistic MU-MIMO Decoding Problems. Research Square preprint (Version 1) (22 September 2021). 22 Saha, D., Birdsall, T.G.: Quadrature-quadrature phase-shift keying. IEEE Transactions on Communications 37(5), 437–448 (1989). 23 Sreedhara, S., Roychowdhury, J., Wabnig, J., Srinath, K.P.: Digital Emulation of Oscillator Ising Machines. In: Proc. IEEE DATE. pp.1–2 (2023). 24 Sreedhara, S., Roychowdhury, J., Wabnig, J., Srinath, P.K.: MU-MIMO Detection Using Oscillator Ising Machines. In: Proc. ICCAD. pp.1–9 (2023). 25 Sreedhara, S., Roychowdhury, J., Wabnig, J., Srinath, P.K.: MU-MIMO detection using oscillator Ising machines. In: IEEE / ACM Int. Conf. on Computer Aided Design, ICCAD 2023, San Francisco, CA, USA, October 28 - Nov. 2, 2023. pp. 1–9. IEEE (2023). 26 T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu and others: A Coherent Ising machine for 2000- node Optimization Problems. Science 354(6312), 603–606 (2016). 27 T. Wang and J. Roychowdhury: OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. In: arXiv:1903.07163 (2019). 28 T. Wang and J. Roychowdhury: OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. In: Proc. UCNC. LNCS sublibrary: Theoretical 38 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) computer science and general issues, Springer (June 2019), preprint available at arXiv:1903.07163 [cs.ET]. 29 Wang, T., Roychowdhury, J.: Oscillator-based Ising Machine. arXiv:1709.08102 (2017). 30 Wang, T., Roychowdhury, J.: OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. In: Proc. UCNC. LNCS sublibrary: Theoretical computer science and general issues, Springer (June 2019), preprint available at arXiv:1903.07163 [cs.ET]. 31 Wang, T., Wu, L., Nobel, P., Roychowdhury, J.: Solving combinatorial optimisation problems using oscillator based Ising machines. Natural Computing pp. 1–20 (April 2021). 32 Z. Bian, F. Chudak, R. Israel, B. Lackey, W. G. Macready and A. Roy: Discrete optimization using quantum annealing on sparse Ising models. Frontiers in Physics 2, 56 (2014). 39 4883-0360-9781v.1

Claims

Attorney Docket # 00495-0023 (B2024-136) CLAIMS: What is claimed is:

1. A spin coupling unit for use with an Ising machine comprising:(a) a first sampling circuit operatively connected to a first oscillator and asecond oscillator that: (i) receives as a first input a first waveform from the first oscillator;(ii) receives as a second input a second waveform from the secondoscillator; and (iii) provides as an output a first state value based on sampling of the second waveform at a transition of the first waveform; and (b) a first current source operatively connected to the first sampling circuitthat: (i) receives as an input the first state value from the first sampling circuit;and (ii) provides a first current to the first oscillator based on the first statevalue and a programmable first conductance parameter.

2. The spin coupling unit of claim 1, wherein the first sampling circuit is a flip-flop.

3. The spin coupling unit of claim 1, wherein the programmable firstconductance parameter is obtained from a register.

4. The spin coupling unit of claim 1, wherein the programmable firstconductance parameter is based on a first coupling weight of the first oscillator and the second oscillator.

5. An Ising machine comprising:(a) a first oscillator that generates a first waveform;40 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (b) a second oscillator that generates a second waveform; and(c) a plurality of spin coupling units, including a first spin coupling unit and asecond spin coupling unit, wherein the first spin coupling unit comprises: (i) a first sampling circuit operatively connected to the first oscillator andthe second oscillator that: (a) receives as a first input the first waveform from the first oscillator;(b) receives as a second input the second waveform from the secondoscillator; and (c) provides as an output a first state value based on sampling of thesecond waveform at a transition of the first waveform; and (ii) a first current source operatively connected to the first samplingcircuit that: (a) receives as an input the first state value from the first samplingcircuit; and (b) provides a first current to the first oscillator based on the first statevalue and a programmable first conductance parameter; and wherein the second spin coupling unit comprises: (i) a second sampling circuit operatively connected to the first oscillatorand the second oscillator that: (a) receives as a third input the second waveform from the secondoscillator; (b) receives as a fourth input the first waveform from the firstoscillator; and (c) provides as an output a second state value based on sampling of thefirst waveform at a transition of the second waveform; and 41 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (ii) a second current source operatively connected to the second samplingcircuit that: (a) obtains the second state value from the second sampling circuit;and (b) provides a second current to the second oscillator based on thesecond state value and a programmable second conductance parameter, wherein the first oscillator and the second oscillator are interdependent.

6. The Ising machine of claim 5, wherein the Ising machine further comprises:(d) a third oscillator that generates a third waveform;and wherein the plurality of spin coupling units further includes a third spin coupling unit and a fourth spin coupling unit.

7. The Ising machine of claim 6, wherein the third spin coupling unit comprises:(i) a third sampling circuit operatively connected to the first oscillator andthe third oscillator that: (a) receives as a fifth input the first waveform from the first oscillator;(b) receives as a sixth input the third waveform from the thirdoscillator; and (c) provides as an output a third state value based on sampling of thethird waveform at a transition of the first waveform; and (ii) a third current source operatively connected to the third samplingcircuit that: (c) receives as an input the third state value from the third samplingcircuit; and 42 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (d) provides a third current to the first oscillator based on the thirdstate value and a programmable third conductance parameter; and wherein the fourth spin coupling unit comprises: (i) a fourth sampling circuit operatively connected to the first oscillatorand the third oscillator that: (a) receives as a seventh input the third waveform from the thirdoscillator; (b) receives as an eighth input the first waveform from the firstoscillator; and (c) provides as an output a fourth state value based on sampling of thefirst waveform at a transition of the third waveform; and (ii) a fourth current source operatively connected to the fourth samplingcircuit that: (a) obtains the fourth state value from the fourth sampling circuit; and(b) provides a fourth current to the third oscillator based on the fourthstate value and a programmable fourth conductance parameter, wherein the first oscillator and the third oscillator are interdependent.

8. The Ising machine of claim 7, wherein the second oscillator and the thirdoscillator are independent.

9. The Ising machine of claim 7, wherein the plurality of spin coupling unitsfurther includes a fifth spin coupling unit and a sixth coupling unit, wherein the fifth spin coupling unit comprises: (i) a fifth sampling circuit operatively connected to the second oscillatorand the third oscillator that: 43 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (a) receives as a ninth input the second waveform from the secondoscillator; (b) receives as a tenth input the third waveform from the thirdoscillator; and (c) provides as an output a fifth state value based on sampling of thethird waveform at a transition of the second waveform; and (ii) a fifth current source operatively connected to the fifth samplingcircuit that: (a) receives as an input the fifth state value from the fifth samplingcircuit; and (b) provides a fifth current to the second oscillator based on the fifthstate value and a programmable fifth conductance parameter; and wherein the sixth coupling unit comprises: (i) a sixth sampling circuit operatively connected to the second oscillatorand the third oscillator that: (a) receives as an eleventh input the third waveform from the thirdoscillator; (b) receives as a twelfth input the first waveform from the secondoscillator; and (c) provides as an output a sixth state value based on sampling of thesecond waveform at a transition of the third waveform; and (ii) a sixth current source operatively connected to the sixth samplingcircuit that: (c) obtains the sixth state value from the sixth sampling circuit; and44 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) (d) provides a sixth current to the third oscillator based on the sixthstate value and a programmable sixth conductance parameter, wherein the second oscillator and the third oscillator are interdependent.

10. The Ising machine of any of claims 5, 7, 8, or 9, wherein the first samplingcircuit is a flip-flop.

11. The Ising machine of any of claims 5, 7, 8, or 9, wherein the second samplingcircuit is a flip-flop.

12. The Ising machine of any of claims 7-9, wherein the third sampling circuit is aflip-flop.

13. The Ising machine of any of claims 7-9, wherein the fourth sampling circuit isa flip-flop.

14. The Ising machine of claim 9, wherein the fifth sampling circuit is a flip-flop.

15. The Ising machine of claim 9, wherein the sixth sampling circuit is a flip-flop.

16. The Ising machine of any of claims 5, 7, 8, or 9, wherein the programmablefirst conductance parameter is obtained from a register.

17. The Ising machine of any of claims 5, 7, 8, or 9, wherein the programmablesecond conductance parameter is obtained from a register.

18. The Ising machine of any of claims 7-9, wherein the programmable thirdconductance parameter is obtained from a register.

19. The Ising machine of any of claims 7-9, wherein the programmable fourthconductance parameter is obtained from a register.

20. The Ising machine of claim 9, wherein the programmable fifth conductanceparameter is obtained from a register.

21. The Ising machine of claim 9, wherein the programmable sixth conductanceparameter is obtained from a register. 45 4883-0360-9781v.1Attorney Docket # 00495-0023 (B2024-136) 22. The Ising machine of any of claims 5, 7, 8, or 9, wherein the programmablefirst conductance parameter is based on a coupling weight of the first oscillator and the second oscillator.

23. The Ising machine of any of claims 5, 7, 8, or 9, wherein the programmablesecond conductance parameter is based on a coupling weight of the first oscillator and the second oscillator.

24. The Ising machine of any of claims 7-9, wherein the programmable thirdconductance parameter is based on a coupling weight of the first oscillator and the third oscillator.

25. The Ising machine of any of claims 7-9, wherein the programmable fourthconductance parameter is based on a coupling weight of the first oscillator and the third oscillator.

26. The Ising machine of claim 9, wherein the programmable fifth conductanceparameter is based on a coupling weight of the second oscillator and the third oscillator.

27. The Ising machine of claim 9, wherein the programmable sixth conductanceparameter is based on a coupling weight of the second oscillator and the third oscillator. 46 4883-0360-9781v.1

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