Coupled-Field Dynamical Relaxation for QUBO and Ising Optimizations

Abstract

This work presents a classical theoretical framework in which combinatorial optimization emerges from the nonlinear relaxation of coupled real-valued phase fields governed by a global Lyapunov energy functional. Each computational element (CF-bit) evolves in a bistable periodic potential while pairwise interactions encode problem-specific couplings, enabling gradient-descent minimization of QUBO and Ising objective functions. The key contribution is an explicit global energy functional from which all dynamics are derived, guaranteeing monotonic energy descent under damping. This distinguishes the approach from several existing oscillator-based Ising architectures where the governing dynamics contain non-gradient terms and an explicit global Lyapunov functional has not been derived in their standard formulations. Numerical simulations on instances up to 20 bits demonstrate deterministic phase-locking convergence, with optional transient noise improving the exploration of rugged landscapes. While limited in scale and not overcoming NP-hardness, this work provides a conceptual framework showing how discrete optimization can emerge from continuous classical dynamics with a mathematically transparent energy structure.

1. Introduction

Quadratic Unconstrained Binary Optimization (QUBO) problems arise in logistics, medical imaging, portfolio optimization, robotics, scheduling, and machine learning. They also appear in many fields including combinatorial optimization [1,2], statistical physics models [3], and discrete optimization formulations [4,5]. The standard QUBO objective seeks to minimize $f\left(x\right)=x^T Q x$ over binary variables $\mathrm{x}\in {\left\{0, 1\right\}}^N$, which can be equivalently expressed as an Ising Hamiltonian $H=\sum _{ij}{J}_{ij}{s}_i{s}_j+\sum _i{h}_i{s}_i$ with spins s_i ∈ {−1, +1}. Since general QUBO/Ising optimization is NP-hard, practical systems rely on specialized heuristics or hardware accelerators.

Quantum annealers (D-Wave) and coherent optical Ising machines implement energy minimization through quantum dynamics or nonlinear photonics [6,7,8,9,10], but require cryogenic conditions or ultrastable lasers, limiting accessibility and scalability. This work introduces a fully classical, deterministic architecture operating at room temperature with no quantum coherence requirement, yet retaining the essential property needed for optimization: a multi-element system that naturally evolves toward energy minima.

Recent work has also explored classical analog optimization systems based on nonlinear dynamics. These include simulated bifurcation solvers, oscillator-based Ising machines, and analog electronic implementations of combinatorial optimization. In particular, simulated bifurcation methods introduced by Goto and collaborators as well as subsequent extensions have demonstrated efficient classical dynamical approaches for solving Ising-type problems using nonlinear oscillators and second-order dynamics. These developments highlight the broader trend toward dynamical system-based optimization and motivate the search for architectures with mathematically transparent energy structures and convergence properties.

Novel Contributions and Scope

This work contributes a theoretical framework with four distinguishing features that are not present in existing oscillator-based Ising machines:

Unified Lyapunov Energy Functional: CF-bit dynamics are derived from a single explicit energy functional $E_{total}=\sum \sum _{i}V\left({\phi }_i\right)+{\sum }_{ij}{J}_{ij}cos\left({\phi }_i-{\phi }_j\right)$, enabling analytical convergence guarantees. Existing optical and electronic oscillator machines do not provide a closed-form global Lyapunov function.

Binary Encoding in the Potential Itself: The binary state arises from minima of the intrinsic potential $V\left(\phi \right)=-\cos \left(2\phi \right)$, not from external thresholding or gain saturation. This enables a clean algebraic mapping from QUBO coefficients to coupling strengths.

Deterministic Lyapunov Dynamics: The CF system guarantees a monotonic energy descent dE/dt ≤ 0 under damping, unlike coherent Ising machines where gain saturation and noise injection are intrinsic. Stochastic noise is not required for stability or final convergence but may optionally enhance exploration.

Direct Algebraic QUBO Mapping: The transformation from QUBO matrix Q to CF couplings {J_ij, h_i} is derived analytically from the energy functional, not heuristically fit as in some optical implementations.

Scope and Limitations: This work is presented as a theoretical and conceptual contribution emphasizing dynamical structure and physical interpretability. Numerical demonstrations are limited to instances with N ≤ 20 bits. The framework does not overcome NP-hardness and may converge to local minima for frustrated systems. Larger-scale behavior, physical coupling topology costs, and competitive benchmarking against state-of-the-art solvers remain important directions for future research but lie beyond the scope of the present study.

2. Coupled-Fields Theory and the CF-Bit

2.1. Derivation from Energy Functional

A CF-bit is a classical nonlinear oscillator with an internal phase state ${\phi }_i$ evolving in a bistable periodic potential. We begin from a Lyapunov-like energy functional for a single CF-bit:

$$E_{single}=V\left(\phi \right)+(m/2){(d\phi /dt)}^2$$

(1)

where the potential is defined as V(φ) = −cos(2φ), a periodic double-well potential with minima at φ = 0 and φ = π corresponding to the two binary states of the CF-bit, and m represents effective inertia inherited from underlying field dynamics.

The choice of the cosine potential is not unique; any periodic bistable potential with equivalent minima structure would produce qualitatively similar binary dynamics.

The equation of motion follows from the Euler-Lagrange formulation with linear damping:

$$m d^2\phi /d{t}^2+\alpha d\phi /dt=-dV/d\phi$$

(2)

Substituting V(φ) = −cos(2φ) yields:

$$m d^2\phi /d{t}^2+\alpha d\phi /dt=-2 sin\left(2\phi \right)$$

(3)

This second-order differential equation describes a damped nonlinear oscillator. For the binary case (n = 2 in the general formulation $V\left(\phi \right)=-cos\left(n\phi \right))$, the potential (Figure 1) exhibits exactly two stable minima at φ = 0 and φ = π, corresponding to logical states 0 and 1 (see Figure 1).

A conceptual implementation of a CF-bit oscillator is shown in Figure 2.

2.2. Network-Level Lyapunov Stability

For a network of N coupled CF-bits, the total energy functional is:

$$E_{total}={\Sigma }_i V({\phi }_i)+\left({\displaystyle \frac{m}{2}}\right){\Sigma }_i {(d{\phi }_i/dt)}^2+{\sum }_{i<j}{J}_{ij}cos\left({\phi }_i-{\phi }_j\right)+{\Sigma }_i{h}_i{\phi }_i$$

(4)

This serves as a global Lyapunov function. Computing its time derivative:

$$dE/dt={\Sigma }_i\left[\right(dV/d{\phi }_i+\partial E_{coupling}/\partial {\phi }_i\left)\right(d{\phi }_i/dt)+m(d^2{\phi }_i/d{t}^2\left)\right(d{\phi }_i/dt\left)\right]$$

(5)

Using the equation of motion:

$$m d^2{\phi }_i/d{t}^2=-(dV/d{\phi }_i)-(\partial E_{coupling}/\partial {\phi }_i)-\alpha (d{\phi }_i/dt)$$

(6)

we obtain:

dE/dt = −α Σ_i (dφ_i/dt)² ≤ 0

(7)

This confirms monotonic energy descent for α > 0, guaranteeing convergence to fixed points. This Lyapunov property distinguishes the CF architecture from coherent Ising machines (CIMs) and oscillator-based Ising machines (OIMs), where dynamics include non-gradient terms (gain saturation, noise) and no global Lyapunov functional has been established [10,11,12].

3. Interaction Between CF-Bits and Ising Mapping

CF-bits interact through coupling terms depending on phase differences:

$$E_{coupling}={\sum }_{i<j}{J}_{ij}cos\left({\phi }_i-{\phi }_j\right)$$

(8)

Near the stable phases φ_i ∈ {0, π}, expanding to second order around these minima and defining binary variables $b_i$ via ${\phi }_i\approx b_i \pi$ (with $b_i\in \left\{0, 1\right\}$), the coupling energy reduces to:

$$E_{coupling}\approx {\Sigma }_{i<j}{J}_{ij}cos(b_i \pi -b_j \pi )={\Sigma }_{i<j}{J_{ij}\left(-1\right)}^{(b_i+b_j)}$$

(9)

Using the Ising transformation $s_i=2b_i$ − 1 (mapping {0, 1} to {−1, +1}), this becomes:

$$E_{Ising}={\Sigma }_{i<j}{s}_i{s}_j+{\Sigma }_i{h}_i{s}_i$$

(10)

demonstrating that the CF network energy naturally reduces to an Ising Hamiltonian. Local fields h_i can be encoded through bias terms added to the individual CF-bit potentials. This provides the formal justification for mapping QUBO coefficients directly to CF coupling strengths.

4. Mapping QUBO to CF Networks

A QUBO problem seeks to minimize $f\left(x\right)=x^T Q x$ with $x\in {\left\{0, 1\right\}}^N$. Using the transformation $s_i=2x_i-1$, the QUBO objective becomes equivalent to the Ising Hamiltonian:

$$H_{Ising}=\left({\displaystyle \frac{1}{4}}\right){\Sigma }_i {\Sigma }_j Q_{ij} s_i s_j+\left({\displaystyle \frac{1}{2}}\right){\Sigma }_i\left({\Sigma }_j Q_{ij}\right)s_i+constant$$

(11)

Identifying the coupling matrix and local fields:

$$\begin{gathered} J_{ij}={\displaystyle \frac{Q_{ij}}{4}} \left(i\ne j\right) \\ {h}_i=\left({\displaystyle \frac{1}{2}}\right){\Sigma }_j Q_{ij} \end{gathered}$$

(12)

The diagonal $elements Q_{ii}$ represent linear contributions in the QUBO objective and therefore map directly to the local bias fields $h_i$. In practice the Q matrix is first symmetrized as $Q_{sym}={\displaystyle \frac{Q+Q^T}{2}}$, after which the diagonal entries are incorporated into the local field terms. This ensures that the resulting CF energy functional preserves the exact objective value of the original QUBO formulation, making the mapping lossless.

This mapping is lossless: every QUBO instance produces a unique CF coupling matrix whose minima correspond exactly to QUBO minima. Since each CF-bit has exactly two stable attractors (φ = 0, π), the transformation guarantees one-to-one correspondence between the QUBO solutions and the CF equilibrium configurations.

Explicit 3 × 3 Example

To illustrate the mapping concretely, consider the 3 × 3 QUBO matrix Q:

$$Q=\left[\begin{array}{ccc}2& -1& 0\\ -1& 3& -2\\ 0& -2& 1\end{array}\right]$$

(13)

The QUBO cost function is $f\left(x\right)=x^T Q x$. Applying the binary-to-Ising transformation $s_i=2x_i-1$ and expanding yields an Ising energy with couplings and fields determined analytically from Q entries. For this specific matrix:

$$\begin{gathered} J_{12}=-1/4, J_{13}=0, J_{23}=-1/2 \\ {h}_1=1/2, h_2=0, h_3=-1/2 \end{gathered}$$

These Ising parameters map directly to the CF network energy functional, defining the physical couplings and bias coefficients. The minima of E_CF are in one-to-one correspondence with the minima of the original QUBO function f(x).

This mapping is illustrated schematically in Figure 3, where QUBO coefficients are translated into CF coupling strengths and local bias fields.

5. Dynamics and Convergence

The time evolution of each CF-bit follows:

$$m d^2{\phi }_i/d{t}^2+\alpha d{\phi }_i/dt=-2sin\left(2{\phi }_i\right)+{\Sigma }_j{J}_{ij}sin\left({\phi }_i-{\phi }_j\right)-h_i$$

(14)

For numerical simulations, the network was integrated using a fourth-order Runge–Kutta scheme with fixed time step dt = 0.01. The damping parameter α = 1.1 was chosen to ensure a monotonic energy decrease while allowing sufficient transient oscillation to traverse shallow minima. Convergence was defined as $\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\displaystyle \frac{d{\phi }_i}{dt}}|<\epsilon$, with $\epsilon ={10}^{-3}$.

Key dynamical features:

Parallel evolution: All CF-bits update simultaneously, providing massive analog parallelism.

Energy descent: Damping guarantees dE/dt ≤ 0, ensuring relaxation toward energy minima.

Deterministic operation: Noise is not intrinsic to stability or late-time phase locking. However, optional transient stochastic perturbations can enhance the exploration of rugged landscapes.

Multi-minima exploration: Large initial oscillations enable the system to bypass shallow minima during early evolution.

Sensitivity tests were performed by varying the standard deviation of the initial velocity distribution in the range σ = 0.01–0.2. The qualitative convergence behavior and final success rates were found to be largely insensitive to this parameter within this range. Larger values increase early exploration of the energy landscape but do not significantly affect the final convergence properties once damping dominates the dynamics.

6. Simulation Results

We present numerical demonstrations on three problem classes. All simulations used the parameters listed in

Table 1. Standard simulation parameters used throughout this study.

Parameter	Value
Integration method	4th-order Runge–Kutta
Time step (dt)	0.01
Damping coefficient (α)	1.1
Initial phase range	[0, 2π] uniform random
Initial velocity distribution	Gaussian, σ = 0.05
Convergence threshold (ε)	10−3
QUBO coefficient normalization	[−1, 1]

below.

6.1. 3-Bit Illustrative Example

A 3-bit QUBO with Q given in Equation (12) was encoded into a 3-CF-bit system. The system evolved deterministically from random initial conditions to a final configuration φ_final ≈ (π, π, 0), corresponding to binary solution x = (1, 1, 0). This matches one of the global minima found by brute-force enumeration, validating the mapping.

Figure 4 shows the energy convergence (top panel) and phase trajectories (bottom panel). The CF energy decreases monotonically from its initial value to a stable minimum within approximately 150 iterations, demonstrating the Lyapunov descent property.

6.2. Random 20-Bit QUBO Instances

To probe larger systems, twenty dense 20-bit QUBO instances were generated with random symmetric matrices Q. For each instance, the corresponding Ising parameters were computed and mapped to a 20-CF-bit network. The system was integrated until convergence or until a maximum of 500 iterations.

Results are summarized in

Table 2. Performance on 20-bit random dense QUBO instances. CF Final Energy shows the energy at convergence; “Global Optimum?” indicates whether this matched the exact minimum found by exhaustive search.

Instance	Optimal Energy	CF Final Energy	Global Optimum?	Iterations
1	−23	−23	Yes	164
2	−17	−16	No	188
3	−21	−21	Yes	152
4	−19	−19	Yes	171
5	−25	−24	No	193
6	−18	−18	Yes	142
7	−22	−21	No	201
8	−20	−20	Yes	167
9	−24	−24	Yes	158
10	−16	−15	No	207
11	−22	−22	Yes	149
12	−23	−22	No	195
13	−20	−20	Yes	161
14	−21	−21	Yes	173
15	−18	−17	No	185
16	−26	−26	Yes	159
17	−19	−19	Yes	147
18	−25	−24	No	203
19	−22	−22	Yes	166
20	−21	−21	Yes	172

Summary: Global optimum reached in 15/20 instances (75%). Near-optimal solutions (within one energy level) in the remaining 5/20 (25%). Mean convergence: 171 iterations. This behavior is consistent with analog Ising-type dynamics exhibiting monotonic relaxation with occasional local minima.

. The CF network reached the global optimum (verified by brute-force enumeration) in 15 out of 20 cases (75% success rate). The remaining five instances converged to local minima one energy level above the global minimum. Mean convergence time was 171 iterations (σ = 22 iterations).

This performance is consistent with other analog Ising-type solvers and demonstrates that CF dynamics provide an effective heuristic for mid-size QUBO instances. The deterministic relaxation properties ensure stable convergence, while the continuous phase space allows exploration of the energy landscape during transient evolution.

6.3. Structured MaxCut Instances

Structured QUBO instances derived from MaxCut problems on graphs with 10–20 vertices were tested using a custom-developed CF simulation framework. For these problems, the CF network exhibited rapid initial energy descent, followed by stable phase-locking. Final bit assignments corresponded to cuts matching optimal MaxCut values (for small graphs verified by exhaustive search) or high-quality solutions comparable to established heuristics for larger graphs.

Increasing initial oscillation amplitude or adjusting the damping parameter improved escape from shallow local minima, analogous to temperature schedule tuning in simulated annealing. These structured problems demonstrate that CF dynamics successfully reproduce behavior expected from physical Ising hardware on both random and structured optimization instances.

6.4. Computational Cost and Scaling

Each integration step requires O(N²) operations for dense coupling matrices (evaluating all pairwise interactions) or O(N·k) for sparse instances with k neighbors per CF-bit. For the 20-bit problems that were tested, typical convergence required 150–200 iterations, giving a total computational cost of approximately 30,000–40,000 pairwise evaluations.

These costs reflect numerical integration of continuous dynamics rather than optimized algorithmic implementations. The comparisons shown in

Table 3. Qualitative comparison on 20-bit QUBO instances. Runtimes reflect software simulations on standard CPU, not projected hardware execution times.

Method	Global Optima Found	Mean Runtime (ms)	Notes
CF Network	15/20 (75%)	~45	Numerical integration
Simulated Annealing	16/20 (80%)	~120	CPU implementation
Random Local Search	8/20 (40%)	~15	Greedy descent
Exhaustive Search	20/20 (100%)	~850,000	Exponential scaling

should be interpreted as illustrative rather than definitive benchmarks. The simulations were performed using research implementations intended to demonstrate the dynamical behavior of the CF system. A comprehensive benchmarking study against optimized QUBO solvers (e.g., advanced simulated annealing implementations, tabu search, or recent dynamical optimization methods) would require standardized benchmark suites and carefully optimized implementations, which lies beyond the scope of this initial conceptual study.

The numerical experiments presented here focus on relatively small problem instances (N ≤ 20) because they allow exact verification of the global optimum through exhaustive search. This ensures that the correctness of the QUBO-to-CF mapping and the convergence behavior of the dynamical system can be validated unambiguously. The purpose of these experiments is therefore to illustrate the dynamical mechanism rather than to provide large-scale benchmarking. Systematic studies on larger instances (N ≥ 100) are an important direction for future work and will require optimized implementations of the CF dynamics.

The CF approach achieves solution quality comparable to simulated annealing with faster convergence (measured in integration steps), though this comparison reflects research code implementations rather than optimized production solvers. Systematic scaling analysis beyond N = 20 and comparison with state-of-the-art optimization libraries remain important directions for future work.

As system size increases, three factors influence scalability: (i) the computational cost of evaluating pairwise couplings, (ii) the convergence time of the dynamical system, and (iii) the probability of reaching global minima in increasingly complex energy landscapes. For dense QUBO matrices the interaction computation scales as O(N²), while sparse coupling structures reduce this cost to O(Nk) where k is the average connectivity. In hardware implementations based on physical oscillator networks, these interactions may be realized in parallel, potentially mitigating the effective computational scaling relative to purely digital simulations.

7. Comparison with Alternative Optimization Approaches

7.1. Relation to Existing Oscillator-Based Ising Machines

Several prior approaches use nonlinear phase dynamics for optimization, including LC oscillator networks [11], MEMS-based Ising solvers, and coherent optical machines [8,9,10]. The CF architecture differs in three key respects:

CF dynamics are derived from an explicit global Lyapunov functional $E_{total}$, enabling analytical convergence guarantees and transparent energy-descent interpretation.

The CF potential directly embeds binary states through intrinsic bistability V(φ) = −cos(2φ), yielding a clean algebraic mapping to the Ising Hamiltonian without external thresholding.

The architecture operates deterministically without requiring gain saturation, stochastic noise injection, or optical coherence for stability or final convergence.

These distinctions clarify the theoretical contribution: the CF model provides a classical, room-temperature framework with mathematically transparent dynamics and direct QUBO-to-coupling mapping.

From a dynamical system perspective, the CF architecture belongs to a broader class of oscillator-based optimization methods in which the solution of a combinatorial problem emerges from the collective dynamics of coupled nonlinear elements. These include optical coherent Ising machines, electronic oscillator networks, and simulated bifurcation systems. The distinguishing feature of the CF formulation is that the entire dynamics are derived from an explicit global Lyapunov functional, ensuring that the system evolves through deterministic energy descent under damping. This property provides a mathematically transparent interpretation of the optimization process as gradient relaxation on a well-defined energy landscape.

7.2. Non-Existence of Lyapunov Functional in CIM/OIM Architectures

A central distinction between the CF architecture and coherent Ising machines (CIMs) or oscillator-based Ising machines (OIMs) is the existence of a global Lyapunov functional governing the dynamics.

CIM dynamics based on degenerate optical parametric oscillators (DOPOs) evolve according to equations that include gain-saturation and pump-depletion terms [8]. These terms cannot be expressed as gradients of any global potential function E. As noted in [8], CIM dynamics are fundamentally non-gradient, relying on stochastic amplitude bifurcation to reach low-energy states. Similarly, OIM systems based on coupled LC oscillators include phase-normalization dynamics and amplitude-regulating terms that violate gradient structure [10,11].

The consequence is that CIM and OIM trajectories may temporarily increase effective Ising energy, and that convergence relies on noise, bifurcation, or gain-control mechanisms rather than guaranteed monotonic relaxation. In contrast, the CF architecture possesses an explicit Lyapunov functional whose gradient fully determines the dynamics, ensuring dE/dt ≤ 0 at all times under damping.

This structural property, which to our knowledge is not demonstrated for optical or electronic oscillator Ising machines in their standard formulations, provides the CF framework with mathematically transparent convergence behavior and enables design principles based on energy-landscape engineering.

7.3. Relation to Classical Neural Energy Models

Early neural energy models such as Hopfield networks [13] and Cohen-Grossberg systems [14] demonstrated that gradient-based dynamics can perform combinatorial optimization by descending a Lyapunov energy landscape. More recent developments in classical and dynamical optimization frameworks, including coupled-field formulations, bifurcation-based methods, and oscillator-based approaches, further extend these ideas [15,16,17,18,19]. The CF model shares this gradient-descent principle but differs in implementation:

Hopfield networks use threshold units with discrete states, while CF-bits are continuous phase oscillators that naturally explore the energy landscape during transient evolution.

The CF architecture derives binary computation from intrinsic potential bistability V(φ) = −cos(2φ), whereas Hopfield networks impose discrete thresholds externally.

CF dynamics include second-order (inertial) terms, enabling richer transient behavior and natural frequency-based coupling mechanisms.

The CF framework can thus be viewed as a classical oscillator-based realization of energy-descent optimization, bridging concepts from neural computation and analog physical systems.

8. Convergence Properties and Role of Noise

8.1. Deterministic Lyapunov Convergence

Under purely deterministic damped dynamics (no stochastic perturbations), the CF system exhibits monotonic energy descent $dE/dt=-\alpha {\Sigma }_i{(d{\phi }_i/dt)}^2\le 0$. This guarantees convergence to a fixed point corresponding to a local (or global) minimum of the energy landscape. However, as with all gradient-descent methods on non-convex landscapes, deterministic dynamics may become trapped in the nearest local minimum.

8.2. Noise-Assisted Exploration

To enhance the exploration of rugged energy landscapes, optional transient stochastic perturbations can be introduced:

$$m d^2{\phi }_i/d{t}^2+\alpha d{\phi }_i/dt=-dE/d{\phi }_i+\sigma (t)·{\xi }_i(t)$$

(15)

where ${\xi }_i\left(t\right)$ is white Gaussian noise and σ(t) is a time-dependent noise amplitude. The noise amplitude is annealed according to $\sigma \left(t\right)={\sigma }_0 exp(-t/\tau )$, starting at ${\sigma }_0\approx 0.1–0.5$ and decaying with the time constant τ chosen to match the typical relaxation time of the system.

From a dynamical system perspective, the CF framework is related to synchronization models such as the Kuramoto oscillator network, in which phase differences between oscillators determine collective behavior. However, unlike the Kuramoto model, the CF architecture includes an explicit bistable potential and a global Lyapunov functional governing the entire network dynamics. The stochastic extension introduced above is mathematically analogous to Langevin dynamics in which deterministic relaxation is supplemented by thermal noise, enabling barrier crossing in non-convex energy landscapes.

During the exploratory regime (large σ), the noise temporarily breaks Lyapunov monotonicity, enabling barrier crossing and escape from shallow local minima. As σ → 0 in the late-time regime, the system transitions back to deterministic convergence, ensuring stable final phase-locking. This approach combines the stability guarantees of deterministic Lyapunov dynamics with enhanced global search capability.

Figure 5 illustrates this behavior, while the noise-assisted case Figure 6 (fluctuating curve): the deterministic case (smooth curve) shows strict energy monotonicity, while the noise-assisted case Figure 6 (fluctuating curve), exhibits temporary energy increases during exploration followed by convergence to a deeper minimum as noise is reduced.

8.3. Comparison with Coherent Ising Machines

In coherent Ising machines (CIMs), stochastic noise from quantum vacuum fluctuations and amplifier noise is intrinsic to the system dynamics and required for optimization performance [8]. The CIM phase dynamics can be written schematically as dφ_i/dt = F_i({φ_j}) + ξ_i(t), where ξ_i represents unavoidable stochastic processes. No global Lyapunov functional exists and convergence is fundamentally probabilistic.

In contrast, the CF model separates deterministic convergence (guaranteed by the Lyapunov functional) from optional noise-assisted exploration (controllable and transient). This design choice enables stable late-time phase locking while retaining the ability to enhance global search when needed. The noise amplitude and annealing schedule are external control parameters rather than intrinsic physical constraints.

9. Applications

The CF architecture is applicable to combinatorial optimization problems that can be formulated as QUBO or Ising instances, including:

Logistics and routing (vehicle routing, traveling salesman);

Portfolio optimization and financial risk minimization;

Scheduling and resource allocation;

Graph problems (MaxCut, graph coloring, maximum independent set);

Machine learning (training binary neural networks, feature selection);

VLSI design and circuit layout optimization.

The specific advantage of the CF approach lies in its deterministic convergence guarantees, room-temperature operation, and transparent energy-landscape structure, which may facilitate analysis and design of problem-specific coupling architectures.

10. Discussion and Future Directions

This work introduces a coupled-field dynamical framework in which discrete optimization emerges from the continuous relaxation of phase fields governed by an explicit global energy functional. The key theoretical contributions are (i) derivation of dynamics from a single Lyapunov function enabling analytical convergence analysis, (ii) natural algebraic mapping from QUBO to CF couplings, and (iii) deterministic operation without intrinsic noise requirements.

Important directions for future research include:

Systematic scaling analysis up to N = 100–1000 bits to characterize performance degradation and identify problem classes where CF dynamics remain effective.

Computational complexity characterization: theoretical analysis of iteration count vs. problem size and coupling density.

Physical implementation feasibility studies examining electronic, mechanical, or photonic oscillator realizations and their engineering constraints.

Hybrid classical-CF algorithms combining CF relaxation with discrete search heuristics.

Analysis of coupling topology costs for dense vs. sparse QUBO matrices and the development of approximation schemes for hardware-constrained connectivity.

Application-specific architectures optimized for particular problem classes (e.g., MaxCut, SAT, scheduling).

10.1. Limitations

The CF architecture does not overcome NP-hardness and may converge to local minima for frustrated systems with conflicting constraints. Numerical demonstrations are limited to N ≤ 20 bits; behavior at larger scales requires further investigation. Physical implementation of fully dense coupling matrices scales as O(N²) in component count, potentially limiting scalability without sparse approximations or hierarchical architectures.

These limitations are characteristic of analog Ising-type solvers and define important trade-offs between problem size, coupling density, solution quality, and hardware complexity.

The present work focuses primarily on the theoretical structure of the CF dynamical framework rather than large-scale benchmarking. Systematic scalability studies on instances with N ≥ 100, including time-to-solution (TTS) metrics and comparison with modern solvers, constitute an important direction for future work and will require optimized implementations beyond the exploratory simulations presented here.

10.2. Potential Physical Implementations

Several physical platforms could potentially realize the CF architecture. One possibility is networks of nonlinear electronic oscillators, where LC resonators combined with nonlinear reactive elements generate the bistable phase potential required for CF-bit operation. Coupling between oscillators could be implemented using programmable analog coupling circuits or digitally controlled feedback networks.

Another possible implementation pathway involves MEMS or nano-mechanical resonators with controllable coupling interactions, which naturally exhibit phase dynamics similar to those described by the CF equations. Photonic oscillator networks and parametric resonators represent additional candidates for realizing the required nonlinear dynamics.

Practical implementations must address engineering constraints such as coupling precision, noise levels, oscillator frequency stability, and scalability of interconnection networks. These considerations are analogous to challenges encountered in other analog Ising-type hardware architectures and will be an important topic for future experimental studies.

11. Conclusions

We have introduced a coupled-field dynamical framework in which discrete optimization problems are represented as energy landscapes governing the relaxation of interacting phase fields. By constructing an explicit global energy functional and deriving overdamped gradient dynamics, the framework admits a Lyapunov description guaranteeing monotonic energy descent and stable late-time phase locking.

This structure provides a physically transparent alternative to oscillator-based Ising approaches whose dynamics include non-gradient terms and for which no global Lyapunov functional has been established. The analysis clarifies that stochastic perturbations are not intrinsic to stability or final convergence but can be introduced transiently to enhance exploration of complex landscapes.

Numerical results on small instances (up to 20 bits) illustrate the dynamical mechanism and validate the QUBO-to-CF mapping. While these demonstrations focus on conceptual validation rather than large-scale benchmarking, they show how discrete optimization behavior can emerge naturally from continuous nonlinear dynamics governed by a global energy functional.

While the present study is theoretical in nature and limited in scale, it highlights how discrete computational behavior can arise from continuous nonlinear dynamics governed by a single energy functional. Extensions to larger problem instances, systematic performance studies, and physical realizations constitute important directions for future work.