Coupled-Field Dynamical Relaxation for QUBO and Ising Optimizations

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.