General scaling law for noise thresholds in deterministic analog combinatorial optimization solvers

Establish whether the noise-tolerance thresholds observed in deterministic analog combinatorial optimization solvers universally obey scaling laws similar to those found in the chaotic amplitude control Ising solver and the analog k-SAT solver; specifically, determine if the hard noise-thresholds for solution-finding capability exhibit an approximate polynomial (algebraic) dependence on the problem size N across diverse solver architectures, problem classes, and noise models.

Background

The paper analyzes the impact of noise on two representative deterministic analog combinatorial optimization solvers: the chaotic amplitude control (CAC) algorithm for Ising problems and a continuous-time dynamical system for k-SAT. For both, the authors numerically identify hard noise thresholds that demarcate regimes of successful solution-finding from regimes where solutions cannot be found.

For the CAC Ising solver, the noise thresholds decrease with problem size following an approximate algebraic (polynomial) scaling, robust across dense and sparse interaction matrices and under both white and colored noise. For the k-SAT solver, a hard noise threshold also exhibits polynomial scaling with N; additionally, a soft noise threshold appears at an N-independent noise level that filters a growing fraction of instances as N increases.

Based on these findings, the authors explicitly conjecture that similar scaling laws should hold more generally for deterministic analog combinatorial optimization solvers beyond the specific algorithms and settings studied.