用复数相位模拟二元变量，显著提升组合优化求解精度。
Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization
- 将二元变量建模为复平面上的波状相位，平滑非凸能量景观。
- 在160×160的QUBO问题中噪声达0.25时仍实现零误差，稀疏编码在σ=0.15下完美恢复。
- 该隐式正则化机制可移植至传统实值框架，适合高精度组合优化场景。
我们提出一种受物理启发的连续松弛框架，显著提升NP难组合优化问题的求解效果，包括无约束二次二值优化（QUBO）、二值稀疏编码及带已知解的Ising模型。通过将离散二元变量参数化为复单位圆上的波状状态，天然平滑了高度非凸的能量景观。研究发现，以复数相位表示二元变量会激发一种隐式正则化机制，促进收敛至离散状态。提取该机制后，即使在标准实值优化框架中显式使用，也能带来显著性能提升。实验表明，该正则化使基态收敛率远超传统实值方法：在160×160大规模QUBO任务中，当噪声水平σ=0.25时仍实现零误差；在欠定稀疏编码中，σ=0.15时达到完美恢复。该求解器在11个严格设计的带已知解基准测试中成功恢复8个精确基态配置，验证其强大鲁棒性。
原文摘要 · Abstract (English)
We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.
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