转移的量子近似优化算法参数记住惩罚尺度：约束量子优化的λ共振定律
Transferred QAOA Parameters Remember the Penalty Scale: A $λ$-Resonance Law for Constrained Quantum Optimization
AI总结：
研究量子近似优化算法参数转移问题，发现当约束编码为惩罚时，训练角度能记住惩罚权重λ。证明了可行子空间概率质量F(λ)是特定三角多项式，得出转移可行性的共振特性等结论，通过实验证实，还重新解释了基于惩罚的QAOA失败模式。
AI中文摘要：
在小实例上训练一次量子近似优化算法（QAOA）的变分角度并在大实例上复用，即参数转移，是突破精确模拟限制的标准方法。现有文献几乎完全通过结构相似性解释其成功。我们发现一个新的独立轴，当约束编码为惩罚时，它控制转移：训练角度记住其训练实例的惩罚权重λ。对于任何具有整数值违反计数的标量成本函数，我们证明在深度为p的任意固定QAOA角度(β,γ)下，可行子空间上的概率质量F(λ)是λ的有限实三角多项式，其角频率位于由训练的γ生成的整数格上。由此立即得出三个结果：转移可行性在部署惩罚与训练惩罚匹配处是一个共振峰值；共振宽度按1/(v_max∑k|γ_k|)缩放，所以低|γ|角度集系统性地更具可转移性；曲线在间距2π/γ_k处呈现复兴峰值。我们通过对20量子比特多用户资源分配QUBO的精确态矢实验证实了这三个预测。该定理与角度如何获得无关，适用于任何整数惩罚QUBO，将基于惩罚的QAOA广泛报道的失败模式重新解释为确定性、可预测的相位干扰而非能量调谐问题。
英文摘要：
Training the variational angles of the Quantum Approximate Optimization Algorithm once on a small instance and reusing them on larger ones, known as parameter transfer, is the standard route past the exact-simulation wall. Existing literature explains its success almost entirely through structural similarity. We identify a new, independent axis that governs transfer whenever constraints are encoded as penalties: the trained angles memorize the penalty weight $λ$ of their training instance. For any scalarized cost function with an integer-valued violation count, we prove that at arbitrary fixed QAOA angles $(β,γ)$ of depth $p$, the probability mass $F(λ)$ on the feasible subspace is a finite real trigonometric polynomial in $λ$ whose angular frequencies lie on an integer lattice generated by the trained $γ$'s. Three consequences follow immediately: transfer feasibility is a resonance peaked where the deployment penalty matches the training penalty; the resonance width scales as $1/(v_{max}\sum_k|γ_k|)$, so low-$|γ|$ angle sets are systematically more transferable; and the curve exhibits revival peaks at spacings $2π/γ_k$. We confirm all three predictions by exact statevector experiments on a 20-qubit multi-user resource-allocation QUBO. The theorem is independent of how the angles were obtained and applies to any integer-penalty QUBO, recasting a widely reported failure mode of penalty-based QAOA as deterministic, predictable phase interference rather than an energetic tuning problem.
