AI 中文总结
该研究针对约束优化问题，提出几何感知的多项式时间量子近似方案，引入FPRASq与Heavy-Hitter QAOA算法，在IBM Eagle r3处理器上的硬件实验验证了其性能优于QOptlib参考结果。
AI 中文摘要
带噪声的量子采样器何时能生成具备性能保证的端到端多项式时间优化算法？基于约束增强型量子近似优化算法（Constraint-Enhanced QAOA）的有限深度与有限采样次数保证，我们证明最优集上的逆多项式理想概率，结合独立采样、多项式时间可行性修复与评分，可生成精确命中的 fully polynomial randomized approximation scheme（FPRASq）。该保证在实例依赖窗口内可抵御设备噪声。对于有效电路深度线性于层数与问题规模乘积的情况，若保留理想最优质量的逆深度分数，采样次数复杂度将增加问题规模的一次幂。超出该窗口时，确定性修复可保证可行性，且当诱导目标函数膨胀受控制时，能提供实例依赖的近似保证。所得NP-HQ算法符合Chen-Cotler-Huang-Li预言机模型。对于任意NP难核可容许的承诺族，若多项式时间经典采样器能重现其逆多项式最优重叠，将意味着NP包含于BPP，即便采用相同修复且可完美访问约束结构，故分离性源于采样分布的生成。我们进一步提出Heavy-Hitter QAOA，其在保留这些条件保证的同时，将保留的候选集与经典后处理成本降低问题规模的一次幂。在IBM Eagle r3处理器上的硬件实验覆盖了含多达100个逻辑变量的实例，且匹配或优于所有测试的QOptlib参考路径。
英文摘要
When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.
