AI 中文总结
针对最优多项式相交问题，在满足特定条件下，通过相干成员预言机访问，利用确定性完全列表解码等方法，给出用于傅里叶定义分布$P_u$的有界误差多项式时间量子采样器，实现对不同极限速率的改进及高概率满意解。
AI 中文摘要
最优多项式相交（OPI）是一个结构化优化问题，解码量子干涉测量（DQI）对其有一个由半圆定律支配的满意度保证。Sun和Wootters表明，对于素域上的平衡OPI，傅里叶定义的分布$P_u$从极限速率$0.6225$起给出严格的最坏情况改进，从速率$0.7496$起给出渐近完美解，并询问$P_u$是否能被高效采样。对于满足其指数条件且严格低于对偶约翰逊半径的里德 - 所罗门OPI参数，我们回答了这个问题。在相干成员预言机访问下，我们给出了一个用于$P_u$的有界误差多项式时间量子采样器。理想电路在成功时精确采样$P_u$，有限精度实现可达到任何规定的逆多项式总变差误差。因此，每个固定极限速率$0.6225\leq r<1$都比DQI半圆值有严格的最坏情况改进，每个极限速率$r\geq3/4$都以高概率有满意度为$1 - o(1)$的解。该算法使用确定性完全列表解码相干地对每个综合征类中所有低权重误差的幅度求和。完全里德 - 所罗门列表解码和Sun - Wootters分母估计使列表大小和后选择开销为多项式。在并行和独立工作中，Horinaga和Yamakawa在素幂域上获得了最坏情况OPI算法，并在每个严格高于$3/4$的固定速率下实现了精确满意度。
英文摘要
Optimal Polynomial Intersection (OPI) is a structured optimization problem for which Decoded Quantum Interferometry (DQI) attains a satisfaction guarantee governed by the semicircle law. Sun and Wootters recently showed that, for balanced OPI over prime fields, a Fourier-defined distribution $P_u$ gives a strict worst-case improvement from limiting rate $0.6225$ onward and asymptotically perfect solutions from rate $0.7496$ onward, and asked whether $P_u$ can be sampled efficiently. We answer this question for Reed--Solomon OPI parameters satisfying their exponent condition strictly below the dual Johnson radius. Under coherent membership-oracle access, we give a bounded-error polynomial-time quantum sampler for $P_u$. The ideal circuit samples $P_u$ exactly conditioned on success, while a finite-precision implementation achieves any prescribed inverse-polynomial total-variation error. Consequently, every fixed limiting rate $0.6225\le r<1$ admits a strict worst-case improvement over the DQI semicircle value, and every limiting rate $r\ge 3/4$ admits solutions of satisfaction $1-o(1)$ with high probability. The algorithm coherently sums the amplitudes of all low-weight errors in each syndrome class using deterministic complete list decoding. Complete Reed--Solomon list decoding and the Sun--Wootters denominator estimate make the list size and postselection overhead polynomial. In concurrent and independent work, Horinaga and Yamakawa obtain worst-case OPI algorithms over prime-power fields and exact satisfaction at every fixed rate strictly above $3/4$.
Comments24 pages, 1 figure
