发表机构
USRA Research Institute for Advanced Computer Science（USRA高级计算机科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义量子近似复杂度类BQ-APX、BQ-PTAS和BQ-FPTAS，在NP不等于BQP假设下证明严格层级，并展示基于分解和离散对数问题的量子优势，但MaxCut等常见问题的差距仍开放。
AI 中文摘要
经典近似复杂度研究在多项式时间计算中能保证的解质量。类APX、PTAS和FPTAS区分了固定近似比、近似到任意固定精度以及运行时间也随逆精度多项式增长的近似方案。它们的随机对应类是R-APX、R-PTAS和R-FPTAS。我们定义了有界误差量子对应类BQ-APX、BQ-PTAS和BQ-FPTAS。成员资格要求存在一个均匀量子算法，对每个输入，以至少2/3的概率返回一个可行的经典解，且该解达到所声称的近似比。得分（目标值）必须能被经典高效计算。运行时间包括参数选择、制备、测量、解码和重复。许多量子优化方法被启发式地使用，仅靠高基准分数并不能建立这些保证。我们进一步为对数、多项式和指数近似因子建立了条件层级。假设NP $\subsetneq$ BQP，量子类形成严格层级。基于素数分解和离散对数的问题给出了条件性的量子-经典分离。认证最大阶问题有一个精确量子算法，而任何保证每个输入至少达到逆多项式近似比的随机多项式时间算法都将导致高效分解。离散对数拟合问题有一个精确量子算法和一个确定性的二分之一近似，但任何对二分之一的固定改进都将为安全素数离散对数问题提供随机多项式时间算法。我们的结果表明，在明确的复杂度假设下，量子计算可以改进最坏情况下的近似保证。对于MaxCut或MaxSAT等常见问题的量子-经典差距仍然开放。
英文摘要
Classical approximation complexity asks what solution quality can be guaranteed with polynomial-time computation. The classes APX, PTAS, and FPTAS distinguish a fixed approximation ratio, approximation to any fixed accuracy, and approximation schemes whose running time is also polynomial in inverse accuracy. Their randomized counterparts are R-APX, R-PTAS, and R-FPTAS. We define bounded-error quantum counterparts BQ-APX, BQ-PTAS, and BQ-FPTAS. Membership requires a uniform quantum algorithm that, on every input, returns a feasible classical solution achieving at least the claimed approximation ratio with probability at least 2/3. Scores (objective values) must be efficiently classically computable. Running time includes parameter selection, preparation, measurement, decoding, and repetition. Many quantum optimization methods are used heuristically, and high benchmark scores alone do not establish these guarantees. We further establish a conditional hierarchy for logarithmic, polynomial, and exponential approximation factors. Assuming NP $\nsubseteq$ BQP, the quantum classes form a strict hierarchy. Problems based on prime factorization and discrete logarithms give conditional quantum-classical separations. Certified Maximum Order has an exact quantum algorithm, while any randomized polynomial-time algorithm guaranteeing at least an inverse-polynomial approximation ratio on every input would yield efficient factoring. Discrete-Logarithm Fitting has an exact quantum algorithm and a deterministic one-half approximation, but any fixed improvement over one half would give a randomized polynomial-time algorithm for the safe-prime discrete logarithm problem. Our results show, under explicit complexity assumptions, that quantum computation can improve worst-case approximation guarantees. A quantum-classical gap for common problems such as MaxCut or MaxSAT remains open.
