发表机构
University of Washington; Sandia National Laboratories（华盛顿大学; 桑迪亚国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为量子查询复杂度证明了所有函数的强选择性直接积定理，并推广了列表解码直接积定理至部分布尔函数，通过新的乘法对手表述统一了相关结果。
AI 中文摘要
针对特定函数的量子强直接积定理已为人所知近二十年。这些结果已推广到函数计算和态生成的通用结果。这些结果的证明使用了乘法对手方法的某个版本，该方法不能自然地推广到关系。标准强直接积定理适用于算法必须正确回答每个给定问题的情况。先前的工作将其推广到等价阈值直接积定理，该定理要求回答所有问题，但仅要求大多数答案正确。我们关注两个进一步的推广。强选择性直接积适用于算法根据从查询中学到的信息自适应地从大列表中选择要回答的问题。这种推广是关系型的，并且对于证明时间-空间权衡很有用。我们证明了所有函数的量子强选择性直接积定理，使用一种新的针对关系的乘法对手表述，该表述满足强选择性直接积性质，同时足够强大以捕获由负权重对手证明的函数的任何查询下界。这在以前即使是无需选择性的情况下也是未知的。第二个推广是Ben-David和Blais为经典随机查询复杂度引入的列表解码直接积问题。这些允许算法产生一个大的可能输出向量列表，使得其中一个完全正确。他们证明了这样的定理对经典随机复杂度的所有布尔函数成立。我们证明了该定理对所有部分布尔函数的量子类比。我们表明，强列表解码直接积定理由乘法对手的一个特例蕴含，我们通过一种新的归约表明，对于任何布尔值函数，该特例可以从负权重对手获得。
英文摘要
Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a relational formulation of the multiplicative adversary method by Jeffery and Zur which we prove, via an equivalent formulation, satisfies a strong selective direct product property and captures any query lower bound for functions proven by negative-weights adversaries. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean-valued functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.
Comments64 pages, 3 figures, submitted to QIP 2027
