发表机构
University of Maryland; Graduate School of Mathematics, Nagoya University; The University of British Columbia（马里兰大学; 名古屋大学数学研究院; 不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次给出布尔矩阵乘积验证的量子查询上界Õ(n^{17/12})，并证明Ω(n^{5/4})下界，通过等价性分析实现半径与直径判定问题的多项式分离。
AI 中文摘要
我们证明了布尔矩阵乘积验证（BMPV）的量子查询复杂度的第一个非平凡上界，回答了量子查询复杂度中一个长期悬而未决的问题。对于n×n矩阵，我们的上界为Õ(n^{17/12})，改进了Buhrman和Špalek（SODA 2006）使用Grover搜索得到的标准O(n^{3/2})界。我们通过展示Ω(n^{5/4})的下界来补充这一结果，该下界优于Childs、Kimmel和Kothari（ESA 2012）先前已知的最佳下界ÕΩ(n^{19/18})。我们的方法集中于与正交向量（OV）的联系，该问题询问一个包含n个维度为n的布尔向量的索引列表是否包含两个具有不相交支撑的向量。特别地，我们证明了OV与BMPV之间的等价性，并为OV建立了上述界。我们还证明了BMPV的一个变体的紧致ÕΘ(n^{3/2})界，该变体询问乘积是否包含给定的行向量。这些结果共同意味着，在判定图是否具有半径至多2和直径至多2的量子查询复杂度之间存在多项式分离。
英文摘要
We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification ($\mathsf{BMPV}$), answering a longstanding open question in quantum query complexity. For $n\times n$ matrices, our upper bound is $\widetilde O(n^{17/12})$, improving on the standard $O(n^{3/2})$ bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an $Ω(n^{5/4})$ lower bound, which improves over the previous best known lower bound of $\widetildeΩ(n^{19/18})$ by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors ($\mathsf{OV}$), which asks whether an indexed list of $n$ Boolean vectors of dimension $n$ contains two vectors with disjoint supports. In particular, we prove equivalences between $\mathsf{OV}$ and $\mathsf{BMPV}$ and establish the above bounds for $\mathsf{OV}$. We also prove a tight $\widetilde Θ(n^{3/2})$ bound for a variant of $\mathsf{BMPV}$ that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.
Comments38 pages
