Improved Quantum Query Bounds for Boolean Matrix Product Verification

2026-09-30

AI Summary

This study addresses the long-standing open problem of determining the quantum query complexity of Boolean matrix product verification, for which conventional Grover search techniques have proven insufficient to surpass existing bounds. By establishing an equivalence between this problem and the orthogonal vectors problem, the proposed method systematically refines the analytical framework for query complexity through the integration of quantum query complexity theory and reduction techniques. The primary contribution lies in deriving the first non-trivial upper bound of O(n^{17/12}) and lower bound of Ω(n^{5/4}), thereby breaking the limitations of Grover search and establishing tight bounds. Furthermore, this work achieves a polynomial separation for graph radius and diameter decision problems, offering a novel paradigm for related research in quantum algorithms.

Abstract

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.

Problem

• Boolean Matrix Product Verification
• Quantum Query Complexity
• Orthogonal Vectors
• Lower Bound
• Upper Bound

Innovation

• Boolean Matrix Product Verification
• Quantum Query Complexity
• Orthogonal Vectors
• Grover Search
• Polynomial Separation