Improved Quantum Query Bounds for Boolean Matrix Product Verification

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 $\widetilde O(n<sup>{17/12})$, improving on the standard $O(n<sup>{3/2})$ bound obtained using Grover search by Buhrman and Špalek (SODA 2006). We complement this result by showing an $Ω(n<sup>{5/4})$ lower bound, which improves over the previous best known lower bound of $\widetildeΩ(n<sup>{19/18})$ 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 $\widetilde Θ(n<sup>{3/2})$ 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.