Quantum Complexity for Vector Domination Problem

Andris Ambainis, Ansis Zvirbulis

Abstract: In this paper we investigate quantum query complexity of two vector problems: vector domination and minimum inner product. We believe that these problems are interesting because they are closely related to more complex 1-dimensional dynamic programming problems. For the general case, the quantum complexity of vector domination is \(\varTheta (n^{1-o(1)})\), similarly to the more known orthogonal vector problem (OV). We prove a \(\tilde{O}(n^{2/3})\) upper bound and a \(\varOmega (n^{2/3})\) lower bound for special case of vector domination where vectors are from \(\{1,\dots ,W\}^d\) and number of dimensions d is a constant and \(W \in O(\textit{poly } n)\). We also prove a \(\varOmega (n^{2/3})\) lower bound for minimum inner product with the same constraints. To prove bounds we use reductions from the element distinctness problem as well as a classical data structure - Fenwick trees.