Quantum Complexity for Vector Domination Problem

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.