Quantum query complexity with matrix-vector products

TL;DR

This paper investigates the capabilities of quantum algorithms in learning matrix properties via matrix-vector queries, revealing both limitations and exponential speedups for specific problems.

Contribution

It establishes the equivalence of different quantum query models involving matrix-vector operations and compares their power to classical algorithms.

Findings

Quantum algorithms do not outperform classical ones for computing trace, determinant, or rank.

Quantum algorithms can exponentially outperform classical algorithms for problems like parity and detecting duplicate rows or columns.

Models based on matrix-vector, vector-matrix, and vector-matrix-vector queries are equivalent in quantum settings.

Abstract

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.