Quantum Algorithms to Matrix Multiplication

TL;DR

This paper develops efficient quantum algorithms for matrix multiplication, focusing on input/output challenges, and introduces new techniques to improve quantum data handling and extend algorithm applicability.

Contribution

It proposes three quantum algorithms for matrix multiplication, with the swap test method being the most efficient, and extends techniques for quantum data input/output handling.

Findings

Swap test-based algorithm achieves complexity $ ilde{O}(n^2/\epsilon)$ with classical input/output.

Extended swap test for parallel quantum data processing.

New methods for efficient quantum state preparation of classical data.

Abstract

In this paper, we study quantum algorithms of matrix multiplication from the viewpoint of inputting quantum/classical data to outputting quantum/classical data. The main target is trying to overcome the input and output problem, which are not easy to solve and many quantum algorithms will encounter, to study matrix operations in quantum computer with high efficiency. And solving matrix multiplication will be the first step. We propose three quantum algorithms to matrix multiplication based on swap test, SVE and HHL. From the point of making fewer assumptions, swap test method works the best than the other two. We also show that the quantum algorithm of matrix multiplication with classical input and output data by swap test achieves the best complexity $\widetilde{O}(n^2/\epsilon)$ with no assumptions. This is proved by giving an efficient quantum algorithm in polynomial time to solve the input problem, that is to prepare the quantum states of the classical data efficiently. Other contributions of this paper include: (1). Extending swap test to a more general form that is suitable to deal with quantum data in parallel, which will have further applications in other matrix operations. (2). Generalizing SVE technique such that it applies to any matrix (not just Hermitian) directly only with quantum data. (3). Proposing two new efficient quantum algorithms to prepare quantum states of classical data, which solves the input problem efficiently than other quantum algorithms.