Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices

TL;DR

This paper introduces a quantum algorithm for solving non-square linear systems that leverages multi-row iteration, offering exponential speedup and improved convergence over previous quantum algorithms for square matrices.

Contribution

It presents the first quantum multi-row iteration algorithm for non-square matrices, with explicit circuit design and proven faster convergence.

Findings

Quantum algorithm achieves $O(K \, \log m)$ complexity, exponentially faster than classical methods.

Algorithm converges faster than previous quantum one-row iteration algorithms.

Suitable for inconsistent systems and quadratic optimization problems.

Abstract

In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing non-square matrices. Towards this kind of problems defined by $ Ax = b $ where $ A $$ \in\mathbb{R}^{m \times n} $, we propose a quantum algorithm inspired by the classical multi-row iteration method and provide an explicit quantum circuit based on the quantum comparator and Quantum Random Access Memory (QRAM). The time complexity of our quantum multi-row iteration algorithm is $ O(K \log m) $, with $ K $ representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version. Based on the convergence of the classical multi-row iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.