Efficient Quantum Algorithms for Analyzing Large Sparse Electrical Networks

TL;DR

This paper introduces two quantum algorithms for efficiently analyzing large sparse electrical networks, capable of computing key electrical properties with polynomial time complexity based on network parameters.

Contribution

The paper presents two novel quantum algorithms for electrical network analysis, demonstrating their optimal polynomial time complexity and applicability to large sparse networks.

Findings

Algorithms compute voltages, currents, and resistances efficiently.

Time complexity depends polynomially on network parameters.

Polynomial dependence on inverse spectral gap is proven necessary.

Abstract

Analyzing large sparse electrical networks is a fundamental task in physics, electrical engineering and computer science. We propose two classes of quantum algorithms for this task. The first class is based on solving linear systems, and the second class is based on using quantum walks. These algorithms compute various electrical quantities, including voltages, currents, dissipated powers and effective resistances, in time $\operatorname{poly}(d, c, \operatorname{log}(N), 1/\lambda, 1/\epsilon)$, where $N$ is the number of vertices in the network, $d$ is the maximum unweighted degree of the vertices, $c$ is the ratio of largest to smallest edge resistance, $\lambda$ is the spectral gap of the normalized Laplacian of the network, and $\epsilon$ is the accuracy. Furthermore, we show that the polynomial dependence on $1/\lambda$ is necessary. This implies that our algorithms are optimal up to polynomial factors and cannot be significantly improved.