# Efficient Quantum Algorithms for Analyzing Large Sparse Electrical Networks. (arXiv:1311.1851v9 [quant-ph] UPDATED)

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.