Quantum algorithm for systems of linear equations with exponentially improved dependence on precision

TL;DR

This paper presents a quantum algorithm for solving linear systems that significantly reduces the complexity's dependence on the precision parameter, leveraging Fourier and Chebyshev series techniques to improve efficiency.

Contribution

The authors develop a new quantum algorithm that exponentially improves the dependence on precision compared to previous methods, avoiding quantum phase estimation.

Findings

Running time polynomial in log(1/ε), exponential improvement over previous algorithms.

Maintains similar dependence on matrix size and condition number as prior algorithms.

Uses Fourier and Chebyshev series to implement operators without phase estimation.

Abstract

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x}=\vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time $\mathrm{poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive.