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

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 $\mathrm{poly}(\log N, 1/ε)$, where $ε$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/ε)$, 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 $ε$ is prohibitive.

v1: 28 pages; v2: 31 pages, minor change to title, various minor changes and clarifications in response to referee comments