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 wellconditioned, 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.02306
 Bibcode:
 2015arXiv151102306C
 Keywords:

 Quantum Physics
 EPrint:
 v1: 28 pages