%0 Journal Article %J SIAM Journal on Computing %D 2017 %T Quantum algorithm for systems of linear equations with exponentially improved dependence on precision %A Andrew M. Childs %A Robin Kothari %A Rolando D. Somma %X

Harrow, Hassidim, and Lloyd showed that for a suitably specified N×N matrix A and N-dimensional vector b⃗ , there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations Ax⃗ =b⃗ . If A is sparse and well-conditioned, their algorithm runs in time poly(logN,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.

%B SIAM Journal on Computing %V 46 %P 1920-1950 %8 2017/12/21 %G eng %U http://epubs.siam.org/doi/10.1137/16M1087072 %N 6 %R 10.1137/16M1087072