Summary: A. W. Harrow et al. [“Quantum algorithm for linear systems of equations”, Phys. Rev. Lett. 103, No. 13, Article ID 150502, 4 p. (2009; doi:10.1103/PhysRevLett.103.150502)] showed that for a suitably specified \(N \times N\) matrix \(A\) and an \(N\)-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.