  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.
