2017/01/01 by Andrew M. Childs, Robin Kothari, Rolando D. Somma · 89 citations
Computer Science · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Numerical Methods and Algorithms
paper · doi:10.1137/16m1087072
Harrow, Hassidim, and Lloyd [Phys. Rev. Lett., 103 (2009), 150502] showed that for a suitably specified N × N matrix A and an N-dimensional vector b, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations Ax = b. If A is sparse and well-conditioned, their algorithm runs in time 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.