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arXiv · 1511.02306

Quantum algorithm for systems of linear equations with exponentially improved dependence on precision

Andrew M. Childs·Robin Kothari·Rolando D. Somma
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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 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.

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Andrew M. Childs, Robin Kothari, Rolando D. Somma. 2015-11-07. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. https://doi.org/10.1137/16m1087072

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