Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
Andrew M. Childs, Robin Kothari, Rolando D. Somma
TL;DR
This work improves quantum algorithms for the Quantum Linear Systems Problem by achieving a poly(log 1/ε) dependence on precision, exponentially better than the prior poly(1/ε) scaling. It presents two complementary approaches to implement A^{-1} as a linear combination of easily realizable unitaries: a Fourier-series-based method using Hamiltonian simulation and a Chebyshev-polynomial-based quantum walk. Both routes, combined with a linear-combination-of-unitaries (LCU) framework, yield gate- and query-efficient algorithms under the same sparsity/conditioning assumptions as HHL, while introducing variable-time amplitude amplification to push the condition-number dependence to nearly linear. The resulting algorithms are applicable as fast subroutines in broader quantum linear algebra tasks and potentially advantageous in quantum simulations and PDE solvers where precision requirements are stringent.
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 $\mathrm{poly}(\log N, 1/ε)$, 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.