High-precision quantum algorithms for partial differential equations

Andrew M. Childs; Jin-Peng Liu; Aaron Ostrander

doi:10.22331/q-2021-11-10-574

High-precision quantum algorithms for partial differential equations

Andrew M. Childs^1,2,3, Jin-Peng Liu^1,2,4, and Aaron Ostrander^1,2,5

¹Joint Center for Quantum Information and Computer Science, University of Maryland, MD 20742, USA
²Institute for Advanced Computer Studies, University of Maryland, MD 20742, USA
³Department of Computer Science, University of Maryland, MD 20742, USA
⁴Department of Mathematics, University of Maryland, MD 20742, USA
⁵Department of Physics, University of Maryland, MD 20742, USA

Published:	2021-11-10, volume 5, page 574
Eprint:	arXiv:2002.07868v2
Doi:	https://doi.org/10.22331/q-2021-11-10-574
Citation:	Quantum 5, 574 (2021).

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Abstract

Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

► BibTeX data

@article{Childs2021highprecision,
  doi = {10.22331/q-2021-11-10-574},
  url = {https://doi.org/10.22331/q-2021-11-10-574},
  title = {High-precision quantum algorithms for partial differential equations},
  author = {Childs, Andrew M. and Liu, Jin-Peng and Ostrander, Aaron},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {574},
  month = nov,
  year = {2021}
}

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