



@article {2543,
	title = {High-precision quantum algorithms for partial differential equations},
	journal = {Quantum 5, 574},
	volume = {5},
	year = {2021},
	month = {11/4/2021},
	abstract = {<p>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 poly(1/ε), where ε 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 poly(d,log(1/ε)), 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.\&nbsp;</p>},
	doi = {https://doi.org/10.22331/q-2021-11-10-574},
	url = {https://arxiv.org/abs/2002.07868},
	author = {Andrew M. Childs and Jin-Peng Liu and Aaron Ostrander}
}
