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arXiv · 1901.00961

Quantum spectral methods for differential equations

Andrew M. Childs·Jin-Peng Liu
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Abstract

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear equations or linear differential equations with complexity $\mathrm{poly}(\log d)$. While several of these algorithms approximate the solution to within $ε$ with complexity $\mathrm{poly}(\log(1/ε))$, no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity $\mathrm{poly}(\log d, \log(1/ε))$.

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Andrew M. Childs, Jin-Peng Liu. 2019-01-04. Quantum spectral methods for differential equations. https://doi.org/10.1007/s00220-020-03699-z

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