%0 Journal Article %J Commun. Math. Phys. %D 2020 %T Quantum spectral methods for differential equations %A Andrew M. Childs %A Jin-Peng Liu %X

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 poly(logd). While several of these algorithms approximate the solution to within ε with complexity 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 poly(logd,log(1/ε)).

%B Commun. Math. Phys. %V 375 %P 1427-1457 %8 2/18/2020 %G eng %U https://arxiv.org/abs/1901.00961 %R https://doi.org/10.1007/s00220-020-03699-z