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Journal Article
Quantum Spectral Methods for Differential Equations
2020
Overview
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
(
log
d
)
. 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
(
log
d
,
log
(
1
/
ϵ
)
)
.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V,Springer
