Improved quantum algorithms for linear and nonlinear differential equations
ArXiv(2022)
摘要
We present substantially generalized and improved quantum algorithms over
prior work for inhomogeneous linear and nonlinear ordinary differential
equations (ODE). Specifically, we show how the norm of the matrix exponential
characterizes the run time of quantum algorithms for linear ODEs opening the
door to an application to a wider class of linear and nonlinear ODEs. In Berry
et al., (2017), a quantum algorithm for a certain class of linear ODEs is
given, where the matrix involved needs to be diagonalizable. The quantum
algorithm for linear ODEs presented here extends to many classes of
non-diagonalizable matrices. The algorithm here is also exponentially faster
than the bounds derived in Berry et al., (2017) for certain classes of
diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear
differential equations using Carleman linearization (an approach taken recently
by us in Liu et al., (2021)). The improvement over that result is two-fold.
First, we obtain an exponentially better dependence on error. This kind of
logarithmic dependence on error has also been achieved by Xue et al., (2021),
but only for homogeneous nonlinear equations. Second, the present algorithm can
handle any sparse, invertible matrix (that models dissipation) if it has a
negative log-norm (including non-diagonalizable matrices), whereas Liu et al.,
(2021) and Xue et al., (2021) additionally require normality.
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关键词
quantum algorithms,nonlinear differential equations
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