Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An1, Di Fang2,3,5, and Lin Lin2,4,5

1Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland, College Park, MD 20742, USA
2Department of Mathematics, University of California, Berkeley, CA 94720, USA
3Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720, USA
4Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
5Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA

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We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

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