DQC Seminar Series: Time-dependent Hamiltonian Simulation: Quantum Algorithm and Superconvergence

Abstract: Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence, with an error preconstant independent of the number of spatial grids. The proof of superconvergence is based on semiclassical analysis that is of independent interest.

Bio: Di Fang is an Assistant Professor of Mathematics at Duke University, and a member of Duke Quantum Center. Previously, she was a Morrey Assistant Professor at the Department of Mathematics, University of California, Berkeley, and a Simons Quantum Postdoctoral Fellow at the Simons Institute for the Theory of Computing, hosted by Prof. Lin Lin and Prof. Umesh Vazirani. She earned her PhD in mathematics from University of Wisconsin-Madison. Her expertise lies in the applied and numerical analysis of differential equations, particularly with applications to quantum algorithms and quantum dynamics.