Abstract

Hamiltonian simulation on quantum computers is one of the primary candidates for demonstration of quantum advantage. A central tool in Hamiltonian simulation is the matrix exponential. While uniform polynomial approximations (Chebyshev), best polynomial approximations, and unitary but asymptotic rational approximations (Padé) are well known and are extensively used in computational quantum mechanics, there was an important gap which has now been filled by the development of the theory and algorithms for unitary rational best approximations. This class of approximants leads to geometric numerical integrators with excellent approximation properties. In the second part of the talk I will talk about time-dependent Hamiltonians for many-body two-level systems, including a quantum algorithm for their simulation and some (classical) optimal control algorithms for quantum gate design.