UNIFORM OBSERVABLE ERROR BOUNDS OF TROTTER FORMULAE FOR THE SEMICLASSICAL SCHRÖDINGER EQUATION

Known as the no fast-forwarding theorem in quantum computing (see, e.g., Theorem 3 in [D. W. Berry et al., Comm. Math. Phys., 270 (2007), pp. 359-371], Theorem 5 in [A. M. Childs, Comm. Math. Phys., 294 (2010), pp. 581-603], and [R. Kothari, Efficient Simulation of Hamiltonians, M.S. thesis, University of Waterloo, 2010]), the simulation time for the Hamiltonian evolution is typically \scrO(⃦H⃦t), which essentially states that one cannot go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schr\"odinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant h ≪ 1), the operator norm of the Hamiltonian is \scrO(h⁻¹). We show that the number of Trotter steps used for the observable evolution can be \scrO(1), that is, to simulate some observables of the Schr\"odinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds in [C. Lasser and C. Lubich, Acta Numer., 29 (2020), pp. 229-401] to uniform-in-h observable error bounds. This is, to our knowledge, the first uniform observable error bound for the semiclassical Schr\"odinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, and of the underlying dynamics, and sheds light on going across the scale for quantum dynamics simulation.