Ish Dhand, University of Calgary
Abstract
Trotter-Suzuki ordered operator-exponential decomposition finds use in
quantum control, simulation of physical systems and in algorithms for
finding the ground state of quantum many-body systems. The decomposition has well-defined error bounds in theory but is unstable in practice; higher-order approximations should provide exponentially exact solutions but yield approximations with error that grows exponentially beyond a certain order. We investigate the stability of the Trotter-Suzuki
decomposition scheme for ordered operator exponentials in the context of
finite-precision computation. Finite precision is a result of finite computational space on a classical Turing machine and of finiteness of gate set on a quantum circuit. We show that that numerical error in computation causes instabilities in the Lie-Trotter-Suzuki product formulae for operator exponentials for approximation of exp(iHλ) for λ large as compared to 1/|H|, where |*| is the operator norm. To address the problem of approximating an operator exponential to within a specified error, we present an algorithm to decide whether a Trotter-Suzuki expansion is possible on a computer with given numerical precision and then find the corresponding optimal decomposition. This algorithm would enable the simulation, control and analysis of more complex physical systems using the same computational resources.