Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

Abstract

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension and show that the π8 gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on t qutrit π8 gates that are followed by Clifford evolution, and show that this only requires 3t2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.