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 π/8 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 π/8 gates that are followed by Clifford evolution, and show that this only requires 3^t/2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼-3pt 3^0.8 t for 10⁻² precision) for any number of π/8 gates to full precision.