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