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

Quantum 5, 494 (2021).

https://doi.org/10.22331/q-2021-07-05-494

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 $simhspace{-3pt} 3^{0.8 t}$ for $10^{-2}$ precision) for any number of $frac{pi}{8}$ gates to full precision.