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

We apply the periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension using results from p-adic number theory. We derive the Wigner-Weyl-Moyal (WWM) formalism with higher order h-bar corrections representing contextual corrections to non-contextual Clifford operations. We characterize the stationary phase critical points as a quantum resource injecting contextuality and show that this resource allows for the replacement of the p^2t points that represent t magic state Wigner functions on p-dimensional qudits by ≤ p^t points. We find that the π/8 gate introduces the smallest higher order h-bar correction possible, requiring the lowest number of additional critical points compared to the Clifford gates. We then establish a relationship between the stabilizer rank of states and their number of critical points and exploit the stabilizer rank decomposition of two qutrit π/8 gates 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 calculating 3^t/2+1 critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm for any number of π/8 gates to full precision.