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

Quantum Journal version

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• Improved Strong Simulation of Universal Quantum Circuits
• Noncontextual Pauli Hamiltonians