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

Lucas Kocia1 and Peter Love2

1National Institute of Standards and Technology, Gaithersburg, Maryland, 20899, U.S.A.
2Department of Physics, Tufts University, Medford, Massachusetts 02155, U.S.A.

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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.

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