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 π8 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 π8 gates that are followed by Clifford evolution, and show that this only requires 3t2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.
Highlights
This paper aims to bridge the gap between efficient strong simulation qubit methods, which use low stabilizer rank representations of magic states, and oddprime-dimensional qudit methods, which have relied on Wigner negativity in past studies
It would be interesting to see if there is a similar result to Theorem 3 that deals with approximate stabilizer rank and if there is some such “approximate” analog to the discrete stationary phase method that is useful for weak simulation
We found that the usage of higher order uniformizations through the stationary phase method is more efficient than using negativity for π/8 gate magic states, at least in the manner that has so far been tried
Summary
This paper aims to bridge the gap between efficient strong simulation qubit methods, which use low stabilizer rank representations of magic states, and oddprime-dimensional qudit methods, which have relied on Wigner negativity in past studies. Without appealing to the continuous case, we use a periodized version of Taylor’s theorem to develop the stationary phase method in the discrete setting. +1 for calculations to full precision while the best alternative qutrit algorithm scales as ∼30.8t for calculations to 10−2 precision This allows the stationary phase method to be more useful for intermediate-sized circuits. Related approaches include recent proposals to use non-Gaussianity as a resource in continuous (infinite dimensional) quantum mechanics such as optics [23,24,25], as well as the well-established field dealing with semiclassical propagators in continuous systems [7, 26, 27].
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