Stationary phase method in discrete Wigner functions and classical simulation of quantum circuits

Presenting Author: Lucas Kocia, National Institute of Standards and Technology, Maryland
Contributing Author(s): Peter Love

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 \(hbar\) corrections representing contextual corrections to non-contextual Clifford operations. We apply this formalism to a subset of unitaries that include diagonal gates such as the \({\pi}/{8}\) gates. 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 \(\le p^{t}\) points. We find that the \({\pi}/{8}\) gate introduces the smallest higher order \(hbar\) 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 the number of critical points and exploit the stabilizer rank decomposition of two qutrit \({\pi}/{8}\) gates to develop a classical strong simulation of a single qutrit marginal on \(t\) qutrit \({\pi}/{8}\) gates that are followed by Clifford evolution, and show that this only requires calculating \(3^{\frac{t}{2}+1}\) critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm for any number of \({\pi}/{8}\) gates to full precision.

Read this article online: https://arxiv.org/abs/1810.03622

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