# Discrete Wigner Function Derivation of the Aaronson-Gottesman Tableau Algorithm

## TL;DR

This paper introduces a discrete Wigner function-based simulation algorithm for odd-dimensional qudits that matches the efficiency of the Aaronson-Gottesman algorithm, leveraging symplectic structure in phase space.

## Contribution

It derives a new simulation algorithm using discrete Wigner functions for odd qudits, aligning with Aaronson-Gottesman in efficiency and offering insights for qubit extension.

## Key findings

- The Wigner function algorithm has the same complexity as Aaronson-Gottesman.
- Efficiency is due to harmonic evolution in symplectic phase space.
- Differences are linked to group structure and contextuality in qubits.

## Abstract

The Gottesman-Knill theorem established that stabilizer states and operations can be efficiently simulated classically. For qudits with dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-$d$ qudits that has the same time and space complexity as the Aaronson-Gottesman algorithm. We show that the efficiency of both algorithms is due to the harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm and Aaronson-Gottesman are likely due only to the fact that the Weyl-Heisenberg group is not in $SU(d)$ for $d=2$ and that qubits have state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.