  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.
