  We show that the cost of strong simulation of quantum circuits using $t$ $T$
gate magic states exhibits non-trivial reductions on its upper bound for $t=1$,
$t=2$, $t=3$, and $t=6$ with odd-prime-qudits. This agrees with previous
numerical bounds found for qubits. We define simulation cost by the number of
terms that require Gaussian elimination of a $t \times t$ matrix and so capture
the cost of simulation methods that proceed by computing stabilizer inner
products or evaluating quadratic Gauss sums. Prior numerical searchs for qubits
were unable to converge beyond $t=7$. We effectively increase the space
searched for these non-trivial reductions by $>10^{10^4}$ and extend the bounds
to $t=14$ for qutrits. This is accomplished by using the Wigner-Weyl-Moyal
formalism to algebraically find bounds instead of relying on numerics. We find
a new reduction in the upper bound from the $12$-qutrit magic state of
${3^{\sim 0.469t}}$, which improves on the bound obtained from the $6$-qutrit
magic state of ${3^{\sim 0.482t}}$.
