  Bounding the cost of classically simulating the outcomes of universal quantum
circuits to additive error $\delta$ is often called weak simulation and is a
direct way to determine when they confer a quantum advantage. Weak simulation
of the $T$+Clifford gateset is $BQP$-complete and is expected to scale
exponentially with the number $t$ of $T$ gates. We constructively tighten the
upper bound on the worst-case $L_1$ norm sampling cost to next order in $t$
from $\mathcal O(\xi^t \delta^{-2})$ if $\delta^2 \gg \xi^{-t}$ to $\mathcal
O((\xi^t{-}t) \delta^{-2} )$ if $\delta^2 \gg (\xi^t -t)^{-1}$, where $\xi^t =
2^{\sim 0.228 t}$ is the stabilizer extent of the $t$-tensored $T$ gate magic
state. We accomplish this by replacing independent $L_1$ sampling in the
popular SPARSIFY algorithm used in many weak simulators with correlated $L_1$
sampling. As an aside, this result demonstrates that the $T$ gate magic state's
approximate stabilizer state decomposition is not multiplicative with respect
to $t$, for finite values, despite the multiplicativity of its stabilizer
extent. This is the first weak simulation algorithm that has lowered this
bound's dependence on finite $t$ in the worst-case to our knowledge and
establishes how to obtain further such reductions in $t$.
