  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 $\mathcal O(\xi^t \delta^{-2})$ to $\mathcal
O((\xi^t{-}\frac{2{-}\sqrt{2}}{2} t)\delta^{-2})$, 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 minimal $L_1$ stabilizer state norm's
dependence on $t$ for finite values is not multiplicative, 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 $t$.
