Improved Weak Simulation of Universal Quantum Circuits by Correlated $L_1$ Sampling

Abstract: 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 $t$.