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In this work we improve the runtime of recent classical algorithms for strongsimulation of quantum circuits composed of Clifford and T gates. Theimprovement is obtained by establishing a new upper bound on the stabilizerrank of $m$ copies of the magic state$|T\rangle=\sqrt{2}^{-1}(|0\rangle+e^{i\pi/4}|1\rangle)$ in the limit of large$m$. In particular, we show that $|T\rangle^{\otimes m}$ can be exactlyexpressed as a superposition of at most $O(2^{\alpha m})$ stabilizer states,where $\alpha\leq 0.3963$, improving on the best previously known bound $\alpha\leq 0.463$. This furnishes, via known techniques, a classical algorithm whichapproximates output probabilities of an $n$-qubit Clifford + T circuit $U$ with$m$ uses of the T gate to within a given inverse polynomial relative errorusing a runtime $\mathrm{poly}(n,m)2^{\alpha m}$. We also provide improvedupper bounds on the stabilizer rank of symmetric product states$|\psi\rangle^{\otimes m}$ more generally; as a consequence we obtain a strongsimulation algorithm for circuits consisting of Clifford gates and $m$instances of any (fixed) single-qubit $Z$-rotation gate with runtime$\text{poly}(n,m) 2^{m/2}$. We suggest a method to further improve the upperbounds by constructing linear codes with certain properties.

Improved upper bounds on the stabilizer rank of magic states