Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

Abstract: We show that a form of strong simulation for $n$-qubit quantum stabilizer circuits $C$ is computable in $O(s + n^\omega)$ time, where $\omega$ is the exponent of matrix multiplication. Solution counting for quadratic forms over $\mathbb{F}_2$ is also placed into $O(n^\omega)$ time. This improves previous $O(n^3)$ bounds. Our methods in fact show an $O(n^2)$-time reduction from matrix rank over $\mathbb{F}_2$ to computing $p = |\langle \; 0^n \;|\; C \;|\; 0^n \;\rangle|^2$ (hence also to solution counting) and a converse reduction that is $O(s + n^2)$ except for matrix multiplications used to decide whether $p > 0$. The current best-known worst-case time for matrix rank is $O(n^{\omega})$ over $\mathbb{F}_2$, indeed over any field, while $\omega$ is currently upper-bounded by $2.3728\dots$ Our methods draw on properties of classical quadratic forms over $\mathbb{Z}_4$. We study possible distributions of Feynman paths in the circuits and prove that the differences in $+1$ vs. $-1$ counts and $+i$ vs. $-i$ counts are always $0$ or a power of $2$. Further properties of quantum graph states and connections to graph theory are discussed.