Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

TL;DR

This paper presents an efficient method to simulate stabilizer quantum circuits and solve quadratic form problems over finite fields using matrix multiplication techniques, significantly improving computational bounds.

Contribution

It introduces an $O(n^ ext{omega})$-time approach to simulate stabilizer circuits and solve quadratic forms, reducing complexity compared to previous methods, and establishes reductions between matrix rank and quantum amplitude calculations.

Findings

Simulation of stabilizer circuits in $O(n^ ext{omega})$ time

Quadratic form solution counting in $O(n^ ext{omega})$ time

Reduction from matrix rank over $ ext{F}_2$ to quantum amplitude computation

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.