Various algorithms have been developed to simulate quantum circuits on classical hardware. Among the most prominent are approaches based on \emph{ stabilizer decompositions} and \emph{ tensor network contraction}. In this work, we present a unified framework that bridges these two approaches, placing them under a common formalism. Using this, we present two new algorithms to simulate an $n$-qubit circuit $C$: one that runs in $\tilde{ O}(T^{ \mathsf{ tw}(C)})$ time and the other in $\tilde{ O}(T^{ \gamma\cdot \mathsf{ tw}(C)})$ time, where $\mathsf{ tw}(C)$ and $\mathsf{ rw}(C)$ refer to the the tree-width and rank-width, respectively, of a tensor network associated to $C$, $T$ is the number of non-Clifford gates in $C$, and $\gamma \approx 3.42$. The proposed algorithms are simple, only require a linear amount of memory, are trivially parallelizable, and interact nicely with ZX-diagram simplification routines. Furthermore, we introduce the refined complexity measures \emph{ focused tree-width} and \emph{ focused rank-width}, which are always at least as efficient as their standard equivalent; these can be directly applied within our simulation algorithms, allowing for a more precise upper bound on the run time.
Categories: quant-ph