Unifying Graph Measures and Stabilizer Decompositions for the Classical Simulation of Quantum Circuits

TL;DR

This paper unifies stabilizer decompositions and tensor network contraction into a common framework, introducing new algorithms for classical simulation of quantum circuits based on graph measures like tree-width and rank-width.

Contribution

It presents a unified formalism for two main quantum simulation approaches and introduces algorithms with complexity bounds based on graph-theoretic measures.

Findings

Algorithms run in O(T^{tw(C)}) and O(T^{b3 d7 tw(C)}) time

Algorithms are simple, memory-efficient, and parallelizable

Introduction of focused tree-width and focused rank-width for better bounds

Abstract

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.