Improved Graph Formalism for Quantum Circuit Simulation

TL;DR

This paper introduces a new graph-based formalism and algorithms to enhance the classical simulation of stabilizer quantum circuits, achieving faster inner product computations and better state representations.

Contribution

It presents a canonical form for stabilizer states based on graph states and an improved algorithm for graph state stabilizer simulation, advancing classical quantum circuit simulation methods.

Findings

Faster inner product computation from O(n^3) to O(nd^2)

A canonical form for stabilizer states based on graph states

Limitations on reducing quadratic runtime for controlled-Pauli Z gates

Abstract

Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a new way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulation and establish limitations on reducing the quadratic runtime of applying controlled-Pauli $Z$ gates. We do so by creating a simpler formula for combining two Pauli-related stabilizer states into one. Third, to better understand the linear dependence of stabilizer states, we characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our novel controlled-Pauli $Z$ algorithm, we improve runtime for inner product computation from $O(n^3)$ to $O(nd^2)$ where $d$ is the maximum degree of the graph.