Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus

Abstract

The ZX-calculus is an algebraic formalism that allows quantum computations to be simplified via a small number of simple graphical rewrite rules. Recently, it was shown that, when combined with a family of "sum-over-Cliffords" techniques, the ZX-calculus provides a powerful tool for classical simulation of quantum circuits. However, for several important classical simulation tasks, such as computing the probabilities associated with many measurement outcomes of a single quantum circuit, this technique results in reductions over many very similar diagrams, where much of the same computational work is repeated. In this paper, we show that the majority of this work can be shared across branches, by developing reduction strategies that can be run parametrically on diagrams with boolean free parameters. As parameters only need to be fixed after the bulk of the simplification work is already done, we show that it is possible to perform the final stage of classical simulation quickly utilising a high degree of GPU parallelism. Using these methods, we demonstrate an average speedup factor of 78.3+-10.2 for certain classical simulation tasks vs. the non-parametric approach.

In Proceedings QPL 2025, arXiv:2508.13619

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