Efficient Classical Simulation of Low-Rank-Width Quantum Circuits Using ZX-Calculus

In this paper, we introduce an algorithm for contracting (i.e.numerically evaluating) ZX-diagrams whose computational complexity is governed by rank-width, a structural graph parameter that behaves nicely under ZX rewrite rules.Our method evaluates a graph-like ZX-diagram in Õ(4 R ) time, given its rank-decomposition of width R. By constructing suitable rank-decompositions, we provide a classical simulation algorithm for quantum circuits whose runtime is bounded by both Õ(2 n ) and Õ(2 t/2 ), where n is the number of qubits and t is the number of non-Clifford phase gates.Thus, our method is no slower than either the naïve state-vector simulation or the elementary stabiliser decomposition baseline, and in practice can be even faster when the reduced ZX-diagram has low rank-width.We benchmark our simulation algorithm against Quimb, a popular tensor contraction library, and observe substantial reductions in floating-point operations, often by several orders of magnitude, for random and structured non-Clifford circuits and for random ZX-diagrams.