Unveiling a Hidden Percolation Transition in Monitored Clifford Circuits: Inroads from ZX Calculus

We revisit the measurement-induced phase transition (MPT) in Clifford circuits, which are both classically simulable and exhibit critical behavior widely believed to be distinct from classical percolation theory, using ZX calculus. We analyze the MPT in a dynamical model composed of controlled not (cnot) gates, SWAP gates, identity gates, and Bell-pair measurements, respectively, arranged randomly in a brickwork pattern. Our circuits exhibit a transition that is seemingly distinct from classical percolation based on standard arguments, that is in line with the prevailing understanding in the field. In contrast, by employing ZX-calculus based simplification techniques, we unveil a hidden percolation transition within the circuit structure. We demonstrate that the classical percolation transition in the ZX-simplified network coincides with the MPT observed through mutual information. Our findings suggest that the MPT in Clifford circuits is, in fact, controlled by a classical percolation transition in disguise.