2026/08/04 by Yu-Xuan Zhang, Yu-Xiang Zhang
Physics and Astronomy · #quant-ph #cond-mat.stat-mech
22 pages, 13 figures
arxiv created 2026/08/04 · arxiv updated 2026/08/05
Monitored random Clifford circuit is a paradigmatic platform for exploring non-equilibrium quantum many-body dynamics using quantum-information methods. It is well-known for exhibiting a measurement-induced phase transition (MIPT) between volume-law and area-law phases of bipartite entanglement. In this Article, we develop a graph-state based framework that grants direct access to the typical output states of monitored random Clifford circuits. We first show that, in the large-N limit, where N denotes qubit number, the graph representations of random stabilizer states converge to the Erdős--Rényi random graph ensemble G(N,1/2). This observation allows us to resolve the open problem of Greenberger--Horne--Zeilinger (GHZ) entanglement in random stabilizer states. We derive analytically the mean GHZ content, ⟨ g3⟩=1.204 for even N and 1.325 for odd N. For monitored one dimensional (1D) circuits in the volume-law phase, we uncover an emergent dense subgraph of the form G(Nsub,1/2) in the output-state graphs. This implies that the output state of a monitored circuit is equivalent to an output of a unitary circuit on Nsub qubits, weakly perturbed by the remaining N-Nsub qubits carrying little entanglement. This result directly accounts for the quantum error-correcting capability of the volume-law phase. We further identify a clustering effect in the spatial distribution of the dense subgraph along the 1D qubit chain, and reproduce it with an infection-recovery toy model that exhibits a measurement-induced absorbing-state phase transition. Finally, we locate the critical point of the MIPT at pc = 0.1608 through a mean-field argument on the graph, in excellent agreement with the numerical result pc≈ 0.16.