Typical Output States of Monitored Random Clifford Circuits: A Graph-Theoretic Approach

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 Erd\H{o}s--R\'{e}nyi 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, $\langle g_3\rangle=1.204$ 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 $G(N_{\mathrm{sub}},1/2)$ 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 $N_{\mathrm{sub}}$ qubits, weakly perturbed by the remaining $N-N_{\mathrm{sub}}$ 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 $p_c = 0.1608$ through a mean-field argument on the graph, in excellent agreement with the numerical result $p_c\approx 0.16$.