Typical Output States of Monitored Random Clifford Circuits: A Graph-Theoretic Approach
In many-body physics, an explicit wavefunction often provides the most thorough understanding, yet for monitored random Clifford circuit it has remained missing. Here we develop a framework that grants direct access to the typical output states. As every stabilizer state is local-Clifford-equivalent to a graph state, the graph adjacency matrix, a classical bit matrix, provides a complete description of the quantum state. In the large-$N$ limit, we show that these graphs converge to the Erd\H{o}s--R\'{e}nyi random graph $G(N,1/2)$, which allows us to resolve the open problem of Greenberger--Horne--Zeilinger (GHZ) entanglement generated by deep random Clifford circuits. We obtain analytically the mean GHZ content $\langle g_3\rangle=1.204$ for even $N$ and $1.325$ for odd $N$. For monitored Clifford circuits with a 1D layout, we uncover an emergent Erd\H{o}s--R\'{e}nyi subgraph $G(N_{\mathrm{sub}},1/2)$ in the output states of the volume-law phase, where $N_{\mathrm{sub}}/N\approx \sqrt{1-p/p_c}$ with $p$ the measurement rate and $p_c$ the critical point of the measurement-induced phase transition (MIPT). The output state is thus equivalent to the output of an unmonitored random Clifford 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, and implies the same GHZ statistics for the whole volume-law phase. We further identify a clustering effect for qubits in the dense subgraph, which we reproduce with an infection-recovery toy model that exhibits a measurement-induced absorbing-state phase transition. Finally, a mean-field argument on the graph locates the MIPT critical point at $p_c = 0.1608$, in excellent agreement with the numerical value $p_c\approx 0.16$.