受监控随机Clifford电路的典型输出态：一种图论方法
Typical Output States of Monitored Random Clifford Circuits: A Graph-Theoretic Approach
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中文总结 AI 辅助
该研究提出基于图态的框架，通过图论方法探究受监控随机Clifford电路，解析推导了随机稳定子态的平均GHZ含量，确定测量诱导相变临界点，揭示了体积律相的涌现稠密子图等特性。
中文摘要 AI 辅助
受监控随机Clifford电路是利用量子信息方法探究非平衡量子多体动力学的典型平台，以展现二分纠缠的体积律相与面积律相之间的测量诱导相变（MIPT）而闻名。本文中，我们开发了一种基于图态的框架，可直接获取受监控随机Clifford电路的典型输出态。我们首先证明，在量子比特数N的大N极限下，随机稳定子态的图表示收敛到Erdős–Rényi随机图系综G(N,1/2)。这一发现使我们能够解决随机稳定子态中Greenberger–Horne–Zeilinger（GHZ）纠缠这一开放问题，我们解析推导得到平均GHZ含量，对于偶数N为⟨g₃⟩=1.204，对于奇数N为1.325。对于体积律相中的一维（1D）受监控电路，我们在输出态图中发现了形式为G(N_sub,1/2)的涌现稠密子图，这意味着受监控电路的输出态等价于N_sub个量子比特上的幺正电路输出，仅受携带极少纠缠的剩余N−N_sub个量子比特的微弱扰动，该结果直接解释了体积律相的量子纠错能力。我们进一步识别了一维量子比特链上稠密子图空间分布的聚类效应，并通过展现测量诱导吸收态相变的感染-恢复玩具模型重现了该效应。最后，我们通过对图的平均场论证确定MIPT的临界点为p_c = 0.1608，与数值结果p_c≈0.16高度吻合。
英文摘要
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 Erdős--Rényi 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 Erdős--Rényi 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 $p_c\approx 0.16$.
