Probing the Spacetime Structure of Entanglement in Monitored Quantum Circuits with Graph Neural Networks

Abstract

Global entanglement in quantum many-body systems is inherently nonlocal, raising the question of whether it can be inferred from local observations. We investigate this problem in monitored quantum circuits, where projective measurements generate classical records distributed across spacetime. Using graph neural networks (GNNs), we represent individual quantum trajectories as directed spacetime graphs and reconstruct the half-chain entanglement entropy from local measurement data alone. Because information propagates through the network via local message passing, the architecture directly controls the spacetime region over which correlations can be aggregated. By systematically varying this accessible scale -- through network depth and hierarchical spacetime coarse-graining -- we probe how much measurement information is required to reconstruct global entanglement. We find that prediction accuracy improves as the accessible spacetime region grows and that results from different architectures collapse when expressed in terms of an effective spacetime scale combining depth and coarse-graining. These results demonstrate that the information required to reconstruct global entanglement is organized in spacetime scales and show that graph-based learning architectures provide a controlled operational framework for probing how global quantum correlations emerge from local measurement data.

Figures (7)

• Figure 1: Overview of the learning framework for predicting entanglement in monitored quantum circuits. (A) A one-dimensional monitored quantum circuit is mapped to a directed spacetime graph $G$, where nodes correspond to spacetime events $(i,t)$ and edges encode the causal structure of the circuit through worldline and gate-induced connections. Each node carries features derived from the measurement record, including a measurement indicator, measurement outcome, normalized time coordinate, and positional information. (B) Single-scale graph neural network architecture. Stacking $K$ directed GraphSAGE message-passing layers enlarges the causal receptive field linearly with depth, corresponding to a $K$-layer causal light cone on the spacetime graph. (C) RG-inspired hierarchical architecture. Iterative $2\times2$ spacetime blocking generates progressively coarser graphs, enabling the network to access larger effective spacetime regions and increasing the receptive field exponentially with the number of coarse-graining levels. (D) Regression task. Node embeddings at the final time slice are pooled separately over the left and right halves of the chain to obtain global representations, which are combined by a multilayer perceptron (MLP) to predict the normalized half-chain von Neumann entropy $s=S_{N/2}/(N/2)$. (E) Prediction and training objective. The model prediction $\hat{s}$ is trained to match the target entropy using the mean squared error loss $\mathcal{L}=(\hat{s}-s)^2$.
• Figure 2: Baseline behavior of the monitored circuit and prediction accuracy of the graph neural network. Results are obtained using the single-scale GNN with depth $K=6$. (Top) Normalized half-chain entropy $s = S_{N/2}/(N/2)$ as a function of measurement rate $p$ for system sizes $N=14$ (main panel) and $N=16$ (inset). Black circles show the exact values obtained from quantum circuit simulations, while red squares denote neural-network predictions. The dashed vertical line indicates the approximate location of the measurement-induced transition at $p_c \approx 0.17$. (Bottom) Root-mean-square prediction error (RMSE) as a function of measurement rate for several system sizes. The prediction error is largest in the weak-measurement regime where entanglement is extensive and long-range correlations dominate, and decreases significantly at larger $p$ where frequent measurements suppress entanglement growth.
• Figure 3: Depth dependence of the single-scale graph neural network. (Top) Root-mean-square prediction error (RMSE) as a function of network depth $K$ for several measurement rates, shown for system size $N=14$. The inset shows the corresponding results for the unseen system size $N=16$, demonstrating that the depth dependence generalizes beyond the system sizes used during training. (Bottom) Prediction error as a function of measurement rate for several network depths at $N=14$. The dashed vertical line indicates the approximate measurement-induced transition $p_c\approx0.17$. The prediction error is largest in the weak-measurement regime and decreases with increasing depth, reflecting the need to integrate information over larger spacetime scales when entanglement is extensive. Insets again show the corresponding behavior for the unseen system size $N=16$.
• Figure 4: Comparison between single-scale and hierarchical graph neural network architectures and their dependence on the accessible spacetime scale. (a) Prediction error as a function of measurement rate for the single-scale architecture with depth $K=6$ and for two hierarchical architectures RG$(2,2)$ and RG$(2,2,2)$ at system size $N=14$. All models have comparable numbers of message-passing layers but access different effective spacetime scales. The dashed vertical line indicates the approximate measurement-induced transition $p_c\approx0.17$. (b) Prediction error plotted as a function of the effective spacetime scale $\ell_{\mathrm{eff}}$ defined in Sec. \ref{['subsec:rg_hgnn']}. Each point corresponds to a different architecture and measurement rate. Colors indicate the measurement rate $p$. The error decreases systematically with increasing accessible scale, largely independent of the architectural details. (c) Scaling of the prediction error with accessible scale for a representative measurement rate $p=0.2$. Blue circles show the single-scale architecture for depths $K=1\ldots8$, while red squares show the hierarchical architectures. The dashed curve indicates a power-law guide $\varepsilon \sim \ell_{\mathrm{eff}}^{-0.35}$.
• Figure 5: Same as Fig \ref{['fig:fig3']}, scaling of prediction error with accessible spacetime scale for an unseen system size $N=24$. Exact data were generated using tensor-network simulations. The network was trained only on systems with $N\leq14$.