Post-Selection-Free Decoding of Measurement-Induced Area-Law Phases via Neural Networks

Abstract

Monitored quantum circuits host a rich variety of exotic non-equilibrium phases. Among the most representative examples are measurement-induced phase transitions between distinct area-law entangled states. However, because these transitions are characterized by specific entanglement quantities such as mutual information or topological entanglement entropy that are nonlinear functionals of the density matrix, their experimental observation requires multiple identical quantum trajectories via post-selection, which becomes exponentially unfeasible for large systems. Here, we leverage modern machine learning tools to address this challenge. We devise a neural network architecture combining a convolutional neural network with an attention mechanism, and use raw measurement outcomes directly as input to classify trivial, long-range entangled, and symmetry-protected topological phases. We show that the system's relaxation to a steady-state phase manifests as a sharp convergence in the classifier's accuracy, entirely bypassing the need for quantum state reconstruction. We systematically study the performance of our network as a function of sample size, input data, spatial and temporal constraints, and system size scalability. Our results demonstrate that this approach is robust and post-selection free, offering a practical pathway for experimentally probing measurement-induced phases.

Figures (5)

• Figure 1: Circuit architecture and phase diagram. (a) Schematic of the measurement-only brickwork circuit applied to $L=6$ qubits over three time steps. Each step features three sequential layers of measurements: single-qubit $X$ operators shown in purple, nearest-neighbor two-qubit $ZZ$ operator in blue, and three-qubit $ZXZ$ operators on adjacent triplets in green. (b) Ternary phase diagram defined on the simplex $\gamma_{X}+\gamma_{ZZ}+\gamma_{ZXZ}=1$. The purple, blue, and green shaded regions correspond to the trivial, long range entangled, and SPT phases, respectively, with dashed lines representing the critical boundaries between them. Stars at the three vertices mark the phase training points. Test parameter locations are indicated by red circles for outer points and orange squares for inner points. The color bar indicates the fraction of independent training runs that agree on the majority label.
• Figure 2: Schematic of the neural network architecture. Raw $X$, $ZZ$, and $ZXZ$ measurement records for $N$ trajectories are reshaped to align with the circuit's brickwork periodicity. Three parallel CNN branches independently process these inputs, followed by spatial global average pooling to extract feature sequences. These features are concatenated, projected through a linear layer, and compressed via a temporal readout. Finally, a pooling attention module aggregates the $N$ trajectory representations into a single vector, which is mapped via a softmax layer to produce $y(N)$, a probability distribution over the trivial, LR, and SPT phases. Bracketed notation (e.g.,$[...]$) is provided at each intermediate stage to explicitly indicate the shape of the data tensors.
• Figure 3: Dependence of classification accuracy on trajectory count and input data type. (a) Classification accuracy $P$ as a function of the number of measurement trajectories $M$. Results are shown for $N=25$ across discrete values of $M\in\{100, 500, 1000, 4000, 7500, 10000, 30000, 50000\}$. (b) Classification accuracy $P$ across different combinations of input measurement channels. Blue and orange denote inner and outer test points, respectively. Error bars represent the standard error across 10 trained models from different random initializations.
• Figure 4: Dependence of classification accuracy on evolution time and spatial subsystem size. (a) Classification accuracy $P$ as a function of the total evolution time $T$, evaluated at discrete times $T \in \{0.5L, L, 1.5L, 2L, 4L, 6L\}$ with $L=12$. The inset shows the half chain entanglement entropy $S$ versus $T$ at $\gamma_{X}=\gamma_{ZXZ}=0.3$. Dashed vertical lines indicate the specific evaluation times plotted in the main panel. (b) Classification accuracy $P$ versus spatial subsystem size $L_{A} \in \{2, 4, 6, 8, 10, 12\}$. In both panels, blue and orange markers correspond to inner and outer test points, respectively. Error bars denote the standard error of the mean across 10 independent trained models from different random initializations.
• Figure 5: Transfer learning scalability and impact of the attention mechanism. (a) Classification accuracy $P$ versus system size $L$ for a network trained on $L = 12$ and $T=36$ and tested on larger systems without retraining ($h_{1}=8$, $h_{2}=5$). (b) Classification accuracy $P$ for the model with (dark) and without (light) the attention mechanism with $h_{1}=8$. In both panels, blue and orange markers correspond to inner and outer test points, respectively. Error bars denote the standard error of the mean across 10 independent trained models.