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arXiv:2604.03550 · Quantum Science and Technology

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

Hui Yu, Jiangping Hu, Shi-Xin Zhang

quant-phSubmitted Apr 4, 2026• Announced Apr 7, 2026• 0 citations
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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.

Comments
9 pages,5 figures

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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.
Open quantum systems have been shown to host a plethora of exotic dynamical phases. Measurement-induced entanglement phase transitions in monitored quantum systems are a striking example of this phenomena. However, naive realizations of such phase transitions requires an exponential number of repetitions of the experiment which is practically unfeasible on large systems. Recently, it has been proposed that these phase transitions can be probed locally via entangling reference qubits and studying their purification dynamics. In this work, we leverage modern machine learning tools to devise a neural network decoder to determine the state of the reference qubits conditioned on the measurement outcomes. We show that the entanglement phase transition manifests itself as a stark change in the learnability of the decoder function. We study the complexity and scalability of this approach in both Clifford and Haar random circuits and discuss how it can be utilized to detect entanglement phase transitions in generic experiments.
Measurement-induced phase transitions (MIPTs) epitomize new intellectual pursuits inspired by the advent of quantum hardware and the emergence of discrete and programmable circuit dynamics. Nevertheless, experimentally observing this transition is challenging, often requiring non-scalable protocols, such as post-selecting measurement trajectories or relying on classical simulations. We introduce a scalable data-centric approach using Quantum Attention Networks (QuAN) to detect MIPTs without requiring post-selection or classical simulation. Applying QuAN to dynamics generated by Haar random unitaries and weak measurements, we first demonstrate that it can pinpoint MIPTs using their interpretation as "learnability" transitions, where it becomes possible to distinguish two different initial states from the measurement record, locating a phase boundary consistent with exact results. Motivated by sample efficiency, we consider an alternative "phase recognition" task-classifying weak- and strong-monitoring data generated from a single initial state. We find QuAN can provide an efficient and noise-tolerant upper bound on the MIPT based on measurement data alone by coupling Born-distribution-level (inter-trajectory) and dynamical (temporal) attention. In particular, our inspection of the inter-trajectory scores of the model trained with minimal sample size processing test data confirmed that QuAN paid special attention to the tail of the distribution of the Born probabilities at early times. This reassuring interpretation of QuAN's learning implies the phase-recognition approach can meaningfully signal MIPT in an experimentally accessible manner. Our results lay the groundwork for observing MIPT on near-term quantum hardware and highlight attention-based architectures as powerful tools for learning complex quantum dynamics.
Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE--the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity, our method reveals a learnability transition with increasing circuit depth: below a threshold the MIE can be effectively learned with resources that grow only polynomially with system size, whereas above it the required resources grow exponentially. This computational phase transition coincides with the breakdown of efficient classical simulation of the underlying quantum state. We further observe signatures of this transition on current noisy quantum devices. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.
Abstract We explore the interplay of quantum computing and machine learning to advance experimental protocols for observing measurement-induced phase transitions (MIPTs) in quantum devices. In particular, we focus on trapped ion monitored circuits and apply the cross entropy benchmark recently introduced by Li et al (2023 Phys. Rev. Lett. 130 220404), which can mitigate the post-selection problem. By doing so, we reduce the number of projective measurements—the sample complexity—required per random circuit realization, which is a critical limiting resource in real devices. Since these projective measurement outcomes form a classical probability distribution, they are suitable for learning with a standard machine learning generative model. In this paper, we use a recurrent neural network to learn a representation of the measurement record for a native trapped-ion MIPT, and show that using this generative model can substantially reduce the number of measurements required to accurately estimate the cross entropy. This illustrates the potential of combining quantum computing and machine learning to overcome practical challenges in realizing quantum experiments.
Measurements can drive quantum many-body systems into nontrivial steady states and induce interesting dynamical phase transitions, rendering measurement-only quantum circuits a useful platform for exploring quantum many-body phases beyond those of equilibrium Hamiltonian systems. Here we study a class of long-range measurement-only quantum circuits with competing two-qubit and three-qubit measurements. We demonstrate that these circuits exhibit rich steady-state structure and uncover a strong influence of the measurement range on the resulting phases. In particular, states with symmetry-protected topological (SPT) order can emerge with sufficiently short-range measurements beyond the nearest-neighbor limit. These states feature robust topological edge modes, which can also be detected from circuit dynamics. With longer-range measurements, an extended parameter regime emerges in which conventional order parameters are suppressed while spatial correlations remain nontrivial. Moreover, we show that in this circuit model sufficiently long-range measurements can produce significant entanglement with scaling beyond an area law despite the absence of any unitary evolution.
Measurement-induced entanglement phase transitions in monitored quantum circuits have stimulated activity in a diverse research community. However, the study of measurement-induced dynamics, due to the requirement of exponentially complex postselection, has been experimentally limited to small or specially designed systems that can be efficiently simulated classically. We present a solution to this outstanding problem by introducing a scalable protocol in U(1)U(1)U(1) symmetric circuits that facilitates the observation of entanglement phase transitions \emph{directly} from experimental data, without detailed assumptions of the underlying model or benchmarking with simulated data. Thus, the method is applicable to circuits which do not admit efficient classical simulation and allows a reconstruction of the full entanglement entropy curve with minimal theoretical input. Our approach relies on adaptive circuits and a steering protocol to approximate pure-state trajectories with mixed ensembles, from which one can efficiently filter out the subsystem U(1)U(1)U(1) charge fluctuations of the target trajectory to obtain its entanglement entropy. The steering protocol replaces the exponential costs of postselection and state tomography with a scalable overhead which, for fixed accuracy εεε and circuit size LLL, scales as Ns∼L5/2/ε\mathcal{N}_s\sim L^{5/2}/εNs​∼L5/2/ε.