Open quantum systems, interacting with their environment, exhibit a rich variety of dynamical phases, a notable example being measurement-induced entanglement phase transitions (MIPTs). In these systems, the interplay between unitary evolution and quantum measurements leads to a phase transition between a pure phase (area-law entanglement) and a mixed phase (volume-law entanglement). Experimentally probing these transitions using traditional methods is challenging because it requires an exponentially large (in system size) number of repeated experiments to reconstruct the density matrix conditioned on measurement outcomes. This exponential scaling arises from the intrinsic randomness in the measurement outcomes. The need to average over exponentially many measurement outcomes poses a significant obstacle for experimental verification of MIPTs in large systems. Recent work has shown that MIPTs can be probed locally via the purification dynamics of entangled reference qubits. By entangling a reference qubit with a subsystem of the main system and monitoring its entanglement entropy, the phase transition can be identified. However, this approach still relies on calculating the density matrix of the reference qubits conditioned on the measurement outcomes of the main system which can be computationally complex. In the past, such decoders have only been analytically derived and implemented for specific circuit types like stabilizer circuits. This study leverages machine learning techniques, specifically neural networks, to overcome these challenges. The authors propose a neural network decoder to efficiently learn the mapping from measurement outcomes to the reference qubit's density matrix. The central hypothesis is that the efficiency of this neural network decoder, in terms of the number of training samples required to reach a certain accuracy, will change drastically at the phase transition point. This approach allows for probing MIPTs in more general and larger quantum systems.

The paper builds upon previous research demonstrating the existence and characteristics of MIPTs in various quantum systems. Studies on measurement-induced entanglement transitions in monitored quantum systems using random quantum circuits with interspersed measurements have highlighted the competition between unitary and non-unitary dynamics leading to area-law and volume-law entanglement behavior. The close connection between MIPTs and quantum error correction has been established, offering a framework for understanding the underlying mechanisms. The concept of using purification dynamics of ancilla reference qubits to probe these transitions, originally introduced in Ref. [38], serves as a crucial foundation for this work. However, existing analytical decoders were limited to specific circuit classes, highlighting the need for a more general approach. The paper also draws from recent successful applications of machine learning in quantum science, particularly in quantum error correction and decoder optimization, to motivate the proposed neural network approach.

The authors employ a brickwork structure for their quantum circuits, consisting of L qubits and seven time steps. Each time step involves layers of two-qubit random unitary gates (Clifford gates for a significant part of the analysis, later extending to Haar random gates) followed by single-site Pauli-Z measurements with probability p. The measurement rate p serves as a control parameter, inducing a phase transition between volume-law and area-law entanglement as it crosses a critical value pc. A reference qubit is entangled with a central qubit in the system. The goal is to determine the state of this reference qubit (represented by its Bloch vector components σx, σy, σz) given the measurement outcomes M1 from the main circuit. This can be formulated as a decoder function F_c(M1) = (σx, σy, σz). Instead of analytically solving for this decoder function, the authors utilize a machine learning approach. They frame the problem as a probabilistic classification task, where a neural network is trained to predict the reference qubit's state (or, for Clifford circuits, the measurement outcome along the purification axis) given the measurement outcomes M1 as input. The neural network architecture used is a convolutional neural network (CNN), chosen for its efficiency in processing spatially local features. The training data consists of pairs (M1, m), where M1 represents the measurement record and m is the reference qubit's measurement outcome. The paper investigates the complexity of learning the decoder function by analyzing the number of circuit runs (N) required to train the neural network to a specified accuracy. It considers three learning schemes: (1) Conditional learning, where circuits are selected based on their purification time tp, and only measurements within a time window are used; (2) Light-cone learning, where only measurement outcomes within a light cone around the reference qubit are considered; and (3) Unconditional learning, where all measurement outcomes are used regardless of the purification time. The analysis focuses on characterizing the minimum number of training samples M required to reach a given error ε, and how this depends on the system parameters (p, L, tp). The effectiveness of the method is first demonstrated using numerical simulations of Clifford circuits, followed by generalization to Haar random circuits using density matrix tomography. The analysis of the temporal behavior of the learnability allows the estimation of the critical properties of the phase transition. The decay rate of the average entanglement entropy of the reference qubit, S0(t), is analyzed, using a scaling form based on critical exponents z and v, and pc is estimated by data collapse using these exponents.

The key findings of the paper demonstrate the successful application of neural network decoders for efficient probing of measurement-induced phase transitions. 1. **Learnability Phase Transition:** The paper shows a clear correlation between the learnability of the neural network decoder and the measurement-induced phase transition. The difficulty of training the neural network changes significantly around the critical point pc, marking the transition between the area-law and volume-law phases. 2. **Complexity Analysis:** The authors perform a detailed analysis of the complexity of the learning task. They observe that the number of training samples needed scales exponentially with the purification time tp in both the pure and mixed phases. This scaling is less severe when only measurement outcomes within the light-cone of the reference qubit are considered. For a fixed purification time, the required number of samples appears largely independent of the system size for larger system sizes. 3. **Critical Exponent Estimation:** By analyzing the temporal decay rate of the average entanglement entropy of the reference qubit, the authors estimate critical exponents characterizing the phase transition. The obtained estimates of pc, v, and z are consistent, within error margins, with those obtained from other methods, although more precise measurements are needed to distinguish this transition from the percolation transition. 4. **Scalability:** The paper demonstrates the scalability of the neural network decoder. Training the network on smaller circuits effectively predicts the behavior of larger circuits. This is consistent with the approximate locality of the purification dynamics, which is captured in the light-cone learning scheme. This implies that the decoder does not need the entire history of measurement, only that within the light-cone. 5. **Generalization to Haar Random Circuits:** The method is generalized to Haar random circuits, showing that the neural network decoder can be adapted to learn the density matrix of the reference qubit even without the simplifying features of Clifford circuits. The observed behaviour in the averaged entanglement entropy for Haar circuits further supports the existence of the two phases.

This work significantly advances the experimental accessibility of MIPTs by offering a scalable and efficient method. The use of neural networks to decode the reference qubit's state based on measurement outcomes overcomes the exponential scaling limitations of traditional approaches. The strong correlation between the learnability of the decoder and the phase transition provides a practical and potentially experimentally realizable way to identify the critical point. The scalability demonstrated through the light-cone learning scheme is crucial for applying the method to large systems. The results on Haar random circuits show the general applicability of the approach beyond the simpler case of Clifford circuits. The findings contribute to a deeper understanding of MIPTs and their connection to quantum error correction. The method provides a new tool for probing the critical properties of these transitions, offering insights into their dynamical exponents. This approach could have broad implications for understanding open quantum systems and developing quantum technologies.

This study presents a novel approach for probing measurement-induced phase transitions using neural network decoders. The method successfully addresses the exponential scaling limitations of traditional approaches by efficiently learning the mapping between measurement outcomes and the reference qubit's state. The observed learnability phase transition and the estimation of critical exponents demonstrate the effectiveness of the approach. The scalability and generalization to Haar random circuits highlight the broad applicability of this method for studying MIPTs in diverse quantum systems. Future research could focus on improving the neural network architecture for even greater efficiency, exploring applications to other types of MIPTs, incorporating experimental noise models, and investigating the connection to unsupervised learning techniques.

The accuracy of the critical exponent estimations relies on the assumptions of the scaling form and the underlying conformal symmetry. The analysis uses approximate methods for data collapse, introducing uncertainties in the values of the critical exponents. Additionally, the efficiency of the neural network decoder may depend on the specific architecture and training parameters. The computational cost of training the neural network can still be significant for very large systems, although the light-cone learning approach significantly mitigates this.