Learning measurement-induced phase transitions using attention

Abstract

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.