Learning the phases of monitored quantum dynamics - II

In the monitored dynamics with entangling evolution subjected to mid-circuit measurements, a measurement-induced phase transition (MIPT) can be characterized by the learnability of quantum information extracted from the trajectories. The probabilistic nature of measurement makes it challenging to observe MIPT without relying on a non-scalable protocol, such as post-selecting measurement trajectories. We propose a post-selection-free approach that utilizes Quantum Attention Networks (QuAN)[1] to detect MIPT under generic Haar random unitaries and weak measurements. QuAN, which leverages the attention mechanism's power to drive large language models, is an efficient classical machinery to process measurement trajectories. QuAN is designed to access high-order moments of bit-string distribution while maintaining permutation invariance. We demonstrate that QuAN can witness MIPT, predicting the phase boundary that aligns with exact results. A sample complexity study highlights the potential for QuAN to learn MIPT from experimental data without post-selection, as it requires only a small number of samples readily accessible with current experimental platforms.