A Unified Blockwise Measurement Design for Learning Quantum Channels and Lindbladians via Low-Rank Matrix Sensing

Quantum superoperator learning is a pivotal task in quantum information science, enabling accurate reconstruction of unknown quantum operations from measurement data. We propose a robust approach based on the matrix sensing techniques for quantum superoperator learning that extends beyond the positive semidefinite case, encompassing both quantum channels and Lindbladians. We first introduce a randomized measurement design using a near-optimal number of measurements. By leveraging the restricted isometry property (RIP), we provide theoretical guarantees for the identifiability and recovery of low-rank superoperators in the presence of noise. Additionally, we propose a blockwise measurement design that restricts the tomography to the sub-blocks, significantly enhancing performance while maintaining a comparable scale of measurements. We also provide a performance guarantee for this setup. Our approach employs alternating least squares (ALS) with acceleration for optimization in matrix sensing. Numerical experiments validate the efficiency and scalability of the proposed methods.