Efficient Learning Algorithms for Noisy Quantum State and Process Tomography

Abstract

Efficiently characterizing large quantum states and processes is a central yet notoriously challenging task in quantum information science, as conventional tomography methods typically require resources that grow exponentially with system size. Here, we introduce a provably efficient and structure-agnostic learning framework for noisy $n$-qubit quantum circuits under generic noise with arbitrary noise strength. We first develop a sample-efficient learning algorithm for unital noisy quantum states. Building on this result, we extend the framework to quantum process tomography, obtaining a unified protocol applicable to both unital and non-unital channels. The resulting approach is input-agnostic and does not rely on assumptions about specific input distributions. Our theoretical analysis shows that both state and process learning require only polynomially many samples and polynomial classical post-processing in the number of qubits, while achieving near-unit success probability over ensembles generated by local random circuits. Numerical simulations of two-dimensional Hamiltonian dynamics further demonstrate the accuracy and robustness of the approach, including for structured circuits beyond the random-circuit setting assumed in the theoretical analysis. These results provide a scalable and practically relevant route toward characterizing large-scale noisy quantum devices, addressing a key bottleneck in the development of quantum technologies.