Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix

Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for learning an unknown n-qubit quantum process N. We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix of an arbitrary N, (b) predicting expectation values of bounded Pauli-sparse observables measured on the output of an arbitrary N upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown N with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in-n many copies of the Choi state of N, and even time-efficiently in the case of (b). In contrast, any learner without quantum memory requires exponentially-in-n many queries, even when querying N on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have non-local all-to-all interactions, from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.