Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix

Abstract

Learning about physical systems from quantum-enhanced experiments can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for state learning, quantum process learning is less understood. We establish an exponential quantum advantage for learning an unknown n -qubit quantum process \(\mathcal {N}\) . We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix (PTM) of an arbitrary \(\mathcal {N}\) , (b) predicting expectation values of Pauli-sparse observables measured on the output of an arbitrary \(\mathcal {N}\) upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary observables measured on the output of an unknown \(\mathcal {N}\) with sparse PTM 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 \(\mathcal {N}\) . In contrast, any learner without quantum memory requires exponentially-in- n many queries, even when using adaptively designed experiments. In proving this separation, we extend existing shadow tomography bounds from states to channels. Moreover, we combine PTM learning with polynomial interpolation to learn arbitrary Hamiltonians from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.