Power and limitations of learning quantum dynamics incoherently

Quantum process learning is emerging as an important tool to study quantum systems. While studied extensively in coherent frameworks, where the target and model system can share quantum information, less attention has been paid to whether the dynamics of quantum systems can be learned without the system and target directly interacting. Such incoherent frameworks are practically appealing since they open up methods of transpiling quantum processes between the different physical platforms without the need for technically challenging hybrid-entanglement schemes. Here, we provide bounds on the sample complexity of learning unitary processes incoherently by analyzing the number of measurements that are required to emulate well-established coherent learning strategies. We prove that if arbitrary measurements are allowed, then any efficiently representable unitary can be efficiently learned within the incoherent framework; however, when restricted to shallow-depth measurements, logarithmic depth unitaries can be learned but there exist linear-depth unitaries that are unlearnable. We demonstrate our incoherent learning algorithm for low-entangling unitaries by successfully learning a 16-qubit unitary on ibmq_kolkata, and further demonstrate the scalability of our proposed algorithm through extensive numerical experiments.