Active Learning of Hamiltonians

Querying a quantum system produces measurement outcomes which originate from the Hamiltonian dynamics of the system and its environment. Learning a Hamiltonian from a class that best fits these observations is the Hamiltonian tomography problem. Prior work has focused on estimation and offline optimal experiment design. Here, we consider an active learner that is given an initial set of training examples and the ability to interactively query the quantum system to generate new training data. The goal is to then minimize the training data required for Hamiltonian learning. To this end, we present an efficient active learning algorithm based on Fisher information and assess its performance on recalibrating superconducting qubit systems based on the cross-resonance (CR) gate. The CR Hamiltonian has up to nine parameters and admits queries involving input state, measurement operator and interaction time. Practical challenges include exponential growth of number of parameters with number of qubits, and modeling different noise sources such as readout, and imperfect pulse shapes. We show that we can achieve a constant 30% reduction in queries compared to a uniformly random approach. We also describe a regime where we achieve Heisenberg limited convergence rate during learning.