  Recently, tremendous progress has been made in the field of quantum science
and technologies: different platforms for quantum simulation as well as quantum
computing, ranging from superconducting qubits to neutral atoms, are starting
to reach unprecedentedly large systems. In order to benchmark these systems and
gain physical insights, the need for efficient tools to characterize quantum
states arises. The exponential growth of the Hilbert space with system size
renders a full reconstruction of the quantum state prohibitively demanding in
terms of the number of necessary measurements. Here we propose and implement an
efficient scheme for quantum state tomography using active learning. Based on a
few initial measurements, the active learning protocol proposes the next
measurement basis, designed to yield the maximum information gain. We apply the
active learning quantum state tomography scheme to reconstruct different
multi-qubit states with varying degree of entanglement as well as to ground
states of the XXZ model in 1D and a kinetically constrained spin chain. In all
cases, we obtain a significantly improved reconstruction as compared to a
reconstruction based on the exact same number of measurements and measurement
configurations, but with randomly chosen basis configurations. Our scheme is
highly relevant to gain physical insights in quantum many-body systems as well
as for benchmarking and characterizing quantum devices, e.g. for quantum
simulation, and paves the way for scalable adaptive protocols to probe,
prepare, and manipulate quantum systems.
