Keywords: quantum many-body theory, quantum systems away from equilibrium, geometry of quantum states, quantum phase transitions, quantum adiabaticity, information geometry AbstractThe manifold of ground states of a family of quantum Hamiltonians can be endowed with a quantum geometric tensor whose singularities signal quantum phase transitions and give a general way to define quantum phases. In this paper, we show that the same information-theoretic and geometrical approach can be used to describe the geometry of quantum states away from equilibrium. We construct the quantum geometric tensor Q μν for ensembles of states that evolve in time and study its phase diagram and equilibration properties. If the initial ensemble is the manifold of ground states, we show that the phase diagram is conserved, that the geometric tensor equilibrates after a quantum quench, and that its time behavior is governed by out-of-time-order commutators (OTOCs). We finally demonstrate our results in the exactly solvable Cluster-XY model.

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Learning the unknown Hamiltonian governing the dynamics of a quantum many-body system is a challenging task. In this manuscript, we propose a possible strategy based on repeated measurements on a single time-dependent state. We prove that the accuracy of the learning process is maximized for states that are delocalized in the Hamiltonian eigenbasis. This implies that delocalization is a quantum resource for Hamiltonian learning, that can be exploited to select optimal initial states for learning algorithms. We investigate the error scaling of our reconstruction with respect to the number of measurements, and we provide examples of our learning algorithm on simulated quantum systems.

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