Quantum Circuits as a Dynamical Resource to Learn Nonequilibrium Long-Range Order

Equilibrium statistical ensembles impose stringent constraints on phases of quantum matter. For example, the Mermin-Wagner theorem prohibits long-range order in low-dimensional systems beyond the ground state. Here, we show that quantum circuits can learn states of matter with long-range order that are inaccessible in equilibrium. We construct variational quantum circuits that generate symmetry-broken and symmetry-protected topological states with long-range order in one-dimensional systems at finite energy density, where equilibrium states are typically featureless. Importantly, the learned states can exhibit unconventional features with enhanced metrological properties such as a quantum Fisher information close to a GHZ state, but robust against local measurements. Our work establishes coherent quantum dynamics as a powerful resource for engineering nonequilibrium phases of matter, opening a path toward a broader dynamical scope of quantum order beyond the constraints of equilibrium ensembles.