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

Abstract

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.

Figures (4)

• Figure 1: (A) Schematic illustration of the symmetry-constrained variational quantum circuit optimization setup. (B) Evolution of the order parameter $\chi$ as a function of training iterations. As an inset, the final value of the order parameter. (C) Spectral support of a trained state in the energy eigenbasis.
• Figure 2: Average post-measurement entanglement entropy of the post-training states as a function of the number of measurements applied to the state for different system sizes $N$. GHZ state for reference, which has the same behavior independent of system size.
• Figure 3: (A) Training curve of the string order parameter $\langle \mathcal{O}_{ij}\rangle$ as a function of the number of epochs for different system sizes $N$. (B) Support of the trained state at $N = 12$ in the energy eigenbasis. Eigenstates colored by their corresponding $\langle \mathcal{O}_{ij}\rangle$ values.
• Figure 4: (A) Spectrum analysis for trained effective Hamiltonians for the SPT case. The inset shows the $4$-fold degeneracies of each eigenstate. (B) Histogram of trained weights. Vertical red lines indicate the angles where the gates are Clifford.