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Adaptive quantum circuits, in which unitary operations, measurements, and feedback are used to steer quantum many-body systems, provide an exciting opportunity to generate interesting dynamical steady states. We introduce an adaptive quantum dynamics with continuous symmetry where unitary operations, measurements, and local unitary feedback are used to drive ordering. In this setting, we find a pure steady state hosting symmetry-breaking order, which is the ground state of a gapless, local Hamiltonian. We explore the dynamical properties of the approach to this steady state. We find that this steady-state order is fragile to perturbations, even those that respect the continuous symmetry.
Hauser et al. (2024) studied this question.