Slow Relaxation in a Glassy Quantum Circuit

Quantum circuits have become a powerful tool in the study of many-body quantum physics, providing insights into both fast-thermalizing chaotic and non-thermalizing integrable many-body dynamics. In this work, we explore a distinct intermediate class - glassy quantum systems - where thermalization occurs, but over very long timescales. We introduce and analyze a Floquet random quantum circuit that can be tuned between glassy and fully ergodic behavior through a single adjustable parameter. This circuit can be understood as the unitary analog of the block Rosenzweig-Porter model, which is defined by a Hamiltonian. Using an effective field theory for random quantum circuits, we analyze the correlations between quasienergy eigenstates and thereby determine the time evolution of the disorder-averaged density matrix. In the intermediate regime the circuit displays a two-step thermalization process: an initial relaxation within weakly coupled sectors followed by a later, global thermalization. We also show that the ramp of the spectral form factor is enhanced by a factor of the number of sectors in the glassy regime, and at early times in the intermediate regime. These results indicate that quantum circuits provide an ideal platform for the exploration of nontrivial thermalization dynamics in many-body quantum systems, offering deeper insights into quantum thermalization.