Hierarchy of Symmetric Phases in Random Hybrid Quantum Circuits

Measurement-induced phase transitions are novel non-equilibrium phenomena that have yielded fruitful insights into the role of measurement in quantum information. Distinct phases reflect different ensembles of state entanglement structures emerging from individual trajectories through a quantum circuit. Of interest is the relationship between the properties of these phases and resource states considered in measurement-based quantum computing. The robustness of the latters' entanglement structures, which typically reflect (subsystem) topological symmetry, have been considered in a variety of ways in the past. Recently it was shown that the 1d cluster state belongs to a distinguishable phase in a random hybrid quantum circuit ensemble. The phase transition exhibits the same percolation universality as that occurring between generic unitary dynamics and measurements. To probe the sensitivity of the transition to the presence symmetry, we investigate the 2d cluster state for three different levels of symmetry-respecting unitary dynamics interspersed by measurements: arbitrary 5-qubit Clifford unitaries, symmetry-respecting-, and subsystem-symmetry respecting Cliffords. Our work on these random hybrid circuit models may have implications for hierarchies of transitions in computational power.