Entanglement Transitions in Noisy Quantum Circuits on Trees

Decoherence is ubiquitous, and poses a significant impediment to the observation of quantum phenomena, such as the measurement-induced entanglement phase transition (MIPT). In this work, we study entanglement transitions in quantum circuits on trees, subject to both noise and measurements. We uncover a rich phase diagram that describes the ability of a tree quantum circuit to retain quantum or classical information in the presence of decoherence. By developing a mapping between the dynamics of information on the tree to a classical Markov process -- also defined on the tree -- we obtain exact solutions to the entanglement transitions displayed by the circuit under various noise and measurement strengths. Moreover, we find a host of phenomena, including the MIPT, which are \textit{robust} to decoherence. The analytical tractability facilitated by the method developed in this paper showcases the first example of an exactly solvable noise-robust MIPT, and holds promise for studies on broader, tree-like circuits.