Observing a purification phase transition with a trapped ion quantum computer*

Many-body open quantum systems balance internal dynamics against decoherence from interactions with an environment. Here, we explore this balance via random quantum circuits implemented on a trapped-ion quantum computer, where the system evolution is represented by unitary gates with interspersed projective measurements. As the measurement rate is varied, a purification phase transition is predicted to emerge at a critical point akin to a fault-tolerent threshold. We probe the "pure'' phase, where the system is rapidly projected to a deterministic state conditioned on the measurement outcomes, and the "mixed'' or "coding'' phase, where the initial state becomes partially encoded into a quantum error correcting codespace.  We find evidence of the two phases and show numerically that, with modest system scaling, critical properties of the transition emerge.