Polynomial-time experimental protocol for observation of measurement-induced entanglement phase transition

Recent theoretical work has highlighted a unique phase transition in quantum systems subjected to alternating sequences of entangling gates and randomly spaced local measurements. In the absence of measurements, or when they are sparse, the system evolves into a highly entangled, "volume-law", state, but when the probability of measurement per step passes a certain critical value, the steady state becomes a localized state with "area-law" entanglement. While there has been great interest in theoretical analysis of these transitions, analytical approaches are mostly limited to a small subset of entangling gates, the Clifford group. Experimentally, this transition is difficult to observe: as the entanglement cannot be seen at the ensemble level, so experiments must generally be repeated until the set of measurement results repeats itself, a likelihood that vanishes exponentially with the number of measurements. Here, we present an experimental algorithm that scales polynomially with the number of qubits, allowing us to observe and measure the transition on available NISQ devices. This allows the use of any kind of gates in the entangling layer, expanding the results beyond the currently limited regime that is analytically solvable.