Shortest Route to Non-Abelian Topological Order on a Quantum Processor

A highly coveted goal is to realize emergent non-Abelian gauge theories and their anyonic excitations, which encode decoherence-free quantum information. While measurements in quantum devices provide new hope for scalably preparing such long-range entangled states, existing protocols using the experimentally established ingredients of a finite-depth circuit and a single round of measurement produce only Abelian states. Surprisingly, we show there exists a broad family of non-Abelian states- namely those with a Lagrangian subgroup-which can be created using these same minimal ingredients, bypassing the need for new resources such as feed forward. To illustrate that this provides realistic protocols, we show how D4 non-Abelian topological order can be realized, e.g., on Google's quantum processors using a depth-11 circuit and a single layer of measurements. Our work opens the way toward the realization and manipulation of non-Abelian topological orders, and highlights counterintuitive features of the complexity of non-Abelian phases.