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

Abstract: 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 $D_4$ 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 towards the realization and manipulation of non-Abelian topological orders, and highlights counter-intuitive features of the complexity of non-Abelian phases.