Coherently moving nonabelian anyons in quantum double models: theory and experiment

The ability to coherently move nonabelian anyons is an important prerequisite for fusion-tree-based topological quantum computation. We present new results on the minimal resources needed to move and fuse nonabelian anyons in quantum double models, which are topological code with group-valued qudits on a lattice. Under geometrically local unitary circuits, we argue that moving nonabelian anyons coherently requires a linear-depth circuit and an ancilla qudit. However, the resource costs can be reduced by allowing measurement and k-local gates. With measurement, solvable anyons can be moved using finite-depth adaptive circuits. Notably, with k-local unitary circuits under mild conditions, nonabelian anyons can be moved without using ancillas, countering the conventional wisdom that an ancilla is needed. In contrast to moving, fusing anyons is argued to require an ancilla even with k-local unitary circuits, separating the resource complexity of moving anyons from that of fusing anyons. We also comment on applications of coherent moving protocols in the recent experimental work by Iqbal et al on the realization of S3 quantum double and demonstration of universal gate set by braiding and fusion of nonabelian anyons.