  Proposals to measure non-Abelian anyons in a superconductor by quantum
interference of vortices suffer from the predominantly classical dynamics of
the normal core of an Abrikosov vortex. We show how to avoid this obstruction
using coreless Josephson vortices, for which the quantum dynamics has been
demonstrated experimentally. The interferometer is a flux qubit in a Josephson
junction circuit, which can nondestructively read out a topological qubit
stored in a pair of anyons --- even though the Josephson vortices themselves
are not anyons. The flux qubit does not couple to intra-vortex excitations,
thereby removing the dominant restriction on the operating temperature of
anyonic interferometry in superconductors.
