Efficient quantum communications with coherent state fingerprints over multiple channels

We provide an example of a communication model and a distributed task, for which there exists a realistic quantum protocol that is asymptotically more efficient than any classical protocol, both in the communication and the information resources. For this, we extend a recently proposed coherent state mapping for quantum communication protocols, study the use of coherent state fingerprints over multiple channels, and show their role in the design of an efficient quantum protocol for estimating the Euclidean distance of two real vectors within a constant factor.