Computing scalar products via a two-terminal quantum transmission line

Abstract

The scalar product of two vectors with $K$ real components can be computed using two quantum channels, that is, information transmission lines in the form of spin-1/2 XX chains. Each channel has its own $K$-qubit sender and both channels share a single two-qubit receiver. The $K$ elements of each vector are encoded in the pure single-excitation initial states of the senders. After time evolution, a bi-linear combination of these elements appears in the only matrix element of the second-order coherence matrix of the receiver state. An appropriate local unitary transformation of the extended receiver turns this combination into a renormalized version of the scalar product of the original vectors. The squared absolute value of this scaled scalar product is the intensity of the second-order coherence which consequently can be measured, for instance, employing multiple-quantum NMR. The unitary transformation generating the scalar product of two-element vectors is presented as an example.