Quantum similarity estimation with hybrid discrete-continuous encoding: algorithm and application to atomtronics precision measurements

We introduce a quantum similarity estimation algorithm which employs an entangled discrete-continuous representation and a modified SWAP-test protocol using conditional displacement and controlled-SWAP operations. The corresponding exact form of the similarity metric is analytically derived and its feasible circuit implementation is provided. We apply this result to interferometric images, which are numerically generated from an experimentally viable atomtronics setup. A ring-shaped potential, confining a localized Bose-Einstein condensate (BEC), is considered where the BEC manifests fractional revivals during its confined time-evolution. We perform a precise identification of nonlinearity, originated from interatomic interactions, which can be tuned by Feshbach resonance and thus, the presented method paves the possibility for a magnetic field sensor. Moreover, we perform quantum simulation with Shot-noise of the proposed quantum circuit with public benchmark datasets and analyse the resource and computational complexities of the proposed algorithm.