Quantum learning and universal quantum matching machine

Suppose that three kinds of quantum systems are given in some unknown states $|f{〉}^{\otimes N},$ $|g_1{〉}^{\otimes K},$ and $|g_2{〉}^{\otimes K},$ and we want to decide which template state $|g_1〉$ or $|g_2〉,$ each representing the feature of the pattern class ${\mathcal{C}}_1$ or ${\mathcal{C}}_2,$ respectively, is closest to the input feature state $|f〉.$ This is an extension of the pattern matching problem into the quantum domain. Assuming that these states are known a priori to belong to a certain parametric family of pure qubit systems, we derive two kinds of matching strategies. The first one is a semiclassical strategy that is obtained by the natural extension of conventional matching strategies and consists of a two-stage procedure: identification (estimation) of the unknown template states to design the classifier (learning process to train the classifier) and classification of the input system into the appropriate pattern class based on the estimated results. The other is a fully quantum strategy without any intermediate measurement, which we might call as the universal quantum matching machine. We present the Bayes optimal solutions for both strategies in the case of $K=1,$ showing that there certainly exists a fully quantum matching procedure that is strictly superior to the straightforward semiclassical extension of the conventional matching strategy based on the learning process.