Quantum learning and universal quantum matching machine

Abstract

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 \textit{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 \textit{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 is a semiclassical strategy which 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 (\textit{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 {\it 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 which is strictly superior to the straightforward semiclassical extension of the conventional matching strategy based on the learning process.

11 pages, RevTeX, 3 figures

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