Recommendation systems with the quantum k-NN and Grover algorithms for data processing

In this article, we discuss the implementation of a quantum recommendation system that uses a quantum variant of the k-nearest neighbours algorithm and the Grover algorithm to search for a specific element in an unstructured database. In addition to the presentation of the recommendation system as an algorithm, the article also shows the main steps in construction of a suitable quantum circuit for realisation of a given recommendation system. The computational complexity of individual calculation steps in the recommendation system is also indicated. The verification of the correctness of the proposed system is analysed as well, indicating an algebraic equation describing the probability of success of the recommendation. The article also shows numerical examples presenting the behaviour of the recommendation system for two selected cases.