Quantum Algorithms for Similarity Measurement Based on Euclidean Distance

Similarity measurement is a fundamental problem that arise both on its own and as a key subroutine in more complex tasks, such as machine learning. However, in classical algorithms, the time used to similarity measurement usually increases exponentially as the amount of data and the number of data dimensions increase. In this paper, we presented three quantum algorithms based on Euclidean distance to measure the similarity between data sets. In the proposed algorithms, some special unitary operations are utilized to construct imperative quantum states from quantum random access memory. Then, a badly needed result for estimating the similarity between data sets, can be got by performing projective measurements. Furthermore, it is shown that these algorithms can achieve the exponential acceleration of the classical algorithm in the quantity or the dimension of data.