Quantum k -medoids algorithm using parallel amplitude estimation

Abstract

Quantum computing is a promising new paradigm that can provide viable solutions to highcomplexity problems. k-medoids algorithm is a powerful clustering method ubiquitously used in data mining, image processing, pattern recognition, etc. The core of k-medoids is to perform cluster assignment and centre update, which are time-consuming for large datasets. Aïmeur et al. proposed a quantum k-medoids algorithm [E. Aïmeur, G. Brassard, and S. Gambs, Machine Learning 90, 261 (2013)] by quantizing the centre update. Nevertheless, it has a query complexity O(N 3/2 ) for one iteration, which is computationally expensive for a large N where N is the number of points. In this paper, we propose a complete quantum algorithm for k-medoids. Specifically, in cluster assignment, we devise a quantum subroutine to calculate the Manhattan distance between any two points and then assign all points to the closest centre in parallel, which is faster than what is achievable classically. In centre update, for a cluster, we use parallel amplitude estimation to calculate the average distance of each point to all the others. It makes our algorithm polynomially faster than Aïmeur et al.'s algorithm whose sum of distances of each point to all the others is computed by adding the distances one by one. Our quantum k-medoids algorithm, with time complexity O(N 1/2 ), achieves a polynomial speedup in N compared to the existing one.