Quantum Spectral Clustering through a Biased Phase Estimation Algorithm

Abstract

In this brief paper, we go through the theoretical steps of the spectral clustering on quantum computers by employing the phase estimation and the amplitude amplification algorithms. We discuss circuit designs for each step and show how to obtain the clustering solution from the output state. In addition, we introduce a biased version of the phase estimation algorithm which significantly speeds up the amplitude amplification process. The complexity of the whole process is analyzed: it is shown that when the circuit representation of a data matrix of order N is produced through an ancilla based circuit in which the matrix is written as a sum of L number of Householder matrices; the computational complexity is bounded by O(2 m LN ) number of quantum gates. Here, m represents the number of qubits (e.g., 6) involved in the phase register of the phase estimation algorithm.

Quantum density peak clustering

“…Clustering algorithms thus need to separate unlabeled data into different classes (or clusters) without any external labeling and supervision. Solutions to the clustering problem based on quantum computing have been proposed for a long time, resorting to a plethora of different strategies (Aïmeur et al 2007;Yu et al 2010;Li et al 2011;Aïmeur et al 2013;Lloyd et al 2013;Otterbach et al 2017;Bauckhage et al 2017;Daskin 2017;Kerenidis et al 2019;Li and Kais 2021;Kerenidis and Landman 2021;Pires et al 2021). For example, in Ref.…”