A quantum algorithm of K-means toward practical use

Abstract

Clustering problems in image recognition are recurrent in unsupervised machine learning. The K-means algorithm is a simple and popular algorithm for solving the clustering problem. However, for example, the calculation of the distance between the cluster center and data is time-consuming in the centroid calculation of the K-means algorithm, in particular, for large data sizes. In the present paper, we investigate the possibility of quantum computation to speedup K-means algorithm for large data sizes. We describe a quantum-enhanced K-means algorithm from which centroid calculations are removed. For mean and distance calculations of vector data, we propose a quantum subroutine based on quantum entanglement with a potential speedup to make the speed of the proposed subroutine comparable to its classical counterpart. The proposed K-means algorithm is evaluated on three datasets: synthetic, Iris, and image datasets. The numerical experimental results show that the clustering performance of the proposed algorithm is comparable to that of the classical K-means algorithm.

Appendices

An implementation of the classical K-means algorithm

A classical K-means algorithm can be implemented in Python as follows⁶:

An implementation of Q-MIN

A quantum subroutine qsub_min for Q-MIN for two qubits is implemented in Python and Qiskit [2] as follows:

Detailed derivation of Eq. 8

The state $\vert \psi _{3} \rangle$ is derived as follows:

$$\begin{aligned} \begin{aligned} \vert \psi _{3} \rangle&= \cos {(t)} (I \cos {(t)} + i H^{\otimes m} \sin {(t)}) \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle \\&\quad - i \sin {(t)} \frac{1}{\sqrt{M+1}} (I \cos {(t)} + i H^{\otimes m} \sin {(t)}) \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle \\&= \cos ^{2}{(t)} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle + i \cos {(t)} \sin {(t)} H^{\otimes m} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle \\&\quad - i \frac{\sin {(t)} \cos {(t)}}{\sqrt{M+1}} \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle + \frac{\sin ^{2}{(t)}}{\sqrt{M+1}} H^{\otimes m} \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle \\&= \cos ^{2}{(t)} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle + i \frac{\cos {(t)} \sin {(t)}}{\sqrt{M+1}} \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{0} \rangle \\&\quad - i \frac{\sin {(t)} \cos {(t)}}{\sqrt{M+1}} \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle + \frac{\sin ^{2}{(t)}}{\sqrt{M+1}} \sum _{j = 0}^{M} \frac{1}{\sqrt{M+1}} \sum _{k = 0}^{M} (-1)^{j \cdot k} \vert k \rangle \vert \varvec{x}_{j} \rangle \\&= \cos ^{2}{(t)} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle + i \frac{\cos {(t)} \sin {(t)}}{\sqrt{M+1}} \sum _{j = 0}^{M} \vert j \rangle \vert \varvec{x}_{0} \rangle - i \frac{\sin {(t)} \cos {(t)}}{\sqrt{M+1}} \vert 0 \rangle \vert \varvec{x}_{0} \rangle \\&\quad - i \frac{\sin {(t)} \cos {(t)}}{\sqrt{M+1}} \sum _{j = 1}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle + \frac{\sin ^{2}{(t)}}{M+1} \sum _{j = 0}^{M} \vert 0 \rangle \vert \varvec{x}_{j} \rangle \\&\quad + \frac{\sin ^{2}{(t)}}{M+1} \sum _{j = 0}^{M} \sum _{k = 1}^{M} (-1)^{j \cdot k} \vert k \rangle \vert \varvec{x}_{j} \rangle \\&= \cos ^{2}{(t)} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle + i \frac{\cos {(t)} \sin {(t)}}{\sqrt{M+1}} \sum _{j = 1}^{M} \vert j \rangle \vert \varvec{x}_{0} \rangle \\&\quad - i \frac{\sin {(t)} \cos {(t)}}{\sqrt{M+1}} \sum _{j = 1}^{M} \vert j \rangle \vert \varvec{x}_{j} \rangle + \frac{\sin ^{2}{(t)}}{M+1} \sum _{j = 0}^{M} \vert 0 \rangle \vert \varvec{x}_{j} \rangle + \frac{\sin ^{2}{(t)}}{M+1} \sum _{j = 0}^{M} \sum _{k = 1}^{M} (-1)^{j \cdot k} \vert k \rangle \vert \varvec{x}_{j} \rangle \\&= \left( \cos ^{2}{(t)} + \frac{\sin ^{2}{(t)}}{M + 1} \right) \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{0} \rangle + \frac{\sin ^{2}{(t)}}{M + 1} \sum _{j = 1}^{M} \vert 0 \rangle ^{\otimes m} \vert \varvec{x}_{j} \rangle \\&\quad + i \frac{\cos {(t)} \sin {(t)}}{\sqrt{M + 1}} \sum _{j = 1}^{M} \vert j \rangle (\vert \varvec{x}_{0} \rangle - \vert \varvec{x}_{j} \rangle ) + \frac{\sin ^{2}{(t)}}{M + 1} \sum _{j = 0}^{M} \sum _{k = 1}^{M} (-1)^{j \cdot k} \vert k \rangle \vert \varvec{x}_{j} \rangle . \end{aligned} \end{aligned}$$

(13)

Implementation of the proposed algorithm

A quantum subroutine qsub_md for the proposed algorithm for $m = 2$ and $d = 2$ is implemented in Python and Qiskit as follows:

Fig. 10

Clustering results for the synthetic dataset

Fig. 11

Clustering results for the Iris dataset

Fig. 12

Clustering results of C-KM for the image dataset

Fig. 13

Clustering results of Q-KM for the image dataset

Fig. 14

Clustering results of Q-KM* for the image dataset

Implementation of the proposed quantum-enhanced K-means algorithm

Using the quantum subroutines (qsub_md and qsub_min), the proposed algorithm is implemented as follows:

Other results for synthetic, Iris, and image datasets