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Variational Quantum and Quantum-Inspired Clustering

Pablo Bermejo, Roman Orus

TL;DR

A quantum algorithm for clustering data based on a variational quantum circuit that allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum devices.

Abstract

Here we present a quantum algorithm for clustering data based on a variational quantum circuit. The algorithm allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum (NISQ) devices. The idea of the algorithm relies on reducing the clustering problem to an optimization, and then solving it via a Variational Quantum Eigensolver (VQE) combined with non-orthogonal qubit states. In practice, the method uses maximally-orthogonal states of the target Hilbert space instead of the usual computational basis, allowing for a large number of clusters to be considered even with few qubits. We benchmark the algorithm with numerical simulations using real datasets, showing excellent performance even with one single qubit. Moreover, a tensor network simulation of the algorithm implements, by construction, a quantum-inspired clustering algorithm that can run on current classical hardware.

Variational Quantum and Quantum-Inspired Clustering

TL;DR

A quantum algorithm for clustering data based on a variational quantum circuit that allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum devices.

Abstract

Here we present a quantum algorithm for clustering data based on a variational quantum circuit. The algorithm allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum (NISQ) devices. The idea of the algorithm relies on reducing the clustering problem to an optimization, and then solving it via a Variational Quantum Eigensolver (VQE) combined with non-orthogonal qubit states. In practice, the method uses maximally-orthogonal states of the target Hilbert space instead of the usual computational basis, allowing for a large number of clusters to be considered even with few qubits. We benchmark the algorithm with numerical simulations using real datasets, showing excellent performance even with one single qubit. Moreover, a tensor network simulation of the algorithm implements, by construction, a quantum-inspired clustering algorithm that can run on current classical hardware.
Paper Structure (5 equations, 3 figures)

This paper contains 5 equations, 3 figures.

Figures (3)

  • Figure 1: [Color online] Clustering results for one qubit: (a) Iris dataset with three labels, (b) 3 gaussian blobs, (c) 3 gaussian blobs, (d) 4 gaussian blobs. Colors between true test and prediction do not necessarily match, since for the prediction the labelling is generated automatically by the algorithm.
  • Figure 2: [Color online] Clustering results for several qubits: (a) 2 qubits and 2 gaussian blobs, (b) 8 qubits and 6 gaussian blobs, (c) 10 qubits and 5 blobs, (d) 3 qubits and 3 gaussian blobs. Colors between true test and prediction do not necessarily match, since for the prediction the labelling is generated automatically by the algorithm.
  • Figure 3: Three variational quantum circuits for one epoch: (a) one qubit, (b) two qubits, and (3) three qubits. For qubit $i$, the set of variational angles is $\{ \theta_i \}$, and the set of angles for the initial rotations is $\{ \rho_i^{init} \}$.