  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.
