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arXiv:2206.09893 · Scientific Reports

Variational quantum and quantum-inspired clustering

Pablo Bermejo, Román Orús

Published Aug 16, 2023• 19 citations
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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 devices. The idea of the algorithm relies on reducing the clustering problem to an optimization, and then solving it via a Variational Quantum Eigensolver 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.

DOI
10.1038/s41598-023-39771-6

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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 devices. The idea of the algorithm relies on reducing the clustering problem to an optimization, and then solving it via a Variational Quantum Eigensolver 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.
This article introduces a novel approach to perform the simulation of a single qubit quantum-inspired algorithm using laser beams. Leveraging the polarization states of photonic qubits, and inspired by variational quantum eigensolvers, we develop a variational quantum-inspired algorithm implementing a clustering procedure following the approach proposed by some of us in SciRep 13, 13284 (2023). A key aspect of our research involves the utilization of non-orthogonal states within the photonic domain, harnessing the potential of polarization schemes to reproduce unitary circuits. By mapping these non-orthogonal states into polarization states, we achieve an efficient and versatile quantum information processing unit which serves as a clustering device for a diverse set of datasets.
Current universal quantum computers have a limited number of noisy qubits. Because of this, it is difficult to use them to solve large-scale complex optimization problems. In this paper we tackle this issue by proposing a quantum optimization scheme where discrete classical variables are encoded in non-orthogonal states of the quantum system. We develop the case of non-orthogonal qubit states, with individual qubits on the quantum computer handling more than one bit classical variable. Combining this idea with Variational Quantum Eigensolvers (VQE) and quantum state tomography, we show that it is possible to significantly reduce the number of qubits required by quantum hardware to solve complex optimization problems. We benchmark our algorithm by successfully optimizing a polynomial of degree 8 and 15 variables using only 15 qubits. Our proposal opens the path towards solving real-life useful optimization problems in today's limited quantum hardware.
Abstract Noisy intermediate-scale quantum (NISQ) devices are cutting-edge technology expected to demonstrate potential and advantages of quantum computing over classical computing. Its low number of qubits and imperfection from noises restrict running full-scale quantum algorithms on such devices; however, quantum advantages can still be obtained. To achieve quantum advantages from NISQ devices, the hybrid quantum-classical algorithms were introduced. Quantum approximate optimization algorithm (QAOA) is a variational hybrid algorithm, which utilizes a NISQ device as a sub-unit for specific tasks and performs most calculations on a classical computer. QAOA provides an approximate solution, with arbitrary precision as the number of operations increases, for optimization problems. In this work we investigate the possibility of applying QAOA to a clustering problem and compare its performance with the classical k-means algorithm. It turns out that the weights in graph connectivity can degrade the algorithm operation and make it more difficult to approximate the solution. We also benchmark the QAOA by comparing the approximated solutions with the exact one obtained from a classical clustering algorithm.
Current quantum hardware prohibits any direct use of large classical datasets. Coresets allow for a succinct description of these large datasets and their solution in a computational task is competitive with the solution on the original dataset. The method of combining coresets with small quantum computers to solve a given task that requires a large number of data points was first introduced by Harrow [arXiv:2004.00026]. In this paper, we apply the coreset method in three different well-studied classical machine learning problems, namely Divisive Clustering, 3-means Clustering, and Gaussian Mixture Model Clustering. We provide a Hamiltonian formulation of the aforementioned problems for which the number of qubits scales linearly with the size of the coreset. Then, we evaluate how the variational quantum eigensolver (VQE) performs on these problems and demonstrate the practical efficiency of coresets when used along with a small quantum computer. We perform noiseless simulations on instances of sizes up to 25 qubits on CUDA Quantum and show that our approach provides comparable performance to classical solvers.
While full-scale quantum computers remain years away from practical deployment, quantum-inspired algorithms executing on classical hardware have emerged as a promising approach for solving computationally intractable problems in large-scale data analytics. This paper presents a comprehensive framework for applying quantum-inspired tensor network algorithms to complex data problems including high-dimensional clustering, feature selection, and recommendation system matrix factorization. We introduce the Quantum-Inspired Algorithm Library (QIAL), a scalable implementation that seamlessly integrates with Apache Spark and other big data frameworks. Through extensive experimentation on datasets ranging from 1 million to100\mathbf{1 0 0}100million records, we demonstrate that our approach achieves 2.6x speedup compared to classical algorithms while maintaining superior accuracy (average improvement of 2.1% across five benchmark datasets) and reduced memory footprint (up to52%\mathbf{5 2 \%}52%reduction). Our tensor network-based clustering algorithm shows particular promise for NP-hard combinatorial optimization problems that are currently intractable for conventional big data systems. The proposed framework opens new avenues for practical quantum-inspired computing in enterprise-scale data analytics applications.
The advent of noisy-intermediate scale quantum computers has introduced the exciting possibility of achieving quantum speedups in machine learning tasks. These devices, however, are composed of a small number of qubits and can faithfully run only short circuits. This puts many proposed approaches for quantum machine learning beyond currently available devices. We address the problem of compressing classical data into efficient representations on quantum devices. Our proposed methods allow both the required number of qubits and depth of the quantum circuit to be tuned. We achieve this by using a correspondence between matrix-product states and quantum circuits and further propose a hardware-efficient quantum circuit approach, which we benchmark on the Fashion-MNIST dataset. Finally, we demonstrate that a quantum circuit-based classifier can achieve competitive accuracy with current tensor learning methods using only 11 qubits.