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arXiv:2206.07852

Performance analysis of coreset selection for quantum implementation of K-Means clustering algorithm

Fanzhe Qu, Sarah Erfani, Muhammad Usman

Published Jun 16, 2022• 0 citations
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Abstract

Quantum computing is anticipated to offer immense computational capabilities which could provide efficient solutions to many data science problems. However, the current generation of quantum devices are small and noisy, which makes it difficult to process large data sets relevant for practical problems. Coreset selection aims to circumvent this problem by reducing the size of input data without compromising the accuracy. Recent work has shown that coreset selection can help to implement quantum K-Means clustering problem. However, the impact of coreset selection on the performance of quantum K-Means clustering has not been explored. In this work, we compare the relative performance of two coreset techniques (BFL16 and ONESHOT), and the size of coreset construction in each case, with respect to a variety of data sets and layout the advantages and limitations of coreset selection in implementing quantum algorithms. We also investigated the effect of depolarisation quantum noise and bit-flip error, and implemented the Quantum AutoEncoder technique for surpassing the noise effect. Our work provides useful insights for future implementation of data science algorithms on near-term quantum devices where problem size has been reduced by coreset selection.

DOI
10.48550/arxiv.2206.07852

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Quantum computing is anticipated to offer immense computational capabilities which could provide efficient solutions to many data science problems. However, the current generation of quantum devices are small and noisy, which makes it difficult to process large data sets relevant for practical problems. Coreset selection aims to circumvent this problem by reducing the size of input data without compromising the accuracy. Recent work has shown that coreset selection can help to implement quantum K-Means clustering problem. However, the impact of coreset selection on the performance of quantum K-Means clustering has not been explored. In this work, we compare the relative performance of two coreset techniques (BFL16 and ONESHOT), and the size of coreset construction in each case, with respect to a variety of data sets and layout the advantages and limitations of coreset selection in implementing quantum algorithms. We also investigated the effect of depolarisation quantum noise and bit-flip error, and implemented the Quantum AutoEncoder technique for surpassing the noise effect. Our work provides useful insights for future implementation of data science algorithms on near-term quantum devices where problem size has been reduced by coreset selection.
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Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigated using this paradigm to perform k-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We used numerical simulations to compare the performance of this approach to classical k-means clustering. We were able to find data sets with which coresets work well relative to random sampling and where QAOA could potentially outperform standard k-means on a coreset. However, finding data sets where both coresets and QAOA work well—which is necessary for a quantum advantage over k-means on the entire data set—appears to be challenging.
kkk-Clustering in Rd\mathbb{R}^dRd (e.g., kkk-median and kkk-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality nnn, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for kkk-clustering in Rd\mathbb{R}^dRd with O~(nkd3/2)\tilde{O}(\sqrt{nk}d^{3/2})O~(nk​d3/2) query complexity. Our coreset reduces the input size from nnn to poly(kε−1d)\mathrm{poly}(kε^{-1}d)poly(kε−1d), so that existing ααα-approximation algorithms for clustering can run on top of it and yield (1+ε)α(1 + ε)α(1+ε)α-approximation. This eventually yields a quadratic speedup for various kkk-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Ω(nk)Ω(\sqrt{nk})Ω(nk​) queries in order to achieve even O(1)O(1)O(1)-approximation for kkk-clustering.
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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.
Abstract Recent work has proposed solving the k‐means clustering problem on quantum computers via the Quantum Approximate Optimization Algorithm (QAOA) and coreset techniques. Although the current method demonstrates the possibility of quantum k‐means clustering, it does not ensure high accuracy and consistency across a wide range of datasets. The existing coreset techniques are designed for classical algorithms, and there is no quantum‐tailored coreset technique designed to boost the accuracy of quantum algorithms. This study proposes solving the k‐means clustering problem with the variational quantum eigensolver (VQE) and a customized coreset method, the Contour coreset, which is formulated with a specific focus on quantum algorithms. Extensive simulations with synthetic and real‐life data demonstrated that the VQE+Contour Coreset approach outperforms existing QAOA+Coreset k‐means clustering approaches with higher accuracy and lower standard deviation. This research demonstrates that quantum‐tailored coreset techniques can remarkably boost the performance of quantum algorithms compared to generic off‐the‐shelf coreset techniques.