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Application of HHL-Based Quantum Support Vector Machine Algorithm for Private Equity Fund Decision-Making Support During Leveraged Buyout Funding Process

Dmitry Garmash

Published Jan 1, 2025• 0 citations
Publisher
DOI
10.2139/ssrn.5257794

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Abstract We investigated the potential application of quantum computing using the Kronecker kernel to pairwise classification and have devised a way to apply the Harrow-Hassidim-Lloyd (HHL)-based quantum support vector machine algorithm. Pairwise classification can be used to predict relationships among data and is used for problems such as link prediction and chemical interaction prediction. However, in pairwise classification using a Kronecker kernel, it is very costly to calculate the Kronecker product of the kernel matrices when there is a large amount of data. We found that the Kronecker product of kernel matrices can be represented more efficiently in time and space in quantum computing than that in classical computing. We also found that a pairwise classifier can be effectively trained by applying the HHL-based quantum support vector machine algorithm to the Kronecker kernel matrix. In an experiment comparing a classical algorithm with a quantum support vector machine with a Kronecker kernel run on a quantum computing simulator, the misclassification rate of the latter was almost the same as that of the former for the same pairwise classification problem in some cases. This indicates that a quantum support vector machine with a Kronecker kernel algorithm can achieve accuracy equivalent to that of the classical algorithm more efficiently and scalably. This finding paves the way for applying quantum machine learning to predicting relationships in large-scale data.

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Abstract We compare two quantum approaches that use support vector machines for multi-class classification on a reduced Sloan Digital Sky Survey () dataset: the quantum kernel-based QSVM and the Harrow-Hassidim-Lloyd least-squares SVM (HHL LS-SVM). Both one-vs-rest and two-step hierarchical classification schemes were implemented. The QSVM involves angle encoding of ten features, two unitary operator blocks consisting of rotational operator gates, and a projective measurement that projects the final state to the zero state. The HHL-based method involves solving a system of linear equations using the HHL algorithm and using the solution in a support vector machine approach. The results indicate that the QSVM outperforms HHL LS-SVM in most cases. HHL LS-SVM performs somewhat competitively in selected cases, particularly when isolating galaxies (majority), however, it also performs poorly in others, especially when isolating QSOs (minority). Comparisons with classical SVMs confirm that quantum and classical methods achieve broadly similar performance, with classical models performing slightly ahead overall. Scaling analysis reveals a trade-off: QSVM performance suffers from quadratic scaling with the number of samples and features, but benefits from explicit feature representation during training, while HHL LS-SVM scales essentially constantly, with moderate fluctuations, but suffers from limited representative elements. The HHL-based method is also highly noise-sensitive. These results suggest that QSVM performs better overall under the experimental conditions considered here and is likely to be more amenable to near-term hardware execution in terms of circuit depth and structure, while the more efficient scaling of HHL LS-SVM makes it a useful option for larger datasets beyond the NISQ era.

arXiv:2509.10190•EPJ Quantum Technology•Mar 13, 2026•3 citations
Abstract We investigated the potential application of quantum computing using the Kronecker kernel to pairwise classification and have devised a way to apply the Harrow-Hassidim-Lloyd (HHL)-based quantum support vector machine algorithm. Pairwise classification can be used to predict relationships among data and is used for problems such as link prediction and chemical interaction prediction. However, in pairwise classification using a Kronecker kernel, it is very costly to calculate the Kronecker product of the kernel matrices when there is a large amount of data. We found that the Kronecker product of kernel matrices can be represented more efficiently in time and space in quantum computing than that in classical computing. We also found that a pairwise classifier can be effectively trained by applying the HHL-based quantum support vector machine algorithm to the Kronecker kernel matrix. In an experiment comparing a classical algorithm with a quantum support vector machine with a Kronecker kernel run on a quantum computing simulator, the misclassification rate of the latter was almost the same as that of the former for the same pairwise classification problem in some cases. This indicates that a quantum support vector machine with a Kronecker kernel algorithm can achieve accuracy equivalent to that of the classical algorithm more efficiently and scalably. This finding paves the way for applying quantum machine learning to predicting relationships in large-scale data.
Abstract We compare two quantum approaches that use support vector machines for multi-class classification on a reduced Sloan Digital Sky Survey () dataset: the quantum kernel-based QSVM and the Harrow-Hassidim-Lloyd least-squares SVM (HHL LS-SVM). Both one-vs-rest and two-step hierarchical classification schemes were implemented. The QSVM involves angle encoding of ten features, two unitary operator blocks consisting of rotational operator gates, and a projective measurement that projects the final state to the zero state. The HHL-based method involves solving a system of linear equations using the HHL algorithm and using the solution in a support vector machine approach. The results indicate that the QSVM outperforms HHL LS-SVM in most cases. HHL LS-SVM performs somewhat competitively in selected cases, particularly when isolating galaxies (majority), however, it also performs poorly in others, especially when isolating QSOs (minority). Comparisons with classical SVMs confirm that quantum and classical methods achieve broadly similar performance, with classical models performing slightly ahead overall. Scaling analysis reveals a trade-off: QSVM performance suffers from quadratic scaling with the number of samples and features, but benefits from explicit feature representation during training, while HHL LS-SVM scales essentially constantly, with moderate fluctuations, but suffers from limited representative elements. The HHL-based method is also highly noise-sensitive. These results suggest that QSVM performs better overall under the experimental conditions considered here and is likely to be more amenable to near-term hardware execution in terms of circuit depth and structure, while the more efficient scaling of HHL LS-SVM makes it a useful option for larger datasets beyond the NISQ era.