Publication Details
Keywords: quantum machine learning, variational quantum circuits, hierarchical learning, QCNN, barren plateaus, data re-uploading, NISQ
Abstract
Variational quantum classifiers are promising for machine learning on noisy intermediate-scale quantum (NISQ) hardware, but flat circuits that act on all input features at once need many qubits, accumulate gate errors and suffer vanishing gradients as they widen. This paper proposes Hierarchical Quantum Learning (HQL), a two-level framework that splits a high-dimensional input into correlation-coherent feature groups, processes each group in a narrow 4-qubit module built from data re-uploading and quantum-convolutional pooling, and fuses the module outputs in a trainable quantum aggregator optimised end to end. HQL was implemented with an autograd statevector simulator and compared with three classical learners, a quantum-kernel SVM, a hardware-efficient VQC and a quantum convolutional neural network (QCNN) on three binary tasks with 30–64 features over five random splits. With identical inputs, HQL reached 69.6% accuracy on a 64-feature benchmark versus 62.4% for the best flat circuit, and 92.5% versus 90.3% on handwritten digits, using circuits half as wide. Its gradient variance stayed near 0.11 up to 32 qubits, while that of a hardware-efficient circuit fell 63.4-fold between 4 and 16 qubits. Classical baselines remained the most accurate on all tasks, which we report openly. HQL offers a practical, trainable route to variational learning on modular quantum processors.
References
- [1]. J. Preskill, “Quantum computing in the NISQ era and beyond,” Quantum, vol. 2, p. 79, 2018, doi: 10.22331/q-2018-08-06-79. [2]. J. Biamonte et al., “Quantum machine learning,” Nature, vol. 549, pp. 195–202, 2017, doi: 10.1038/nature23474. [3]. M. Cerezo et al., “Variational quantum algorithms,” Nat. Rev. Phys., vol. 3, pp. 625–644, 2021, doi: 10.1038/s42254-021-00348-9. [4]. J. R. McClean et al., “Barren plateaus in quantum neural network training landscapes,” Nat. Commun., vol. 9, Art. 4812, 2018, doi: 10.1038/s41467-018-07090-4. [5]. M. Cerezo et al., “Cost function dependent barren plateaus in shallow parametrized quantum circuits,” Nat. Commun., vol. 12, Art. 1791, 2021, doi: 10.1038/s41467-021-21728-w. [6]. I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks,” Nat. Phys., vol. 15, pp. 1273–1278, 2019, doi: 10.1038/s41567-019-0648-8. [7]. E. Grant et al., “Hierarchical quantum classifiers,” npj Quantum Inf., vol. 4, Art. 65, 2018, doi: 10.1038/s41534-018-0116-9. [8]. A. Pesah et al., “Absence of barren plateaus in quantum convolutional neural networks,” Phys. Rev. X, vol. 11, Art. 041011, 2021, doi: 10.1103/PhysRevX.11.041011. [9]. T. Hur, L. Kim, and D. K. Park, “Quantum convolutional neural network for classical data classification,” Quantum Mach. Intell., vol. 4, Art. 3, 2022, doi: 10.1007/s42484-021-00061-x. [10]. K. K. Gajula, “Quantum-Enhanced Cryptographic Protocols for Ultra-Secure Data Transmission in Next-Generation Networks,” Journal of Dalian University of Technology, vol. 31, no. 12, 2024. [11]. K. K. Gajula, “Advancing Post-Quantum Cryptography: Novel Algorithmic Models and Security Implementations,” Goya Journal, vol. 18, no. 12, pp. 594–605, 2025. [12]. V. HavlĂÄŤek et al., “Supervised learning with quantum-enhanced feature spaces,” Nature, vol. 567, pp. 209–212, 2019, doi: 10.1038/s41586-019-0980-2. [13]. A. PĂ©rez-Salinas et al., “Data re-uploading for a universal quantum classifier,” Quantum, vol. 4, p. 226, 2020, doi: 10.22331/q-2020-02-06-226. [14]. M. Schuld, R. Sweke, and J. J. Meyer, “Effect of data encoding on the expressive power of variational quantum-machine-learning models,” Phys. Rev. A, vol. 103, Art. 032430, 2021, doi: 10.1103/PhysRevA.103.032430. [15]. S. Y.-C. Chen and S. Yoo, “Federated quantum machine learning,” Entropy, vol. 23, no. 4, Art. 460, 2021, doi: 10.3390/e23040460. [16]. R. Shaydulin and S. M. Wild, “Importance of kernel bandwidth in quantum machine learning,” Phys. Rev. A, vol. 106, Art. 042407, 2022, doi: 10.1103/PhysRevA.106.042407. [17]. K. Mitarai et al., “Quantum circuit learning,” Phys. Rev. A, vol. 98, Art. 032309, 2018, doi: 10.1103/PhysRevA.98.032309. [18]. J. Bowles, S. Ahmed, and M. Schuld, “Better than classical? The subtle art of benchmarking quantum machine learning models,” arXiv:2403.07059, 2024. [19]. M. Schuld et al., “Evaluating analytic gradients on quantum hardware,” Phys. Rev. A, vol. 99, Art. 032331, 2019, doi: 10.1103/PhysRevA.99.032331. [20]. W. N. Street, W. H. Wolberg, and O. L. Mangasarian, “Nuclear feature extraction for breast tumor diagnosis,” in Proc. SPIE 1905, 1993, pp. 861–870, doi: 10.1117/12.148698. [21]. E. Alpaydin and C. Kaynak, “Optical recognition of handwritten digits,” UCI Machine Learning Repository, 1998, doi: 10.24432/C50P49. [22]. K. K. Gajula, “A Hybrid Interpretable AI Framework for Detecting Financial Fraud in Dynamic Transaction Networks,” International Journal of Communication Networks and Information Security, 2022. [23]. T. Peng et al., “Simulating large quantum circuits on a small quantum computer,” Phys. Rev. Lett., vol. 125, Art. 150504, 2020, doi: 10.1103/PhysRevLett.125.150504.