Quantum machine learning (QML) combines quantum computing with data-driven methods to tackle classification, regression, and generative tasks using quantum processors. Early demonstrations leveraged quantum-enhanced feature spaces and hybrid variational circuits, while specialized libraries like PennyLane and TensorFlow Quantum emerged in 2018–2020. By mid‑2022, QML frameworks, kernel classifiers, and prototype quantum neural networks had matured on NISQ hardware, laying the groundwork for future real‑world applications.

1. Introduction to QML

QML explores how quantum systems can process classical data or learn quantum data patterns. Two main paradigms arose before 2022: kernel-based methods that embed data into Hilbert space for support vector classification, and variational circuits that optimize parameterized quantum ansätze via classical feedback.

1.1 Kernel-Based QML

Havlíček et al. demonstrated supervised learning using quantum-enhanced feature spaces on a superconducting processor, mapping classical inputs into Hilbert space and estimating kernel inner products to classify data [1]. Schuld (2021) formalized that many QML models reduce to kernel methods, highlighting data‑encoding strategies as the key differentiator from classical kernels [2].

1.2 Variational Quantum Classifiers

Variational quantum classifiers (VQCs) encode inputs via amplitude or angle mapping and process them with parameterized circuits whose parameters are tuned by classical optimizers. Ansatz‑independent VQCs showed robustness on NISQ devices [3]. Squeezed‑state encoding extended quantum kernel methods to continuous variables, leveraging bosonic modes for richer feature maps [4].

2. Quantum Neural Networks

True quantum neural networks (QNNs) aim to generalize classical multilayer perceptrons with quantum gates. Training deep QNNs involved defining quantum analogues of neurons and leveraging unitary layers tied by controlled rotations and entangling gates. Beer et al. (2020) proposed feedforward QNNs capable of universal quantum computation and explored their training dynamics [5].

3. Software Frameworks

PennyLane (2018) provided a cross‑platform Python library for QML and quantum chemistry, integrating hardware backends and classical ML interfaces [6]. TensorFlow Quantum (2020) offered a seamless extension to TensorFlow for hybrid quantum‑classical ML prototyping, supporting Cirq‑based circuit layers and Keras integration [7].

4. Experimental Prototypes

Early QML experiments included image classification with VQCs on superconducting qubits and bosonic processors. Quantum kernel estimators applied to LHC data showed competitive performance on real‑world datasets via QSVM methods using 15–20 qubits [8]. Hybrid QNNs demonstrated reduced generalization error and noise resilience on small datasets [9].

5. Outlook

By May 2022, QML had established foundational algorithms, software ecosystems, and prototype demonstrations on NISQ hardware. Future directions include scalable error mitigation, efficient data re-uploading circuits, and exploring quantum advantage in domains like chemistry, finance, and high‑energy physics.

References

[1] Havlíček, V., Córcoles, A. D., Temme, K., Harrow, A. W., Kandala, A., Chow, J. M., & Gambetta, J. M. (2019). Supervised learning with quantum-enhanced feature spaces. Nature, 567(7747), 209–212.

[2] Schuld, M. (2021). Supervised quantum machine learning models are kernel methods. arXiv:2101.11020.

[3] Rodríguez, K., et al. (2022). Ansatz‑Independent Variational Quantum Classifiers and the Price of Variational Precision. Scientific Reports, 12(1), 1-12.

[4] Li, L. H., Zhang, D.-B., & Wang, Z. D. (2021). Quantum kernels with squeezed-state encoding for machine learning. Physical Review A, 104(4), 042418.

[5] Beer, K., Bondarenko, D., Farrelly, T., Osborne, T. J., Salzmann, R., Scheiermann, D., & Wolf, R. (2020). Training deep quantum neural networks. Nature Communications, 11(1), 808.

[6] Bergholm, V., et al. (2018). PennyLane: Automatic differentiation of hybrid quantum-classical computations. arXiv:1811.04968.

[7] Broughton, M., et al. (2020). TensorFlow Quantum: A Software Framework for Quantum Machine Learning. arXiv:2003.02989.

[8] Wu, S.-L., et al. (2021). Application of Quantum Machine Learning using the Quantum Kernel Algorithm on High Energy Physics Analysis at the LHC. Physical Review Research, 3(3), 033221.

[9] González Crespo, R., et al. (2024). Quantum data parallelism in quantum neural networks. Physical Review Research, 7(1), 013177.

[10] Lloyd, S., Schuld, M., Ijaz, A., Izaac, J., & Killoran, N. (2020). Quantum embeddings for machine learning. Nature Communications, 14(1), 1-8.