Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces

Abstract

Encoding classical data into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms to be performed on near-term intermediate-scale quantum computers. The crucial idea is using the quantum Hilbert space as a quantum-enhanced feature space in machine learning models. While the quantum feature map has demonstrated its capability when combined with linear classification models in some specific applications, its expressive power from the theoretical perspective remains unknown. We prove that the machine learning models induced from the quantum-enhanced feature space are universal approximators of continuous functions under typical quantum feature maps. We also study the capability of quantum feature maps in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that machine learning algorithms based on quantum feature maps can handle a broad class of machine learning tasks. In light of this, one can design a quantum machine learning model with more powerful expressivity.

arXiv commentMain (6 pages, 2 figures); Supplemental material (13 pages, 1 figure); Close to the published version; T. G. and Q. H. T. contributed equally to this work. K.N. and Q.H.T. were supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Nos. JPMXS0118067394 and JPMXS0120319794

References in corpus (20)

• Quantum Computing in the NISQ era and beyond
• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Supervised learning with quantum enhanced feature spaces
• A Quantum Approximate Optimization Algorithm
• Quantum machine learning in feature Hilbert spaces
• Quantum Circuit Learning
• Circuit-centric quantum classifiers
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Power of data in quantum machine learning
• Data re-uploading for a universal quantum classifier
• A rigorous and robust quantum speed-up in supervised machine learning
• Continuous-variable quantum neural networks
• Robust data encodings for quantum classifiers
• Harnessing disordered quantum dynamics for machine learning
• Supervised quantum machine learning models are kernel methods
• Machine learning quantum states in the NISQ era
• Unitary quantum perceptron as efficient universal approximator
• The theory of the quantum kernel-based binary classifier
• Quantum reservoir computing: a reservoir approach toward quantum machine learning on near-term quantum devices

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