New perspectives on quantum kernels through the lens of entangled tensor kernels

Abstract

Quantum kernel methods are one of the most explored approaches to quantum machine learning. However, the structural properties and inductive bias of quantum kernels are not fully understood. In this work, we introduce the notion of entangled tensor kernels - a generalization of product kernels from classical kernel theory - and show that all embedding quantum kernels can be understood as an entangled tensor kernel. We discuss how this perspective allows one to gain novel insights into both the unique inductive bias of quantum kernels, and potential methods for their dequantization.

20 pages main text, 5 page appendix, lots of figures. Comments and feedback very welcome

References in corpus (22)

• The density-matrix renormalization group in the age of matrix product states
• Variational Quantum Algorithms
• Supervised learning with quantum enhanced feature spaces
• Quantum machine learning in feature Hilbert spaces
• Parameterized quantum circuits as machine learning models
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Power of data in quantum machine learning
• A rigorous and robust quantum speed-up in supervised machine learning
• Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Efficient decomposition of quantum gates
• A positive tensor network approach for simulating open quantum many-body systems
• Training Quantum Embedding Kernels on Near-Term Quantum Computers
• Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks
• Purifications of multipartite states: limitations and constructive methods
• Matrix product operators and states: NP-hardness and undecidability
• Classical surrogates for quantum learning models
• Quantum Kernel Methods under Scrutiny: A Benchmarking Study
• Efficient quantum circuits for arbitrary sparse unitaries
• Potential and limitations of random Fourier features for dequantizing quantum machine learning
• On the expressivity of embedding quantum kernels
• Neural quantum kernels: training quantum kernels with quantum neural networks