[Paper Review] Supervised quantum machine learning models are kernel methods

Maria Schuld|arXiv (Cornell University)|Jan 26, 2021

Quantum Computing Algorithms and ArchitectureComputer Science28 references171 citations

TL;DR

The paper shows that supervised quantum models can be formulated as classical kernel methods, with quantum kernels defined by inner products of data-encoding density matrices, and argues that kernel-based training can outperform variational approaches under certain conditions.

ABSTRACT

With near-term quantum devices available and the race for fault-tolerant quantum computers in full swing, researchers became interested in the question of what happens if we replace a supervised machine learning model with a quantum circuit. While such "quantum models" are sometimes called "quantum neural networks", it has been repeatedly noted that their mathematical structure is actually much more closely related to kernel methods: they analyse data in high-dimensional Hilbert spaces to which we only have access through inner products revealed by measurements. This technical manuscript summarises and extends the idea of systematically rephrasing supervised quantum models as a kernel method. With this, a lot of near-term and fault-tolerant quantum models can be replaced by a general support vector machine whose kernel computes distances between data-encoding quantum states. Kernel-based training is then guaranteed to find better or equally good quantum models than variational circuit training. Overall, the kernel perspective of quantum machine learning tells us that the way that data is encoded into quantum states is the main ingredient that can potentially set quantum models apart from classical machine learning models.

Motivation & Objective

• Clarify the bridge between quantum machine learning and kernel theory.
• Show that quantum models are linear in a high-dimensional feature space defined by density matrices.
• Demonstrate that training reduces to kernel-based optimization in an RKHS.
• Highlight the importance of data encoding in determining model expressivity and learning behavior.

Proposed method

• Define data-encoding feature maps as mappings x -> ρ(x) in the space of density matrices with Hilbert-Schmidt inner product.
• Define quantum kernel κ(x, x') = tr[ρ(x) ρ(x')] = |⟨φ(x')|φ(x)⟩|^2 and show it is positive definite.
• Show the equivalence between quantum models and linear models in the RKHS induced by the quantum kernel.
• Apply the representer theorem to express optimal models as f_opt(x) = ∑ α_m tr[ρ(x^m) ρ(x)].
• Argue that kernel-based training yields models with measurements that lie in the subspace spanned by training data, and discuss convexity benefits.
• Compare kernel-based training with variational approaches and explain when one might be preferable.

Experimental results

Research questions

• RQ1 Can quantum models be reformulated as classical kernel methods via a quantum kernel?
• RQ2 How does data encoding into quantum states (the feature map) determine the kernel and the model’s expressivity?
• RQ3 What are the benefits and trade-offs of kernel-based training versus variational circuit training for quantum models?
• RQ4 How does the representer theorem constrain the form of optimal quantum measurements?
• RQ5 Under what conditions do quantum kernels provide computational or expressivity advantages over classical kernels?

Key findings

• Quantum models can be expressed as linear models in a feature space of density matrices, with predictivity governed by the quantum kernel κ(x, x').
• The RKHS associated with the quantum kernel contains exactly the same functions as the space of quantum models, enabling kernel-based analysis.
• Optimal quantum measurements for typical cost functions can be written as kernel expansions in the training data, requiring only M degrees of freedom (coefficients α_m).
• Kernel-based training reduces the search for the best model to an M-dimensional optimization problem, which is convex for common loss functions, contrasting with non-convex variational training.
• Training with a quantum kernel can guarantee at least as good or better minima than variational training, at the cost of needing pairwise kernel evaluations.
• The data-encoding strategy largely determines expressivity and potential quantum advantages, emphasizing the central role of encoding over circuit design alone.