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On the expressivity of embedding quantum kernels

Elies Gil-Fuster, Jens Eisert, Vedran Dunjko

TL;DR

The paper investigates how expressive embedding quantum kernels (EQKs) are for quantum machine learning. It proves universal expressivity of EQKs when efficiency is not constrained, and then establishes efficient universality for two important kernel families: shift-invariant kernels via random Fourier features (RFF) and composition kernels via pre-processing (RFF_pp), showing that both can be realized as EQKs with polynomial resources and, in favorable conditions, polynomial runtime. It further demonstrates that the projected quantum kernel belongs to the composition family and thus admits efficient EQK realization. The work introduces constructive schemes (C2QE, QRFF, QRFF_pp) and provides dimension/run-time scaling results under regularity and sampling assumptions, while outlining open questions about broader kernel classes and time-efficient implementations. These findings substantiate the reach of EQKs and guide future exploration of quantum kernel expressivity beyond EQKs.

Abstract

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.

On the expressivity of embedding quantum kernels

TL;DR

The paper investigates how expressive embedding quantum kernels (EQKs) are for quantum machine learning. It proves universal expressivity of EQKs when efficiency is not constrained, and then establishes efficient universality for two important kernel families: shift-invariant kernels via random Fourier features (RFF) and composition kernels via pre-processing (RFF_pp), showing that both can be realized as EQKs with polynomial resources and, in favorable conditions, polynomial runtime. It further demonstrates that the projected quantum kernel belongs to the composition family and thus admits efficient EQK realization. The work introduces constructive schemes (C2QE, QRFF, QRFF_pp) and provides dimension/run-time scaling results under regularity and sampling assumptions, while outlining open questions about broader kernel classes and time-efficient implementations. These findings substantiate the reach of EQKs and guide future exploration of quantum kernel expressivity beyond EQKs.

Abstract

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.
Paper Structure (22 sections, 18 theorems, 86 equations, 4 figures, 5 algorithms)

This paper contains 22 sections, 18 theorems, 86 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Kernel methods
  5. Embedding quantum kernels
  6. Related work
  7. The universality of quantum feature maps
  8. The universality of efficient shift-invariant embedding quantum kernels
  9. Composition kernel and projected quantum kernel
  10. Outlook
  11. Time efficient EQKs
  12. Non-stationary kernels and indefinite kernels
  13. The variational QML lens
  14. Conclusion
  15. Towards diagonalizing integral kernel operators
  16. ...and 7 more sections

Key Result

Theorem 1

Let $k\colon\mathcal{X}\times\mathcal{X}\Tilde{o}\mathbb{R}$ be a kernel function. Then, for any $\varepsilon>0$ there exists $n\in\mathbb{N}$ and a data embedding $\rho_n$ onto the Hilbert space of quantum states of $n$ qubits such that for almost all $x,x'\in\mathcal{X}$.

Figures (4)

  • Figure 1: Illustration of the main question of this paper. Embedding Quantum Kernels (EQKs) have the form of an explicit inner product on the Hilbert space of quantum density matrices, which is evaluated using a quantum circuit. The box "Kernel functions" indicates that EQKs correspond to an inner product of feature vectors on a Hilbert space. The box "Efficient Quantum functions" restricts EQKs to functions that can be evaluated using a quantum computer in polynomial time, for instance these would include preparing a state-dependent state $\Tilde{\rho}(x,x')$ and then measuring the expectation value of an observable $\mathcal{M}$ on the data-dependent state. The box "Efficient Embedding Quantum Kernels" then clearly lives in the intersection of the two other boxes. The question we address here is then whether EQKs do cover the whole intersection. Said otherwise, can every efficient quantum kernel function be expressed as efficient EQKs? Or, on the contrary, do there exist efficient quantum kernels which are not expressible as efficient EQKs?
  • Figure 2: Schematic of the different ingredients that form embedding quantum kernels. The data input is mapped onto the "quantum feature space" of quantum density operators via a quantum embedding. There the Embedding Quantum Kernel is defined as the Hilbert-Schmidt inner product of pairs of quantum features.
  • Figure 3: Venn diagram with set relations of different classes of kernels and quantum kernels. Each of the arrows represents a reduction found in this manuscript, they should be read as "for any element of the first set, there exists an element of the second set which is a good approximation." In the case of Theorem \ref{['thm:approx_universality']}, elements are individual functions. In every other case, elements are sequences of kernel functions, for which the notions of efficiency make sense. In summary, we find that efficient embedding quantum kernels (EQK) can approximate two important classes of kernels: shift-invariant and composition kernels.
  • Figure 4: Conceptual sketch of three constructions to approximate kernel functions as Embedding Quantum Kernels (EQKs). The three parts correspond to different kernel families: (a) General kernels refers to any PSD kernel function, as introduced in Definition \ref{['def:kernel']}; (b) Shift-invariant kernels are introduced in Definition \ref{['def:shift-invariant']}; and (c) Composition kernels are a new family that we introduce in Section \ref{['s:composition']}. The boxes refer to routines taken either from the existing literature (Mercer, from Corollary A.3 in Appendix A; and RFF, from Theorem \ref{['thm:RFF_existence']} originally in Ref. rahimi2007random) or introduced in this manuscript (Algorithm \ref{['alg:QEPIP']}; QRFF, as Algorithm \ref{['alg:QRFF']}; RFF_pp, as Algorithm \ref{['alg:RFF_PP']}; and QRFF_pp, as Algorithm \ref{['alg:QRFF_PP']}). The details for each of the three families are elucidated in the corresponding sections: the universality of EQKs for general kernels is explained in Section \ref{['s:universality']}, with Theorem \ref{['thm:approx_universality']}; the universality of efficient EQKs for shift-invariant kernels appears in Section \ref{['s:shift-invariant']}, studied formally in Corollaries \ref{['cor:secondderivative']} and \ref{['cor:time_efficient']}; finally, composition kernels are introduced in Section \ref{['s:composition']}, with Proposition \ref{['prop:composition']} stating their efficient approximation as EQKs, and Proposition \ref{['prop:projected']} confirming that composition kernels contain the so-called projected quantum kernels presented in Refs. huang2021powerhuang2022provably.

Theorems & Definitions (40)

  • Definition 1: Kernel function
  • Definition 2: Embedding quantum kernel (EQK)
  • Theorem 1: Approximate universality of finite-dimensional quantum feature maps
  • Lemma 2: Correctness and runtime of Algorithm \ref{['alg:QEPIP']}
  • Lemma 3: Euclidean inner products
  • proof : Proof of Lemmas \ref{['l:mappingqs']} and \ref{['l:innerprodqs']}
  • proof : Proof of Theorem \ref{['thm:approx_universality']}
  • Definition 3: Shift-invariant kernel function
  • Theorem 4: Bochner Rudin1990fourier
  • Theorem 5: Random Fourier features, Claim 1 in Ref. rahimi2007random
  • ...and 30 more