Further Reading: Quantum Kernels

Tagged Tier 1 (confident it exists and recommended) and Tier 2 (real and worth seeking, but verify the current version or URL).

This is the best-founded literature in Part VI. The kernel formulation is analytically tractable in a way variational circuits are not, and it shows: the theorems here are sharper and the negative results are more useful.

The foundational papers

Havlíček, Córcoles, Temme, Harrow, Kandala, Chow, and Gambetta, "Supervised learning with quantum-enhanced feature spaces" (2019), Nature 567, 209. Introduces both the variational classifier of Chapter 33 and the kernel estimator of this chapter, and frames them as alternatives. Read it for that framing, which is exactly the choice §34.3 measures. Tier 1.
Schuld and Killoran, "Quantum machine learning in feature Hilbert spaces" (2019), Physical Review Letters 122, 040504. The clean statement of the feature-map picture: encoding data into a quantum state is a feature map, and the kernel is the overlap. The conceptual paper for §34.1. Tier 1.
Schuld, "Supervised quantum machine learning models are kernel methods" (2021), arXiv:2101.11020. Argues that every variational quantum model is secretly a kernel method. If right, it explains §34.5's finding that this chapter's kernel and Chapter 33's variational model land in the same place — they were never as different as the architectures suggest. Exercise 34.33's subject, and the most conceptually important item here. Tier 1.

The proven separation

Liu, Arunachalam, and Temme, "A rigorous and robust quantum speed-up in supervised machine learning" (2021), Nature Physics 17, 1013. A learning problem, built on discrete-log structure, where a quantum kernel provably beats any classical learner under standard assumptions. This is where QML's rigorous results live, and it is a kernel result for a reason: a fixed feature map is analytically tractable. Read it for what a rigorous separation actually requires, and notice how specific the constructed problem is. Tier 1.
Huang et al., "Power of data in quantum machine learning" (2021), Nature Communications 12, 2631. Characterizes when a classical learner with data access can match a quantum model — and when it cannot. The complement to Liu et al.: one paper says a separation exists, the other says how rare it is. Tier 1.

Concentration

The failure mode §34.6 measured, and got the regime wrong about.

Thanasilp, Wang, Cerezo, and Holmes, "Exponential concentration in quantum kernel methods" (2022/2024), Nature Communications. The systematic treatment: what causes concentration (expressibility, entanglement, global measurements, noise), how fast, and what it does to generalization. Read this before designing any feature map — and read it against §34.6's correction, because it is careful about exactly the regime distinction I got wrong. Tier 1.
Kübler, Buchholz, and Schölkopf, "The inductive bias of quantum kernels" (2021), NeurIPS. Argues that expressive quantum kernels need exponentially much data to generalize, which is the generalization-side statement of the same problem. Read alongside the concentration paper; together they are the strongest critique of the method. Tier 1.
Chapter 32's and Chapter 16's barren-plateau reading. Exercise 34.23 asks whether concentration and barren plateaus are the same phenomenon; these are the sources for arguing it. Tier 1.

Classical kernel methods

You cannot evaluate a quantum kernel without knowing what it is being compared against.

Any standard treatment of SVMs and the kernel trick — Schölkopf and Smola's Learning with Kernels, or the relevant chapters of Bishop or Hastie–Tibshirani–Friedman. The RBF kernel that beat the quantum one by seven standard errors is not a strawman, and understanding why it is so strong is the point. Tier 1.
scikit-learn's documentation on SVC(kernel='precomputed'). The interface this chapter uses, and the reason a quantum kernel drops into mature classical tooling with no modification. Tier 1.
Literature on Nyström approximation and random-features methods. How classical practice handles the $n^2$ Gram matrix at scale — which is §34.7's problem, already solved on the classical side. Whether the same tricks apply to quantum kernels is a live question. Tier 2.

On benchmarking

Bowles, Ahmed, and Schuld on benchmarking quantum machine learning models (2024). Chapter 33's recommendation, and it covers kernel methods too. The systematic version of §34.5. Tier 1.
Chapter 33 §33.3's replication protocol. Ten independent datasets and splits, with an error bar. This chapter uses it throughout because Chapter 33 learned why. Tier 1.

Backward references

Chapter 32 — the input problem, and the barren-plateau/shot-budget product that concentration reproduces.
Chapter 33 — the variational alternative, the replication protocol, and the inference bill.
Chapter 24 — the $1/\epsilon^2$ shot wall that turns concentration into a quadratic bill.

Forward references

Chapter 35 — where the data is already quantum, the encoding cost vanishes, and the kernel is an overlap between states you already have.

Where to go next. If one thing: Schuld's "Supervised quantum machine learning models are kernel methods." It reframes everything in Part VI — if variational models are kernel methods, then §34.5's tie between this chapter's kernel and Chapter 33's classifier is not a coincidence but a prediction.

If two: add Thanasilp et al. on exponential concentration. It is the rigorous version of §34.6, it is careful about exactly the regime distinction that tripped up my prediction, and it tells you which feature maps are safe before you build one.

Then Chapter 35, which closes Part VI on the case where the arithmetic changes — quantum data, no encoding cost, and no classical alternative to lose to.