quantum_kernel_classification

When Do Quantum Kernels Help?

An empirical and diagnostic study of the Havlíček et al. (2019) ZZ feature-map quantum kernel, implemented from scratch in NumPy (no quantum SDK) and benchmarked against classical kernels (RBF, polynomial, linear) under one shared model-selection protocol.

Full write-up: paper/main.pdf (NeurIPS-format, double-blind)

Key finding

The quantum kernel is not a general-purpose upgrade over classical kernels — whether it helps depends entirely on whether the data's structure matches its feature map.

Dataset	Quantum (ZZ)	Best classical	Majority baseline
Synthetic, engineered to match the feature map (n=4)	98.3%	48.3%	50.0%
Synthetic, engineered to match the feature map (n=6)	73.3%	53.3%	50.0%
MiniBooNE particle identification (real physics data, PCA-4)	71.7%	81.7%	71.7%
MiniBooNE particle identification (real physics data, PCA-6)	71.7%	86.7%	71.7%

On data engineered so its labels are a function of the same feature map, the quantum kernel dominates every classical kernel. On real particle-physics data (MiniBooNE, Fermilab) with no such engineered relationship, it collapses exactly to the majority-class baseline at every qubit count tested, while classical kernels reach 81–87%. We diagnose why using kernel-target alignment and kernel-value concentration statistics — see the paper for the full analysis.

Decision regions on the 2-qubit engineered dataset

Only the quantum kernel's feature space matches how the labels were generated; classical kernels with smoother inductive biases cannot resolve the same (highly non-convex) true boundary:

Sample efficiency

Project structure

src/
  quantum_kernel.py        exact NumPy simulator of the ZZ feature-map fidelity kernel
  test_quantum_kernel.py   validates the simulator against an independent brute-force implementation
  synthetic_data.py        engineered "quantum-advantage" dataset generator
  datasets.py               MiniBooNE loader (PCA + [0, 2pi) scaling)
  kernels.py                classical kernels + normalization
  model_selection.py        shared CV / train / test protocol for all kernels
  run_experiments.py        main benchmark -> results/benchmark.json
  figures.py                learning curves + decision-boundary figure -> figures/
  optimize_quantum_kernel.py     Optuna search over R, C -> results/quantum_kernel_hpo.json
  export_datasets_for_julia.py   dumps datasets/splits for the Julia HPO script
julia/
  QuantumKernel.jl          independent Julia reimplementation of the simulator (FWHT-based)
  test_quantum_kernel.jl    same sanity checks, ported to Julia
  optimize_quantum_kernel.jl  random search over R and C (LIBSVM.jl), reads julia/data/
  data/                       datasets/splits exported by src/export_datasets_for_julia.py
results/
  benchmark.json             raw accuracy / alignment numbers for every kernel and dataset
figures/                     PNGs used in the paper
paper/
  main.tex, references.bib   NeurIPS-format paper source
  main.pdf                   compiled paper

Reproducing the results

Requires Python with numpy, scipy, scikit-learn, matplotlib.

cd src
python test_quantum_kernel.py   # sanity checks: unitarity, PSD kernel, brute-force cross-check
python run_experiments.py       # main benchmark -> ../results/benchmark.json
python figures.py               # -> ../figures/*.png

To recompile the paper (requires a LaTeX distribution, e.g. MiKTeX or TeX Live):

cd paper
pdflatex main.tex && bibtex main && pdflatex main.tex && pdflatex main.tex

Method summary

• Quantum kernel: k(x,x') = |<phi(x)|phi(x')>|^2 for the "full entanglement" ZZ feature map, simulated exactly (not sampled) via a batched Walsh–Hadamard transform + vectorized diagonal-phase update — no quantum library required, and fast enough (< 2s) to run every experiment in this repo on a laptop CPU. Independently reimplemented in Julia (julia/QuantumKernel.jl) using a different transform algorithm (in-place FWHT butterfly) and RNG; on 200 random 6-qubit inputs the two implementations' kernel matrices agree to within 5.6e-15 (machine precision).
• Positive control: synthetic datasets whose labels are the sign of a fixed observable measured in a Haar-random rotated basis of the same feature map, following Havlíček et al. (2019) — separable by the quantum kernel by construction.
• Negative control: MiniBooNE particle identification (UCI/OpenML), a real high-energy-physics benchmark with no relationship to the feature map.
• All kernels are normalized to unit diagonal and driven through the same cross-validated SVM protocol, so comparisons isolate the kernel choice.