QEGK: Entangled Group-Equivariant Quantum Kernels

A simulator-based research prototype for symmetry-aware image kernels. Small image patches are encoded into quantum-inspired feature states, compared with a fidelity-style kernel, and that kernel is averaged over a finite symmetry group (rotations / reflections) so the similarity score is invariant to those transformations. The group-averaged kernel is benchmarked against matched classical kernels on low-data, symmetry-heavy tasks.

Simulator-based research prototype. No quantum-advantage claim.

Core idea

• Each input is mapped to a normalized feature state via a parameterized encoding.
• Two inputs are compared by a fidelity kernel — the squared overlap of their states.
• To build in symmetry, the kernel is averaged over a finite group $G$ of input transformations (for example the eight rotations and reflections of a square).
• The resulting kernel feeds a standard kernel SVM (classification) or a nearest-neighbour anomaly score, and is compared against classical kernels.

Mathematical sketch

Encode an input $x$ into a unit feature state and compare two inputs by their squared overlap (a fidelity kernel):

$$|\phi(x)\rangle = U(x)\,|0\rangle, \qquad k(x,y) = \bigl|\langle \phi(x)\,|\,\phi(y)\rangle\bigr|^{2}.$$

Make the kernel invariant to a finite symmetry group $G$ by averaging over the group orbit of both inputs:

$$k_{G}(x,y) \;=\; \frac{1}{|G|^{2}} \sum_{g,h \in G} \bigl|\langle \phi(g\!\cdot\! x)\,|\,\phi(h\!\cdot\! y)\rangle\bigr|^{2}.$$

By construction $k_G$ is two-sided $G$-invariant, i.e. $k_G(g!\cdot! x,,y)=k_G(x,y)$ for every $g \in G$.

Selected visuals and results

Exploratory matched-baseline comparison on small simulator tasks. Each point is one task; the dashed line is parity. Results are mixed.

Selected qualitative example inputs and their transformations.

How to run

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

python scripts/run_all_experiments.py --quick
pytest -q

Results status

• Exploratory, simulator-based, small scale.
• Results are mixed: the group-averaged quantum kernel matches or exceeds matched classical kernels on some symmetry-heavy, low-data tasks, and not on others.
• No quantum advantage is claimed. No state-of-the-art claim is made.

Limitations

• Statevector simulator only, few qubits — classically tractable by design.
• Synthetic / small datasets chosen to probe symmetry and low-data regimes, not for external validity.
• No hardware noise model; finite-shot estimation would degrade the kernel.
• Classical kernel methods are the only baselines; deep models are out of scope.