Speaker
Description
Quantum kernel methods represent one of the most promising paradigms for achieving quantum advantage in machine learning. However, the design of expressive quantum feature maps remains largely heuristic, with little theoretical guidance from the geometry of data or the topology of the quantum state space. In this work, we introduce Topological Variational Quantum Kernels (TVQK) a new family of quantum kernels that explicitly encode the topological structure of classical data through a differentiable, scaleadaptive feature map. Our framework embeds data into quantum states via a parameterized unitary that depends on a continuous scale parameter τ, which probes the simplicial structure of the data manifold at variable resolutions. The kernel is defined as the fidelity between these topologicallyinformed quantum states, and the scale parameter is optimized end-to-end via quantum kernel alignment. We establish three foundational theorems: (i) the kernel is positive semidefinite for all τ; (ii) it is Lipschitz-stable under input perturbations, with robustness against depolarizing noise that scales linearly with qubit count; and (iii) it achieves exponential separation between topologically distinct data classes that are indistinguishable by any fixed-scale classical or quantum kernel. This work provides a principled bridge between topology, quantum information geometry, and kernel learning, opening new avenues for theoretically-grounded quantum machine learning architectures.
| Topic | Quantum kernel methods and their applications |
|---|---|
| Additional Topic | Theoretical foundations of quantum learning |