Geodesics of Quantum Feature Maps on the Space of Quantum Operators
Andrew Vlasic
TL;DR
The paper addresses how real-world data, viewed as a smooth Riemannian manifold, deforms when encoded into the space of quantum operators, specifically a subset $U(M)$ of $\mathrm{SU}(2^N)$. It develops a ground-up, rigorous Riemannian framework for the codomain of quantum feature maps arising from Hamiltonian encodings, deriving Tangent spaces, a compatible metric, curvature, and notions of volume and harmonic maps. Key contributions include showing a bijection between geodesics on the data manifold and on the quantum-operator codomain, providing closed-form curvature expressions, and constructing explicit frames and covectors that reveal how data geometry influences quantum circuits. These results offer a principled way to analyze information loss and data-induced deformation in quantum circuits, with implications for selecting encoding schemes and understanding layer-wise geometric effects on performance. Overall, the work supplies a mathematical bridge between data geometry and quantum circuit deformation, enabling geometry-aware design and analysis of quantum feature maps.
Abstract
This manuscript rigorously displays how the Riemannian structure of a point cloud--taking the manifold hypothesis--impacts the encoded subspace of quantum gates, exhibiting a direct effect of data on quantum circuits. Selecting a scheme to encode real-world data onto a quantum circuit, or quantum feature map, is an essential step in quantum machine learning. There have been many proposed encoding schemes and proposed techniques to test the efficacy of a map. However, very few techniques address how the data is 'deformed' when mapped to quantum state space--complex projective space--or the state of quantum operators--the Lie group of (special) unitary operators--and the potential downstream effects on an algorithm. This paper takes the assumption that a point cloud is a smooth Riemannian manifold and establishes a rigorous computational/theoretical framework to study how an encoding scheme deforms this geometry once mapped to the space of quantum operators, $\SU(2^N)$. Since the Riemannian manifold structure of the codomain of a quantum feature map has yet to be formalized, this rigorous framework is required to ensure the validity of analysis. Using a ground-up approach, we mathematically establish a Riemannian geometry for a general class of Hamiltonian quantum feature maps--which describes the vast majority of encoding methods--that are induced from a given Euclidean embedded manifold. We then establish analytic and computational tangible forms for curvature, volume, frames and coframes, and harmonic maps, which is the main tool for deformation analysis. Interestingly, the form of vector fields, and by extension, the form of covector fields, shows the interconnection between the change of a path on the real embedded manifold influences the change of a path in the respective subspace of special unitary operators, revealing the effect data has on a quantum circuit.