Hamiltonian Quantum Feature Maps

• Hamiltonian Quantum Feature Maps are techniques that embed classical or structured data into quantum states using Hamiltonian dynamics and Lie algebra principles.
• They leverage geometric and Riemannian properties to preserve data structure in quantum representations, facilitating nonlinear kernel methods for classification and regression.
• These maps support protocols like ground state evolution and time-evolution encoding, offering high expressivity and capacity for complex data and simulation tasks.

Hamiltonian Quantum Feature Maps are a class of quantum feature maps that embed classical or structured data into quantum states or quantum operators through dynamics or parameterization governed by Hamiltonians, often with the aim of encoding task-relevant structure, physics, or complexity that is inaccessible to conventional classical feature encodings. They are increasingly prominent in quantum machine learning and quantum simulation, bridging the formalism of quantum theory, differential geometry, representation theory, and modern supervised and unsupervised learning protocols.

1. Mathematical Structure of Hamiltonian Quantum Feature Maps

A Hamiltonian quantum feature map can be formalized as the mapping

$$U(p) = \exp \left( L(p) \right), \qquad L(p) = \sum_k f_k(p) L_k,$$

where each $f_k: M \to \mathbb{R}$ is a smooth function on the embedded data manifold $M$, and $L_k \in \mathfrak{su}(2^N)$ are fixed (typically linearly independent) skew-Hermitian operators (Vlasic, 2 Sep 2025). This construction is underpinned by Lie theoretic principles: for more general circuits consisting of products of exponentials, the Baker–Campbell–Hausdorff (BCH) formula shows that such circuits correspond to a single exponential with a generator composed from the original constituent terms. In practical implementations, the encoding often arises from or is inspired by physically meaningful Hamiltonians, such as those encountered in quantum many-body systems, spin models, or graph-based couplings (Albrecht et al., 2022, Umeano et al., 10 Apr 2024).

The tangent space at a point $U(p)$ in the codomain $U(M)$—the image of $M$ under the map—can be expressed as

$$T_{U(p)}U(M) = U(p) \cdot \mathfrak{su}_{\mathcal{L}},$$

where $\mathfrak{su}_{\mathcal{L}}$ denotes the Lie algebra generated by all derivatives of the $L_k$. This identifies how infinitesimal changes in the input are mapped into directions in operator space, with nontrivial commutator structures leading to more intricate local geometry.

2. Geometric and Riemannian Properties

The codomain $U(M)$ of a Hamiltonian quantum feature map inherits a Riemannian metric from its construction in operator space. One standard form is

$$g(H, K) = \frac{1}{2N} [\operatorname{Tr}(H^\dagger K) + \operatorname{Tr}(K^\dagger H)],$$

which is real and symmetric and allows the development of the full machinery of Riemannian geometry including Levi–Civita connection, sectional curvature, Ricci curvature, and scalar curvature (Vlasic, 2 Sep 2025). When the exponentiated operators $L_k$ commute (as in simple angle-encoding maps), the induced metric is flat (zero sectional curvature). In contrast, noncommuting terms (as in certain interaction-based encodings like IQP or XY Hamiltonians) produce nonconstant and nonzero curvature, leading to a “warped” geometry in the quantum representation of the data manifold.

A central result is the one-to-one correspondence between geodesics on $M$ and geodesics in $U(M)$: for any geodesic $\gamma$ on $M$, the curve $t \mapsto U(\gamma(t))$ is a geodesic in $U(M)$ with respect to the induced metric. This ensures that the geometric notion of shortest paths or interpolations is preserved under the map, facilitating analyses of feature space distances, curvature, and ultimately the expressive power of the map for quantum learning (Vlasic, 2 Sep 2025).

3. Hamiltonian Quantum Feature Maps in Data Embedding Protocols

Hamiltonian feature maps have been realized in various quantum machine learning protocols, notably in ground state-based embeddings and time evolution-based encodings.

• Ground state-based maps: A parameterized Hamiltonian $H_1(x)$ is programmed by the classical data $x$. Adiabatic evolution transforms a simple initial state (ground state of $H_0$) into the ground state $|\psi_G(x)\rangle$ of $H_1(x)$, producing the feature embedding. Mathematically, this protocol is governed by

$H(t; x) = (1-t/T) H_0 + (t/T) H_1(x),$

and the mapping is realized as

$$x \mapsto |\psi_G(x)\rangle = U_T(x) |\psi_0\rangle,$$

with $U_T(x)$ a time-ordered exponential (Umeano et al., 10 Apr 2024). The process may be Trotterized for actual digital implementation.

• Time-evolution driven maps and Hamiltonian kernels: In graph machine learning settings, a classical graph $G$ is embedded via spatial arrangement of neutral atoms or spins and encoded into a Hamiltonian whose dynamics “write” the graph structure into the resultant quantum state (Albrecht et al., 2022). The feature vector may be based on excitation probabilities, time-dependent observables, or refined expectation values. For general tasks, features can be expectation values of the form $\operatorname{Tr}[e^{-i t H}\rho]$, which, when collected for various $t$, serve as truncated Hamiltonian Fourier series representatives ("Hamiltonian Fourier features") (Morohoshi et al., 23 Apr 2025).

4. Expressivity, Capacity, and Mode Spectra

Analysis of the frequency and mode spectra generated by Hamiltonian feature maps provides insight into their capacity and expressivity:

• For ground state-based embeddings, the Fourier-like mode spectrum is described by combinations of eigenvalues of the data-dependent Hamiltonians, with the degree of the spectrum growing at least polynomially and potentially exponentially with the system size (number of qubits), i.e., the “model capacity” can be very high (Umeano et al., 10 Apr 2024).
• However, these spectra typically exhibit massive degeneracies, and the weighting coefficients of different modes are highly structured, which can constrain the set of truly independent features (hence, actual expressivity).
• In contrast, rotation-based or simple parameterized quantum models possess nondegenerate mode spectra, reflecting a more limited, controlled set of representable functions of the data.

For tasks where the mapping $x \mapsto |\psi(x)\rangle$ or $$H \mapsto \{ \operatorname{Tr}(e^{-i t H} \rho) \}_{t}$$ is tailored to encode structure inaccessible to classical feature maps, the Hamiltonian construction may facilitate quantum advantage in learning, under the assumption that the induced feature space geometry or kernel is not classically simulable (Albrecht et al., 2022, Ahmad et al., 2021, Umeano et al., 10 Apr 2024, Morohoshi et al., 23 Apr 2025).

5. Kernel Methods, Measurement, and Classification Protocols

The quantum kernel underlying most Hamiltonian quantum feature maps is determined by the overlap or distance in Hilbert space or operator space, such as

$$K(x, x') = |\langle \psi(x) | \psi(x') \rangle|^2 \quad \text{or} \quad K(G, G') = \exp( - \mathrm{JS}(\mathcal{P}, \mathcal{P}') ),$$

where $\mathrm{JS}$ denotes Jensen–Shannon divergence between excitation histograms $\mathcal{P}$ and $\mathcal{P}'$ (Albrecht et al., 2022).

Practical protocols leverage these kernels in classical algorithms such as support vector machines (QSVM) for classification tasks. The capacity of the kernel, conditioned on its non-classical simulability, can reveal highly nonlinear or global structures in the data, leading to superior separability in feature space, provided that the data encoding (circuit depth, entangling structure) is suitably expressive or matches the intrinsic data geometry (Ahmad et al., 2021, Albrecht et al., 2022).

In recent Hamiltonian classifier architectures (Tiblias et al., 13 Apr 2025), the data are used to parameterize the Hamiltonian directly, typically decomposing $H_\phi(x)$ into a sum of Pauli strings for efficient measurement, so that

$$H_\phi(x) = \sum_j \alpha_j(x) P_j, \qquad f_{\theta, \phi}(x) = \sigma( \langle \psi_\theta | H_\phi(x) | \psi_\theta \rangle ),$$

bypassing explicit amplitude encoding and achieving logarithmic scaling in both qubits and quantum gates.

6. Geometry, Information Retention, and Implications for Learning

The metric and curvature properties of the induced operator manifold $U(M)$ reflect how information about the original data geometry is retained or deformed under the feature map:

• Flat geometry (zero sectional curvature), as found in commuting/angle encoding schemes, suggests minimal warping of the data manifold and possibly limited quantum enhancement.
• Nonconstant curvature, induced by noncommuting Hamiltonian terms as in IQP-type encodings, produces more complex, potentially highly expressive feature spaces but may also risk information loss or state concentration (Vlasic, 2 Sep 2025).

The correspondence between geodesics in $M$ and in $U(M)$ ensures that distance-based reasoning or interpolation in the data space is preserved, providing a concrete mathematical framework for quantifying how quantum embeddings capture data relationships.

This geometric perspective allows practitioners to diagnose or optimize encoding schemes: if the imposed Riemannian structure aligns with problem-specific requirements (e.g., preserving neighborhood relationships, augmenting global distinguishing power), the quantum model may display enhanced learning capability. Conversely, excessive curvature or feature space warping may impede interpretability or learning efficacy.

7. Representative Examples and Application Scenarios

Several concrete application domains illustrate the deployment and utility of Hamiltonian quantum feature maps:

Context/Protocol	Hamiltonian Feature Map Principle / Task	Reference
Quantum harmonic/anharmonic oscillator	Matrix discretization and mapping of Hamiltonians to quantum circuits	(Miceli et al., 2018)
Classification of graph-structured molecules	Encoding spatial graph layouts as parameters in neutral atom Hamiltonians	(Albrecht et al., 2022)
Regression of $\operatorname{Tr}[f(H)\rho]$	Fourier-feature-based mapping of Hamiltonians for supervised learning	(Morohoshi et al., 23 Apr 2025)
Large-scale text/image classification	Parameterize input Hamiltonian, measure expectation, Pauli string decomposition	(Tiblias et al., 13 Apr 2025)

These implementations demonstrate flexibility: the Hamiltonian can encode quantum many-body structure, graph connectivity, time/frequency information, or be tuned for computational efficiency (e.g., via Pauli decomposition to match NISQ hardware constraints).

• (Miceli et al., 2018) Quantum Computation and Visualization of Hamiltonians using Discrete Quantum Mechanics and IBM QISKit
• (Ahmad et al., 2021) Quantum Machine Learning with HQC Architectures using non-Classically Simulable Feature Maps
• (Albrecht et al., 2022) Quantum Feature Maps for Graph Machine Learning on a Neutral Atom Quantum Processor
• (Kwon et al., 2023) Feature Map for Quantum Data in Classification
• (Umeano et al., 10 Apr 2024) Ground state-based quantum feature maps
• (Tiblias et al., 13 Apr 2025) An Efficient Quantum Classifier Based on Hamiltonian Representations
• (Morohoshi et al., 23 Apr 2025) Learning functions of Hamiltonians with Hamiltonian Fourier features
• (Vlasic, 2 Sep 2025) Geodesics of Quantum Feature Maps on the space of Quantum Operators

These results collectively establish Hamiltonian quantum feature maps as a geometrically and physically grounded method for quantum data embedding, with properties that are precisely quantified, implementation strategies adapted to a variety of settings, and suitability for applications where classical encoding or simulation is inadequate. The interplay between Lie algebraic structure, induced geometry, and expressive capacity frames ongoing research in optimizing feature map choice for scalable quantum machine learning and simulation.