Fourier analysis reveals how quantum classifier design shapes outputs
Fourier Analysis of Parametrized Interactive Quantum Classifiers
Machine Learning
Summary
Quantum classifiers process information differently by interacting a key quantum bit with its surroundings, but how their design impacts results was unclear. The authors worked out exact formulas showing how the parameters controlling this interaction affect the classifier’s output like a combination of waves. This wave view helps create more flexible quantum classifiers capturing complex patterns in data. Tests on multiple datasets confirm that their new designs often improve how well these classifiers identify categories. The study also finds that overall similarity of quantum states does not always predict classification success.
What this means in practice
- •For quantum algorithm developers: Design quantum classifiers with tunable Fourier components to improve classification of complex nonlinear data patterns.
- •For quantum software engineers: Implement generalized Hamiltonian encodings in interactive quantum models for enhanced flexibility and accuracy in quantum machine learning tasks.
Authors
Fábio Novaes, Fernando M. de Paula Neto, João V. M. Cardoso
Abstract
Interactive Quantum Classifiers (IQCs) constitute a family of quantum machine learning models inspired by open quantum systems, in which the interaction between a target qubit and an environment is described by a Hamiltonian. Previous works introduced alternative Hamiltonian parameterizations and showed empirically that they can improve classification performance, but the role of these parameters in the resulting classifier remains poorly understood. In this work, we derive a closed-form expression for the reduced quantum channel generated by a parametrized IQC with a single target qubit. The analytical solution explicitly reveals how the Hamiltonian parameters control the constant, sine, and cosine components of the classifier output, establishing a Fourier interpretation of the induced feature map. This analysis motivates a generalized family of Hamiltonian encodings, including matrix-parameterized environmental Hamiltonians whose Fourier components depend on linear combinations of input features, thereby enabling non-separable Fourier structures. Numerical experiments on synthetic and real-world datasets show that the proposed models can improve classification performance on several nonlinear benchmarks. The generalized matrix encoding achieves the strongest aggregate performance in the evaluated benchmark, while a simpler four-parameter extension often attains comparable performance with substantially fewer trainable parameters. We additionally characterize the generated state ensembles using the standard fidelity-based expressibility measure, finding that global expressibility does not directly predict classification performance. Our results provide an analytical characterization of parametrized Hamiltonians in Interactive Quantum Classifiers and establish Fourier analysis as a useful framework for understanding and designing open-system-inspired quantum learning models.