Quantum Spectral Model: Data Reuploading with Input-Conditioned Frequency Support

2026-07-24

AI Summary

Existing quantum machine learning models struggle to explicitly capture the spectral structure—such as eigenvalues and subspaces—of input matrices, limiting their ability to exploit the intrinsic geometry of data. This work proposes the Quantum Spectral Model (QSM), which directly maps input matrices to the generator of a data-encoding unitary operator, thereby embedding spectral information explicitly and constructing a truncated Fourier representation at the output. The model innovatively employs the input-dependent spectral gap as a phase carrier and uses spectral subspaces to determine expansion coefficients, introducing an analytically tractable inductive bias. Three QSM variants are developed based on symmetric, global-block, and local-block Hamiltonian designs, incorporating data re-uploading and input-conditioned frequencies. Experiments demonstrate that QSM achieves state-of-the-art average test accuracy on Pendigits and two synthetic spectral tasks, with the local-block variant excelling on Pendigits and the global-block variant performing best on synthetic tasks.

Abstract

A central design principle in modern machine learning and artificial intelligence is to align a model's inductive bias with the structure of its input data. For matrix-valued inputs, relevant matrix-level relationships can be characterised through spectral values and spectral subspaces; however, common coordinate-wise rotation-gate data-encoding unitaries used in most quantum machine learning models do not explicitly construct such a matrix-level representation. We introduce Quantum Spectral Models (QSMs), in which we construct the generator of the data-encoding unitary directly from each input matrix. We study three QSM variants based on symmetric, global block, and non-overlapping patch-local block Hamiltonians. Their outputs admit truncated Fourier representations in which input-dependent spectral gaps supply candidate phase carriers, while spectral subspaces help determine their coefficients. We evaluate the QSMs and comparison quantum models on two matrix representations of Pendigits and two controlled synthetic tasks defined by spectral statistics. At the largest evaluated circuit depth, QSM variants lead the tested quantum models in mean test accuracy across all four benchmarks. The patch-local QSM leads on Pendigits, whereas the global block-Hamiltonian QSM leads on the controlled spectral tasks. Ablations show a task-dependent reversal: subspace-preserving controls perform better on Pendigits, whereas spectral-value-only controls lead among the tested ablations on the synthetic tasks. Together, these results shed new light on quantum machine-learning model design by showing how input-conditioned spectral representations can provide an analysable inductive bias, while offering a broader perspective on structure-aware model design in machine learning and artificial intelligence.

Problem

• quantum machine learning
• spectral representation
• inductive bias
• matrix-valued inputs
• data encoding

Innovation

• Quantum Spectral Models
• input-conditioned Hamiltonian
• spectral representation
• inductive bias
• data reuploading