Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning

TL;DR

This paper introduces a spectral framework that links quantum circuit architecture to its expressive capacity and trainability, enabling analysis without data or optimization dependence.

Contribution

It presents a data-agnostic, architecture-based matrix framework that explicitly relates circuit structure, feature correlations, and training kernel geometry in quantum machine learning.

Findings

The framework analytically reconstructs kernel structure from circuit design.

Correlations between features arise from shared parameter-induced harmonics.

Circuit architecture influences expressivity independently of data.

Abstract

Parametrised quantum circuits are a central framework for near term quantum machine learning. However, it remains challenging to determine in advance how architectural choices, such as encoding strategies, gate placement, and entangling structure, influence both the expressive capacity of the model and its trainability during optimisation. We introduce a data-agnostic framework, one requiring no knowledge of a training dataset or optimisation trajectory, that maps a broad family of circuits into a single architecture matrix built over learnable features and parameters. We show that this framework provides an explicit link between circuit structure, the correlations among learnable features, and the geometry of training kernels through the factorisation of each of these objects as quadratic forms in terms of these matrices. We show how correlations between learnable features arise from shared parameter-induced harmonics generated by non-commuting gate-observable interactions during Heisenberg back-propagation, and how these correlations are encoded directly in the architecture matrix. From this perspective, kernel structure and coefficient statistics can be reconstructed analytically from circuit design alone, without reference to a dataset or optimisation trajectory. The resulting framework makes circuit-induced structure explicit, separating architectural effects from data-dependent ones, and provides a principled foundation for analysing and comparing parametrised quantum circuits based on intrinsic, design-level signatures.