Kyle James Stuart Campbell

Parametrised quantum circuits are a leading model for near-term quantum machine learning, yet it remains difficult to predict—without training on data—how a circuit’s design shapes what it can learn and how trainable it will be. We introduce a data-agnostic representation that maps a broad family of circuits into a single architecture matrix built from a joint harmonic expansion over inputs and parameters. This matrix provides an explicit, interpretable link between circuit structure, the correlations among learnable features, and the geometry of training kernels. 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.