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Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning

Kyle James Stuart Campbell, Luigi Del Debbio, Petros Wallden

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.

Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning

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.

Paper Structure

This paper contains 112 sections, 11 theorems, 139 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Summary of results
  4. Contributions
  5. Organisation
  6. Preliminaries
  7. PQC model and re-uploading structure
  8. Quantum Fourier models and encoder-accessible input harmonics
  9. Commuting phase-encoder assumption
  10. Difference sets and the accessible frequency set
  11. Reality condition
  12. Parameter harmonics for single-use Pauli-rotation ansätze
  13. Heisenberg back-propagation through a single Pauli rotation
  14. Trigonometric-polynomial structure and character expansion on $\mathbb{T}^m$
  15. Single-use support restriction
  16. ...and 97 more sections

Key Result

Proposition 4.1

Let $a_\omega(\theta)=\sum_{k\in K} C_{\omega k}e^{ik\cdot\theta}$ and $\theta\sim\mathrm{Unif}(\mathbb{T}^m)$. Then: In components, for $\omega,\mu\in\Omega$, $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: Composite summary for the YZY circuit with entangling gates and the associated variance and correlation structure for $R_X$ encoding. See Appendix \ref{['appendix:results']} for additional results for other circuits.
  • Figure 2: Mean off-diagonal correlation for increasing depth across the range of circuits tested, with both $R_X$ and $R_Y$ encoding. Note that Circuits 16 and 17 with $R_Y$ encoders and Circuit 15 with $R_X$ encoder stabilise to a non-trivial ($\gtrsim 10^{-2}$), mean correlation value.
  • Figure 4: Composite summary for the YZY circuit without entangling gates and the associated variance and correlation structure for $R_Y$ encoding. See Section \ref{['sec:numerics']} for setup, see \ref{['subsubsec:results_var']} for discussion of the variances, and see \ref{['subsubsec:results_corr']} for discussion of the correlation matrices.
  • Figure 5: Composite summary for Circuit 16 and the associated variance and correlation structure for $R_Y$ encoding. See Section \ref{['sec:numerics']} for setup, see \ref{['subsubsec:results_var']} for discussion of the variances, and see \ref{['subsubsec:results_corr']} for discussion of the correlation matrices.
  • Figure 6: Composite summary for Circuit 17 and the associated variance and correlation structure for $R_Y$ encoding. See Section \ref{['sec:numerics']} for setup, see \ref{['subsubsec:results_var']} for discussion of the variances, and see \ref{['subsubsec:results_corr']} for discussion of the correlation matrices.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Proposition 4.1: Coefficient moments from $C$
  • proof : Proof sketch
  • Lemma C.1: Difference-frequency expansion
  • proof
  • Proposition C.1: Encoder-accessible harmonic set
  • proof : Proof sketch
  • Definition C.1: Path set and redundancy
  • Lemma D.1: Orthogonality of torus characters
  • proof
  • Proposition D.1: Moments from $C$
  • ...and 16 more