# Techniques: Constrained and Vanishing Expressivity of Quantum Fourier Models

## What the paper contributes (one paragraph)
This paper analyses the expressivity of parameterised quantum circuits (PQCs) used for machine learning when they are viewed as truncated Fourier series in the classical input, with frequencies set by the encoding Hamiltonians. It introduces the notion of frequency redundancy, the number of distinct eigenvalue paths that generate a given frequency, and proves that the variance of each Fourier coefficient is controlled by that redundancy. Under exact 2-design, approximate 2-design, and brickwise local 2-design assumptions on the trainable layers, it gives exact expressions and upper bounds for these variances. From these it establishes a vanishing expressivity phenomenon, where low-redundancy Fourier coefficients concentrate exponentially in the qubit count, and a generic Parseval-style bound on the coefficient vector. The results expose an encoding-dependent inductive bias and inform classical surrogate (random Fourier feature) dequantisation. Findings are confirmed numerically.

## Techniques offered
- **Fourier coefficient variance characterisation for PQCs (redundancy-based expressivity analysis)** - computes exact values and upper bounds on the variance of each Fourier coefficient of a quantum model as a function of its frequency redundancy, under 2-design, approximate 2-design, and local 2-design assumptions on the trainable layers, supplying an encoding-dependent measure of effective expressivity.
  - guarantee: statistical bound (exact variance expression and upper bounds, holding on average over trainable parameters sampled uniformly)
  - quote: "we study the concentration of Fourier coefficients by computing their variance under different assumptions about the trainable unitaries distribution."
- **Vanishing expressivity diagnosis** - identifies which Fourier coefficients suffer exponentially decaying variance as the number of qubits grows, giving a diagnostic for exponential concentration at the level of individual spectral components rather than the whole model.
  - guarantee: statistical bound (exponential decay of variance, order one over b to the n for some b greater than one, for low-redundancy frequencies)
  - quote: "we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially as the number of qubits grows."
- **Fourier norm bound (Parseval-style coefficient bound)** - a generic constraint bounding the 2-norm of the Fourier coefficient vector by the largest eigenvalue of the observable, valid for any Hamiltonian-encoding quantum Fourier model independent of the trainable distribution.
  - guarantee: static proof (deterministic upper bound holding for all parameters and any real-valued spectrum)
  - quote: "Moreover, we establish another constraint on the 2-norm of the Fourier coefficients vector that holds for any quantum Fourier Model."
- **Redundancy frequency distribution for RFF-based dequantisation** - proposes the redundancy-induced probability distribution over the spectrum as the frequency-sampling distribution for building a classical random Fourier feature surrogate of a quantum model.
  - guarantee: none (heuristic distribution choice supported by numerics, not an analytical surrogate-accuracy guarantee)
  - quote: "With our results, we can propose the frequency distribution given by the redundancies as described in section \ref{sec:Framework} as a natural distribution that encodes the bias in the quantum model."

## Where it could apply
- Target primitive(s): VQE, QAOA, and variational quantum-circuit models more broadly; specifically quantum reuploading / quantum Fourier models for supervised learning
- Target application group(s): quantum_ml, platforms_tooling
- Code family / hardware assumptions: none (gate-model PQCs with Hamiltonian data encoding; results extend to subspace-preserving circuits by replacing the Hilbert-space dimension with the subspace dimension)

## Caveats
This is primarily a theory and characterisation paper on quantum machine learning expressivity, not an accreditation, decoding, or error-correction result. The variance and vanishing-expressivity guarantees hold only on average over uniformly sampled trainable parameters that form (approximate) 2-designs, so they do not transfer directly to a specific trained model; the approximate 2-design bounds apply only to single-uploading-layer circuits, and estimating the epsilon distance to a 2-design is not efficient in practice. The dequantisation distribution is a heuristic supported by numerics rather than a proven surrogate guarantee.