# Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

arXiv: https://arxiv.org/abs/2605.05942 · HTML: https://mlsanity.com/paper/2605.05942/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 23 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 10 of 10)

- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Spectral Bias in Variational Quantum Machine Learning](https://mlsanity.com/paper/2506.22555/) (2025)
- [Spectral methods: crucial for machine learning, natural for quantum computers?](https://mlsanity.com/paper/2603.24654/) (2026)
- [Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks](https://mlsanity.com/paper/2402.14515/) (2024)

## Abstract (arXiv)

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging N qubits with L encoding layers each -- sharing encoding budget E = NL -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape (N,L) at fixed E. We identify structural rank deficiency of the coefficient matching Jacobian J as the mechanism responsible. For serial single-qubit architectures, we prove rank(J) ≤ 2L+1 regardless of parameter count P, with dim(ker J) ≥ P-(2L+1) growing without bound -- a phenomenon we term structural gradient starvation: a growing fraction of parameters become structurally decoupled from the loss as P increases at fixed L. Parallel architectures avoid this via independent phase trajectories, ensuring σmin(J(par)) > 0 generically for P ≤ 2E+1, so no parameter lies in ker J. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves R2 ≥ 0.95 with 1.6--2.2× fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.