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

TL;DR

This paper investigates how the architecture shape of variational quantum circuits affects their trainability, revealing structural limitations in serial designs and advantages of parallel architectures, with implications for parameter efficiency.

Contribution

It introduces the concept of structural gradient starvation in serial architectures and demonstrates how parallel designs mitigate this issue, improving trainability and parameter efficiency.

Findings

Serial architectures exhibit rank deficiency in the Jacobian, leading to decoupled parameters.

Parallel architectures maintain full rank Jacobian, avoiding gradient starvation.

Adding feature map layers enhances the Jacobian spectrum more effectively than adding trainable blocks.

Abstract

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 $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{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 $\sigma_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, 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 $R^2 \geq 0.95$ with $1.6$--$2.2\times$ 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.