Emergent Effective Depth in Quantum Machine Learning: Empirical Gradient Attenuation Scaling in Noisy Variational Quantum Algorithms

Abstract

Noisy Intermediate-Scale Quantum (NISQ) devices impose structural constraints on the training of Variational Quantum Algorithms (VQAs). In the absence of full error correction, each quantum gate introduces a non-zero probability of error that accumulates with circuit depth, while gradient estimation through finite sampling adds additional statistical variability. As a result, convergence depends on a delicate balance between physical coherence and estimation variance. In this setting, the depth of the variational ansatz not only determines model expressivity, but also its operational feasibility under noise. Without an explicit characterization of the interaction between error accumulation, number of measurements (shots), and control strategies, increasing experimental resources may fail to improve performance and can even become counterproductive. In this work, we empirically analyze how circuit depth affects the practical signal of the gradient under depolarizing noise. Across multiple configurations ($q=4-7$ qubits), we observe a pattern consistent with an effective exponential attenuation of the coherent gradient contribution, characterized by a per-layer rate $\epsilon_{\text{eff}}$. The estimated magnitude of $\epsilon_{\text{eff}}$ depends on circuit complexity and remains largely independent of the measurement budget, indicating that it primarily reflects architectural and physical factors rather than sampling effects. The analysis of adaptive control strategies further suggests that accuracy gains are constrained by gradient attenuation, rather than scaling significantly with circuit depth. However, a crossmode comparison across three distinct training strategies reveals that, while accuracy improvements remain modest, adaptive perturbations substantially improve the statistical detectability of the attenuation signal, acting as diagnostic probes rather than mere optimization heuristics. Taken together, these results suggest that the usable depth of a variational ansatz can be interpreted as an emergent property of the effective signal-to-noise regime, governed by a measurable structural parameter whose ranking across circuit configurations is preserved independently of the training mode. This framework enables principled comparison of circuit architectures in terms of their effective trainability under noise, providing guidance for architectural and control design in Quantum Machine Learning under NISQ conditions.