Quantifying the Hadamard Resilience Law: Discovery of the Coherence Gap in NISQ-Era Classifiers

We report on a fundamental disparity between stochastic noise models and algorithmic performance in NISQ-era classifiers. Utilizing the ibm_kingston processor, we characterize the “Kingston Constant” ($\kappa \approx 0.07$), representing a 93% signal magnitude collapse. Despite this decay, we demonstrate that the Hadamard Test Perceptron maintains a 93.9% MNIST accuracy, validating our proposed Hadamard Resilience Law. However, a systemic divergence – the “Coherence Gap” ($\mathrm{\Delta }\rho \approx 0.91$) – emerges at high feature depths ($N=256$), where physical hardware collapses while stochastic simulations remain resilient. This gap identifies coherent phase errors, rather than depolarizing noise, as the primary barrier to scaling quantum linear layers. Furthermore, experimental results on the ibm_kingston processor reveal a “Coherence Wall” at $N=256$, where circuit depth ($D\approx 10k$) exceeds the hardware’s resilient depth limit ($D_{max}\approx 3.5k$). We provide a refined hardware-aware model that accounts for this coherence-induced signal decay, establishing a predictive boundary for robust quantum linear layers on current NISQ devices.