Quantifying the Hadamard Resilience Effect and the Coherence Gap in NISQ-Era Classifiers

Abstract

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 show via a hardware-calibrated stochastic digital twin that a Hadamard Test based Perceptron maintains a 93.9% accuracy for the MNIST dataset, validating our proposed Hadamard Resilience Effect under affine depolarization. Through parametric simulation, we establish a critical gate-noise threshold at $\lambda_{\text{crit}} \approx 0.65$ (corresponding to a critical signal retention $\alpha_{\text{crit}} \approx 0.35$) where topological rank preservation breaks down. Furthermore, physical execution at high feature depths ($N = 256$) reveals a systemic divergence---the ``Coherence Gap'' ($\Delta\rho \approx 0.91$)---where physical hardware classification accuracy collapses to 53.0\% due to a ``Coherence Wall'' at a circuit depth ($D \approx 10\text{k}$) exceeding the hardware's resilient depth limit ($D_{\text{max}} \approx 3.5\text{k}$). This gap is consistent with coherent phase errors and crosstalk being the dominant unmodeled contributors under the tested hardware configuration, establishing a predictive operational boundary for NISQ classification architectures.