Decoherence as Defence and the Magnitude of Noise Regularisation: A Rigorous N -Qubit Theory of Stochastic Quantum Neural Networks for Adversarially Robust Network Intrusion Detection

# Decoherence as Defence and the Magnitude of Noise Regularisation: A Rigorous N -Qubit Theory of Stochastic Quantum Neural Networks for Adversarially Robust Network Intrusion Detection
Source: [https://arxiv.org/abs/2606.24219](https://arxiv.org/abs/2606.24219)
[View PDF](https://arxiv.org/pdf/2606.24219)

> Abstract:Stochastic quantum neural networks \(SQNNs\) encode neuronal activations as qubits, synaptic topology as entanglement, and neural noise through a Lindblad master equation\. A recent conference study applied a ring\-entangled SQNN to collaborative intrusion detection and reached three conclusions: ring entanglement is \\emph\{essential\} for non\-local anomaly detection; an adversarial\-resilience bound holds but is \\emph\{conservative\}; and the depolarising channel \\emph\{fails\} to act as a dropout\-style regulariser, behaving instead as output noise\. It left open whether a per\-gate stochastic deactivation \(\`\`true quantum dropout''\) could regularise where the depolarising channel could not, and whether the loose robustness bound could be replaced by a predictive theory\. This paper resolves both and extends the framework to real data and to neutral\-atom hardware\. We give an $N$\-qubit formulation through the stochastic master equation and its vectorised Liouvillian, and prove a \\emph\{decoherence\-contraction theorem\}: a depolarising channel of strength $\\gamma$ over $L$ entangling layers contracts every weight\-$w$ Pauli read\-out by a factor $\(1\-4\\gamma/3\)^\{wL\}$ \(for the weight\-$1$ read\-out used here, $\(1\-4\\gamma/3\)^\{L\}$\); building on the general noise\-as\-defence result of Du et al\., we make this quantitative and operational for intrusion detection\. On the real NSL\-KDD dataset under white\-box FGSM and PGD attacks, a depolarising SQNN trained with the channel is, over seven seeds under strong $\\ell\_\\infty$/$\\ell\_2$ attacks, significantly more robust than the noiseless circuit \($\\ell\_\\infty$ PGD\-$20$, $p=0\.04$, large effect\) and, critically, never suffers the catastrophic robustness collapse that the noiseless model and gradient\-trained classical detectors \(which fall from $95\\%$ to $47\\%$\) do, cutting robustness variance roughly twofold; we show this robustness arises from a noise\-reshaped training boundary rather than from attack\-time gradient contraction\. For generalisation, we derive an adaptive\-penalty formula showing that per\-gate dropout implements a curvature\-weighted $L\_2$ penalty $\\tfrac\{p\(1\-p\)\}\{2\}\\sum\\theta^2\\partial^2\_\\theta L$ in weight space, maximised at $p=1/2$, whereas depolarising noise implements an output\-space penalty\. A $30$\-seed study confirms the formula's quantitative prediction: both mechanisms reduce the train\-test gap by a small but statistically significant margin \($\\approx\\\!0\.01$; $p<10^\{\-4\}$ and $p=0\.004$\), are statistically indistinguishable from each other, and the effect is concentrated where overfitting is largest; increasing the dropout rate past $1/2$ does not help, as the formula predicts\. The single\-seed dichotomy of prior work does not survive replication\. We close with a neutral\-atom realisation and a feasibility\-by\-$N$ analysis\.

## Submission history

From: Gautier\-Edouard FILARDO \[[view email](https://arxiv.org/show-email/db8f8f62/2606.24219)\] \[via CCSD proxy\] **\[v1\]**Tue, 23 Jun 2026 07:06:56 UTC \(455 KB\)