Fault-Tolerant Quantum Communication over r-Robust Graph Topologies under Adversarial and Noisy Conditions

The realization of scalable quantum communication networks faces critical challenges from decoherence, noise, and adversarial attacks. This paper presents a comprehensive framework for fault-tolerant quantum communication using \textit{r}-robust graph topologies, which ensure strong connectivity and resilience in the presence of both random failures and Byzantine adversaries. We formalize \textit{r}-robustness in single-layer and multiplex quantum networks and derive theoretical bounds on fidelity, consensus success, and error propagation. A hybrid quantum-classical protocol is proposed that leverages multi-path entanglement and quantum certificate-based identity enforcement. Extensive simulations using Qiskit and NetworkX show that \textit{r}-robust networks improve average entanglement fidelity by up to 26\%, achieve over 95\% consensus success under 30\% adversarial nodes, and exponentially suppress error rates. Compared to ring, scale-free, and random graphs, \textit{r}-robust topologies provide superior robustness with moderate resource overhead. Our results establish a graph-theoretic foundation for constructing secure, resilient, and NISQ-compatible quantum networks capable of sustaining high-performance communication in noisy and adversarial environments.