Probing Threshold Behavior of Adaptive Cascaded Quantum Codes Under Variable Biased Noise for Practical Fault-Tolerant Quantum Computing

This paper proposes a resource optimized cascaded quantum surface repetition code architecture integrated with a Union Find (UF) enhanced hybrid decoder, which suppresses biased noise and improves the scalability of quantum error correction through synergistic inner outer quantum code collaboration. The hybrid architecture employs inner quantum repetition codes for local error suppression and outer rotated quantum surface codes for topological robustness, reducing auxiliary quantum qubits by 12.5% via shared stabilizers and compact lattice embedding. An optimized UF decoder employing path compression and adaptive cluster merging achieves near-linear time complexity O(nα(n)), outperforming minimum-weight perfect matching (MWPM) decoders O(n2.5). Under Z-biased noise η=10, simulations demonstrate a 28.2% error threshold, 2.6% higher than standard quantum surface codes, and 15% lower logical error rates via dynamic boundary expansion. At code distance d=7, resource savings reach 9.3% with maximum relative error below 8.5%, fulfilling fault-tolerance criteria. The UF decoder exhibits 38% threshold advantage over MWPM at low bias η≈10−3 and 15% less degradation at high noise p=0.5, enabling scalable real-time decoding. This framework bridges theoretical thresholds with practical resource constraints, offering a noise-adaptive QEC solution for near-term quantum devices including photonic quantum systems referenced in the paper’s background on repetition cat qubits.