Tapestry of dualities in decohered quantum error correction codes

Tapestry of dualities in decohered quantum error correction codes

February 1, 2024 | Kaixiang Su, Zhou Yang, and Chao-Ming Jian
This paper explores the dualities among statistical models that characterize the Rényi entropies of decohered quantum error correction (QEC) codes, particularly Calderbank-Shor-Steane (CSS) codes under Pauli noise. The study reveals a "tapestry of dualities" woven by rich relationships among these models. For CSS codes with bit-flip and phase-flip errors, each Rényi entropy is captured by a pair of dual statistical models with randomness. For R = 2, 3, and ∞, additional dualities map between the two error types, relating the critical error rates of the decoherence-induced phase transitions. For CSS codes with "em symmetry" between X-type and Z-type stabilizers, these dualities become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by S_{R=2,3,∞}. For general stabilizer codes, the statistical models and their duality relations are constructed, showing general dualities between Pauli noise with different error rates. The paper also discusses the implications of these dualities for understanding the robustness of QEC codes against errors and the effect of decoherence on topological orders. The results are illustrated with examples such as the decohered 3D toric code and the decohered X-cube model. The study provides a framework for analyzing the behavior of error-corrupted mixed states in stabilizer codes under decoherence, highlighting the rich structure of dualities in the context of QEC and topological quantum matter.This paper explores the dualities among statistical models that characterize the Rényi entropies of decohered quantum error correction (QEC) codes, particularly Calderbank-Shor-Steane (CSS) codes under Pauli noise. The study reveals a "tapestry of dualities" woven by rich relationships among these models. For CSS codes with bit-flip and phase-flip errors, each Rényi entropy is captured by a pair of dual statistical models with randomness. For R = 2, 3, and ∞, additional dualities map between the two error types, relating the critical error rates of the decoherence-induced phase transitions. For CSS codes with "em symmetry" between X-type and Z-type stabilizers, these dualities become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by S_{R=2,3,∞}. For general stabilizer codes, the statistical models and their duality relations are constructed, showing general dualities between Pauli noise with different error rates. The paper also discusses the implications of these dualities for understanding the robustness of QEC codes against errors and the effect of decoherence on topological orders. The results are illustrated with examples such as the decohered 3D toric code and the decohered X-cube model. The study provides a framework for analyzing the behavior of error-corrupted mixed states in stabilizer codes under decoherence, highlighting the rich structure of dualities in the context of QEC and topological quantum matter.
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Understanding Tapestry of dualities in decohered quantum error correction codes