Tapestry of dualities in decohered quantum error correction codes

Abstract

Quantum error correction (QEC) codes protect quantum information from errors due to decoherence. Many of them also serve as prototypical models for exotic topological quantum matters. Investigating the behavior of the QEC codes under decoherence sheds light on not only the codes' robustness against errors but also new out-of-equilibrium quantum phases driven by decoherence. The phase transitions, including the error threshold, of the decohered QEC codes can be probed by the systems' Rényi entropies $S_R$ with different Rényi indices $R$. In this paper, we study the general construction of the statistical models that characterize the Rényi entropies of QEC codes decohered by Pauli noise. We show that these statistical models can be organized into a "tapestry" woven by rich duality relations among them. For Calderbank-Shor-Steane (CSS) codes with bit-flip and phase-flip errors, we show that each Rényi entropy is captured by a pair of dual statistical models with randomness. For $R=2,3,\infty$, there are additional dualities that map between the two error types, relating the critical bit-flip and phase-flip error rates of the decoherence-induced phase transitions in the CSS codes. For CSS codes with an "$em$ symmetry" between the $X$-type and the $Z$-type stabilizers, the dualities with $R=2,3,\infty$ become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by $S_{R=2,3,\infty }$. For general stabilizer codes decohered by generic Pauli noise, we also construct the statistical models that characterize the systems' entropies and obtain general duality relations between Pauli noise with different error rates.

22+16 pages, 11 figures

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• Topological Quantum Distillation
• Dynamically Generated Logical Qubits
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Duality in finite-dimensional spin glasses
• (Four) Dual Plaquette 3D Ising Models

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• Revisiting Nishimori multicriticality through the lens of information measures
• Quantum Strong-to-Weak Spontaneous Symmetry Breaking in Decohered One Dimensional Critical States
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