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.

arXiv comment22+16 pages, 11 figures

References in corpus (25)

• Fault-tolerant quantum computation by anyons
• Topological quantum memory
• Quantum Error Correction for Quantum Memories
• Suppressing quantum errors by scaling a surface code logical qubit
• Probing Topological Spin Liquids on a Programmable Quantum Simulator
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory
• A Unified and Generalized Approach to Quantum Error Correction
• Dynamically Generated Logical Qubits
• Quantum criticality under decoherence or weak measurement
• Strong Resilience of Topological Codes to Depolarization
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Diagnostics of mixed-state topological order and breakdown of quantum memory
• Statistical mechanical models for quantum codes with correlated noise
• Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory
• Coherent error threshold for surface codes from Majorana delocalization
• Monte Carlo study of very weak first-order transitions in the three-dimensional Ashkin-Teller model
• Non-Standard Finite-Size Scaling at First-Order Phase Transitions
• Higher-form Symmetries under Weak Measurement
• Duality in finite-dimensional spin glasses
• Exotic Symmetry Breaking Properties of Self-Dual Fracton Spin Models
• (Four) Dual Plaquette 3D Ising Models

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• Coherent information as a mixed-state topological order parameter of fermions
• Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order
• Anomalous distribution of magnetization in an Ising spin glass with correlated disorder
• Mixed-State Topological Order under Coherent Noise
• Revisiting Nishimori multicriticality through the lens of information measures
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