On Error Thresholds for Pauli Channels: Some answers with many more questions

2026-03-04

AI Summary

This study addresses the error threshold problem for Pauli channels, focusing on the non-additive behavior of small stabilizer codes and their concatenated structures near the threshold. Building upon the DiVincenzo–Shor–Smolin coset weight enumerator framework, the work combines numerical optimization with stabilizer code construction techniques to derive closed-form expressions for the coset weight enumerators of concatenated phase- and bit-flip repetition codes. The authors design specific Pauli channels that maximize non-additivity at the hashing point and identify several new short-length concatenated stabilizer codes exhibiting significant non-additive gains. They provide lower-bound estimates of their thresholds, reveal both positive and negative examples of non-additivity—including counterintuitive phenomena—and thereby establish a foundation for further research into lower bounds on Pauli channel error thresholds.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.

Problem

• error thresholds
• Pauli channels
• stabilizer codes
• non-additivity
• concatenated codes

Innovation

• Pauli channels
• error threshold
• non-additivity
• stabilizer codes
• coset weight enumerators