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

TL;DR

This paper investigates error thresholds for Pauli channels using coset weight enumerators, revealing new stabilizer codes with significant non-additivity and providing insights into optimizing thresholds for concatenated codes.

Contribution

It introduces new concatenated stabilizer codes exhibiting non-additivity, derives a closed-form for coset weight enumerators, and estimates thresholds for large concatenated codes.

Findings

Identification of stabilizer codes with significant non-additivity

Closed-form expression for coset weight enumerators of certain codes

Estimated error thresholds for large concatenated repetition codes

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.