Cylindrical and M\"obius Quantum Codes for Asymmetric Pauli Errors

TL;DR

This paper introduces cylindrical and M"obius topological quantum codes designed for asymmetric Pauli errors, demonstrating their superior performance over standard surface codes with MWPM decoding.

Contribution

It proposes novel topological quantum codes based on fiber bundle structures, specifically cylindrical and M"obius codes, for better handling asymmetric errors.

Findings

Cylindrical and M"obius codes outperform standard surface codes with MWPM decoding.

The codes are constructed using algebraic chain complexes formalism.

Theoretical upper bounds for logical error rates are provided.

Abstract

In the implementation of quantum information systems, one type of Pauli error, such as phase-flip errors, may occur more frequently than others, like bit-flip errors. For this reason, quantum error-correcting codes that handle asymmetric errors are critical to mitigating the impact of such impairments. To this aim, several asymmetric quantum codes have been proposed. These include variants of surface codes like the XZZX and ZZZY surface codes, tailored to preserve quantum information in the presence of error asymmetries. In this work, we propose two classes of Calderbank, Shor and Steane (CSS) topological codes, referred to as cylindrical and M\"obius codes, particular cases of the fiber bundle family. Cylindrical codes maintain a fully planar structure, while M\"obius codes are quasi-planar, with minimal non-local qubit interactions. We construct these codes employing the algebraic chain complexes formalism, providing theoretical upper bounds for the logical error rate. Our results demonstrate that cylindrical and M\"obius codes outperform standard surface codes when using the minimum weight perfect matching (MWPM) decoder.