Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions

Journal Article · 08 August 2024 · PRX Quantum

DOI: https://doi.org/10.1103/PRXQuantum.5.030328 · OSTI ID:2427424

Liang, Zijian ; Xu, Yijia; Iosue, Joseph T. ; Chen, Yu-An

In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to ${\mathbb{Z}}_d$ qudits, including instances where $d$ is a nonprime number. This capability allows the identification of topological orders that differ from the ${\mathbb{Z}}_d$ toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for ${\mathbb{Z}}_p$ qudits (with $p$ being a prime) are equivalent to finite copies of ${\mathbb{Z}}_p$ toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024

• Sponsoring Organization: USDOE
• OSTI ID: 2427424
• Journal Information: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 3 Vol. 5; ISSN 2691-3399; ISSN PQRUAG
• Publisher: American Physical SocietyCopyright Statement
• Country of Publication: United States
• Language: English