CSS codes from the Bruhat order of Coxeter groups

2026-03-16

AI Summary

This work proposes a systematic method for constructing CSS quantum error-correcting codes with controllable stabilizer weights and favorable distance properties, leveraging Coxeter groups and their Bruhat partial order. By interpreting Bruhat intervals as face posets of regular CW complexes, the authors construct associated chain complexes and convert them into CSS codes, thereby achieving fine-grained control over stabilizer weights for the first time. A novel weight-reduction technique is introduced to handle sparse high-weight stabilizers, alongside a four-term chain complex structure incorporating elementary checks. The approach yields explicit code constructions—including $[6006,924,\{\leq14,\leq7\}]$, $[22880,3432,\{\leq8,\leq16\}]$, and a highly non-uniform $[571,199,\{5,5\}]$—demonstrating both the efficacy and flexibility of the framework.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group elements named after Bruhat. The Bruhat order is known to provide a link to algebraic topology -- it doubles as a face poset capturing the inclusion relations of the $p$-dimensional cells of a regular CW~complex and that is what makes it interesting for QEC code design. Assisted by the Bruhat face poset interval structure unique to Coxeter groups I show that the corresponding chain complexes can be turned into multitudes of CSS codes. Depending on the approach, I obtain CSS codes (and their families) with controlled stabilizer weights, for example $[6006, 924, \{{\leq14},{\leq7}\}]$ (stabilizer weights~14 and 9) and $[22880,3432,\{{\leq8},{\leq16}\}]$ (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as $[571,199,\{5,5\}]$. For the latter, I develop a weight-reduction method to deal with rare heavy stabilizers. Finally, I show how to extract four-term (length three) chain complexes that can be interpreted as CSS codes with a metacheck.

Problem

• CSS codes
• Coxeter groups
• Bruhat order
• quantum error correction
• stabilizer weights

Innovation

• CSS codes
• Bruhat order
• Coxeter groups
• chain complexes
• stabilizer weight reduction