CSS codes from the Bruhat order of Coxeter groups

TL;DR

This paper introduces a novel method to generate CSS quantum error-correcting codes from Coxeter groups using the Bruhat order, resulting in codes with controlled and diverse stabilizer weights, including irregular distributions.

Contribution

It develops a new approach to construct CSS codes from Coxeter group structures and Bruhat order, enabling control over stabilizer weights and code parameters.

Findings

Generated CSS codes with stabilizer weights up to 14 and 16.

Created codes with irregular stabilizer weight distributions.

Developed a weight-reduction method for heavy stabilizers.

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.