CSS codes from the Bruhat order of Coxeter groups
This work provides a method for designing quantum error-correcting codes with tailored properties, which is incremental as it builds on known geometric structures from Coxeter groups.
The authors tackled the problem of generating families of CSS codes with controlled parameters by leveraging the Bruhat order of Coxeter groups, resulting in codes with specific stabilizer weights such as [6006, 924, {≤14,≤7}] and [22880,3432, {≤8,≤16}].
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.