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CSS codes from the Bruhat order of Coxeter groups

Kamil Bradler

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.

CSS codes from the Bruhat order of Coxeter groups

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 -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 (stabilizer weights~14 and 9) and (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as . 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.
Paper Structure (19 sections, 8 theorems, 60 equations, 15 figures, 4 tables, 1 algorithm)

This paper contains 19 sections, 8 theorems, 60 equations, 15 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. CSS code construction
  3. Trivial codes
  4. Non-trivial codes
  5. $S^1$ (crown) decomposition
  6. The case of $S^0$ and $S^2$ decompositions
  7. Other CSS code constructions -- random splicing and chain complex folding
  8. Code examples obtained by splicing or folding
  9. Stabilizer weight reduction
  10. Conclusions and open problems
  11. Math background
  12. Discrete groups
  13. Coxeter groups
  14. Partially ordered sets and Bruhat order
  15. Bruhat order on Coxeter group
  16. ...and 4 more sections

Key Result

Lemma 1

Let $(W,w_b,w_t,p)_{3}$ be a length two chain complex such that and $\ell([w_b,w_t])\geq5$. Further, let $\euQ \subseteq l_p$ be a poset layer of rank $p$. Whenever $\euX\subseteq l_{p-1}$ and $\euZ\subseteq l_{p+1}$ are such that for all $x\in\euX$ and $z\in\euZ$, either $(x,z) \subseteq \euQ$ or $ and define a CSS code where any $X$ and $Z$ check overlap on either zero or two qubits.

Figures (15)

  • Figure 1: The Bruhat order of the Weyl group $A_3$, \ref{['eq:A3presentation']}. The open interval $(w_b,w_t)$ is a face poset of a regular CW-cellulated sphere $S^d$ for $d=4$. The vertices can be seen as $(p-1)$-cells ($1\leq p\leq d+1=5$) ordered by inclusion or as Coxeter group elements $w=s_{i_1}\dots s_{i_p}\in A_3$ of the reduced length $\ell(w)=p$ related by a reflection. The edges are covering relations of the Bruhat order.
  • Figure 2: The Bruhat poset of $A_3$ from \ref{['fig:A3BruhatPoset']} in a geometrically friendly way as a truncated octahedron (also known as a permutahedron in combinatorics) with some additional inner-face edges whose origin are non-simple reflections. The orange balls are the black vertices and the rods connecting them are the covering relations.
  • Figure 3: The Tanner graphs of the $(A_3,\mathop{{\mathrm{id}}}\nolimits,s_1s_2s_3s_1s_2s_1,3)_{3}$ CSS code (consisting of layers $l_{2},l_3$ and $l_4$ in \ref{['fig:A3BruhatPoset']}) with different highlighted $S^2$ posets in magenta. The connected bottom and top elements $\hat{b}$ and $\hat{t}$ from neighboring layers are not depicted. The corresponding CW complexes are depicted in \ref{['fig:A3S2decompositionCW']}.
  • Figure 4: $S^2$ regular CW complexes corresponding to the magenta subposets in Fig. \ref{['fig:A3S2decompositionTanner']} with the same color coding and numbering: red vertices (say $X$ checks), black edges (data qubits) and blues faces ($Z$ checks) including the ambient ones. The CW complex on the left is recognizable as a tetrahedron. Also cf. the classified $S^2$ CW complexes number 1 and 3 in \ref{['fig:hultman']}.
  • Figure 5: The Bruhat/weak order of the reducible Coxeter group $C_2^{\times4}$. The open interval $(w_b,w_t)$ of length four is a face poset $\euF(S^2)$.
  • ...and 10 more figures

Theorems & Definitions (24)

  • Definition 1
  • Example
  • Example
  • Example
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • Definition 2
  • ...and 14 more