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Generalizing Quantum Tanner Codes

Olai Å. Mostad, Eirik Rosnes, Hsuan-Yin Lin

TL;DR

This paper broadens quantum Tanner codes by using group actions on finite sets, Schreier graphs, and square complexes to generate new families of asymptotically good quantum LDPC codes. It introduces commuting Schreier graphs and a square-complex framework to produce CSS codes via Tanner constructions, with a main equivalence theorem linking these square complexes to commuting graphs through a swapping condition. The approach subsumes the prior Cayley-graph based qTanner codes and clarifies when non-Cayley families can arise, offering a path to novel code families. Overall, the work lays a spectral- and combinatorics-based foundation for constructing robust quantum LDPC codes with potential practical impact for fault-tolerant quantum computation.

Abstract

In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Zémor, which contains a construction of asymptotically good quantum LDPC codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to group actions on finite sets, Schreier graphs, and a family of square complexes which is the largest possible in a certain sense. Furthermore, we discuss how the proposed generalization opens up the possibility of finding other families of asymptotically good quantum codes.

Generalizing Quantum Tanner Codes

TL;DR

This paper broadens quantum Tanner codes by using group actions on finite sets, Schreier graphs, and square complexes to generate new families of asymptotically good quantum LDPC codes. It introduces commuting Schreier graphs and a square-complex framework to produce CSS codes via Tanner constructions, with a main equivalence theorem linking these square complexes to commuting graphs through a swapping condition. The approach subsumes the prior Cayley-graph based qTanner codes and clarifies when non-Cayley families can arise, offering a path to novel code families. Overall, the work lays a spectral- and combinatorics-based foundation for constructing robust quantum LDPC codes with potential practical impact for fault-tolerant quantum computation.

Abstract

In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Zémor, which contains a construction of asymptotically good quantum LDPC codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to group actions on finite sets, Schreier graphs, and a family of square complexes which is the largest possible in a certain sense. Furthermore, we discuss how the proposed generalization opens up the possibility of finding other families of asymptotically good quantum codes.
Paper Structure (15 sections, 16 equations, 3 figures)

This paper contains 15 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Graphs
  5. Graph Expansion
  6. Tanner Codes and Quantum CSS Codes
  7. Square Complexes
  8. Proposed Generalized Construction
  9. Example
  10. New Construction
  11. General Case
  12. Symmetric Labeling Set
  13. Connection With Previous Constructions
  14. Equivalent Characterizations
  15. Asymptotically Good Quantum Codes

Figures (3)

  • Figure 1: Our construction (green) generalizes the quantum Tanner codes (qTc) from LeverrierZemor22_1 (brown). We also consider a slightly less general construction (blue), and in \ref{['sec:equic-char']} we show that there are codes not from our construction that also reasonably may be called general quantum Tanner codes (red).
  • Figure 2: The Petersen graph is shown to the left. On the right is a Schreier graph commuting with the Petersen graph. They are labeled by $\set A = \{a_0,a_1,a_2\}$ and $\set B = \{b_0,b_1\}$, respectively, where $a_0^{-1}=a_0$, $a_1^{-1}=a_2$, and $b_0^{-1}=b_1$.
  • Figure 3: The depicted graph is considered in \ref{['ex:red_nonempty']}. It is $4$-regular with edges depicted as arrows. An edge has the indicated label in the local view of the source of the arrow depicting it. An edge labeled $(a_0, b_0)$ in one of its local views is labeled $(a_1, b_0)$ in the other, and similarly the "inverse" of $(a_0, b_1)$ is $(a_1, b_1)$.