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New topological subsystem codes from semi-regular tessellations

Eduardo Brandani da Silva, Evandro Mazetto Brizola

TL;DR

The paper advances topological quantum error-correcting codes by constructing new topological subsystem codes from semi-regular Euclidean and hyperbolic tessellations. It introduces a generalized hypergraph $\Gamma_h$ construction that yields a gauge group via loop operators and stabilizers from hypercycles, ensuring topological properties under conditions $(C_1)$–$(C_3)$ through a careful colorability and intersection framework. Four families of codes are derived, with explicit parameter formulas expressed in terms of Euler characteristic $\chi$, vertex/face counts, and tessellation data, including both Euclidean and hyperbolic regimes. These codes offer locally measurable, 2-local syndromes and flexible parameter regimes, broadening the landscape of fault-tolerant quantum computing architectures based on semi-regular tessellations.

Abstract

In this work, we present new constructions for topological subsystem codes using semi-regular Euclidean and hyperbolic tessellations. They give us new families of codes, and we also provide a new family of codes obtained through an already existing construction, due to Sarvepalli and Brown. We also prove new results that allow us to obtain the parameters of these new codes.

New topological subsystem codes from semi-regular tessellations

TL;DR

The paper advances topological quantum error-correcting codes by constructing new topological subsystem codes from semi-regular Euclidean and hyperbolic tessellations. It introduces a generalized hypergraph construction that yields a gauge group via loop operators and stabilizers from hypercycles, ensuring topological properties under conditions – through a careful colorability and intersection framework. Four families of codes are derived, with explicit parameter formulas expressed in terms of Euler characteristic , vertex/face counts, and tessellation data, including both Euclidean and hyperbolic regimes. These codes offer locally measurable, 2-local syndromes and flexible parameter regimes, broadening the landscape of fault-tolerant quantum computing architectures based on semi-regular tessellations.

Abstract

In this work, we present new constructions for topological subsystem codes using semi-regular Euclidean and hyperbolic tessellations. They give us new families of codes, and we also provide a new family of codes obtained through an already existing construction, due to Sarvepalli and Brown. We also prove new results that allow us to obtain the parameters of these new codes.
Paper Structure (7 sections, 18 theorems, 69 equations, 19 figures, 5 tables)

This paper contains 7 sections, 18 theorems, 69 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Subsystem codes
  3. Topological subsystem codes
  4. A new construction to obtain topological subsystem codes
  5. New topological subsystem codes
  6. Topological subsystem codes from tessellations $\{2p_1,2p_2,4\}$
  7. Topological subsystem codes From tessellation $\{2p_1,4,6\}$

Key Result

Theorem 8

Let $e',e"\in E$ be any two edges of $\Gamma_h$. So, for their respective operators, $K_{e'}$ and $K_{e"}$, we have with $\eta(e',e") = 0$, if $e'$ and $e"$ share an even number of vertices or $e',e"\in E_3$, and otherwise $\eta(e',e") = 1$.

Figures (19)

  • Figure 1: ($a$) Tessellation $\{6,6,6\}$, its dual $\Gamma^*$ and the new tessellation $\overline{\Gamma}$ given by black edges. ($b$) Colored edges of tessellation $\overline{\Gamma}$.
  • Figure 2: Colored edges representing the $\gamma\in\overline{\Gamma}$ subgraph.
  • Figure 3: Colored edges of the operator $S_v^g$.
  • Figure 4: Rank-$3$ edges inserted into face $f\in F$ from the vertices of face $f'$. ($a$) blue rank-$3$ edges and ($b$) green rank-$3$ edges.
  • Figure 5: Hypercycles corresponding to face $f\in F\subseteq F_R$ represented by colored edges. ($a$) Hypercycle $f_{\sigma_1}$, ($b$) hypercycle $f_{\sigma_2}$ and ($c$) Hypercycle $f_{\sigma_3}$.
  • ...and 14 more figures

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 8
  • Remark 9
  • Definition 10
  • ...and 36 more