On the robustness of topological quantum codes: Ising perturbation

TL;DR

This paper investigates the stability of topological quantum codes under Ising perturbations by mapping them to the transverse field Ising model, revealing how lattice geometry affects their robustness.

Contribution

It introduces a detailed mapping from topological codes to the Ising model, enabling analysis of phase transitions and robustness depending on lattice triangulation.

Findings

Robustness varies with lattice triangulation.

Mapping to Ising model identifies phase transition points.

Provides a comparison table for topological code stability.

Abstract

We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse field. Such a mapping is derived by re-writing the initial Hamiltonian in a new basis so that the final model in such a basis has a well-known approximated phase transition point. Specifically, we consider the toric codes and the color codes on some various lattices with Ising perturbation. Our results provide a useful table to compare the robustness of the topological codes and to explicitly show that the robustness of the topological codes depends on triangulation of their underlying lattices.