Topological Color Codes and Two-Body Quantum Lattice Hamiltonians

TL;DR

This paper constructs a ruby lattice model with a 2-body Hamiltonian that encodes topological color codes, revealing novel topological degeneracies, fermionic excitations, and connections to cluster states, advancing quantum error correction and topological quantum computing.

Contribution

It introduces a new 2-body Hamiltonian on a ruby lattice that realizes topological color codes and explores its topological and fermionic properties, linking lattice models to quantum error correction.

Findings

Low energy spectrum described by a many-body Hamiltonian encoding color codes.

High energy excitations exhibit fermionic statistics.

Model shows topological degeneracy and connection to cluster states.

Abstract

Topological color codes are among the stabilizer codes with remarkable properties from quantum information perspective. In this paper we construct a four-valent lattice, the so called ruby lattice, governed by a 2-body Hamiltonian. In a particular regime of coupling constants, degenerate perturbation theory implies that the low energy spectrum of the model can be described by a many-body effective Hamiltonian, which encodes the color code as its ground state subspace. The gauge symmetry $\mathbf{Z}_{2}\times\mathbf{Z}_{2}$ of color code could already be realized by identifying three distinct plaquette operators on the lattice. Plaquettes are extended to closed strings or string-net structures. Non-contractible closed strings winding the space commute with Hamiltonian but not always with each other giving rise to exact topological degeneracy of the model. Connection to 2-colexes can be established at the non-perturbative level. The particular structure of the 2-body Hamiltonian provides a fruitful interpretation in terms of mapping to bosons coupled to effective spins. We show that high energy excitations of the model have fermionic statistics. They form three families of high energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. Also, we use Jordan-Wigner transformation in order to test the integrability of the model via introducing of Majorana fermions. The four-valent structure of the lattice prevents to reduce the fermionized Hamiltonian into a quadratic form due to interacting gauge fields. We also propose another construction for 2-body Hamiltonian based on the connection between color codes and cluster states. We discuss this latter approach along the construction based on the ruby lattice.