II . Quantum Lattice Hamiltonian with Two - Body Interactions 7 A . Color Codes as Topological Stabilizers 7

1 Abstract Topological color codes are among the stabilizer codes with remarkable properties from quantum information perspective. In this paper we construct a lattice, the so called ruby lattice, with coordination number four governed by a 2-body Hamiltonian. In a particular regime of coupling constants, in the triangular limit, 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. Ground state subspace corresponds to vortex-free sector. The gauge symmetry Z 2 × Z 2 of color code could already be realized by identifying three distinct plaquette operators on the lattice. All plaquette operators commute with each other and with the Hamiltonian being integrals of motion. 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 via the coloring of the strings. We discuss it 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. The emergence of invisible charges related to the string-net structure of the model. 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. The corresponding 2-body Hamiltonian encodes cluster state defined on a bipartite lattice as its low energy spectrum, and subsequent selective measurements give rise to the color code model. We discuss this latter approach along the construction based on the ruby lattice.