The paper discusses the potential of anyonic excitations in two-dimensional quantum systems for fault-tolerant quantum computation. The author proposes using these excitations to perform unitary transformations and measurements, which are essential for quantum computing. The key idea is to use the physical properties of anyons, such as non-abelian braiding, to implement quantum gates and perform error correction. The paper begins by introducing the concept of stabilizer quantum codes on a torus, where qubits are placed on the edges of a lattice and stabilizer operators are defined at the vertices and faces. The ground state of the Hamiltonian associated with these operators is protected from errors due to the degeneracy of the ground state space. This degeneracy persists even under small perturbations, making the system robust against local errors. The author then introduces the idea of abelian anyons, which are particles that can move around each other without changing their phase. These particles can be used to perform quantum gates, but their capabilities are limited. The paper also discusses the existence of non-abelian anyons, which can realize more complex quantum gates and are essential for universal quantum computation. The model for non-abelian anyons is constructed using a group algebra, where the Hilbert space is spanned by the elements of a finite group. The Hamiltonian is designed to be robust against local perturbations, and the ground state space is protected from errors. The paper concludes by discussing the algebraic structure of the system, including the classification of elementary excitations and the properties of local operators. Overall, the paper provides a theoretical framework for using anyonic excitations in two-dimensional systems to perform fault-tolerant quantum computation, highlighting the potential of these systems for practical quantum computing applications.The paper discusses the potential of anyonic excitations in two-dimensional quantum systems for fault-tolerant quantum computation. The author proposes using these excitations to perform unitary transformations and measurements, which are essential for quantum computing. The key idea is to use the physical properties of anyons, such as non-abelian braiding, to implement quantum gates and perform error correction. The paper begins by introducing the concept of stabilizer quantum codes on a torus, where qubits are placed on the edges of a lattice and stabilizer operators are defined at the vertices and faces. The ground state of the Hamiltonian associated with these operators is protected from errors due to the degeneracy of the ground state space. This degeneracy persists even under small perturbations, making the system robust against local errors. The author then introduces the idea of abelian anyons, which are particles that can move around each other without changing their phase. These particles can be used to perform quantum gates, but their capabilities are limited. The paper also discusses the existence of non-abelian anyons, which can realize more complex quantum gates and are essential for universal quantum computation. The model for non-abelian anyons is constructed using a group algebra, where the Hilbert space is spanned by the elements of a finite group. The Hamiltonian is designed to be robust against local perturbations, and the ground state space is protected from errors. The paper concludes by discussing the algebraic structure of the system, including the classification of elementary excitations and the properties of local operators. Overall, the paper provides a theoretical framework for using anyonic excitations in two-dimensional systems to perform fault-tolerant quantum computation, highlighting the potential of these systems for practical quantum computing applications.