Toric code

The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice It is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—Z2 topological order(first studied in the context of Z2 spin liquid in 1991). The toric code can also be considered to be a Z2 lattice gauge theory in a particular limit. It was introduced by Alexei Kitaev. The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice It is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—Z2 topological order(first studied in the context of Z2 spin liquid in 1991). The toric code can also be considered to be a Z2 lattice gauge theory in a particular limit. It was introduced by Alexei Kitaev. The toric code gets its name from its periodic boundary conditions, giving it the shape of a torus. These conditions give the model translational invariance, which is useful for analytic study. However, experimental realization requires open boundary conditions, allowing the system to be embedded on a 2D surface. The resulting code is typically known as the planar code. This has identical behaviour to the toric code in most, but not all, cases. The toric code is defined on a two-dimensional lattice, usually chosen to be the square lattice, with a spin-½ degree of freedom located on each edge. They are chosen to be periodic. Stabilizer operators are defined on the spins around each vertex v {displaystyle v} and plaquette (or face) p {displaystyle p} of the lattice as follows, A v = ∏ i ∈ v σ i x , B p = ∏ i ∈ p σ i z . {displaystyle A_{v}=prod _{iin v}sigma _{i}^{x},,,B_{p}=prod _{iin p}sigma _{i}^{z}.} Where here we use i ∈ v {displaystyle iin v} to denote the edges touching the vertex v {displaystyle v} , and i ∈ p {displaystyle iin p} to denote the edges surrounding the plaquette p {displaystyle p} . The stabilizer space of the code is that for which all stabilizers act trivially, hence, A v | ψ ⟩ = | ψ ⟩ , ∀ v , B p | ψ ⟩ = | ψ ⟩ , ∀ p , {displaystyle A_{v}|psi angle =|psi angle ,,,forall v,,,B_{p}|psi angle =|psi angle ,,,forall p,} for any state | ψ ⟩ {displaystyle |psi angle } . For the toric code, this space is four-dimensional, and so can be used to store two qubits of quantum information. This can be proven by considering the number of independent stabilizer operators. The occurrence of errors will move the state out of the stabilizer space, resulting in vertices and plaquettes for which the above condition does not hold. The positions of these violations is the syndrome of the code, which can be used for error correction. The unique nature of the topological codes, such as the toric code, is that stabilizer violations can be interpreted as quasiparticles. Specifically, if the code is in a state | ϕ ⟩ {displaystyle |phi angle } such that, A v | ϕ ⟩ = − | ϕ ⟩ {displaystyle A_{v}|phi angle =-|phi angle } ,