The toric code is a stabilizer code with a qubit on every edge of a two-dimensional lattice on a torus. A vertex check multiplies Pauli operators on its incident edges, while a face check multiplies Pauli operators around its boundary:The checks commute because a vertex and face share either zero or two edges. The code is their common eigenspace, equivalently the space of ground states of
On the torus, gives two redundant checks. The remaining independent checks leave a four-dimensional code space, encoding two logical qubits. Products of Pauli operators around noncontractible loops act on that information. [ref]
A violated vertex or face check is an excitation. Open strings create pairs of excitations at their endpoints; closed noncontractible strings can change a logical state without creating local excitations. The electric and magnetic excitations are bosons with mutual full-braiding phase , while their composite is a fermion. [ref]
Fault-tolerant quantum computation by anyons by A. Yu. Kitaev. Section 2, string operators and anyon statistics
The surface-code anyons are . They obey ; and are bosons with mutual full-braiding phase , while their composite is a fermion.
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The Toric code is a type of topological quantum error-correcting code that was introduced by Alexei Kitaev in 2003. It is designed to protect quantum information from errors that can occur due to decoherence and other noise in quantum systems. The Toric code is notable for its ability to provide fault-tolerant quantum computation and is particularly significant in the field of quantum computing.