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A two-dimensional color code is a topological stabilizer code commonly defined on a trivalent lattice whose faces admit three colors. Place a qubit at each vertex and assign both an X-type and a Z-type stabilizer generator to each face, acting on its boundary vertices.

In the usual allowed lattices, the relevant face overlaps have even size, making X- and Z-type checks commute. Violated checks identify an error syndrome, while operators spanning nontrivial paths or boundaries can act on encoded logical qubits. The geometry and boundary choices determine the number of logical qubits and code distance. Measuring local checks can reveal physical errors without measuring logical amplitudes; the full correction procedure must also address noisy check measurements and decoding.

A surface code is a family of topological stabilizer codes on a two-dimensional cellulation. In a standard square-lattice convention, qubits lie on edges, an X-type check multiplies X on edges meeting a vertex, and a Z-type check multiplies Z around a face.

Vertex and face checks commute because their overlaps contain zero or two edges. A local Pauli operator error changes nearby check signs, producing an error syndrome. Chains of errors have syndromes at their endpoints; a chain connecting appropriate boundaries or winding around a nontrivial cycle can instead implement a logical operation. Locality makes repeated syndrome extraction useful, but a sufficiently long undetected chain can still corrupt the encoded state. The associated surface-code anyon model describes the electric and magnetic check excitations and their mutual braiding.

A topological stabilizer code is a quantum stabilizer code whose local commuting checks are arranged on a spatial lattice and whose logical information is encoded in global structure. Local errors create detectable syndrome patterns; a nontrivial logical operator typically requires a sufficiently extended chain or surface.

The toric code is a central example: products of Pauli operators around vertices and faces define the stabilizers, while noncontractible loops act on encoded qubits. Protection depends on code distance and an appropriate decoder, not on topology alone guaranteeing error-free storage.