stabilizerformalism

Stabilizer formalism is a framework in quantum information theory for describing a broad class of quantum states and quantum error-correcting codes using the Pauli group. It centers on stabilizer groups, which are Abelian subgroups of the n-qubit Pauli group that do not contain the element -I. A stabilizer state is the joint +1 eigenstate of all operators in its stabilizer group; a stabilizer code generalizes this concept to a subspace stabilized by a chosen set of commuting Pauli operators.

Formally, let P_n be the n-qubit Pauli group and S be an Abelian subgroup of P_n that does not include -I. The codespace C is the joint +1 eigenspace of every element of S. If S has r independent generators, the dimension of C is 2^{n-r}, so a stabilizer code encodes k = n - r logical qubits. The code distance d characterizes the code’s ability to detect and correct errors.

In this framework, errors are modeled as Pauli operators up to equivalence, and the error syndrome is obtained by measuring the stabilizer generators. An error E that commutes with all stabilizers acts within the codespace as a logical operation, while an error that anticommutes with at least one stabilizer maps the code space to an orthogonal syndrome subspace, enabling detection and, with sufficient distance, correction.

Key consequences include efficient classical simulation of stabilizer circuits (the Gottesman-Knill theorem) and a unifying language for many quantum error-correcting codes, such as the Steane and 5-qubit codes, as well as topological codes like the surface code. The formalism also underpins the normalizer of S, containing logical operators, and the use of stabilizer states in measurement-based quantum computation.