The Stabilizer Formalism
Describing states by the Pauli operators that fix them yields a compact language for error correction and efficient simulation.
States defined by their symmetries
Instead of listing amplitudes, the stabilizer formalism describes a state by the operators that leave it unchanged. An n-qubit stabilizer state is the unique joint +1 eigenstate of n independent commuting Pauli operators, its stabilizer group. For example, the Bell state Phi+ is stabilized by XX and ZZ. Specifying n generators, each a Pauli string with a sign, replaces specifying 2^n amplitudes.
Efficient representation
Each Pauli generator on n qubits is recorded by 2n bits (X and Z parts) plus a sign bit, so a stabilizer state needs about 2n^2 bits, a polynomial description. Clifford gates, CNOT, Hadamard, and phase, map Paulis to Paulis, so they update the generators in polynomial time. Measurement in the computational basis also updates the stabilizers efficiently. This is the machinery behind the Gottesman-Knill theorem.
- Generators: n commuting, independent Pauli strings
- Bell state Phi+: stabilized by XX and ZZ
- GHZ (3-qubit): stabilized by XXX, ZZI, IZZ
- Clifford gates permute the generators; T gates do not
Quantum error correction
Stabilizer codes are the dominant framework for quantum error correction. The code space is the joint +1 eigenspace of a chosen stabilizer group; errors that anticommute with a generator flip its eigenvalue, producing a detectable syndrome. Measuring the generators reveals which errors occurred without disturbing the encoded data. The surface code, Steane code, and Shor code are all stabilizer codes, differing only in the choice of generators.
Scope
Stabilizer states form a discrete, structured subset of all states, closed under Clifford operations and Pauli measurement. They are not universal: reaching arbitrary states requires non-Clifford resources like magic states. The formalism's power comes precisely from this restriction, giving an efficient, exact description of the operations most relevant to error correction while marking the boundary beyond which classical simulation breaks down.