Stabilizer code

A stabilizer code is defined by a stabilizer subgroup S of the n-qubit Pauli group such that S is Abelian and does not contain −I. The code space is the joint +1 eigenspace of all operators in S; its dimension is 2^k where k = n − rank(S).¹ Errors are detected by measuring the generators of S, which produce a binary syndrome vector. The theory was systematically developed by Gottesman in his 1997 thesis² and extended to the CSS code family by Calderbank, Shor, and Steane.³

The Pauli group on n qubits, P_n, consists of all n-fold tensor products of Pauli matrices {I, X, Y, Z} with phases ±1, ±i. A stabilizer code is specified by a set of n − k independent, commuting generators from P_n. The normalizer N(S) of S in P_n contains operators that preserve the code space; logical operations correspond to cosets of N(S) modulo S.² The code distance d is the minimum weight of a Pauli operator that maps the code space to a distinct, orthogonal subspace, providing a measure of error tolerance.¹

The smallest nontrivial stabilizer code is the [[5,1,3]] perfect code, which encodes one logical qubit into five physical qubits and corrects any single-qubit error.⁴ CSS codes, a subclass of stabilizer codes, are constructed from two classical linear codes and admit transversal gates, simplifying fault-tolerant computation.³ The surface code, a topological stabilizer code on a 2D lattice, is the leading candidate for scalable quantum computing due to its high threshold and local geometry.⁵ Stabilizer codes also underpin measurement-based quantum computation and the Gottesman–Kitaev–Preskill (GKP) code for continuous-variable systems.

Stabilizer codes are equivalent to additive codes over the finite field GF(4), a correspondence that connects them to classical coding theory.³ Non-Pauli stabilizer generalizations exist, such as the conjugate stabilizer formalism for qudits. A subtle edge case occurs when the stabilizer contains operators with phase −1, which would force the code space to be empty; this is avoided by requiring all stabilizer elements to have eigenvalue +1 on the code space. Entanglement-assisted stabilizer codes use pre-shared entanglement to increase the code rate when the stabilizer group is not fully commuting.⁶ Graph states, which are stabilizer states with generators corresponding to a graph, are widely used in quantum networking.

Stabilizer

An operator in the Pauli group that leaves the code space invariant; the set of all such operators forms the stabilizer subgroup.

Pauli group

The group generated by single-qubit Pauli matrices X, Y, Z together with multiplicative phases ±1, ±i, under tensor products.

Logical qubit

A qubit encoded in the protected subspace of a quantum error-correcting code, representing the actual quantum information.

Syndrome

The outcome of measuring the stabilizer generators; a nonzero syndrome indicates the presence of an error.