## Description

Also called a Pauli stabilizer code. An \(((n,2^k,d))\) qubit stabilizer code is denoted as \([[n,k]]\) or \([[n,k,d]]\), where \(d\) is the code's distance. Logical subspace is the joint eigenspace of commuting Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace of a set of \(2^{n-k}\) commuting Pauli operators which do not contain \(-I\). The distance is the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.

Each stabilizer code can be represented by a \((n-k) \times 2n\) check matrix (a.k.a. stabilizer generator matrix) \(H=(A|B)\), where each row \((a|b)\) is the binary symplectic representation of an element from a set of generating elements of the stabilizer group. The check matrix can be brought into standard form via Gaussian elimination [3].

The stabilizer commutation condition can equivalently be stated in the symplectic representation. A pair of \(n\)-qubit stabilizers with symplectic representations \((a|b)\) and \((a^{\prime}|b^{\prime})\) commute iff their symplectic inner product is zero, \begin{align} a \cdot b^{\prime} + a^{\prime}\cdot b = \sum_{j=1}^{n} a_j b^{\prime}_j + a^{\prime}_i b_i = 0~. \tag*{(1)}\end{align} Symplectic representations of stabilizer group elements thus form a self-orthogonal subspace of \(GF(2)^{2n}\) with respect to the symplectic inner product.

## Protection

Detects errors on up to \(d-1\) qubits, and corrects erasure errors on up to \(d-1\) qubits. More generally, define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all Pauli operators that commute with all \(S\in\mathsf{S}\). A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\).

A stabilizer code is geometrically local if the support of the stabilizer generators is bounded by a ball of size independent of \(n\). There is an upper bound of \(d \leq O(L^{D-1})\) on the distance [4] of geometrically local stabilizer codes arranged in a \(D\)-dimensional lattice of length \(L\) with \(n=L^D\).

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Notes

## Parents

- Codeword stabilized (CWS) code — If the CWS set \( \mathcal{W} \) is an abelian group not containing \(-I\), then the CWS code is a stabilizer code.
- XP stabilizer code — The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).
- Stabilizer code
- Quantum Lego code — Qubit stabilizer codes are quantum Lego codes built out of atomic blocks such as the 2-qubit repetition code, single-qubit trivial stabilizer codes, and tensor-products of the \(|0\rangle\) state.

## Children

- Fusion-based quantum computing (FBQC) code — The resource states in FBQC are small stabilizer states, and after fusion measurements, the outputs are stabilizers (conditioned on measurement outcomes).
- Calderbank-Shor-Steane (CSS) stabilizer code — Stabilizer generators can be expressed as either only \(X\)-type or only \(Z\)-type. However, any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[4n,2k,2d]]\) self-orthogonal CSS code, with the mapping preserving geometric locality of a code up to a constant factor [46].
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The HaPPY code is a stabilizer code because it is defined by a contracted network of stabilizer tensors; see Thm. 6 in Ref. [47].
- Cluster-state code — Cluster states are particular qubit stabilizer states defined on a graph.
- Crystalline-circuit code
- EA qubit stabilizer code — EA qubit stabilizer codes utilize additional ancillary qubits in a pre-shared entangled state, but reduce to qubit stabilizer codes when said qubits are interpreted as noiseless physical qubits.
- Majorana stabilizer code — The Majorana stabilizer code is a stabilizer code whose stabilizers are composed of Majorana fermion operators. In addition, any \([[n,k,d]]\) stabilizer code can be mapped into a \([[2n,k,2d]]_{f}\) Majorana stabilizer code [48][46].
- Hierarchical code
- Qubit BCH code — qubit BCH codes constructed via the CSS construction are CSS codes, and the rest are stabilizer codes over \(GF(4)\).
- \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code
- Transverse-field Ising model (TFIM) code
- Stabilizer code over \(GF(4)\)
- Subsystem qubit stabilizer code — Subsystem stabilizer codes reduce to stabilizer codes when there are no gauge qubits.
- Haah cubic code
- Matching code
- Clifford-deformed surface code (CDSC)

## Cousins

- Linear binary code — Qubit stabilizer codes are quantum analogues of binary linear codes.
- Dual linear code — Symplectic representations of stabilizer group elements form a linear code over \(GF(2)\) that is self-orthogonal with respect to the symplectic inner product ([49], Thm. 27.3.6).
- Hamiltonian-based code — Codespace is the ground-state space of the code Hamiltonian, which consists of an equal linear combination of stabilizer generators and which can be made into a commuting projector Hamiltonian.
- Modular-qudit stabilizer code — Modular-qudit stabilizer codes for prime-dimensional qudits \(q=p\) inherit most of the features of qubit stabilizer codes, including encoding an integer number of qudits and a Pauli group with a unique number of generators. Conversely, qubit codes can be extended to modular-qudit codes by decorating appropriate generators with powers. For example, \([[4,2,2]]\) qubit code generators can be adjusted to \(ZZZZ\) and \(XX^{-1} XX^{-1}\). A systematic procedure extending a qubit code to prime-qudit codes involves putting its generator matrix into local-dimension-invariant (LDI) form [50]. Various bounds exist on the distance of the resulting codes [51][52].
- Translationally invariant stabilizer code — Qubit stabilizer codes can be thought of as translationally-invariant stabilizer codes for dimension \(D = 0\), with the lattice consisting of a single site.
- Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [53].
- Cluster-state code — Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into an MBQC protocol [54].
- Movassagh-Ouyang Hamiltonian code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
- Galois-qudit stabilizer code — Galois-qudit stabilizer codes reduce to qubit stabilizer codes for \(q=2\).

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## Page edit log

- Victor V. Albert (2022-09-28) — most recent
- Victor V. Albert (2022-05-19)
- Victor V. Albert (2022-02-16)
- Qingfeng (Kee) Wang (2021-12-07)
- Lane G. Gunderman (2022-02-04)
- Leonid Pryadko (2021-11-02)
- Daniel Gottesman (2021-11-02)
- Victor V. Albert (2021-11-02)

## Cite as:

“Qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qubit_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/qubit_stabilizer.yml.