## 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].

## 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 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 \(n=L^D\).

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Notes

## Parents

- Stabilizer code
- 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 \(N=2\).
- 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

- 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 \([[4n,2k,2d]]\) weakly self-dual CSS code, with the mapping preserving geometric locality of a code up to a constant factor [21].
- Clifford-deformed surface code (CDSC)
- Floquet code — Particular sequences of measurements on this code yield an instantaneous stabilizer group.
- 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.
- Haah cubic code
- 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 [22][21]. However, Pauli- and Majorana-based stabilizer codes have different notions of locality [23] and are thus useful for different physical platforms.
- Matching code
- 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. [24].
- Quantum convolutional code
- Raussendorf-Bravyi-Harrington (RBH) code
- Stabilizer code over \(GF(4)\)
- Transverse-field Ising model (TFIM) code
- XYZ product code
- \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code

## Cousins

- Linear binary code — Qubit stabilizer codes are quantum analogues of binary linear codes.
- 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 [25]. Various bounds exist on the distance of the resulting codes [26][27].
- Movassagh-Ouyang Hamiltonian code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
- Subsystem qubit stabilizer code — Gauge stabilizer codes reduce to stabilizer codes when there are no gauge qubits.
- 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.

## Zoo code information

## References

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## 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.