## Description

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.

Binary symplectic representation: In the binary symplectic representation, the single-qubit identity, \(X\), \(Y\), or \(Z\) Pauli matrices represented using two bits as \((0|0)\), \((1|0)\), \((1|1)\), and \((0|1)\), respectively. In other words, the single-qubit Pauli string \(X^a Z^b\) is converted to the vector \(a|b\). The multi-qubit version follows naturally.

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].

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} The set of all binary symplectic vectors form a symplectic self-orthogonal binary linear code of length \(2n\).

Another correspondence between qubit Pauli matrices and elements of the Galois field \(GF(4)\) yields the one-to-one correspondence between qubit stabilizer codes and trace-Hermitian self-orthogonal additive quaternary codes.

\(GF(4)\) representation: An \(n\)-qubit Pauli stabilizer can be represented as a length-\(n\) quaternary vector using the one-to-one correspondence between the four Pauli matrices \(\{I,X,Y,Z\}\) and the four elements \(\{0,1,\alpha^2,\alpha\}\) of the quaternary field \(GF(4)\).

The sets of \(GF(4)\)-represented vectors for all generators yield a trace-Hermitian self-orthogonal additive quaternary code. This classical code corresponds to the stabilizer group \(\mathsf{S}\) while its trace-Hermitian dual corresponds to the normalizer \(\mathsf{N(S)}\).

Alternative representations include the decoupling representation, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits [4].

Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) [5]. Qubit stabilizer codes can equivalently [6] (see also [7]) be defined using graphs, yielding an analytical form for the codewords [8]. Clifford operations can be realized as operations acting on the corresponding graphs [9].

## Protection

Detects errors on up to \(d-1\) qubits, and corrects erasure errors on up to \(d-1\) qubits.

There is the following analogue of the Knill-Laflamme conditions for qubit stabilizer codes. 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 \(E,F \in {\mathcal{E}}\).

Cleaning lemma: If all logical operators act trivially on some subset of qubits in a stabilizer code, then any logical Pauli operator can be represented on the complementary qubit subset via a stabilizer. More technically, given any subset \(M\) of qubits that is correctable (under erasure), any logical Pauli operator \(P\) can be cleaned off of \(M\) using a stabilizer \(S\) such that \(PS\) is supported on \(M^{\perp}\). More generally, for any \(M\), we have \(g(M)+g(M^{\perp}) = 2k\), where \(g(M)\) is the number of logical-\(X\) and logical-\(Z\) Pauli operators supported fully on \(M\) (up to stabilizers). The Cleaning Lemma was originally proven [10], where an analogous result is states for subsystem codes; see also Ref. [11].

Entropic conditions have been formulated for random projective measurement noise [12].

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Notes

## Parents

- Union stabilizer (USt) code — A stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a USt with only the identity coset representative. Conversely, if the set of coset representatives of a USt form a linear binary code, then they can be absorbed into a stabilizer group that defines the USt.
- XP stabilizer code — The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).
- Modular-qudit stabilizer code — Modular-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes. 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 [74]. Various bounds exist on the distance of the resulting codes [75,76].
- Galois-qudit stabilizer code — Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
- Operator-algebra qubit stabilizer code

## Children

- Crystalline-circuit qubit code
- Low-depth random Clifford-circuit qubit code
- Spacetime circuit 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 [77,78].
- \([[2^r, 2^r-r-2, 3]]\) Gottesman code
- \([[11,1,5]]\) quantum dodecacode
- Transverse-field Ising model (TFIM) code
- Quantum convolutional code
- CPC code
- Fermion-into-qubit code — Fermion-into-qubit codes are qubit stabilizer codes that encode a logical fermionic Hilbert space into a physical space of \(n\) qubits.
- Haah cubic code (CC)
- Hsieh-Halasz (HH) code
- Hsieh-Halasz-Balents (HHB) 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. [79].
- Hermitian qubit code
- \(k\)-orthogonal code
- Cluster-state code — Cluster states are particular qubit stabilizer states defined on a graph. Any qubit stabilizer code is equivalent to a graph code via a single-qubit Clifford circuit [6] (see also [7]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [9][80; Appx. A]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [81].
- 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).
- Hierarchical code
- XYZ product code
- Quantum spatially coupled (SC-QLDPC) code
- Qubit BCH code
- Twist-defect color code
- Matching code
- Clifford-deformed surface code (CDSC)
- Twist-defect surface code
- Three-fermion (3F) Walker-Wang model code

## Cousins

- Codeword stabilized (CWS) code — CWS codes whose underlying classical code is a linear binary code are qubit stabilizer codes containing a cluster-state codeword.
- Linear binary code — Qubit stabilizer codes are the closest quantum analogues of binary linear codes because addition modulo two corresponds to multiplication of stabilizers in the quantum case. Any binary linear code can be thought of as a qubit stabilizer code with \(Z\)-type stabilizer generators [82; Table I]. The stabilizer generators are extracted from rows of the parity-check matrix, while logical \(X\) Paulis correspond to rows of the generator matrix.
- Dual linear code — Qubit stabilizer codes are in one-to-one correspondence with symplectic self-orthogonal binary linear codes of length \(2n\) via the binary symplectic representation.
- Dual additive code — Qubit stabilizer codes are in one-to-one correspondence with trace-Hermitian self-orthogonal additive quaternary codes of length \(n\) via the \(GF(4)\) representation.
- Single-shot code — Any stabilizer code can be single shot if sufficiently non-local high-weight stabilizer generators are used for syndrome measurements. These can be obtained with a Gaussian elimination procedure [83].
- Kerdock code — Kerdock codes can be used to form a subset of stabilizer states, and the corresponding Clifford-group automorphisms of this set form a particular group [84] that is a unitary two-design [85].
- Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [86].
- Spacetime circuit code — Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit stabilizer codes.
- Movassagh-Ouyang Hamiltonian code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
- Error-transmuting code (QETC) — Most QETCs are stabilizer codes: \(\mathsf{C}\) is the subspace stabilised by an abelian subgroup \(\mathsf{S} \subset \mathcal{G}_n\) of the Pauli group on \(n\) qubits.
- 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.
- Qubit CSS code — Qubit CSS codes are qubit stabilizer codes whose stabilizer groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) CSS code, with the mapping preserving geometric locality of a code up to a constant factor [87] (see also [78]). For any non-CSS code \(\mathsf{C}\), there exists a CSS code \(\mathsf{C}^{\prime}\) such that \(\mathsf{C} = DQ\mathsf{C}^{\prime}\), where \(D\) is a diagonal Clifford operator and \(Q\) is an element of an XP stabilizer group [88; Prop. B.3.1].
- Subsystem qubit stabilizer code — Subsystem stabilizer codes reduce to stabilizer codes when there are no gauge qubits.

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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