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
Logical gates implemented via constant-depth quantum circuits of \(D\)-dimensional geometrically local stabilizer codes lie in the \(D\)th level of the Clifford hierarchy [5].With pieceable fault-tolerance, any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [6].
Decoding
The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes). Finding an optimal decoder is \(\#P\)-hard [7].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [8].
Fault Tolerance
With pieceable fault-tolerance, any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [6].Fault-tolerant error correction can be done using Shor error correction [9], which is based on repeated measurements, or Knill error correction, which is based on teleportation [10][11].
Code Capacity Threshold
For correlated Pauli noise, bounds on code capacity thresholds for any stabilizer codes can be obtained by mapping the effect of noise on the code to a statistical mechanical model [12][13][14][15].
Threshold
Computational thresholds against stochastic local noise can be achieved through repeated use of concatenatenation, and can rely on the same small code in every level [16][17][18][19]. The resulting code is highly degenerate, with all but an exponentially small fraction of generators having small weights. Circuit and measurement designs have to take case of the few stabilizer generators with large weights in order to be fault tolerant.
Notes
Tables of bounds and examples of stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Ref. [20], are maintained by M. Grassl at this website.
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.