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.
Qubit stabilizer codes are defined by a stabilizer group \(\mathsf{S}\), a group of commuting Paulis that does not contain the identity and is generated by \(r=n-k\) generators. The table below summarizes the relevant groups and their sizes for a qubit stabilizer code.
purpose | symbol | size |
---|---|---|
stabilizer group | \(\mathsf{S}\) | \(2^{n-k}\) |
code-preserving Paulis | \(\mathsf{N}(\mathsf{S})\) | \(4\cdot 2^{n+k}\) |
logical Paulis | \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) | \(4^{k}\) |
Two qubit stabilizer codes codes are equivalent if the codespace of one code can be mapped into that of the other under a tensor product of elements of the single-qubit Clifford group and a qubit permutation. Equivalence under single-qubit Clifford operations is not the same as the equivalence under a tensor product of arbitrary single-qubit unitary operations [3]. A qubit stabiilzer code is decomposable if there exists a permutation that maps the stabilizer group into a tensor product of two stabilizer groups acting on disjoint sets of qubits.
Symplectic representation: In the 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 symplectic representation of an element from a set of generating elements of the stabilizer group. The check matrix can be brought into standard form (a.k.a. canonical form) via Gaussian elimination [4,5].
A pair of \(n\)-qubit stabilizers with symplectic representation \((a|b)\) 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 quaternary 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 \(\{0,1,\omega^2,\omega\}\) of the quaternary Galois 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 \(\mathsf{N(S)}\). In the case of stabilizer states, the correspondence is between such states and trace-Hermitian self-dual quaternary codes; such codes, and therefore such states, have been classified up to equivalence for \(n \leq 12\) [6,7].
Alternative representations include the decoupling representation, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits [8].
Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) [9].
Protection
Detects errors on up to \(d-1\) qubits, and corrects erasure errors on up to \(d-1\) qubits. There are algorithms to calculate the minimum distance [10–12].
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 [13], where an analogous result is states for subsystem codes; see also Ref. [14].
Entropic conditions have been formulated for random projective measurement noise [15].
Rate
Encoding
Clifford circuits, i.e., those consisting of CNOT, Hadamard, and certain phase gates, using an algorithm [17] based on the Gottesman-Knill theorem [18] or using ZX calculus [19,20] (with the latter providing a unique decomposition [21]).
Destabilizers: A Clifford encoding circuit maps the first \(r = n-k\) qubits to the logical qubits of the code, and the Pauli \(Z\) operators of those first \(r\) qubits are mapped into a set of stabilizer generators. The set of Pauli \(X\) operators of the first \(r\) qubits that are mapped into a set of generators for the destabilizer group [18,22]. Each such generator anticommutes with only one stabilizer generator while commuting with the rest of the stabilizer generators.
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 — XP stabilizer codes reduce to qubit stabilizer codes for \(N=2\).
- Operator-algebra (OA) qubit stabilizer code — An OA qubit stabilizer code storing no classical information and admitting no gauge qubits is a qubit stabilizer code.
- 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 modular-qudit 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 [116]. Various bounds exist on the distance of the resulting codes [117,118].
- True Galois-qudit stabilizer code — True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
Children
- Crystalline-circuit qubit code
- Brown-Fawzi random Clifford-circuit 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 [119][120; Lemma 1].
- \([[2^r, 2^r-r-2, 3]]\) Gottesman code
- \([[7,1,3]]\) bare code
- \([[11,1,5]]\) quantum dodecacode
- \([[6,1,3]]\) Six-qubit stabilizer code
- Transverse-field Ising model (TFIM) code
- Quantum convolutional code
- Coherent-parity-check (CPC) code — CPC codes are a type of stabilizer code. A teleported version of the CPC construction can reduce noise in Clifford circuits with Pauli measurements with at most a three-fold overhead in the number of qubits and gates [58]. There is a simple formula for the probability that a Clifford circuit contains a logical error [59].
- Quantum data-syndrome (QDS) code — QDS codes are stabilizer codes whose stabilizer generators encode extra redundancy (via a linear binary code) so as to protect from syndrome measurement errors.
- 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
- Hermitian qubit code
- Branching MERA 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. [121].
- Six-qubit-tensor holographic code
- Hyperinvariant tensor-network (HTN) code
- \([[6k+2,3k,2]]\) Campbell-Howard code
- \(k\)-orthogonal code
- Cluster-state code — Cluster-state codes are particular qubit stabilizer codes. Any qubit stabilizer code is equivalent to a graph quantum code via a single-qubit Clifford circuit [122] (see also [123,124]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [123][125; Appx. A]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [126].
- 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).
- Purity-testing stabilizer code
- Hierarchical code
- XYZ product code
- Quantum spatially coupled (SC-QLDPC) code
- Qubit BCH code
- Quantum synchronizable 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 [24,127]. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [123][125; Appx. A], any stabilizer code is similarly equivalent to a CWS code.
- 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 [128; 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. States close to the equal superposition of all bitstrings within Hamming distance \(b\) of a binary linear code can be prepared efficiently [129]. Binary linear codes can be used for error-corrected entanglement distillation protocols [82].
- Dual linear code — Qubit stabilizer codes are in one-to-one correspondence with symplectic self-orthogonal binary linear codes of length \(2n\) via the 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 [130].
- \(t\)-design — Stabilizer states on \(n\) qubits form complex projective 3-designs [131], while the Clifford group is a unitary 3-design [132,133].
- Constant-excitation (CE) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [134] that protects against \(d-1\) AD errors [135].
- Amplitude-damping (AD) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [134] that protects against \(d-1\) AD errors [135].
- Barnes-Wall (BW) lattice — Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the Clifford group is related to the symmetry group of the lattice [136].
- Graph quantum code — Graph quantum codes for \(G=\mathbb{Z}_2\) are a subset of qubit stabilizer codes [122]. Any qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_2\) via a single-qubit Clifford circuit [122] (see also [123,124]).
- Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [137].
- Spacetime circuit code — Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit stabilizer codes.
- 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 stabilizer codes can be used to obtain shortened EA qubit stabilizer codes [138].
- Movassagh-Ouyang Hamiltonian code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
- Hybrid stabilizer code — A hybrid stabilizer code storing no classical information reduces to a qubit stabilizer code. Conversely, any qubit stabilizer code can be converted into a hybrid stabilizer code by using some its qubits to store only classical information [139].
- PI qubit code — There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [140].
- Quantum 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.
- 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. Transversal CNOT gates preserve the logical subspace iff a qubit stabilizer code is CSS [84,141]. Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via the Bravyi-Leemhuis-Terhal mapping (a.k.a. the Majorana mapping, named as such because it is done via an intermediate Majorana stabilizer code) [142][120; Corr. 1], which preserves geometric locality of a code up to a constant factor. Any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[2n,2k,\geq d]]\) two-block CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor. For any non-CSS qubit stabilizer 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 where \(Q\) is an element of an XP stabilizer group [143; Prop. B.3.1].'
- Subsystem qubit stabilizer code — Subsystem qubit stabilizer codes reduce to qubit 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