Qubit stabilizer code[1,2] 

Also known as Pauli stabilizer code, Additive quantum code, Additive CWS code, Clifford code.

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 form the joint \(+1\)-eigenspace of a stabilizer group \(\mathsf{S}\), i.e., 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

\(\mathsf{N}(\mathsf{S})/\mathsf{S}\)

\(4^{k}\)

Table I: Groups relevant to qubit stabilizer codes. The normalizer \(\mathsf{N}(\mathsf{S})\) (technically, the centralizer, but these are equivalent for this case) is the group formed by all elements of the \(n\)-qubit Pauli group that commute with all elements in \(\mathsf{S}\). The normalizer is defined so as to include \(i\) and its powers as elements, while the stabilizer group is not.

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.

Code representations

Instead of being represented by a basis of codewords, stabilizer codes can be concisely defined and represented by a presentation of the generators of the stabilizer group. Stabilizer generators can be arranged as rows of a matrix, forming a stabilizer tableau. A set of generators is not unique, and various stabilizer codes admit generators with certain locality properties.

For example, a stabilizer tableau of a two-qubit Bell-state stabilizer state is \(\left[\begin{smallmatrix}Z & Z\\ X & X \end{smallmatrix}\right]\). Another tableau can be defined by letting \(X \to Y\).

Pauli strings, and the corresponding stabilizer tableaus, can be represented in other ways.

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].

ZX calculus is complete, sound, and universal for qubit stabilizer codes [8]. Any stabilizer code can be represented by a ZX canonical form (ZXCF) [9], and there exist two other representations [9,10] that utilize ZX calculus.

Encoder-respecting form: In an encoder-respecting form, each qubit stabilizer code [9] (see also Ref. [11]) is represented by a semi-bipartite graph with \(k\) input and \(n\) output nodes in which the \(k\) input nodes are not connected to each other. Conversion between stabilizer tableaus and graphs is achieved using ZX calculus and takes time that is polynomial in \(n\). Properties of the underlying graph are related to properties of the code [9].

Alternative representations include the decoupling representation, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits [12]. Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) [13].

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 [14–16]. Computing the distance exactly or approximately is generally \(NP\)-complete, and is \(NP\)-hard for non-degenerate codes [17]. Distance approximation and stabilizer weight reduction are approximately optimal strategies for various quantum lights-out (QLO) games that can be played on the codes'' encoder-respecting form [9].

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 [18], where an analogous result is states for subsystem codes; see also Ref. [19].

Entropic conditions have been formulated for random projective measurement noise [20].

The quantum GV bound can be extended into a three-way tradeoff between distance, rate, and stabilizer weight of a qubit stabilizer code [9].

Rate

The hashing bound states that there is a qubit stabilizer code achieving a rate \(R = 1 - H(\mathbf{p})\) for a Pauli noise channel with Pauli error probabilities \(\mathbf{p}=(p_I,p_X,p_Y,p_Z)\), where \(H(\mathbf{p})\) is the entropy of the argument [21; Thm. 23.6.2]. Finite block length bounds and a refinement of the hashing bound have been developed [5].

Encoding

Clifford circuits, i.e., those consisting of CNOT, Hadamard, and certain phase gates, using an algorithm [22] based on the Gottesman-Knill theorem [23] or using ZX calculus [24,25].

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 [23,26]. Each such generator anticommutes with only one stabilizer generator while commuting with the rest of the stabilizer generators.

Circuits obtained by first constructing the CWS form of the code [27,28]. These consist of \(n\) Hadamard gates, a classical encoder which takes at most \(n\) CX gates for a single-qubit encoding code, and at most \(n(n-1)/2\) CZ gates to create the needed graph state.Lindbladian-based dissipative encoding [29,30], for which codespace is steady-state space of a Lindbladian. This does not give a speedup, in terms of scaling with \(n\), over circuit-based encoders [31].

Transversal Gates

All stabilizer codes realize Pauli transformations transversally; for a single logical qubit, these realize a dicyclic subgroup of \(SU(2)\). Several algorithms exist for finding logical Pauli operators [2,32,33].The four-block transversal gate mapping each \(X \to IXXX\) and each \(Z \to IZZZ\) implements the same logical gate on all qubits [2].Transversal logical gates are in a finite level of the Clifford hierarchy, which is shown using stabilizer disjointness [34] (see also [35,36]). Transversal gates for \(n\in\{1,2\}\) are semi-Clifford [37].No stabilizer code can implement a classical universal gate set transversally [38].Fold-transversal gates have been extended from qubit CSS codes to qubit stabilizer codes, and there is an algorithm to determine them from the stabilizer group [39].

Gates

Logical Clifford gates can be performed by physical Clifford circuits that permute logical Pauli operators [33].With pieceable fault-tolerance, any non-degenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [40].Gates in the Clifford hierarchy can be done using gate teleportation, in which a gate can be obtained from a particular magic state (a.k.a. resource state) [41,42]. Such protocols can be made fault tolerant with the help of magic-state distillation [43,44]. There exist various performance metrics for magic-state distillation [45–48]. Gate errors in magic-state distillation protocols can sometimes add up destructively [49]. The Hadamard gate cannot be obtained from a magic state [50]. A magic state arising from a generalized controlled \(Z\) gate is a type of hypergraph state [51–53] (see [54,55]). Magic-state protocol fidelity is upper bounded by the fidelity of protocols that have undergone stabilizer reduction, and there exist non-distillable states outside of the stabilizer octahedron [56,57].Certain operations can be implemented in a fault-tolerant version [58,59] of holonomic quantum computation [60].Magic-state distillation and circuit compilation based on the SWAP test [61].Logical circuit synthesis (LCS) taking in a code and a logical Clifford operation and producing a circuit acting on the physical qubits [62].Clifford stabilizer circuits can be compiled using tableau manipulation [63].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 [64]. There is a simple formula for the probability that a Clifford circuit contains a logical error [65].

Decoding

The size of the circuit extracting the syndrome depends on the weight of its corresponding stabilizer generator. Syndrome extraction circuits can be simulated efficiently using dedicated software (e.g., STIM [66]) and there are many general schemes for generating them [67] (see also [68]). Decoding of qubit stabilizer codes is an approximately optimal strategy for various quantum lights-out (QLO) games that can be played on the codes' encoder-respecting form [9].DiVincenzo-Aliferis syndrome extraction circuits [69].Greedy syndrome measurement schedule [70].Dynamical weight reduction (DWR) scheme in which measurements of smaller-weight Paulis yield the outcome of a larger-weight Pauli via the use of ZX calculus and ancillary qubits [71].MPE decoding, i.e., the process of finding the most likely error, is \(NP\)-complete in general [72,73]. If the noise model is such that the most likely error is the lowest-weight error, then ML decoding is called minimum-weight decoding. Maximum-likelihood (ML) decoding (a.k.a. degenerate maximum-likelihood decoding), i.e., the process of finding the most likely error class (up to degeneracy of errors), is \(\#P\)-complete in general [74].Incorporating faulty syndrome measurements can be done by performing spacetime decoding, i.e., using data from past rounds for decoding syndromes in any given round. If a decoder does not process syndrome data sufficiently quickly, it can lead to the backlog problem [75], slowing down the computation.Splitting decoders [76].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [77].Quantum extension of GRAND decoder [78].Deep neural-network probabilistic decoder [79].Generalized belief propagation (GBP) [80] based on a classical version [81].Integer optimization decoder [82].Autonomous Lindbladian based decoders for codes encoding a single logical qubit [83].For codes encoding a single logical qubit, logical information can be extracted by single-qubit operations and classical communication [84].Correlated decoding can improve performance of Clifford and non-Clifford entangling gates [85].Detector graphs [66,86] and detector error models [87] can be used to design syndrome extraction circuits and logical measurements.Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [88].Degenerate erasure decoder showing near ML decoding for various codes [89].

Fault Tolerance

Gates in the Clifford hierarchy can be done using gate teleportation, in which a gate can be obtained from a particular magic state [41,42]. Such protocols can be made fault tolerant with the help of magic-state distillation [43].Logical Bell measurements can be done transversally, and thus fault tolerantly, by performing bitwise Bell measurements for each pair of qubits (with each member of the pair taken from one of the two code blocks) and processing the result.With pieceable fault-tolerance, any non-degenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of non-transversal fault-tolerant gates [40].Shor error correction [90,91], in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements [92].Generalization of Steane error correction stabilizer codes [93; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [94]), which is based on teleportation [44,95]. A variant of it has been termed the Fibonacci scheme [96].Fault-tolerant error correction using flag qubits for codes satisfying certain conditions [97].GHZ state distillation for Steane error correction [98].Syndrome extraction using flag qubits and classical codes [99].Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [88].Post-selection based algorithm preparing magic state corresponding to arbitrary rotations [100].Code switching can be done using only transversal gates for qubit stabilizer codes [101].Flag-Proxy Networks (FPNs) [70].A logical Pauli can be gauged out to yield a fault-tolerant measurement that requires a qubit overhead linear in the Pauli's support [102].

Code Capacity Threshold

Bounds on code capacity thresholds using ML decoding can be obtained by mapping the effect of noise on the code to a statistical mechanical model [103–106]. The AQEC relative entropy is related to the resulting threshold [107].

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 [108–111]. 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.Entanglement purification protocols with qubit stabilizer codes are related to quantum key distribution (QKD) [112].Certain operations can be implemented in a fault-tolerant version [58,59] of holonomic quantum computation [60].

Notes

Introductions to stabilizer codes can be found in [2,113,114].Tables of bounds and examples of stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Ref. [115], are maintained by M. Grassl at this website. A Magma implementation exists at this website.See Quantum Codes qubit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino [116,117], at this website.Review on magic state distillation [118].There is a correspondence between stabilizer codes and bilocal Clifford entanglement distillation circuits [119].The overlap between any stabilizer codeword and any \(n\)-qubit product state is at most \(2/2^d\) [120; Thm. 2].The stabilizer formalism has been gamified [121].Code can be found via genetic algorithms [16].

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 [122]. Various bounds exist on the distance of the resulting codes [123,124].
  • True Galois-qudit stabilizer code — True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.

Children

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 [28,133]. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [129][131; 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 [134; 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 [135]. Binary linear codes can be used for error-corrected entanglement distillation protocols [88].
  • 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 [136].
  • \(t\)-design — Stabilizer states on \(n\) qubits form complex projective 3-designs [137], while the Clifford group is a unitary 3-design [138,139].
  • Constant-excitation (CE) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [140] that protects against \(d-1\) AD errors [141].
  • Amplitude-damping (AD) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [140] that protects against \(d-1\) AD errors [141].
  • 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 [142].
  • Graph quantum code — Graph quantum codes for \(G=\mathbb{Z}_2\) are a subset of qubit stabilizer codes [128]. Any qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_2\) via a single-qubit Clifford circuit [128] (see also [129,130]).
  • Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [143].
  • 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 [144].
  • 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 [145].
  • PI qubit code — There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [146].
  • 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 [90,147]. 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) [148][126; 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 [149; 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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Zoo Code ID: qubit_stabilizer

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“Qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qubit_stabilizer
BibTeX:
@incollection{eczoo_qubit_stabilizer, title={Qubit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_stabilizer} }
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