Alternative names: Binary stabilizer code, Pauli stabilizer code, Symplectic 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 | \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) | \(4^{k}\) |
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 (see Refs. [3,4]). Equivalence under single-qubit Clifford operations is not the same as equivalence under a tensor product of arbitrary single-qubit unitary operations [5] (see also Ref. [6] about this false LC-LU conjecture). Any qubit stabilizer code is equivalent to a graph quantum code via a single-qubit Clifford circuit [7] (see also [8,9]). There are algorithms that determine whether two stabilizer states are equivalent that work by checking whether their corresponding cluster states are equivalent [10–12].
A qubit stabilizer 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. Otherwise, the code is indecomposable.
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 [13,14].
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 \(n \leq 12\) [15,16]. There is a complete set of invariants characterizing stabilizer states up to equivalence [17,18].
ZX calculus is complete, sound, and universal for qubit stabilizer codes [19]. Any stabilizer code can be represented by a ZX canonical form (ZXCF) [20], and there exist two other representations [20,21] that utilize ZX calculus.
Encoder-respecting form: In an encoder-respecting form, each qubit stabilizer code [20] (see also Ref. [22]) 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 [20].
Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) [23–25]. There are efficient ways to compute their inner products and other functions [26]. The overlap between a stabilizer state and any \(n\)-qubit product state is at most \(2/2^d\) [27; Thm. 2]. Alternative representations include the decoupling representation, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits [28].
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 [29–32]. Computing the distance exactly or approximately is generally \(NP\)-complete, and is \(NP\)-hard for non-degenerate codes [33]. 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 [20].
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 \(E,F \in {\mathcal{E}}\).
There are subtleties with defining degeneracy for non-stabilizer qubit codes with even distance [34], but they are resolved for stabilizer codes. A stabilizer code is a degenerate with respect to \(\mathcal{E}\) if and only if \(E^\dagger F \in \mathsf{N(S)}\) for some Pauli strings \(E,F \in \mathcal{E}\). As a distance-\(d\) code, a stabilizer code is degenerate if it admits a non-identity stabilizer whose weight is lower than the distance [34]. Since that stabilizer is in the normalizer, a stabilizer code is degenerate if and only if it is impure. The pure distance of a qubit stabilizer code is the minimum weight of a non-identity stabilizer, and a qubit stabilizer code is pure if the weight of the lowest-weight stabilizer is the code distance. There is a toolkit for studying degenerate codes for Pauli channels [35]. The quantum GV bound can be extended into a three-way tradeoff between distance, rate, and stabilizer weight of a qubit stabilizer code [20].
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 [36], where an analogous result is states for subsystem codes; see also Ref. [37].
Qubit stabilizer codes can be interleaved to protect against burst errors [38]. Entropic conditions have been formulated for random projective measurement noise [39].
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 [40; Thm. 23.6.2]. Finite block length bounds and a refinement of the hashing bound have been developed [14].Encoding
Clifford circuits, i.e., those consisting of CNOT, Hadamard, and certain phase gates [41], using an algorithm [42] based on the Gottesman-Knill theorem [43] or using ZX calculus [44,45].
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 [43,46]. Each such generator anticommutes with only one stabilizer generator while commuting with the rest of the stabilizer generators.
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 [2,56,57].The four-block transversal gate mapping each \(X \to IXXX\) and each \(Z \to IZZZ\) implements the same logical gate on all qubits [2][58; Exam. 1].Transversal logical gates are in a finite level of the Clifford hierarchy, which is shown using stabilizer disjointness [59] (see also [60,61]). Transversal gates for \(n\in\{1,2\}\) are semi-Clifford [62].No stabilizer code can implement a classical universal gate set transversally [63].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 [64].Computation can be sped up substantially for codes that admit transversal measurements of logical \(X\) and \(Z\) [65]Gates
Logical Clifford gates can be performed by physical Clifford circuits that permute logical Pauli operators [57].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 [66].Non-Clifford gates can be done using gate teleportation, in which a gate can be obtained from a particular magic state (a.k.a. resource state) [67,68]. Such protocols can be made fault tolerant with the help of magic-state distillation [69,70]. There exist various performance metrics for magic-state distillation [71–74] focusing mostly on distilling \(T\) gates. This Clifford+T gate set is self-testable [75]. Gate errors in magic-state distillation protocols can sometimes add up destructively [76]. The Hadamard gate cannot be obtained from a magic state [77]. A magic state arising from a generalized controlled \(Z\) gate is a type of hypergraph state [78–80] (see [81,82]). The Toffoli gate can be distilled from a particular two-qubit state [83]. 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 [84,85].Certain operations can be implemented in a fault-tolerant version [86–89] of holonomic quantum computation [90].Magic-state distillation and circuit compilation based on the SWAP test [91].Logical Clifford synthesis (LCS) taking in a code and a logical Clifford operation and producing a circuit acting on the physical qubits [92].Clifford stabilizer circuits can be compiled using tableau manipulation [93].A teleported version of the CPC construction, the Clifford noise reduction (CliNR) scheme, can reduce noise in Clifford circuits with Pauli measurements with at most a three-fold overhead in the number of qubits and gates [94,95]. There is a simple formula for the probability that a Clifford circuit contains a logical error [96].Logical Trotter circuits for any stabilizer code can be implemented via symplectic transvections [97].Hardware-tailored logical Clifford circuits [98].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 [99]) and there are many general schemes for generating them [100] (see also [101]). Noise can be characterized without interrupting syndrome extraction [102]. 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 [20].DiVincenzo-Aliferis syndrome extraction circuits [103].Greedy syndrome measurement schedule [104].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 [105].Ancilla modes can be used for syndrome extraction instead of ancilla qubits [106], and using two-component cat codes [107] yields fault-tolerant syndrome extraction circuits.Autonomous QEC protocol [108].MPE decoding, i.e., the process of finding the most likely error, is \(NP\)-complete in general [109,110]. 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 [111].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 [112], slowing down the computation.Splitting decoders [113].Trellis decoder, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [114].Quantum extension of GRAND decoder [115].Deep neural-network probabilistic decoder [116].Generalized belief propagation (GBP) [117] based on a classical version [118].Integer optimization decoder [119].Autonomous Lindbladian based decoders for codes encoding a single logical qubit [120].For codes encoding a single logical qubit, logical information can be extracted by single-qubit operations and classical communication [121].Correlated decoding can improve performance of Clifford and non-Clifford entangling gates [122].Detector graphs [99,123] and detector error models [124] 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 [125].Degenerate erasure decoder showing near ML decoding for various codes [126].Tensor-network decoder for non-Markovian noise [127].Delayed-erasure conversion decoder [128].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 [67,68]. Such protocols can be made fault tolerant with the help of magic-state distillation [69]. See review on magic-state distillation [129].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 [66].Shor error correction [130,131] (see also Steane's ancilla factory [132]), in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements [133].Generalization of Steane error correction stabilizer codes [134; Sec. 3.6].Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection [135]), which is based on teleportation [70,136]. A variant of it has been termed the Fibonacci scheme [137].Fault-tolerant error correction using flag qubits for codes satisfying certain conditions [138].GHZ state distillation for Steane error correction [139].Syndrome extraction using flag qubits and classical codes [140].Fault-tolerant constant-depth unencoder transforming logical states into physical states using single-qubit measurements [125].Post-selection based algorithm preparing magic state corresponding to arbitrary rotations [141].Code switching can be done using only transversal gates for qubit stabilizer codes [142].Flag-Proxy Networks (FPNs) [104].A logical Pauli can be gauged out to yield a fault-tolerant measurement that requires a qubit overhead linear in the Pauli's support [143].Automated fault-tolerant circuit synthesis using boolean satisfiability [144].Algorithm for fault-tolerant magic-state initialization [53].Fault tolerance of qubit stabilizer codes has been formalized [145].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 [146–149]. The AQEC relative entropy is related to the resulting threshold [150].Threshold
Certain operations can be implemented in a fault-tolerant version [86,88] of holonomic quantum computation [90].Notes
Introductions to stabilizer codes can be found in [2,151–153].Tables of bounds and examples of stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Ref. [154], 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 [155,156], at this website.See QEC database by S. Burton at this website.Entanglement purification protocols with qubit stabilizer codes are related to quantum key distribution (QKD) [157]. There is a correspondence between stabilizer codes and bilocal Clifford entanglement distillation circuits [158]. Purification protocols using two-way classical channels can exceed the quantum Hamming and quantum Singleton bounds [159].Qubit stabilizer codes can be used to estimate physical Pauli noise up to their pure distance [160], and logical Pauli noise for any correctable physical noise [161].The stabilizer formalism has been gamified [162].Codes can be found via genetic algorithms [32].Stim Python software library for simulating and analyzing qubit stabilizer circuits [99,163].PanQEC Python software library for simulating and visualizing qubit stabilizer codes [164,165].QuantumSavory Julia software library that includes stabilizer formalism tools, symbolic manipulation, and a tool for drawing quantum circuits [166,167].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 [48,168]. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [8][169; 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 [170; 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 [171]. Binary linear codes can be used for error-corrected entanglement distillation protocols [125].
- 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 [172].
- \(t\)-design— Stabilizer states on \(n\) qubits form complex projective 3-designs [173], while the Clifford group is a unitary 3-design [174,175]. The \([[4,2,2]]\) code obstructs the Clifford group from being a 4-design [176].
- \([[2m,2m-2,2]]\) error-detecting code— Stabilizer states on \(n\) qubits form complex projective 3-designs [173], while the Clifford group is a unitary 3-design [174,175]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [176].
- Constant-excitation (CE) code— Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [177] that protects against \(d-1\) AD errors [178].
- Amplitude-damping (AD) code— Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [177] that protects against \(d-1\) AD errors [178].
- Two-component cat code— Ancilla modes can be used for syndrome extraction instead of ancilla qubits [106], and using two-component cat codes [107] yields fault-tolerant syndrome extraction circuits.
- Projective geometry code— \([[n,k,d]]\) qubit stabilizer codes with no weight-one stabilizers are equivalent to particular "quantum" sets of lines in projective space \(PG(n-k-1,2)\) [180,181][179; Thm. 3.7]. This equivalence is stated in the case of pure qubit stabilizer codes with distance two or greater in [182; Thm. 2.6].
- 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 [183].
- Graph quantum code— Graph quantum codes for \(G=\mathbb{Z}_2\) are a subset of qubit stabilizer codes [7]. Any qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_2\) via a single-qubit Clifford circuit [7] (see also [8,9]).
- Perfect-tensor code— The codespace of a qubit stabilizer code with pure distance \(d_{\textnormal{pure}}\) is a \((d_{\textnormal{pure}}-1)\)-uniform space.
- Metrological code— A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [184].
- Dynamical code— Using ZX calculus, an \([[n,k,d]]\) qubit stabilizer code admitting stabilizer generators of weight no more than \(m\) can be Floquetified into an \([[n+\lceil m/2 \rceil+\ell,k,d^{\prime}]]\) dynamical code with single- and two-qubit operations, where \(\ell \leq \log_{2} m\) and \(d^{\prime} \geq d\) [185] (see also Ref. [186]). A more general locality-preserving spacetime concatenation procedure yields a dynamical code out of any qubit stabilizer code by structuring measurement gadgets using low-weight measurements while ensuring the preservation of logical information [187]. In particular, spacetime concatenation reformulates the notion of a dynamical code associated with a stabilizer code in terms of code concatenation for every qubit (spatial concatenation) and measurements between these codes (temporal concatenation), leading to a temporal evolution of the stabilizer state [187]. A matrix rank condition on the bond operators connecting the gadgets, called the Bond-Kernel-Rank Condition, and a strict locality preservation condition (SLPC), along with preservation of the operator algebra of the stabilizer code under the gadget action, preserves fault-tolerance and the spacetime distance of the code [187].
- 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 [188].
- Majorana subsystem stabilizer code— The B\(\mapsto\)F mapping yields Majorana subsystem codes from qubit stabilizer codes such that their gauge groups contain tetrons [190][189; Sec. IV]. The output Majorana subsystem codes can correct odd-weight errors.
- Tetron code— Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[2n,k,2d]]_{f}\) Majorana stabilizer code by concatenating with the tetron code [191][192; Lemma 1]. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [191] and other qubit Hamiltonians on certain graphs [193,194]. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [195].
- 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 [196].
- PI qubit code— There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [197].
- 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 [34,130]. Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code [198][192; 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 [199; Prop. B.3.1]. There is a holographic relation between qubit CSS codes describing CFTs and qubit stabilizer codes describing path integrals over certain topologies [200].
- Subsystem qubit stabilizer code— Subsystem qubit stabilizer codes reduce to qubit stabilizer codes when there are no gauge qubits. An \([[n,k,d]]\) qubit stabilizer code can be converted into an order \([[O(\ell \delta n),k,\Omega(d/w)]]\) subsystem qubit stabilizer code with weight-three gauge operators via the wire-code mapping [201], which uses weight reduction. Here, \(w\) and \(\delta\) are the weight and degree of the input code's Tanner graph, while \(\ell\) is the length of the longest edge of a particular embedding of that graph.
Member of code lists
- Constant-excitation quantum codes and friends
- Hamiltonian-based codes
- Quantum codes
- Quantum codes with a rate
- Quantum codes with code capacity thresholds
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum codes with transversal gates
- Single-shot codes
- Stabilizer codes
Primary Hierarchy
Parents
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 codes reduce to qubit stabilizer codes for \(N=2\).
An OA qubit stabilizer code storing no classical information and admitting no gauge qubits is a qubit 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 [202]. Various bounds exist on the distance of the resulting codes [203,204].
True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
Qubit stabilizer code
Children
Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit stabilizer codes.
A Majorana stabilizer code is a stabilizer code whose stabilizers are composed of Majorana fermion operators. Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[2n,k,2d]]_{f}\) Majorana stabilizer code by concatenating with the tetron code [191][192; Lemma 1]. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [191] and other qubit Hamiltonians on certain graphs [193,194]. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [195].
CPC codes are a type of stabilizer code. A teleported version of the CPC construction, the Clifford noise reduction (CliNR) scheme, can reduce noise in Clifford circuits with Pauli measurements with at most a three-fold overhead in the number of qubits and gates [94,95]. There is a simple formula for the probability that a Clifford circuit contains a logical error [96].
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 codes are qubit stabilizer codes that encode a logical fermionic Hilbert space into a physical space of \(n\) qubits.
The HaPPY code is a stabilizer code because it is defined by a contracted network of stabilizer tensors; see Thm. 6 in Ref. [207].
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 [7] (see also [8,9]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [8][169; Appx. A]. There are algorithms that determine whether two stabilizer states are equivalent that work by checking whether their corresponding cluster states are equivalent [10–12]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [208].
The resource states in FBQC are small stabilizer states, and after fusion measurements, the outputs are stabilizers (conditioned on measurement outcomes).
References
- [1]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction and Orthogonal Geometry”, Physical Review Letters 78, 405 (1997) arXiv:quant-ph/9605005 DOI
- [2]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [3]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [4]
- A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
- [5]
- Z. Ji, J. Chen, Z. Wei, and M. Ying, “The LU-LC conjecture is false”, (2008) arXiv:0709.1266
- [6]
- D. Gross and M. V. den Nest, “The LU-LC conjecture, diagonal local operations and quadratic forms over GF(2)”, (2007) arXiv:0707.4000
- [7]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [8]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [9]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, 45 arXiv:quant-ph/0703112 DOI
- [10]
- A. Dahlberg and S. Wehner, “Transforming graph states using single-qubit operations”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170325 (2018) arXiv:1805.05305 DOI
- [11]
- A. Dahlberg, J. Helsen, and S. Wehner, “How to transform graph states using single-qubit operations: computational complexity and algorithms”, (2018) arXiv:1805.05306
- [12]
- F. Hahn, A. Pappa, and J. Eisert, “Quantum network routing and local complementation”, npj Quantum Information 5, (2019) arXiv:1805.04559 DOI
- [13]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [14]
- D. Ostrev, “Canonical Form and Finite Blocklength Bounds for Stabilizer Codes”, (2024) arXiv:2408.15202
- [15]
- L. E. Danielsen, “On Self-Dual Quantum Codes, Graphs, and Boolean Functions”, (2005) arXiv:quant-ph/0503236
- [16]
- L. E. Danielsen and M. G. Parker, “On the classification of all self-dual additive codes over GF(4) of length up to 12”, Journal of Combinatorial Theory, Series A 113, 1351 (2006) arXiv:math/0504522 DOI
- [17]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Finite set of invariants to characterize local Clifford equivalence of stabilizer states”, Physical Review A 72, (2005) arXiv:quant-ph/0410165 DOI
- [18]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Local invariants of stabilizer codes”, Physical Review A 70, (2004) arXiv:quant-ph/0404106 DOI
- [19]
- M. Backens, “The ZX-calculus is complete for stabilizer quantum mechanics”, New Journal of Physics 16, 093021 (2014) arXiv:1307.7025 DOI
- [20]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2025) arXiv:2411.14448
- [21]
- T. McElvanney and M. Backens, “Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements”, Electronic Proceedings in Theoretical Computer Science 394, 66 (2023) arXiv:2205.02009 DOI
- [22]
- A. T. Hu and A. B. Khesin, “Improved graph formalism for quantum circuit simulation”, Physical Review A 105, (2022) arXiv:2109.10210 DOI
- [23]
- J. Dehaene and B. De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over GF(2)”, Physical Review A 68, (2003) arXiv:quant-ph/0304125 DOI
- [24]
- E. Hostens, J. Dehaene, and B. De Moor, “Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic”, Physical Review A 71, (2005) arXiv:quant-ph/0408190 DOI
- [25]
- M. V. den Nest, “Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond”, (2009) arXiv:0811.0898
- [26]
- H. J. García, I. L. Markov, and A. W. Cross, “On the Geometry of Stabilizer States”, (2017) arXiv:1711.07848
- [27]
- S. Bravyi, D. Lee, Z. Li, and B. Yoshida, “How Much Entanglement Is Needed for Quantum Error Correction?”, Physical Review Letters 134, (2025) arXiv:2405.01332 DOI
- [28]
- Z. Yi, Z. Liang, K. Zhong, Y. Wu, Z. Fang, and X. Wang, “Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes”, (2023) arXiv:2305.17505
- [29]
- T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
- [30]
- L. P. Pryadko, V. A. Shabashov, and V. K. Kozin, “QDistRnd: A GAP package for computing the distance of quantum error-correcting codes”, Journal of Open Source Software 7, 4120 (2022) arXiv:2308.15140 DOI
- [31]
- F. Hernando, G. Quintana-Ortí, and M. Grassl, “Fast Algorithms and Implementations for Computing the Minimum Distance of Quantum Codes”, (2024) arXiv:2408.10743
- [32]
- M. Webster and D. Browne, “Engineering Quantum Error Correction Codes Using Evolutionary Algorithms”, (2024) arXiv:2409.13017
- [33]
- U. Kapshikar and S. Kundu, “On the Hardness of the Minimum Distance Problem of Quantum Codes”, IEEE Transactions on Information Theory 69, 6293 (2023) arXiv:2203.04262 DOI
- [34]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [35]
- G. Smith and J. A. Smolin, “Degenerate Quantum Codes for Pauli Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0604107 DOI
- [36]
- S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
- [37]
- G. Kalachev and S. Sadov, “A linear-algebraic and lattice-theoretical look at the Cleaning Lemma of quantum coding theory”, Linear Algebra and its Applications 649, 96 (2022) arXiv:2204.04699 DOI
- [38]
- S. Kawabata, “Quantum Interleaver: Quantum Error Correction for Burst Error”, Journal of the Physical Society of Japan 69, 3540 (2000) arXiv:quant-ph/0002020 DOI
- [39]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [40]
- M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013) DOI
- [41]
- R. Cleve and D. Gottesman, “Efficient computations of encodings for quantum error correction”, Physical Review A 56, 76 (1997) arXiv:quant-ph/9607030 DOI
- [42]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Graphical quantum Clifford-encoder compilers from the ZX calculus”, (2025) arXiv:2301.02356
- [43]
- S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits”, Physical Review A 70, (2004) arXiv:quant-ph/0406196 DOI
- [44]
- B. Coecke and R. Duncan, “Interacting Quantum Observables”, Lecture Notes in Computer Science 298 DOI
- [45]
- B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
- [46]
- Yoder, Theodore J. "A generalization of the stabilizer formalism for simulating arbitrary quantum circuits." See https://www. scottaaronson. com/showcase2/report/ted-yoder. pdf (2012).
- [47]
- I. Chuang, A. Cross, G. Smith, J. Smolin, and B. Zeng, “Codeword stabilized quantum codes: Algorithm and structure”, Journal of Mathematical Physics 50, (2009) arXiv:0803.3232 DOI
- [48]
- A. Cross, G. Smith, J. A. Smolin, and B. Zeng, “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
- [49]
- J. P. Paz and W. H. Zurek, “Continuous Error Correction”, (1997) arXiv:quant-ph/9707049
- [50]
- J. Dengis, R. König, and F. Pastawski, “An optimal dissipative encoder for the toric code”, New Journal of Physics 16, 013023 (2014) arXiv:1310.1036 DOI
- [51]
- R. König and F. Pastawski, “Generating topological order: No speedup by dissipation”, Physical Review B 90, (2014) arXiv:1310.1037 DOI
- [52]
- G. Baggio, F. Ticozzi, P. D. Johnson, and L. Viola, “Dissipative encoding of quantum information”, Quantum Information and Computation 21, 737 (2021) arXiv:2102.04531 DOI
- [53]
- N. Fazio, M. Webster, and Z. Cai, “Low-overhead Magic State Circuits with Transversal CNOTs”, (2025) arXiv:2501.10291
- [54]
- C. B. Pushpan, T. K. Konar, A. S. De, and A. K. Pal, “Developing universal logical state-purification strategy for quantum error correcting codes”, (2025) arXiv:2502.01393
- [55]
- D. Aasen, J. Haah, M. B. Hastings, and Z. Wang, “Geometrically Enhanced Topological Quantum Codes”, (2025) arXiv:2505.10403
- [56]
- M. M. Wilde, “Logical operators of quantum codes”, Physical Review A 79, (2009) arXiv:0903.5256 DOI
- [57]
- N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
- [58]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [59]
- T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018) arXiv:1710.07256 DOI
- [60]
- B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
- [61]
- J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes”, (2014) arXiv:1409.8320
- [62]
- B. Zeng, X. Chen, and I. L. Chuang, “Semi-Clifford operations, structure ofCkhierarchy, and gate complexity for fault-tolerant quantum computation”, Physical Review A 77, (2008) arXiv:0712.2084 DOI
- [63]
- M. Newman and Y. Shi, “Limitations on Transversal Computation through Quantum Homomorphic Encryption”, (2017) arXiv:1704.07798
- [64]
- H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and Dan E Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, (2025) arXiv:2409.18175
- [65]
- A. G. Fowler, “Time-optimal quantum computation”, (2013) arXiv:1210.4626
- [66]
- T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016) arXiv:1603.03948 DOI
- [67]
- D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
- [68]
- X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000) arXiv:quant-ph/0002039 DOI
- [69]
- S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
- [70]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [71]
- G. Vidal and R. Tarrach, “Robustness of entanglement”, Physical Review A 59, 141 (1999) arXiv:quant-ph/9806094 DOI
- [72]
- M. Heinrich and D. Gross, “Robustness of Magic and Symmetries of the Stabiliser Polytope”, Quantum 3, 132 (2019) arXiv:1807.10296 DOI
- [73]
- J. R. Seddon and E. T. Campbell, “Quantifying magic for multi-qubit operations”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, 20190251 (2019) arXiv:1901.03322 DOI
- [74]
- M. Beverland, E. Campbell, M. Howard, and V. Kliuchnikov, “Lower bounds on the non-Clifford resources for quantum computations”, Quantum Science and Technology 5, 035009 (2020) arXiv:1904.01124 DOI
- [75]
- W. van Dam, F. Magniez, M. Mosca, and M. Santha, “Self-testing of universal and fault-tolerant sets of quantum gates”, Proceedings of the thirty-second annual ACM symposium on Theory of computing 688 (2000) arXiv:quant-ph/9904108 DOI
- [76]
- M. B. Hastings, “Turning Gate Synthesis Errors into Incoherent Errors”, (2016) arXiv:1612.01011
- [77]
- B. D. M. Jones, N. Linden, and P. Skrzypczyk, “The Hadamard gate cannot be replaced by a resource state in universal quantum computation”, Quantum 8, 1470 (2024) arXiv:2312.03515 DOI
- [78]
- M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, “Quantum hypergraph states”, New Journal of Physics 15, 113022 (2013) arXiv:1211.5554 DOI
- [79]
- O. Gühne, M. Cuquet, F. E. S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, and C. Macchiavello, “Entanglement and nonclassical properties of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 47, 335303 (2014) arXiv:1404.6492 DOI
- [80]
- D. W. Lyons, D. J. Upchurch, S. N. Walck, and C. D. Yetter, “Local unitary symmetries of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 48, 095301 (2015) arXiv:1410.3904 DOI
- [81]
- G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
- [82]
- D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
- [83]
- E. Dennis, “Toward fault-tolerant quantum computation without concatenation”, Physical Review A 63, (2001) arXiv:quant-ph/9905027 DOI
- [84]
- E. T. Campbell and D. E. Browne, “Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation”, Physical Review Letters 104, (2010) arXiv:0908.0836 DOI
- [85]
- E. T. Campbell and D. E. Browne, “On the Structure of Protocols for Magic State Distillation”, (2009) arXiv:0908.0838
- [86]
- O. Oreshkov, T. A. Brun, and D. A. Lidar, “Fault-Tolerant Holonomic Quantum Computation”, Physical Review Letters 102, (2009) arXiv:0806.0875 DOI
- [87]
- O. Oreshkov, “Topics in quantum information and the theory of open quantum systems”, (2008) arXiv:0812.4682
- [88]
- O. Oreshkov, T. A. Brun, and D. A. Lidar, “Scheme for fault-tolerant holonomic computation on stabilizer codes”, Physical Review A 80, (2009) arXiv:0904.2143 DOI
- [89]
- O. Oreshkov, T. A. Brun, and D. A. Lidar, “Fault tolerance for holonomic quantum computation”, (2013) arXiv:1312.0165
- [90]
- P. Zanardi and M. Rasetti, “Holonomic quantum computation”, Physics Letters A 264, 94 (1999) arXiv:quant-ph/9904011 DOI
- [91]
- G. Duclos-Cianci and D. Poulin, “Reducing the quantum-computing overhead with complex gate distillation”, Physical Review A 91, (2015) arXiv:1403.5280 DOI
- [92]
- N. Rengaswamy, R. Calderbank, S. Kadhe, and H. D. Pfister, “Logical Clifford Synthesis for Stabilizer Codes”, IEEE Transactions on Quantum Engineering 1, 1 (2020) arXiv:1907.00310 DOI
- [93]
- B. Reid, “A simple method for compiling quantum stabilizer circuits”, (2024) arXiv:2404.19408
- [94]
- N. Delfosse and E. Tham, “Low-cost noise reduction for Clifford circuits”, (2024) arXiv:2407.06583
- [95]
- E. Tham and N. Delfosse, “Optimized Clifford Noise Reduction: Theory, Simulations and Experiments”, (2025) arXiv:2504.13356
- [96]
- D. M. Debroy and K. R. Brown, “Extended flag gadgets for low-overhead circuit verification”, Physical Review A 102, (2020) arXiv:2009.07752 DOI
- [97]
- Z. Chen, J. O. Weinberg, and N. Rengaswamy, “Fault Tolerant Quantum Simulation via Symplectic Transvections”, (2025) arXiv:2504.11444
- [98]
- E. J. Kuehnke, K. Levi, J. Roffe, J. Eisert, and D. Miller, “Hardware-tailored logical Clifford circuits for stabilizer codes”, (2025) arXiv:2505.20261
- [99]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [100]
- G. Zhang and Y. Li, “Time-Efficient Logical Operations on Quantum Low-Density Parity Check Codes”, Physical Review Letters 134, (2025) arXiv:2408.01339 DOI
- [101]
- Y. Ouyang, “Robust projective measurements through measuring code-inspired observables”, npj Quantum Information 10, (2024) arXiv:2402.04093 DOI
- [102]
- M.-X. Huo and Y. Li, “Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction”, New Journal of Physics 19, 123032 (2017) arXiv:1710.03636 DOI
- [103]
- D. P. DiVincenzo and P. Aliferis, “Effective Fault-Tolerant Quantum Computation with Slow Measurements”, Physical Review Letters 98, (2007) arXiv:quant-ph/0607047 DOI
- [104]
- S. Vittal, A. Javadi-Abhari, A. W. Cross, L. S. Bishop, and M. Qureshi, “Flag Proxy Networks: Tackling the Architectural, Scheduling, and Decoding Obstacles of Quantum LDPC codes”, (2024) arXiv:2409.14283
- [105]
- J. C. M. de la Fuente, “Dynamical weight reduction of Pauli measurements”, (2024) arXiv:2410.12527
- [106]
- F. Yamaguchi, K. Nemoto, and W. J. Munro, “Quantum error correction via robust probe modes”, Physical Review A 73, (2006) arXiv:quant-ph/0511098 DOI
- [107]
- S. Puri, A. Grimm, P. Campagne-Ibarcq, A. Eickbusch, K. Noh, G. Roberts, L. Jiang, M. Mirrahimi, M. H. Devoret, and S. M. Girvin, “Stabilized Cat in a Driven Nonlinear Cavity: A Fault-Tolerant Error Syndrome Detector”, Physical Review X 9, (2019) arXiv:1807.09334 DOI
- [108]
- M. Sarovar, C. Ahn, K. Jacobs, and G. J. Milburn, “Practical scheme for error control using feedback”, Physical Review A 69, (2004) arXiv:quant-ph/0402017 DOI
- [109]
- M.-H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes”, Physical Review A 83, (2011) arXiv:1009.1319 DOI
- [110]
- Kuo, Kao-Yueh, and Chung-Chin Lu. "On the hardness of decoding quantum stabilizer codes under the depolarizing channel." 2012 International Symposium on Information Theory and its Applications. IEEE, 2012.
- [111]
- P. Iyer and D. Poulin, “Hardness of decoding quantum stabilizer codes”, (2013) arXiv:1310.3235
- [112]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [113]
- N. Delfosse, A. Paetznick, J. Haah, and M. B. Hastings, “Splitting decoders for correcting hypergraph faults”, (2023) arXiv:2309.15354
- [114]
- H. Ollivier and J.-P. Tillich, “Trellises for stabilizer codes: Definition and uses”, Physical Review A 74, (2006) arXiv:quant-ph/0512041 DOI
- [115]
- D. Cruz, F. A. Monteiro, and B. C. Coutinho, “Quantum Error Correction Via Noise Guessing Decoding”, IEEE Access 11, 119446 (2023) arXiv:2208.02744 DOI
- [116]
- S. Krastanov and L. Jiang, “Deep Neural Network Probabilistic Decoder for Stabilizer Codes”, Scientific Reports 7, (2017) arXiv:1705.09334 DOI
- [117]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [118]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [119]
- R. J. Harris, E. Coupe, N. A. McMahon, G. K. Brennen, and T. M. Stace, “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
- [120]
- O. Shtanko, Y.-J. Liu, S. Lieu, A. V. Gorshkov, and V. V. Albert, “Bounds on Autonomous Quantum Error Correction”, (2023) arXiv:2308.16233
- [121]
- K. Shiraishi, H. Yamasaki, and M. Murao, “Efficient local operations and classical communication extraction of quantum information encoded in stabilizer codes”, Physical Review A 110, (2024) arXiv:2308.14054 DOI
- [122]
- M. Cain, C. Zhao, H. Zhou, N. Meister, J. P. B. Ataides, A. Jaffe, D. Bluvstein, and M. D. Lukin, “Correlated decoding of logical algorithms with transversal gates”, (2025) arXiv:2403.03272
- [123]
- O. Higgott and C. Gidney, “Sparse Blossom: correcting a million errors per core second with minimum-weight matching”, Quantum 9, 1600 (2025) arXiv:2303.15933 DOI
- [124]
- P.-J. H. S. Derks, A. Townsend-Teague, A. G. Burchards, and J. Eisert, “Designing fault-tolerant circuits using detector error models”, (2024) arXiv:2407.13826
- [125]
- Y. Shi, A. Patil, and S. Guha, “Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders”, PRX Quantum 6, (2025) arXiv:2408.06299 DOI
- [126]
- K.-Y. Kuo and Y. Ouyang, “Degenerate quantum erasure decoding”, (2025) arXiv:2411.13509
- [127]
- F. Kobayashi, H. Manabe, G. A. L. White, T. Farrelly, K. Modi, and T. M. Stace, “Tensor-network decoders for process tensor descriptions of non-Markovian noise”, (2024) arXiv:2412.13739
- [128]
- G. Baranes, M. Cain, J. P. B. Ataides, D. Bluvstein, J. Sinclair, V. Vuletic, H. Zhou, and M. D. Lukin, “Leveraging Atom Loss Errors in Fault Tolerant Quantum Algorithms”, (2025) arXiv:2502.20558
- [129]
- C. J. Trout and K. R. Brown, “Magic state distillation and gate compilation in quantum algorithms for quantum chemistry”, International Journal of Quantum Chemistry 115, 1296 (2015) DOI
- [130]
- P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
- [131]
- D. P. DiVincenzo and P. W. Shor, “Fault-Tolerant Error Correction with Efficient Quantum Codes”, Physical Review Letters 77, 3260 (1996) arXiv:quant-ph/9605031 DOI
- [132]
- A. M. Steane, “Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing”, Fortschritte der Physik 46, 443 (1998) arXiv:quant-ph/9708021 DOI
- [133]
- T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction”, Quantum 7, 1075 (2023) arXiv:2208.05601 DOI
- [134]
- Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
- [135]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [136]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [137]
- P. Aliferis and J. Preskill, “Fibonacci scheme for fault-tolerant quantum computation”, Physical Review A 79, (2009) arXiv:0809.5063 DOI
- [138]
- C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
- [139]
- N. Rengaswamy, A. Raina, N. Raveendran, and B. Vasić, “Distilling GHZ States using Stabilizer Codes”, (2022) arXiv:2109.06248
- [140]
- B. Anker and M. Marvian, “Flag Gadgets based on Classical Codes”, (2024) arXiv:2212.10738
- [141]
- H. Choi, F. T. Chong, D. Englund, and Y. Ding, “Fault Tolerant Non-Clifford State Preparation for Arbitrary Rotations”, (2023) arXiv:2303.17380
- [142]
- S. Heußen and J. Hilder, “Efficient fault-tolerant code switching via one-way transversal CNOT gates”, (2024) arXiv:2409.13465
- [143]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
- [144]
- L. Schmid, T. Peham, L. Berent, M. Müller, and R. Wille, “Deterministic Fault-Tolerant State Preparation for Near-Term Quantum Error Correction: Automatic Synthesis Using Boolean Satisfiability”, (2025) arXiv:2501.05527
- [145]
- K. Chen, Y. Liu, W. Fang, J. Paykin, X.-C. Wu, A. Schmitz, S. Zdancewic, and G. Li, “Verifying Fault-Tolerance of Quantum Error Correction Codes”, (2025) arXiv:2501.14380
- [146]
- E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [147]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
- [148]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [149]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions 8, 269 (2021) arXiv:1809.10704 DOI
- [150]
- Y. Zhao and D. E. Liu, “Extracting error thresholds through the framework of approximate quantum error correction condition”, Physical Review Research 6, (2024) arXiv:2312.16991 DOI
- [151]
- A. M. Steane, “Quantum Computing and Error Correction”, (2003) arXiv:quant-ph/0304016
- [152]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [153]
- M. Grassl, “Classical Information Theory and Classical Error Correction”, Lectures on Quantum Information 3 (2006) DOI
- [154]
- M. Grassl, “Searching for linear codes with large minimum distance”, Algorithms and Computation in Mathematics 287 DOI
- [155]
- N. Aydin, P. Liu, and B. Yoshino, “Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions”, (2021) arXiv:2106.12065
- [156]
- N. Aydin, P. Liu, and B. Yoshino, “A Database of Quantum Codes”, (2021) arXiv:2108.03567
- [157]
- R. Matsumoto, “Conversion of a general quantum stabilizer code to an entanglement distillation protocol”, Journal of Physics A: Mathematical and General 36, 8113 (2003) arXiv:quant-ph/0209091 DOI
- [158]
- K. Goodenough, S. de Bone, V. Addala, S. Krastanov, S. Jansen, D. Gijswijt, and D. Elkouss, “Near-Term n to k Distillation Protocols Using Graph Codes”, IEEE Journal on Selected Areas in Communications 42, 1830 (2024) arXiv:2303.11465 DOI
- [159]
- A. Ambainis and D. Gottesman, “The minimum distance problem for two-way entanglement purification”, IEEE Transactions on Information Theory 52, 748 (2006) arXiv:quant-ph/0310097 DOI
- [160]
- T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, “Pauli channels can be estimated from syndrome measurements in quantum error correction”, Quantum 6, 809 (2022) arXiv:2107.14252 DOI
- [161]
- T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, “Learning Logical Pauli Noise in Quantum Error Correction”, Physical Review Letters 130, (2023) arXiv:2209.09267 DOI
- [162]
- R. A. Wolf, “Quantum Error Correction for Kids”, (2024) arXiv:2405.06795
- [163]
- Craig Gidney. Stim. https://github.com/quantumlib/Stim, 2021.
- [164]
- E. Huang, A. Pesah, C. T. Chubb, M. Vasmer, and A. Dua, “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
- [165]
- Eric Huang and Arthur Pesah. PanQEC. https://github.com/panqec/panqec, 2023.
- [166]
- QuantumSavory. https://github.com/QuantumSavory, 2025.
- [167]
- Stefan Krastanov et al., QuantumSavory/QuantumClifford.Jl: V0.9.16 (Zenodo, 2024) DOI
- [168]
- C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
- [169]
- J. Bausch and F. Leditzky, “Error Thresholds for Arbitrary Pauli Noise”, SIAM Journal on Computing 50, 1410 (2021) arXiv:1910.00471 DOI
- [170]
- D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
- [171]
- E. Farhi and S. P. Jordan, “Efficiently constructing a quantum uniform superposition over bit strings near a binary linear code”, (2024) arXiv:2404.16129
- [172]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [173]
- R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
- [174]
- H. Zhu, “Multiqubit Clifford groups are unitary 3-designs”, Physical Review A 96, (2017) arXiv:1510.02619 DOI
- [175]
- Z. Webb, “The Clifford group forms a unitary 3-design”, (2016) arXiv:1510.02769
- [176]
- H. Zhu, R. Kueng, M. Grassl, and D. Gross, “The Clifford group fails gracefully to be a unitary 4-design”, (2016) arXiv:1609.08172
- [177]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [178]
- R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [179]
- S. Ball, A. Centelles, and F. Huber, “Quantum error-correcting codes and their geometries”, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions 10, 337 (2023) arXiv:2007.05992 DOI
- [180]
- Glynn, David G., T. Aaron Gulliver, Johannes G. Maks, and Manish K. Gupta. "The geometry of additive quantum codes." submitted to Springer-Verlag (2004).
- [181]
- S. Ball and P. Puig, “The geometry of non-additive stabiliser codes”, (2021) arXiv:2107.11281
- [182]
- J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini, and F. Pambianco, “The geometry of quantum codes”, Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 6, 53 (2008) DOI
- [183]
- V. Kliuchnikov and S. Schönnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
- [184]
- P. Faist, M. P. Woods, V. V. Albert, J. M. Renes, J. Eisert, and J. Preskill, “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI
- [185]
- B. Rodatz, B. Poór, and A. Kissinger, “Floquetifying stabiliser codes with distance-preserving rewrites”, (2024) arXiv:2410.17240
- [186]
- A. Townsend-Teague, J. Magdalena de la Fuente, and M. Kesselring, “Floquetifying the Colour Code”, Electronic Proceedings in Theoretical Computer Science 384, 265 (2023) arXiv:2307.11136 DOI
- [187]
- Y. Xu and A. Dua, “Fault-tolerant protocols through spacetime concatenation”, (2025) arXiv:2504.08918
- [188]
- D. Ueno and R. Matsumoto, “Explicit method to make shortened stabilizer EAQECC from stabilizer QECC”, (2022) arXiv:2205.13732
- [189]
- S. Kundu and B. W. Reichardt, “Majorana qubit codes that also correct odd-weight errors”, (2023) arXiv:2311.01779
- [190]
- S. Kundu and B. Reichardt, “Majorana subsystem qubit codes that also correct odd-weight errors”, New Journal of Physics 26, 073029 (2024) DOI
- [191]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [192]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [193]
- A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
- [194]
- S. J. Elman, A. Chapman, and S. T. Flammia, “Free Fermions Behind the Disguise”, Communications in Mathematical Physics 388, 969 (2021) arXiv:2012.07857 DOI
- [195]
- R. A. Macedo, C. C. Bellinati, W. B. Fontana, E. C. Andrade, and R. G. Pereira, “Partons from stabilizer codes”, (2025) arXiv:2505.02683
- [196]
- I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI
- [197]
- Y. Ouyang, Y. Jing, and G. K. Brennen, “Measurement-free code-switching for low overhead quantum computation using permutation invariant codes”, (2025) arXiv:2411.13142
- [198]
- J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [199]
- Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.
- [200]
- A. Dymarsky, J. Henriksson, and B. McPeak, “Holographic duality from Howe duality: Chern-Simons gravity as an ensemble of code CFTs”, (2025) arXiv:2504.08724
- [201]
- N. Baspin and D. Williamson, “Wire Codes”, (2024) arXiv:2410.10194
- [202]
- L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
- [203]
- A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
- [204]
- L. G. Gunderman, “Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition”, Physical Review A 105, (2022) arXiv:2110.15274 DOI
- [205]
- S. Yu, J. Bierbrauer, Y. Dong, Q. Chen, and C. H. Oh, “All the stabilizer codes of distance 3”, (2011) arXiv:0901.1968
- [206]
- A. Dymarsky and A. Shapere, “Quantum stabilizer codes, lattices, and CFTs”, Journal of High Energy Physics 2021, (2021) arXiv:2009.01244 DOI
- [207]
- F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
- [208]
- B. J. Brown and S. Roberts, “Universal fault-tolerant measurement-based quantum computation”, Physical Review Research 2, (2020) arXiv:1811.11780 DOI
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