## 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.

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 via Gaussian elimination [3].

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

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

Qubit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_2^n\) [5].

## Protection

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

There is the following analogue of the Knill-Laflamme conditions for qubit stabilizer codes. Define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all Pauli operators that commute with all \(S\in\mathsf{S}\). A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\).

Cleaning lemma: If all logical operators act trivially on some subset of qubits in a stabilizer code, then any logical Pauli operator can be represented on the complementary qubit subset via a stabilizer. More technically, given any subset \(M\) of qubits that is correctable (under erasure), any logical Pauli operator \(P\) can be cleaned off of \(M\) using a stabilizer \(S\) such that \(PS\) is supported on \(M^{\perp}\). More generally, for any \(M\), we have \(g(M)+g(M^{\perp}) = 2k\), where \(g(M)\) is the number of logical-\(X\) and logical-\(Z\) Pauli operators supported fully on \(M\) (up to stabilizers). The Cleaning Lemma was originally proven [6], where an analogous result is states for subsystem codes; see also Ref. [7].

Entropic conditions have been formulated for random projective measurement noise [8].

## Encoding

Clifford circuits, i.e., those consisting of CNOT, Hadamard, and certain phase gates, using an algorithm [9] based on the Gottesman-Knill theorem [10] or using ZX calculus [11,12] (with the latter providing a unique decomposition [13]).

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 [10,14]. 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 — The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).
- 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 [94]. Various bounds exist on the distance of the resulting codes [95,96].
- 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 [97,98].
- \([[2^r, 2^r-r-2, 3]]\) Gottesman 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 [49]. There is a simple formula for the probability that a Clifford circuit contains a logical error [50].
- 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. [99].
- 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 [100] (see also [101,102]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [101][103; Appx. A]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [104].
- Fusion-based quantum computing (FBQC) code — The resource states in FBQC are small stabilizer states, and after fusion measurements, the outputs are stabilizers (conditioned on measurement outcomes).
- Hierarchical code
- XYZ product code
- Quantum spatially coupled (SC-QLDPC) code
- Qubit BCH code
- Twist-defect color code
- Matching code
- Clifford-deformed surface code (CDSC)
- Twist-defect surface code
- Three-fermion (3F) Walker-Wang model code

## Cousins

- Codeword stabilized (CWS) code — CWS codes whose underlying classical code is a linear binary code are qubit stabilizer codes containing a cluster-state codeword [16,105]. Since any stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [101][103; 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 [106; 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 bit strings within Hamming distance \(b\) of a binary linear code can be prepared efficiently [107].
- 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 [108].
- Design — Stabilizer states on \(n\) qubits form complex projective 3-designs [109], while the Clifford group is a unitary 3-design [110,111].
- Constant-excitation (CE) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [112] that protects against \(d-1\) AD errors [113].
- Amplitude-damping (AD) code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [112] that protects against \(d-1\) AD errors [113].
- Barnes-Wall (BW) lattice code — The first lattice shell of a BW lattice over a cyclotomic field is formed by stabilizer states [114].
- Graph quantum code — Graph quantum codes for \(G=\mathbb{Z}_2\) are a subset of qubit stabilizer codes [100]. Any qubit stabilizer code is equivalent to a graph quantum code for \(G=\mathbb{Z}_2\) via a single-qubit Clifford circuit [100] (see also [101,102]).
- Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [115].
- 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 [116].
- 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 [117].
- 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. 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 code \(\mathsf{C}\), there exists a CSS code \(\mathsf{C}^{\prime}\) such that \(\mathsf{C} = DQ\mathsf{C}^{\prime}\), where \(D\) is a diagonal Clifford operator and \(Q\) is an element of an XP stabilizer group [118; Prop. B.3.1].
- Subsystem qubit stabilizer code — Subsystem qubit stabilizer codes reduce to qubit stabilizer codes when there are no gauge qubits.

## References

- [1]
- A. R. Calderbank et al., “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]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [4]
- Z. Yi et al., “Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes”, (2023) arXiv:2305.17505
- [5]
- 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
- [6]
- 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
- [7]
- 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
- [8]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [9]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Graphical quantum Clifford-encoder compilers from the ZX calculus”, (2024) arXiv:2301.02356
- [10]
- S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits”, Physical Review A 70, (2004) arXiv:quant-ph/0406196 DOI
- [11]
- B. Coecke and R. Duncan, “Interacting Quantum Observables”, Automata, Languages and Programming 298 DOI
- [12]
- B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
- [13]
- A. B. Khesin and A. Li, “Equivalence Classes of Quantum Error-Correcting Codes”, (2024) arXiv:2406.12083
- [14]
- Yoder, Theodore J. "A generalization of the stabilizer formalism for simulating arbitrary quantum circuits." See http://www. scottaaronson. com/showcase2/report/ted-yoder. pdf (2012).
- [15]
- I. Chuang et al., “Codeword stabilized quantum codes: Algorithm and structure”, Journal of Mathematical Physics 50, (2009) arXiv:0803.3232 DOI
- [16]
- A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
- [17]
- J. P. Paz and W. H. Zurek, “Continuous Error Correction”, (1997) arXiv:quant-ph/9707049
- [18]
- 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
- [19]
- R. König and F. Pastawski, “Generating topological order: No speedup by dissipation”, Physical Review B 90, (2014) arXiv:1310.1037 DOI
- [20]
- M. M. Wilde, “Logical operators of quantum codes”, Physical Review A 79, (2009) arXiv:0903.5256 DOI
- [21]
- N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
- [22]
- 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
- [23]
- B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
- [24]
- J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes”, (2014) arXiv:1409.8320
- [25]
- 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
- [26]
- M. Newman and Y. Shi, “Limitations on Transversal Computation through Quantum Homomorphic Encryption”, (2017) arXiv:1704.07798
- [27]
- 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
- [28]
- 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
- [29]
- 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
- [30]
- 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
- [31]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [32]
- G. Vidal and R. Tarrach, “Robustness of entanglement”, Physical Review A 59, 141 (1999) arXiv:quant-ph/9806094 DOI
- [33]
- M. Heinrich and D. Gross, “Robustness of Magic and Symmetries of the Stabiliser Polytope”, Quantum 3, 132 (2019) arXiv:1807.10296 DOI
- [34]
- 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
- [35]
- M. Beverland et al., “Lower bounds on the non-Clifford resources for quantum computations”, Quantum Science and Technology 5, 035009 (2020) arXiv:1904.01124 DOI
- [36]
- M. B. Hastings, “Turning Gate Synthesis Errors into Incoherent Errors”, (2016) arXiv:1612.01011
- [37]
- B. D. M. Jones, N. Linden, and P. Skrzypczyk, “The Hadamard gate cannot be replaced by a resource state in universal quantum computation”, (2024) arXiv:2312.03515
- [38]
- M. Rossi et al., “Quantum hypergraph states”, New Journal of Physics 15, 113022 (2013) arXiv:1211.5554 DOI
- [39]
- O. Gühne et al., “Entanglement and nonclassical properties of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 47, 335303 (2014) arXiv:1404.6492 DOI
- [40]
- D. W. Lyons et al., “Local unitary symmetries of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 48, 095301 (2015) arXiv:1410.3904 DOI
- [41]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [42]
- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
- [43]
- O. Oreshkov, T. A. Brun, and D. A. Lidar, “Fault-Tolerant Holonomic Quantum Computation”, Physical Review Letters 102, (2009) arXiv:0806.0875 DOI
- [44]
- 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
- [45]
- P. Zanardi and M. Rasetti, “Holonomic quantum computation”, Physics Letters A 264, 94 (1999) arXiv:quant-ph/9904011 DOI
- [46]
- G. Duclos-Cianci and D. Poulin, “Reducing the quantum-computing overhead with complex gate distillation”, Physical Review A 91, (2015) arXiv:1403.5280 DOI
- [47]
- N. Rengaswamy et al., “Logical Clifford Synthesis for Stabilizer Codes”, IEEE Transactions on Quantum Engineering 1, 1 (2020) arXiv:1907.00310 DOI
- [48]
- B. Reid, “A simple method for compiling quantum stabilizer circuits”, (2024) arXiv:2404.19408
- [49]
- N. Delfosse and E. Tham, “Low-cost noise reduction for Clifford circuits”, (2024) arXiv:2407.06583
- [50]
- D. M. Debroy and K. R. Brown, “Extended flag gadgets for low-overhead circuit verification”, Physical Review A 102, (2020) arXiv:2009.07752 DOI
- [51]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [52]
- M.-H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes”, Physical Review A 83, (2011) arXiv:1009.1319 DOI
- [53]
- 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.
- [54]
- P. Iyer and D. Poulin, “Hardness of decoding quantum stabilizer codes”, (2013) arXiv:1310.3235
- [55]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [56]
- N. Delfosse et al., “Splitting decoders for correcting hypergraph faults”, (2023) arXiv:2309.15354
- [57]
- H. Ollivier and J.-P. Tillich, “Trellises for stabilizer codes: Definition and uses”, Physical Review A 74, (2006) arXiv:quant-ph/0512041 DOI
- [58]
- 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
- [59]
- S. Krastanov and L. Jiang, “Deep Neural Network Probabilistic Decoder for Stabilizer Codes”, Scientific Reports 7, (2017) arXiv:1705.09334 DOI
- [60]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [61]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [62]
- R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
- [63]
- K. Shiraishi, H. Yamasaki, and M. Murao, “Efficient decoding of stabilizer code by single-qubit local operations and classical communication”, (2023) arXiv:2308.14054
- [64]
- M. Cain et al., “Correlated decoding of logical algorithms with transversal gates”, (2024) arXiv:2403.03272
- [65]
- O. Higgott and C. Gidney, “Sparse Blossom: correcting a million errors per core second with minimum-weight matching”, (2023) arXiv:2303.15933
- [66]
- P.-J. H. S. Derks et al., “Designing fault-tolerant circuits using detector error models”, (2024) arXiv:2407.13826
- [67]
- P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
- [68]
- 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
- [69]
- T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction”, Quantum 7, 1075 (2023) arXiv:2208.05601 DOI
- [70]
- Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
- [71]
- 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
- [72]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [73]
- P. Aliferis and J. Preskill, “Fibonacci scheme for fault-tolerant quantum computation”, Physical Review A 79, (2009) arXiv:0809.5063 DOI
- [74]
- C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
- [75]
- N. Rengaswamy et al., “Distilling GHZ States using Stabilizer Codes”, (2022) arXiv:2109.06248
- [76]
- B. Anker and M. Marvian, “Flag Gadgets based on Classical Codes”, (2024) arXiv:2212.10738
- [77]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [78]
- 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
- [79]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [80]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
- [81]
- Y. Zhao and D. E. Liu, “Extracting Error Thresholds through the Framework of Approximate Quantum Error Correction Condition”, (2023) arXiv:2312.16991
- [82]
- E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998) arXiv:quant-ph/9702058 DOI
- [83]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [84]
- J. Preskill, “Reliable quantum computers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998) arXiv:quant-ph/9705031 DOI
- [85]
- P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
- [86]
- 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
- [87]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [88]
- M. Grassl, “Classical Information Theory and Classical Error Correction”, Lectures on Quantum Information 3 (2006) DOI
- [89]
- M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
- [90]
- 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
- [91]
- K. Goodenough et al., “Near-Term n to k Distillation Protocols Using Graph Codes”, IEEE Journal on Selected Areas in Communications 42, 1830 (2024) arXiv:2303.11465 DOI
- [92]
- S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
- [93]
- R. A. Wolf, “Quantum Error Correction for Kids”, (2024) arXiv:2405.06795
- [94]
- L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
- [95]
- A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
- [96]
- 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
- [97]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [98]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [99]
- F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
- [100]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [101]
- 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
- [102]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
- [103]
- J. Bausch and F. Leditzky, “Error Thresholds for Arbitrary Pauli Noise”, SIAM Journal on Computing 50, 1410 (2021) arXiv:1910.00471 DOI
- [104]
- B. J. Brown and S. Roberts, “Universal fault-tolerant measurement-based quantum computation”, Physical Review Research 2, (2020) arXiv:1811.11780 DOI
- [105]
- C. Cafaro, D. Markham, and P. van Loock, “Scheme for constructing graphs associated with stabilizer quantum codes”, (2014) arXiv:1407.2777
- [106]
- D. Bacon and A. Casaccino, “Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes”, (2006) arXiv:quant-ph/0610088
- [107]
- E. Farhi and S. P. Jordan, “Efficiently constructing a quantum uniform superposition over bit strings near a binary linear code”, (2024) arXiv:2404.16129
- [108]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [109]
- R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
- [110]
- H. Zhu, “Multiqubit Clifford groups are unitary 3-designs”, Physical Review A 96, (2017) arXiv:1510.02619 DOI
- [111]
- Z. Webb, “The Clifford group forms a unitary 3-design”, (2016) arXiv:1510.02769
- [112]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [113]
- R. Duan et al., “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [114]
- V. Kliuchnikov and S. Schönnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
- [115]
- P. Faist et al., “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI
- [116]
- D. Ueno and R. Matsumoto, “Explicit method to make shortened stabilizer EAQECC from stabilizer QECC”, (2022) arXiv:2205.13732
- [117]
- 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
- [118]
- Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.

## 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