Description
Also called a Pauli stabilizer code. An \(((n,2^k,d))\) qubit stabilizer code is denoted as \([[n,k]]\) or \([[n,k,d]]\), where \(d\) is the code's distance. Logical subspace is the joint eigenspace of commuting Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace of a set of \(2^{n-k}\) commuting Pauli operators which do not contain \(-I\). The distance is the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.
Binary symplectic representation: Each stabilizer code can be represented by a \((n-k) \times 2n\) check matrix (a.k.a. stabilizer generator matrix) \(H=(A|B)\), where each row \((a|b)\) is the binary symplectic representation of an element from a set of generating elements of the stabilizer group. 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. The check matrix can be brought into standard form via Gaussian elimination [3].
The stabilizer commutation condition can equivalently be stated in the symplectic representation. A pair of \(n\)-qubit stabilizers with symplectic representations \((a|b)\) 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} Binary symplectic representations of stabilizer group elements thus form a self-orthogonal subspace of \(GF(2)^{2n}\) with respect to the symplectic inner product.
Alternative representations include the decoupling representation, in which Pauli strings are represented as vectors over \(GF(2)\) using three bits [4], or the representation over \(GF(4)\) (see stabilizer codes over \(GF(4)\)).
Protection
Detects errors on up to \(d-1\) qubits, and corrects erasure errors on up to \(d-1\) qubits. More generally, define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all 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 \(E,F \in {\mathcal{E}}\).
A stabilizer code is geometrically local if the support of the stabilizer generators is bounded by a ball of size independent of \(n\).
Encoding
Transversal Gates
Gates
Decoding
Fault Tolerance
Code Capacity Threshold
Threshold
Notes
Parents
- Codeword stabilized (CWS) code — If the CWS set \( \mathcal{W} \) is an abelian group not containing \(-I\), then the CWS code is a stabilizer code.
- XP stabilizer code — The XP stabilizer formalism reduces to the Pauli formalism at \(N=2\).
- 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 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 [48]. Various bounds exist on the distance of the resulting codes [49,50].
- Galois-qudit stabilizer code — Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
- Qubit stabilizer operator-algebra quantum error-correcting code
Children
- Crystalline-circuit qubit code
- Low-depth random Clifford-circuit qubit 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 [51,52].
- Cluster-state code — Cluster states are particular qubit stabilizer states defined on a graph. Conversely, any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [53].
- 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).
- \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code
- Transverse-field Ising model (TFIM) code
- 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.
- 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. [54].
- Hierarchical code
- Quantum spatially coupled (SC-QLDPC) code
- Qubit BCH code — Qubit BCH codes constructed via the CSS construction are CSS codes, and the rest are stabilizer codes over \(GF(4)\).
- Qubit CSS code — Stabilizer generators can be expressed as either only \(X\)-type or only \(Z\)-type. However, any \([[n,k,d]]\) stabilizer code can be mapped onto a \([[4n,2k,2d]]\) self-orthogonal CSS code, with the mapping preserving geometric locality of a code up to a constant factor [52].
- Stabilizer code over \(GF(4)\)
- Quantum convolutional code
- Haah cubic code
- X-cube model code
- Matching code
- Clifford-deformed surface code (CDSC)
- Three-fermion (3F) model code
- Subsystem qubit stabilizer code — Subsystem stabilizer codes reduce to stabilizer codes when there are no gauge qubits.
Cousins
- 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.
- Dual linear code — Binary symplectic representations of stabilizer group elements form a linear code over \(GF(2)\) that is self-orthogonal with respect to the symplectic inner product [55; Thm. 27.3.6].
- Metrological code — A joint \(+1\) and \(-1\) eigenstate of a set of stabilizer can form a metrological stabilizer code [56].
- Spacetime circuit code — Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit stabilizer codes.
- Movassagh-Ouyang Hamiltonian code — Many, but not all, Movassagh-Ouyang codes are stabilizer codes.
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]
- S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits”, Physical Review A 70, (2004) arXiv:quant-ph/0406196 DOI
- [6]
- B. Coecke and R. Duncan, “Interacting Quantum Observables”, Automata, Languages and Programming 298 DOI
- [7]
- B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
- [8]
- A. B. Khesin, J. Z. Lu, and P. W. Shor, “Graphical quantum Clifford-encoder compilers from the ZX calculus”, (2023) arXiv:2301.02356
- [9]
- I. Chuang et al., “Codeword stabilized quantum codes: Algorithm and structure”, Journal of Mathematical Physics 50, 042109 (2009) arXiv:0803.3232 DOI
- [10]
- A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
- [11]
- J. P. Paz and W. H. Zurek, “Continuous Error Correction”, (1997) arXiv:quant-ph/9707049
- [12]
- 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
- [13]
- R. König and F. Pastawski, “Generating topological order: No speedup by dissipation”, Physical Review B 90, (2014) arXiv:1310.1037 DOI
- [14]
- 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
- [15]
- B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
- [16]
- J. T. Anderson and T. Jochym-O’Connor, “Classification of transversal gates in qubit stabilizer codes”, (2014) arXiv:1409.8320
- [17]
- 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
- [18]
- 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
- [19]
- M.-H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes”, Physical Review A 83, (2011) arXiv:1009.1319 DOI
- [20]
- 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.
- [21]
- P. Iyer and D. Poulin, “Hardness of decoding quantum stabilizer codes”, (2013) arXiv:1310.3235
- [22]
- H. Ollivier and J.-P. Tillich, “Trellises for stabilizer codes: Definition and uses”, Physical Review A 74, (2006) arXiv:quant-ph/0512041 DOI
- [23]
- D. Cruz, F. A. Monteiro, and B. C. Coutinho, “Quantum Error Correction via Noise Guessing Decoding”, (2023) arXiv:2208.02744
- [24]
- S. Krastanov and L. Jiang, “Deep Neural Network Probabilistic Decoder for Stabilizer Codes”, Scientific Reports 7, (2017) arXiv:1705.09334 DOI
- [25]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [26]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [27]
- P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
- [28]
- T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction”, Quantum 7, 1075 (2023) arXiv:2208.05601 DOI
- [29]
- Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
- [30]
- 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
- [31]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [32]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [33]
- N. Rengaswamy et al., “Distilling GHZ States using Stabilizer Codes”, (2022) arXiv:2109.06248
- [34]
- B. Anker and M. Marvian, “Flag Gadgets based on Classical Codes”, (2022) arXiv:2212.10738
- [35]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [36]
- 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
- [37]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [38]
- 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
- [39]
- 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
- [40]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [41]
- 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
- [42]
- P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
- [43]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [44]
- M. Grassl, “Classical Information Theory and Classical Error Correction”, Lectures on Quantum Information 3 DOI
- [45]
- M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
- [46]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [47]
- K. Goodenough et al., “Near-term \(n\) to \(k\) distillation protocols using graph codes”, (2023) arXiv:2303.11465
- [48]
- L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
- [49]
- A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
- [50]
- 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
- [51]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [52]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [53]
- B. J. Brown and S. Roberts, “Universal fault-tolerant measurement-based quantum computation”, Physical Review Research 2, (2020) arXiv:1811.11780 DOI
- [54]
- 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
- [55]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [56]
- P. Faist et al., “Time-energy uncertainty relation for noisy quantum metrology”, (2022) arXiv:2207.13707
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