## Description

A family of Abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

The construction on closed surfaces (surfaces with boundaries) is called the toric code (planar code [4,5]). There are two types of stabilizers one can put on edges, yielding open (a.k.a. rough or primal) and closed (a.k.a. smooth or dual) boundaries. A mixed boundary consists of an interleaving of the two stabilizer types [6].

## Protection

The original planar code on a square-lattice patch with different boundary conditions on the vertical and horizontal edges is a \([[L^2+(L-1)^2,1,L]]\) CSS code [5]. Code size \(k = 2^{2g}\) for a torus of genus \(g\), and such higher genus surfaces have been investigated [7].

### Topological order and gauge-theory analogy

When treated as ground states of the code Hamiltonian, the code states realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [8]. This order does not persist at nonzero temperature [9,10].

Pauli noise operators can be organized into anyonic strings of the gauge theory, which cause excitations of the ground-state subspace. The inability of local errors to distinguish the codewords translates to the "topologically protected" degeneracy of the ground state, rigorously formulated by the TQO-1 condition. The joint \(+1\)-eigenspace of the \(Z\)-type Paulis corresponds to the subspace that conserves \(\mathbb{Z}_2\) flux, while the joint \(+1\)-eigenspace of \(X\)-type operators corresponds to the subspace that preserves \(\mathbb{Z}_2\) gauge symmetry (a one-form symmetry). Logical Pauli operators correspond to non-contractible Wilson loops in the case of closed boundaries, and to paths connecting different types of boundaries in the case of open boundaries.

Behavior under Hamiltonian \(X\)-type and \(Z\)-type perturbations is related to an anisotropic 3D gauge Higgs model [11,12]. In order to corrupt logical states, any local noise must bring the code state out of the topological order [10].

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Threshold

## Realizations

## Notes

## Parents

- Homological code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
- Twist-defect surface code — Twist-defect surface codes reduce to surface codes when there are no defects.
- Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Lift-connected surface (LCS) code — LCS codes consist of sparsely interconnected stacks of surface codes.
- Modular-qudit surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- Galois-qudit surface code — The Galois-qudit surface code for \(q=2\) reduces to the surface code.

## Children

- Rotated surface code — The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [6,143][144; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [145,146]. The rotated surface code presents certain savings over the original surface code [147].
- Toric code — The toric code is the surface code on a 2D torus.
- 2D hyperbolic surface code

## Cousins

- Layer code — Layer codes are combinations of constant-rate QLDPC codes with surface codes built using lattice surgery.
- Long-range enhanced surface code (LRESC) — LRESCs reduce to planar surface codes when a trivial LDPC code is used in the hypergraph product.
- La-cross code — La-cross codes at \(k=1\) yield the toric (planar surface) code and periodic (open) boundary conditions.
- Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [148].
- Quantum-double code — A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the surface code with a quantum double model [40].
- Hamiltonian-based code — While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [149].
- Self-correcting quantum code — Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons so as to prevent them from spreading [85,87,150–152]
- Asymmetric quantum code — The surface code on a hexagonal lattice is an asymmetric CSS code [153].
- 2D lattice stabilizer code — Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [154–156]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [157].
- Dynamical automorphism (DA) code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
- Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of nine realizations of the \(\mathbb{Z}_2\) 2D surface code.
- X-cube Floquet code — The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes.
- Honeycomb Floquet code — Measurement of each check operator of the honeycomb Floquet code involves two qubits and projects the state of the two qubits to a two-dimensional subspace, which we regard as an effective qubit. These effective qubits form a surface code on a hexagonal superlattice. Electric and magnetic operators on the embedded surface code correspond to outer logical operators of the Floquet code. In fact, outer logical operators transition back and forth from magnetic to electric surface code operators under the measurement dynamics. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [53–55], with the scheme in Ref. [54] being a special case of DWR. Numerical comparisons have been performed [158].
- Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [159,160].
- Majorana surface code — The Majorana surface code is a Majorana stabilizer analogue of the surface code.
- Neural network code — Reinforcement learners can be used to optimize the geometry of the surface code to be more suited to a noise channel [161].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [127,162]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
- Bivariate bicycle (BB) code — Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators.
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes can be concatened with surface codes [163].
- 2D color code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [164–166]. This process can be viewed as an ungauging [167–169,169] of certain symmetries. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [170]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [171].
- Generalized five-squares code — Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [172,173].
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code. There are various ways to embed the surface code into the heavy-hex lattice [174].
- Kitaev honeycomb code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. This code can be obtained from the square-lattice surface code by gauging out the anyon \(em\) [175; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [175; Fig. 12].
- Three-fermion (3F) subsystem code — One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [175; Sec. 7.4].
- Subsystem surface code
- Fracton stabilizer code — Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [176; Eq. (D32)].

## References

- [1]
- A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997) DOI
- [2]
- A. Yu. Kitaev, “Quantum Error Correction with Imperfect Gates”, Quantum Communication, Computing, and Measurement 181 (1997) DOI
- [3]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [4]
- S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary”, (1998) arXiv:quant-ph/9811052
- [5]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
- [6]
- N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
- [7]
- G. Rajpoot, K. Kumari, and S. R. Jain, “Quantum error correction beyond the toric code: dynamical systems meet encoding”, The European Physical Journal Special Topics 233, 1341 (2023) arXiv:2307.04418 DOI
- [8]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI
- [9]
- M. B. Hastings, “Topological Order at Nonzero Temperature”, Physical Review Letters 107, (2011) arXiv:1106.6026 DOI
- [10]
- S. Sang, Y. Zou, and T. H. Hsieh, “Mixed-state Quantum Phases: Renormalization and Quantum Error Correction”, (2023) arXiv:2310.08639
- [11]
- I. S. Tupitsyn et al., “Topological multicritical point in the phase diagram of the toric code model and three-dimensional lattice gauge Higgs model”, Physical Review B 82, (2010) arXiv:0804.3175 DOI
- [12]
- F. J. Wegner, “Duality in generalized Ising models”, (2014) arXiv:1411.5815
- [13]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [14]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
- [15]
- P. Mazurek et al., “Long-distance quantum communication over noisy networks without long-time quantum memory”, Physical Review A 90, (2014) arXiv:1202.1016 DOI
- [16]
- R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation”, New Journal of Physics 9, 199 (2007) arXiv:quant-ph/0703143 DOI
- [17]
- B. J. Brown et al., “Generating topological order from a two-dimensional cluster state using a duality mapping”, New Journal of Physics 13, 065010 (2011) arXiv:1105.2111 DOI
- [18]
- O. Higgott et al., “Optimal local unitary encoding circuits for the surface code”, Quantum 5, 517 (2021) arXiv:2002.00362 DOI
- [19]
- Y.-J. Liu et al., “Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer”, PRX Quantum 3, (2022) arXiv:2110.02020 DOI
- [20]
- R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
- [21]
- J. Joo et al., “Generating and verifying graph states for fault-tolerant topological measurement-based quantum computing in two-dimensional optical lattices”, Physical Review A 88, (2013) arXiv:1207.0253 DOI
- [22]
- S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order”, Physical Review Letters 97, (2006) arXiv:quant-ph/0603121 DOI
- [23]
- J. Eisert and T. J. Osborne, “General Entanglement Scaling Laws from Time Evolution”, Physical Review Letters 97, (2006) arXiv:quant-ph/0603114 DOI
- [24]
- D. Aharonov and Y. Touati, “Quantum Circuit Depth Lower Bounds For Homological Codes”, (2018) arXiv:1810.03912
- [25]
- J. Łodyga et al., “Simple scheme for encoding and decoding a qubit in unknown state for various topological codes”, Scientific Reports 5, (2015) arXiv:1404.2495 DOI
- [26]
- A. Tikku and I. H. Kim, “Circuit depth versus energy in topologically ordered systems”, (2022) arXiv:2210.06796
- [27]
- S. Bravyi et al., “Quantum advantage with noisy shallow circuits”, Nature Physics 16, 1040 (2020) arXiv:1904.01502 DOI
- [28]
- A. Wu et al., “Mapping Surface Code to Superconducting Quantum Processors”, (2021) arXiv:2111.13729
- [29]
- A. Siegel et al., “Adaptive surface code for quantum error correction in the presence of temporary or permanent defects”, Quantum 7, 1065 (2023) arXiv:2211.08468 DOI
- [30]
- K. Yin et al., “Surf-Deformer: Mitigating Dynamic Defects on Surface Code via Adaptive Deformation”, (2024) arXiv:2405.06941
- [31]
- J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016) arXiv:1603.02286 DOI
- [32]
- D. Horsman et al., “Surface code quantum computing by lattice surgery”, New Journal of Physics 14, 123011 (2012) arXiv:1111.4022 DOI
- [33]
- D. Litinski and F. von Oppen, “Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes”, Quantum 2, 62 (2018) arXiv:1709.02318 DOI
- [34]
- D. Litinski, “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery”, Quantum 3, 128 (2019) arXiv:1808.02892 DOI
- [35]
- C. Chamberland and E. T. Campbell, “Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery”, PRX Quantum 3, (2022) arXiv:2109.02746 DOI
- [36]
- D. Litinski, “Magic State Distillation: Not as Costly as You Think”, Quantum 3, 205 (2019) arXiv:1905.06903 DOI
- [37]
- N. P. Breuckmann et al., “Hyperbolic and semi-hyperbolic surface codes for quantum storage”, Quantum Science and Technology 2, 035007 (2017) arXiv:1703.00590 DOI
- [38]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020) arXiv:1806.06078 DOI
- [39]
- B. J. Brown, “A fault-tolerant non-Clifford gate for the surface code in two dimensions”, Science Advances 6, (2020) arXiv:1903.11634 DOI
- [40]
- K. Laubscher, D. Loss, and J. R. Wootton, “Universal quantum computation in the surface code using non-Abelian islands”, Physical Review A 100, (2019) arXiv:1811.06738 DOI
- [41]
- B. Coecke and R. Duncan, “Interacting Quantum Observables”, Automata, Languages and Programming 298 DOI
- [42]
- B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
- [43]
- N. de Beaudrap and D. Horsman, “The ZX calculus is a language for surface code lattice surgery”, Quantum 4, 218 (2020) arXiv:1704.08670 DOI
- [44]
- C. Gidney and A. G. Fowler, “Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation”, Quantum 3, 135 (2019) arXiv:1812.01238 DOI
- [45]
- C. Gidney and A. G. Fowler, “Flexible layout of surface code computations using AutoCCZ states”, (2019) arXiv:1905.08916
- [46]
- J. Gavriel et al., “Transversal Injection: A method for direct encoding of ancilla states for non-Clifford gates using stabiliser codes”, (2022) arXiv:2211.10046
- [47]
- C. Cesare et al., “Adiabatic topological quantum computing”, Physical Review A 92, (2015) arXiv:1406.2690 DOI
- [48]
- Y.-C. Zheng and T. A. Brun, “Fault-tolerant holonomic quantum computation in surface codes”, Physical Review A 91, (2015) arXiv:1411.4248 DOI
- [49]
- P. Zanardi and M. Rasetti, “Holonomic quantum computation”, Physics Letters A 264, 94 (1999) arXiv:quant-ph/9904011 DOI
- [50]
- D. Gottesman and L. L. Zhang, “Fibre bundle framework for unitary quantum fault tolerance”, (2017) arXiv:1309.7062
- [51]
- A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
- [52]
- M. McEwen, D. Bacon, and C. Gidney, “Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics”, Quantum 7, 1172 (2023) arXiv:2302.02192 DOI
- [53]
- R. Chao et al., “Optimization of the surface code design for Majorana-based qubits”, Quantum 4, 352 (2020) arXiv:2007.00307 DOI
- [54]
- C. Gidney, “A Pair Measurement Surface Code on Pentagons”, Quantum 7, 1156 (2023) arXiv:2206.12780 DOI
- [55]
- L. Grans-Samuelsson et al., “Improved Pairwise Measurement-Based Surface Code”, Quantum 8, 1429 (2024) arXiv:2310.12981 DOI
- [56]
- Andrew Landahl, private communication, 2023
- [57]
- A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
- [58]
- J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
- [59]
- J. Edmonds, “Maximum matching and a polyhedron with 0,1-vertices”, Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics 69B, 125 (1965) DOI
- [60]
- F. Barahona et al., “Morphology of ground states of two-dimensional frustration model”, Journal of Physics A: Mathematical and General 15, 673 (1982) DOI
- [61]
- A. Fischer and A. Miyake, “Hardness results for decoding the surface code with Pauli noise”, (2024) arXiv:2309.10331
- [62]
- S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014) arXiv:1405.4883 DOI
- [63]
- N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021) arXiv:1709.06218 DOI
- [64]
- B. A. Galler and M. J. Fisher, “An improved equivalence algorithm”, Communications of the ACM 7, 301 (1964) DOI
- [65]
- J. E. Hopcroft and J. D. Ullman, “Set Merging Algorithms”, SIAM Journal on Computing 2, 294 (1973) DOI
- [66]
- R. E. Tarjan and J. van Leeuwen, “Worst-case Analysis of Set Union Algorithms”, Journal of the ACM 31, 245 (1984) DOI
- [67]
- C. A. Pattison et al., “Improved quantum error correction using soft information”, (2021) arXiv:2107.13589
- [68]
- O. Higgott et al., “Improved decoding of circuit noise and fragile boundaries of tailored surface codes”, (2023) arXiv:2203.04948
- [69]
- T. Chan and S. C. Benjamin, “Actis: A Strictly Local Union–Find Decoder”, Quantum 7, 1183 (2023) arXiv:2305.18534 DOI
- [70]
- A. G. Fowler, “Optimal complexity correction of correlated errors in the surface code”, (2013) arXiv:1310.0863
- [71]
- A. Paler and A. G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”, Quantum 7, 1205 (2023) arXiv:2205.09828 DOI
- [72]
- X. Xu et al., “High-Threshold Code for Modular Hardware With Asymmetric Noise”, Physical Review Applied 12, (2019) arXiv:1812.01505 DOI
- [73]
- Y. Wu and L. Zhong, “Fusion Blossom: Fast MWPM Decoders for QEC”, (2023) arXiv:2305.08307
- [74]
- D. Forlivesi, L. Valentini, and M. Chiani, “Spanning Tree Matching Decoder for Quantum Surface Codes”, IEEE Communications Letters 28, 1509 (2024) arXiv:2405.01151 DOI
- [75]
- K. Sahay et al., “Error correction of transversal CNOT gates for scalable surface code computation”, (2024) arXiv:2408.01393
- [76]
- N. Shutty, M. Newman, and B. Villalonga, “Efficient near-optimal decoding of the surface code through ensembling”, (2024) arXiv:2401.12434
- [77]
- C. Jones, “Improved accuracy for decoding surface codes with matching synthesis”, (2024) arXiv:2408.12135
- [78]
- G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0911.0581 DOI
- [79]
- G. Duclos-Cianci and D. Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”, (2013) arXiv:1304.6100
- [80]
- F. H. E. Watson, H. Anwar, and D. E. Browne, “Fast fault-tolerant decoder for qubit and qudit surface codes”, Physical Review A 92, (2015) arXiv:1411.3028 DOI
- [81]
- J. Wootton, “A Simple Decoder for Topological Codes”, Entropy 17, 1946 (2015) arXiv:1310.2393 DOI
- [82]
- N. Delfosse and G. Zémor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel”, Physical Review Research 2, (2020) arXiv:1703.01517 DOI
- [83]
- A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014) arXiv:1302.2669 DOI
- [84]
- J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
- [85]
- M. Herold et al., “Cellular-automaton decoders for topological quantum memories”, npj Quantum Information 1, (2015) arXiv:1406.2338 DOI
- [86]
- M. Herold et al., “Cellular automaton decoders of topological quantum memories in the fault tolerant setting”, New Journal of Physics 19, 063012 (2017) arXiv:1511.05579 DOI
- [87]
- C.-E. Bardyn and T. Karzig, “Exponential lifetime improvement in topological quantum memories”, Physical Review B 94, (2016) arXiv:1512.04528 DOI
- [88]
- G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017) arXiv:1610.04238 DOI
- [89]
- C. Chamberland and P. Ronagh, “Deep neural decoders for near term fault-tolerant experiments”, Quantum Science and Technology 3, 044002 (2018) arXiv:1802.06441 DOI
- [90]
- Y. Ueno et al., “NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes”, (2022) arXiv:2208.05758
- [91]
- R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020) arXiv:1810.07207 DOI
- [92]
- E. S. Matekole et al., “Decoding surface codes with deep reinforcement learning and probabilistic policy reuse”, (2022) arXiv:2212.11890
- [93]
- K. Meinerz, C.-Y. Park, and S. Trebst, “Scalable Neural Decoder for Topological Surface Codes”, Physical Review Letters 128, (2022) arXiv:2101.07285 DOI
- [94]
- J. Bausch et al., “Learning to Decode the Surface Code with a Recurrent, Transformer-Based Neural Network”, (2023) arXiv:2310.05900
- [95]
- H. Wang et al., “Transformer-QEC: Quantum Error Correction Code Decoding with Transferable Transformers”, (2023) arXiv:2311.16082
- [96]
- P. Das, A. Locharla, and C. Jones, “LILLIPUT: A Lightweight Low-Latency Lookup-Table Based Decoder for Near-term Quantum Error Correction”, (2021) arXiv:2108.06569
- [97]
- N. Delfosse, “Hierarchical decoding to reduce hardware requirements for quantum computing”, (2020) arXiv:2001.11427
- [98]
- S. C. Smith, B. J. Brown, and S. D. Bartlett, “Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction”, Physical Review Applied 19, (2023) arXiv:2208.04660 DOI
- [99]
- G. S. Ravi et al., “Better Than Worst-Case Decoding for Quantum Error Correction”, (2022) arXiv:2208.08547
- [100]
- X. Tan et al., “Scalable surface code decoders with parallelization in time”, (2022) arXiv:2209.09219
- [101]
- L. Skoric et al., “Parallel window decoding enables scalable fault tolerant quantum computation”, Nature Communications 14, (2023) arXiv:2209.08552 DOI
- [102]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [103]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [104]
- K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) (2022) arXiv:2202.06612 DOI
- [105]
- K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
- [106]
- A. Kaufmann and I. Arad, “A blockBP decoder for the surface code”, (2024) arXiv:2402.04834
- [107]
- J. Chen et al., “Improved Belief Propagation Decoding Algorithms for Surface Codes”, (2024) arXiv:2407.11523
- [108]
- H. D. Pfister et al., “Belief Propagation for Classical and Quantum Systems: Overview and Recent Results”, IEEE BITS the Information Theory Magazine 2, 20 (2022) DOI
- [109]
- J. du Crest, M. Mhalla, and V. Savin, “A blindness property of the Min-Sum decoding for the toric code”, (2024) arXiv:2406.14968
- [110]
- A. Benhemou et al., “Minimising surface-code failures using a color-code decoder”, (2024) arXiv:2306.16476
- [111]
- M. Pacenti et al., “Progressive-Proximity Bit-Flipping for Decoding Surface Codes”, IEEE Transactions on Communications 1 (2024) arXiv:2402.15924 DOI
- [112]
- B. Barber et al., “A real-time, scalable, fast and highly resource efficient decoder for a quantum computer”, (2024) arXiv:2309.05558
- [113]
- T. R. Scruby et al., “Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code”, Quantum 6, 721 (2022) arXiv:2012.08536 DOI
- [114]
- C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
- [115]
- S. Huang, T. Jochym-O’Connor, and T. J. Yoder, “Homomorphic Logical Measurements”, (2022) arXiv:2211.03625
- [116]
- D. Litinski and N. Nickerson, “Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections”, (2022) arXiv:2211.15465
- [117]
- A. G. Fowler and S. J. Devitt, “A bridge to lower overhead quantum computation”, (2013) arXiv:1209.0510
- [118]
- H. Bombín et al., “Fault-Tolerant Postselection for Low-Overhead Magic State Preparation”, PRX Quantum 5, (2024) arXiv:2212.00813 DOI
- [119]
- T. Itogawa et al., “Even more efficient magic state distillation by zero-level distillation”, (2024) arXiv:2403.03991
- [120]
- C. Gidney, N. Shutty, and C. Jones, “Magic state cultivation: growing T states as cheap as CNOT gates”, (2024) arXiv:2409.17595
- [121]
- H. Bombin et al., “Unifying flavors of fault tolerance with the ZX calculus”, Quantum 8, 1379 (2024) arXiv:2303.08829 DOI
- [122]
- S. H. Choe and R. Koenig, “Long-range data transmission in a fault-tolerant quantum bus architecture”, (2022) arXiv:2209.09774
- [123]
- D. S. Wang et al., “Threshold error rates for the toric and surface codes”, (2009) arXiv:0905.0531
- [124]
- B. Heim, K. M. Svore, and M. B. Hastings, “Optimal Circuit-Level Decoding for Surface Codes”, (2016) arXiv:1609.06373
- [125]
- A. G. Fowler, “Proof of Finite Surface Code Threshold for Matching”, Physical Review Letters 109, (2012) arXiv:1206.0800 DOI
- [126]
- C. Wang, J. Harrington, and J. Preskill, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory”, Annals of Physics 303, 31 (2003) arXiv:quant-ph/0207088 DOI
- [127]
- R. Raussendorf and J. Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610082 DOI
- [128]
- A. G. Fowler, A. M. Stephens, and P. Groszkowski, “High-threshold universal quantum computation on the surface code”, Physical Review A 80, (2009) arXiv:0803.0272 DOI
- [129]
- M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009) arXiv:0811.0464 DOI
- [130]
- D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, “Surface code quantum computing with error rates over 1%”, Physical Review A 83, (2011) arXiv:1009.3686 DOI
- [131]
- A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014) arXiv:1311.5003 DOI
- [132]
- B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
- [133]
- T. Ohno et al., “Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory”, Nuclear Physics B 697, 462 (2004) arXiv:quant-ph/0401101 DOI
- [134]
- D. Pataki et al., “Coherent errors in stabilizer codes caused by quasistatic phase damping”, Physical Review A 110, (2024) arXiv:2401.04530 DOI
- [135]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [136]
- I. Hesner, B. Hetényi, and J. R. Wootton, “Using Detector Likelihood for Benchmarking Quantum Error Correction”, (2024) arXiv:2408.02082
- [137]
- K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021) arXiv:2104.01180 DOI
- [138]
- G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021) arXiv:2104.04119 DOI
- [139]
- A. Cleland, “An introduction to the surface code”, SciPost Physics Lecture Notes (2022) DOI
- [140]
- K. Fujii, “Quantum Computation with Topological Codes: from qubit to topological fault-tolerance”, (2015) arXiv:1504.01444
- [141]
- A. deMarti iOlius et al., “Decoding algorithms for surface codes”, Quantum 8, 1498 (2024) arXiv:2307.14989 DOI
- [142]
- A. Goswami, M. Mhalla, and V. Savin, “Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit”, (2023) arXiv:2209.06673
- [143]
- H. Bombin and M. A. Martin-Delgado, “Optimal resources for topological two-dimensional stabilizer codes: Comparative study”, Physical Review A 76, (2007) arXiv:quant-ph/0703272 DOI
- [144]
- R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
- [145]
- Nikolas P. Breuckmann, private communication, 2022
- [146]
- Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
- [147]
- A. R. O’Rourke and S. Devitt, “Compare the Pair: Rotated vs. Unrotated Surface Codes at Equal Logical Error Rates”, (2024) arXiv:2409.14765
- [148]
- S. Bravyi et al., “Correcting coherent errors with surface codes”, npj Quantum Information 4, (2018) arXiv:1710.02270 DOI
- [149]
- C. Fernández-González et al., “Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism”, Physical Review Letters 109, (2012) arXiv:1111.5817 DOI
- [150]
- A. Hamma, C. Castelnovo, and C. Chamon, “Toric-boson model: Toward a topological quantum memory at finite temperature”, Physical Review B 79, (2009) arXiv:0812.4622 DOI
- [151]
- S. Chesi, B. Röthlisberger, and D. Loss, “Self-correcting quantum memory in a thermal environment”, Physical Review A 82, (2010) arXiv:0908.4264 DOI
- [152]
- C. Stark et al., “Localization of Toric Code Defects”, Physical Review Letters 107, (2011) arXiv:1101.6028 DOI
- [153]
- C. D. de Albuquerque et al., “Euclidean and Hyperbolic Asymmetric Topological Quantum Codes”, (2021) arXiv:2105.01144
- [154]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
- [155]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [156]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [157]
- Z. Liang et al., “Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions”, PRX Quantum 5, (2024) arXiv:2312.11170 DOI
- [158]
- P. Hilaire et al., “Enhanced Fault-tolerance in Photonic Quantum Computing: Floquet Code Outperforms Surface Code in Tailored Architecture”, (2024) arXiv:2410.07065
- [159]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [160]
- N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
- [161]
- H. P. Nautrup et al., “Optimizing Quantum Error Correction Codes with Reinforcement Learning”, Quantum 3, 215 (2019) arXiv:1812.08451 DOI
- [162]
- R. Raussendorf, J. Harrington, and K. Goyal, “A fault-tolerant one-way quantum computer”, Annals of Physics 321, 2242 (2006) arXiv:quant-ph/0510135 DOI
- [163]
- M. Fang and D. Su, “Quantum memory based on concatenating surface codes and quantum Hamming codes”, (2024) arXiv:2407.16176
- [164]
- B. Yoshida, “Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes”, Annals of Physics 326, 15 (2011) arXiv:1007.4601 DOI
- [165]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [166]
- A. B. Aloshious, A. N. Bhagoji, and P. K. Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”, (2018) arXiv:1804.00866
- [167]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [168]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [169]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [170]
- M. S. Kesselring et al., “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
- [171]
- J. T. Anderson, “Homological Stabilizer Codes”, (2011) arXiv:1107.3502
- [172]
- M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
- [173]
- V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
- [174]
- C. Benito et al., “Comparative study of quantum error correction strategies for the heavy-hexagonal lattice”, (2024) arXiv:2402.02185
- [175]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [176]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI

## Page edit log

- Victor V. Albert (2023-03-29) — most recent
- Marcus P da Silva (2023-03-20)
- Victor V. Albert (2022-09-20)
- Victor V. Albert (2022-06-15)
- Tony Lau (2022-04-02)
- Hassan Shapourian (2022-04-01)
- Victor V. Albert (2022-02-15)
- Philippe Faist (2022-02-11)
- Victor V. Albert (2021-11-05)
- Philippe Faist (2021-11-03)
- Michael Vasmer (2021-11-02)

## Cite as:

“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/surface