## 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). Toric code often either refers to the construction on the two-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the planar code [4–6]. Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

The stabilizers of the surface code on the 2-dimensional torus are generated by star operators \(A_v\) and plaquette operators \(B_p\). Each star operator is a product of four Pauli-\(X\) operators on the edges adjacent to a vertex \(v\) of the lattice; each plaquette operator is a product of four Pauli-\(Z\) operators applied to the edges adjacent to a face, or plaquette, \(p\) of the lattice (Figure I).

The two-dimensional toric code encodes two logical qubits. We denote by \(\overline{X}_i,\overline{Z}_i\) the logical Pauli-\(X\) and Pauli-\(Z\) operator of the \(i\)-th logical qubit. They can are represented by strings of Pauli-\(X\) operators or Pauli-\(Z\) operators that wrap around the torus as shown in Figure I.

## Protection

Toric code on an \(L\times L\) torus is a \([[2L^2,2,L]]\) CSS code. 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 [4].

Coherent physical errors are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed via the Majorana mapping [7] as well as analytical bounds [8].

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Realizations

## Notes

## Parents

- Generalized surface code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
- Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Hypergraph product (HGP) code — Planar (toric) code can be obtained from hypergraph product of two repetition (cyclic) codes [101; Ex. 6].
- Lifted-product (LP) code — A lifted-product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code [102; Appx. B].
- Modular-qudit surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- Galois-qudit topological code — The surface code has been extended to Galois qudits.

## Children

- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is the smallest toric code.
- Rotated surface code — Rotated surface codes can be obtained using the same procedure as for the original surface codes but considering slightly different combinatorial surfaces [6,103] than those considered in the original proposal.

## Cousins

- Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [7].
- 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 [39].
- Translationally invariant stabilizer code — Translation-invariant 2D qubit topological stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [104–106].
- Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of three realizations of the \(\mathbb{Z}_2\) 2D surface code.
- 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 this code, stabilizer measurement circuits consisting of two-body measurements have been designed for the surface code [75,76].
- Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [107,108].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [109,110]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
- Color code — The 3D color code is equivalent to multiple decoupled copies of the 2D surface code [111–113]. 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 hexagonal directions to the stabilizer group and then taking the center of the resulting nonabelian group [114].
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
- 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 \(em\) [115; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [115; Fig. 12].
- Subsystem surface code
- 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 \(e_2m_2\) [115; Sec. 7.4].
- Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [116] that can be compared to the qubit surface code. There is a logical basis for both the toric and double-semion codes where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas some loops appear with a \(-1\) coefficient for the double semion.

## 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]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
- [5]
- S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary”, (1998) arXiv:quant-ph/9811052
- [6]
- N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
- [7]
- S. Bravyi et al., “Correcting coherent errors with surface codes”, npj Quantum Information 4, (2018) arXiv:1710.02270 DOI
- [8]
- J. K. Iverson and J. Preskill, “Coherence in logical quantum channels”, New Journal of Physics 22, 073066 (2020) arXiv:1912.04319 DOI
- [9]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [10]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
- [11]
- 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
- [12]
- 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
- [13]
- O. Higgott et al., “Optimal local unitary encoding circuits for the surface code”, Quantum 5, 517 (2021) arXiv:2002.00362 DOI
- [14]
- 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
- [15]
- R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
- [16]
- 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
- [17]
- 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
- [18]
- D. Aharonov and Y. Touati, “Quantum Circuit Depth Lower Bounds For Homological Codes”, (2018) arXiv:1810.03912
- [19]
- 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
- [20]
- R. König and F. Pastawski, “Generating topological order: No speedup by dissipation”, Physical Review B 90, (2014) arXiv:1310.1037 DOI
- [21]
- 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
- [22]
- A. Tikku and I. H. Kim, “Circuit depth versus energy in topologically ordered systems”, (2022) arXiv:2210.06796
- [23]
- S. Bravyi et al., “Quantum advantage with noisy shallow circuits”, Nature Physics 16, 1040 (2020) arXiv:1904.01502 DOI
- [24]
- J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016) arXiv:1603.02286 DOI
- [25]
- C. Horsman et al., “Surface code quantum computing by lattice surgery”, New Journal of Physics 14, 123011 (2012) arXiv:1111.4022 DOI
- [26]
- 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
- [27]
- D. Litinski, “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery”, Quantum 3, 128 (2019) arXiv:1808.02892 DOI
- [28]
- 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
- [29]
- C. Chamberland and E. T. Campbell, “Circuit-level protocol and analysis for twist-based lattice surgery”, Physical Review Research 4, (2022) arXiv:2201.05678 DOI
- [30]
- H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
- [31]
- A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls”, Communications in Mathematical Physics 313, 351 (2012) arXiv:1104.5047 DOI
- [32]
- A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
- [33]
- H. Zheng, A. Dua, and L. Jiang, “Demonstrating non-Abelian statistics of Majorana fermions using twist defects”, Physical Review B 92, (2015) arXiv:1508.04166 DOI
- [34]
- B. J. Brown et al., “Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code”, Physical Review X 7, (2017) arXiv:1609.04673 DOI
- [35]
- A. Benhemou, J. K. Pachos, and D. E. Browne, “Non-Abelian statistics with mixed-boundary punctures on the toric code”, Physical Review A 105, (2022) arXiv:2103.08381 DOI
- [36]
- D. Litinski, “Magic State Distillation: Not as Costly as You Think”, Quantum 3, 205 (2019) arXiv:1905.06903 DOI
- [37]
- 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
- [38]
- B. J. Brown, “A fault-tolerant non-Clifford gate for the surface code in two dimensions”, Science Advances 6, (2020) arXiv:1903.11634 DOI
- [39]
- 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
- [40]
- 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
- [41]
- A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
- [42]
- J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
- [43]
- 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
- [44]
- F. Barahona et al., “Morphology of ground states of two-dimensional frustration model”, Journal of Physics A: Mathematical and General 15, 673 (1982) DOI
- [45]
- A. G. Fowler, “Optimal complexity correction of correlated errors in the surface code”, (2013) arXiv:1310.0863
- [46]
- A. Paler and A. G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”, (2022) arXiv:2205.09828
- [47]
- Y. Wu and L. Zhong, “Fusion Blossom: Fast MWPM Decoders for QEC”, (2023) arXiv:2305.08307
- [48]
- C. A. Pattison et al., “Improved quantum error correction using soft information”, (2021) arXiv:2107.13589
- [49]
- O. Higgott et al., “Fragile boundaries of tailored surface codes and improved decoding of circuit-level noise”, (2022) arXiv:2203.04948
- [50]
- G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0911.0581 DOI
- [51]
- G. Duclos-Cianci and D. Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”, (2013) arXiv:1304.6100
- [52]
- 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
- [53]
- 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
- [54]
- J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
- [55]
- 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
- [56]
- G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017) arXiv:1610.04238 DOI
- [57]
- 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
- [58]
- R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020) arXiv:1810.07207 DOI
- [59]
- Y. Ueno et al., “NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes”, (2022) arXiv:2208.05758
- [60]
- E. S. Matekole et al., “Decoding surface codes with deep reinforcement learning and probabilistic policy reuse”, (2022) arXiv:2212.11890
- [61]
- N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021) arXiv:1709.06218 DOI
- [62]
- N. Delfosse, “Hierarchical decoding to reduce hardware requirements for quantum computing”, (2020) arXiv:2001.11427
- [63]
- 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
- [64]
- G. S. Ravi et al., “Better Than Worst-Case Decoding for Quantum Error Correction”, (2022) arXiv:2208.08547
- [65]
- X. Tan et al., “Scalable surface code decoders with parallelization in time”, (2022) arXiv:2209.09219
- [66]
- L. Skoric et al., “Parallel window decoding enables scalable fault tolerant quantum computation”, (2023) arXiv:2209.08552
- [67]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, (2022) arXiv:2212.03214
- [68]
- J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.
- [69]
- 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
- [70]
- S. Huang, T. Jochym-O’Connor, and T. J. Yoder, “Homomorphic Logical Measurements”, (2022) arXiv:2211.03625
- [71]
- D. Litinski and N. Nickerson, “Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections”, (2022) arXiv:2211.15465
- [72]
- H. Bombín et al., “Fault-tolerant Post-Selection for Low Overhead Magic State Preparation”, (2022) arXiv:2212.00813
- [73]
- H. Bombin et al., “Unifying flavors of fault tolerance with the ZX calculus”, (2023) arXiv:2303.08829
- [74]
- S. H. Choe and R. Koenig, “Long-range data transmission in a fault-tolerant quantum bus architecture”, (2022) arXiv:2209.09774
- [75]
- R. Chao et al., “Optimization of the surface code design for Majorana-based qubits”, Quantum 4, 352 (2020) arXiv:2007.00307 DOI
- [76]
- C. Gidney, “A Pair Measurement Surface Code on Pentagons”, (2022) arXiv:2206.12780
- [77]
- M. McEwen, D. Bacon, and C. Gidney, “Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics”, (2023) arXiv:2302.02192
- [78]
- 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
- [79]
- J. Roffe et al., “Decoding across the quantum low-density parity-check code landscape”, Physical Review Research 2, (2020) arXiv:2005.07016 DOI
- [80]
- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
- [81]
- R. Fan et al., “Diagnostics of mixed-state topological order and breakdown of quantum memory”, (2023) arXiv:2301.05689
- [82]
- A. Honecker, M. Picco, and P. Pujol, “Universality Class of the Nishimori Point in the 2D±JRandom-Bond Ising Model”, Physical Review Letters 87, (2001) arXiv:cond-mat/0010143 DOI
- [83]
- F. Merz and J. T. Chalker, “Two-dimensional random-bond Ising model, free fermions, and the network model”, Physical Review B 65, (2002) arXiv:cond-mat/0106023 DOI
- [84]
- A. deMarti iOlius et al., “Performance enhancement of surface codes via recursive MWPM decoding”, (2023) arXiv:2212.11632
- [85]
- D. S. Wang et al., “Threshold error rates for the toric and surface codes”, (2009) arXiv:0905.0531
- [86]
- H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012) arXiv:1202.1852 DOI
- [87]
- D. Gottesman, “Keeping One Step Ahead of Errors”, Physics 5, (2012) DOI
- [88]
- T. M. Stace, S. D. Barrett, and A. C. Doherty, “Thresholds for Topological Codes in the Presence of Loss”, Physical Review Letters 102, (2009) arXiv:0904.3556 DOI
- [89]
- N. Nickerson and H. Bombín, “Measurement based fault tolerance beyond foliation”, (2018) arXiv:1810.09621
- [90]
- B. Heim, K. M. Svore, and M. B. Hastings, “Optimal Circuit-Level Decoding for Surface Codes”, (2016) arXiv:1609.06373
- [91]
- A. G. Fowler, “Proof of Finite Surface Code Threshold for Matching”, Physical Review Letters 109, (2012) arXiv:1206.0800 DOI
- [92]
- 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
- [93]
- M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009) arXiv:0811.0464 DOI
- [94]
- A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014) arXiv:1311.5003 DOI
- [95]
- D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
- [96]
- K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021) arXiv:2104.01180 DOI
- [97]
- G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021) arXiv:2104.04119 DOI
- [98]
- S. Xu et al., “Digital simulation of projective non-Abelian anyons with 68 superconducting qubits”, Chinese Physics Letters (2023) arXiv:2211.09802 DOI
- [99]
- A. Goswami, M. Mhalla, and V. Savin, “Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit”, (2023) arXiv:2209.06673
- [100]
- A. Wu et al., “Mapping Surface Code to Superconducting Quantum Processors”, (2021) arXiv:2111.13729
- [101]
- A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0928 DOI
- [102]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [103]
- 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
- [104]
- 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
- [105]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [106]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [107]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [108]
- N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
- [109]
- 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
- [110]
- 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
- [111]
- 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
- [112]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [113]
- 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
- [114]
- M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
- [115]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [116]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 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