## 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].

The original construction can be naturally extended to arbitrary \(D\)-dimensional manifolds [5][6]. Given a cellulation, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces. Such extensions are often called the \(D\)-dimensional surface or \(D\)-dimensional toric codes.

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

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Code Capacity Threshold

## Threshold

## Realizations

## Notes

## Parents

- Calderbank-Shor-Steane (CSS) stabilizer code
- Clifford-deformed surface code (CDSC) — CDSC is generally a non-CSS derivative of the surface code.
- Abelian topological code — 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 [58].

## Children

- Fractal surface code
- Higher-dimensional surface code
- Hyperbolic surface code
- Projective-plane surface code
- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is the smallest toric code.

## Cousins

- Hypergraph product code — Planar (toric) code obtained from hypergraph product of two repetition (cyclic) codes.
- Color code — Color code is equivalent to surface code in several ways [59][60]. For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Double-semion code — There is a logical basis for 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.
- Haah cubic code — The energy of any partial implementation of code 1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
- Honeycomb code — Measurement of each check operator 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.
- 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.
- Modular-qudit surface code — The qudit surface code with \(q=2\) is the surface code.
- Raussendorf-Bravyi-Harrington (RBH) code — Without symmetry protection, one of 2D boundaries of the cubic RBH code is effectively a 2D toric code.
- 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 [61][62][63].

## Zoo code information

## References

- [1]
- A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997). DOI
- [2]
- A. Y. Kitaev, “Quantum Error Correction with Imperfect Gates”, Quantum Communication, Computing, and Measurement 181 (1997). DOI
- [3]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [4]
- S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary”. quant-ph/9811052
- [5]
- “Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002). DOI
- [6]
- G. Zémor, “On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction”, Lecture Notes in Computer Science 259 (2009). DOI
- [7]
- C. Horsman et al., “Surface code quantum computing by lattice surgery”, New Journal of Physics 14, 123011 (2012). DOI; 1111.4022
- [8]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
- [9]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013). DOI; 1301.6588
- [10]
- S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010). DOI; 0909.5200
- [11]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
- [12]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008). DOI; 0712.0348
- [13]
- O. Higgott et al., “Optimal local unitary encoding circuits for the surface code”, Quantum 5, 517 (2021). DOI; 2002.00362
- [14]
- Yu-Jie Liu et al., “Methods for simulating string-net states and anyons on a digital quantum computer”. 2110.02020
- [15]
- J. Dengis, R. König, and F. Pastawski, “An optimal dissipative encoder for the toric code”, New Journal of Physics 16, 013023 (2014). DOI; 1310.1036
- [16]
- J. Łodyga et al., “Simple scheme for encoding and decoding a qubit in unknown state for various topological codes”, Scientific Reports 5, (2015). DOI; 1404.2495
- [17]
- J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016). DOI; 1603.02286
- [18]
- D. Litinski and F. von . Oppen, “Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes”, Quantum 2, 62 (2018). DOI; 1709.02318
- [19]
- D. Litinski, “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery”, Quantum 3, 128 (2019). DOI; 1808.02892
- [20]
- C. Chamberland and E. T. Campbell, “Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery”, PRX Quantum 3, (2022). DOI; 2109.02746
- [21]
- Christopher Chamberland and Earl T. Campbell, “A circuit-level protocol and analysis for twist-based lattice surgery”. 2201.05678
- [22]
- R. Raussendorf and J. Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions”, Physical Review Letters 98, (2007). DOI; quant-ph/0610082
- [23]
- R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation”, New Journal of Physics 9, 199 (2007). DOI; quant-ph/0703143
- [24]
- A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012). DOI; 1208.0928
- [25]
- B. J. Brown et al., “Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code”, Physical Review X 7, (2017). DOI; 1609.04673
- [26]
- D. Litinski, “Magic State Distillation: Not as Costly as You Think”, Quantum 3, 205 (2019). DOI; 1905.06903
- [27]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
- [28]
- B. J. Brown, “A fault-tolerant non-Clifford gate for the surface code in two dimensions”, Science Advances 6, (2020). DOI; 1903.11634
- [29]
- Austin G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time”. 1307.1740
- [30]
- Austin G. Fowler, “Optimal complexity correction of correlated errors in the surface code”. 1310.0863
- [31]
- Alexandru Paler and Austin G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”. 2205.09828
- [32]
- N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021). DOI; 1709.06218
- [33]
- Guillaume Duclos-Cianci and David Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”. 1304.6100
- [34]
- 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). DOI; 1411.3028
- [35]
- S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014). DOI; 1405.4883
- [36]
- A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014). DOI; 1302.2669
- [37]
- M. Herold et al., “Cellular automaton decoders of topological quantum memories in the fault tolerant setting”, New Journal of Physics 19, 063012 (2017). DOI; 1511.05579
- [38]
- G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017). DOI; 1610.04238
- [39]
- C. Chamberland and P. Ronagh, “Deep neural decoders for near term fault-tolerant experiments”, Quantum Science and Technology 3, 044002 (2018). DOI; 1802.06441
- [40]
- R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020). DOI; 1810.07207
- [41]
- F. Merz and J. T. Chalker, “Two-dimensional random-bond Ising model, free fermions, and the network model”, Physical Review B 65, (2002). DOI; cond-mat/0106023
- [42]
- 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). DOI; quant-ph/0207088
- [43]
- H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012). DOI; 1202.1852
- [44]
- T. M. Stace, S. D. Barrett, and A. C. Doherty, “Thresholds for Topological Codes in the Presence of Loss”, Physical Review Letters 102, (2009). DOI; 0904.3556
- [45]
- 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). DOI; quant-ph/0401101
- [46]
- A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014). DOI; 1311.5003
- [47]
- M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009). DOI; 0811.0464
- [48]
- C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020). DOI; 1912.09410
- [49]
- A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021). DOI; 2006.03071
- [50]
- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021). DOI; 2102.13071
- [51]
- Zijun Chen et al., “Exponential suppression of bit or phase flip errors with repetitive error correction”. 2102.06132
- [52]
- Dolev Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”. 2112.03923
- [53]
- Sebastian Krinner et al., “Realizing Repeated Quantum Error Correction in a Distance-Three Surface Code”. 2112.03708
- [54]
- Youwei Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”. 2112.13505
- [55]
- K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021). DOI; 2104.01180
- [56]
- G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021). DOI; 2104.04119
- [57]
- Anbang Wu et al., “Mapping Surface Code to Superconducting Quantum Processors”. 2111.13729
- [58]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971). DOI
- [59]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015). DOI; 1503.02065
- [60]
- Arun B. Aloshious, Arjun Nitin Bhagoji, and Pradeep Kiran Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”. 1804.00866
- [61]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012). DOI; 1103.4606
- [62]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014). DOI; 1107.2707
- [63]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017). DOI; 1607.01387

## Cite as:

“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/surface