Kitaev surface code[1][2][3]


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).

Figure I: Stabilizer generators and logical operators of the 2D surface code on a torus. The star operators \(A_v\) and the plaquette operators \(B_p\) generate the stabilizer group of the toric code. The logical operators are strings that wrap around the torus.

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.


Toric code on an \(L\times L\) torus is a \([[2L^2,2,L]]\) CSS code, and there exists a planar code with \([[L^2,1,L]]\) [7]. More generally, the code distance is related to the homology of the cellulation [8].


Rate depends on the underlying cellulation and manifold. For general 2D manifolds, \(kd^2\leq c(\log k)^2 n\) for some constant \(c\) [9], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [10][11], and (2) surface codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\). Higher-dimensional hyperbolic manifolds (see code children below) yield distances scaling more favorably. Loewner's theorem provides an upper bound for any bounded-geometry surface code [5].


Unitary [12][13][14].Dissipative [15].Stabilizer measurement-based [16].

Transversal Gates

Transversal Pauli gates exist and are based on non-trivial loops on surface. Transversal Clifford gates can be done on folded surface codes [17].


Clifford gates can be implemented via lattice surgery [7][18][19][20], twist-based lattice surgery [21], or braiding defects [22][23][24][25]. Non-Clifford gates require magic state distillation [26], Dehn twists [27], or just-in-time decoding [28].


Minimum weight perfect-matching (MWPM) [8][29] and pipeline MWPM [30][31], a modification accounting for correlations between events.Union-find [32].Renormalization group [33][34].Tensor network [35].Markov-chain Monte Carlo [36].Cellular automaton [37].Machine learning [38][39][40].

Code Capacity Threshold

For correlated Pauli noise, bounds on code capacity thresholds can be obtained by mapping the effect of noise on the code to a statistical mechanical model. The first such threshold, based on the planar code, is \(0.017\%\) [8].\(10.9\%\) (\(10.31\%\)) with pure \(Z\)-dephasing noise for square tiling using tensor-network [41][35] (minimum-weight perfect matching [42]) decoder. \(18.9\%\) with depolarizing noise for square tiling [43].\(50\%\) with erasure errors for square tiling [44].\(3.3\%\) with phenomenological noise for square tiling [45].


\(0.5-1.1\%\) for various error models [46][47].


Distance-two surface codes have been implemented by Andersen et al. [48], Erhard et al. [49], Marques et al. [50], Google Quantum AI [51], and both planar and toric versions by Bluvstein et al. [52]. Distance-three surface code implemented at ETH Zurich [53] and on the Zuchongzhi 2.1 superconducting quantum processor [54]. Signatures of corresponding topological phase of matter detected in superconducting circuits [55] and two-dimensional arrays of Rydberg atoms [56].


Surfmap framework provides a way to stitch the surface code to various superconducting-circuit geometries by assigning each superconducting qubit to be either a physical or ancilla qubit, designing stabilizer measurement circuits, and scheduling stabilizer measurements [57].2D and 3D surface code visualization tool. Tutorials from error-correction perspective by J. Haah and condensed-matter perspective by M. Levin and C. Nayak.




  • 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

Internal code ID: surface

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: surface

Cite as:
“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_surface, title={Kitaev surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Permanent link:


A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997). DOI
A. Y. Kitaev, “Quantum Error Correction with Imperfect Gates”, Quantum Communication, Computing, and Measurement 181 (1997). DOI
A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary”. quant-ph/9811052
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002). DOI
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
C. Horsman et al., “Surface code quantum computing by lattice surgery”, New Journal of Physics 14, 123011 (2012). DOI; 1111.4022
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
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
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010). DOI; 0909.5200
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008). DOI; 0712.0348
O. Higgott et al., “Optimal local unitary encoding circuits for the surface code”, Quantum 5, 517 (2021). DOI; 2002.00362
Yu-Jie Liu et al., “Methods for simulating string-net states and anyons on a digital quantum computer”. 2110.02020
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
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
J. E. Moussa, “Transversal Clifford gates on folded surface codes”, Physical Review A 94, (2016). DOI; 1603.02286
D. Litinski and F. von . Oppen, “Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes”, Quantum 2, 62 (2018). DOI; 1709.02318
D. Litinski, “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery”, Quantum 3, 128 (2019). DOI; 1808.02892
C. Chamberland and E. T. Campbell, “Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery”, PRX Quantum 3, (2022). DOI; 2109.02746
Christopher Chamberland and Earl T. Campbell, “A circuit-level protocol and analysis for twist-based lattice surgery”. 2201.05678
R. Raussendorf and J. Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions”, Physical Review Letters 98, (2007). DOI; quant-ph/0610082
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
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012). DOI; 1208.0928
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
D. Litinski, “Magic State Distillation: Not as Costly as You Think”, Quantum 3, 205 (2019). DOI; 1905.06903
G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
B. J. Brown, “A fault-tolerant non-Clifford gate for the surface code in two dimensions”, Science Advances 6, (2020). DOI; 1903.11634
Austin G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time”. 1307.1740
Austin G. Fowler, “Optimal complexity correction of correlated errors in the surface code”. 1310.0863
Alexandru Paler and Austin G. Fowler, “Pipelined correlated minimum weight perfect matching of the surface code”. 2205.09828
N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algorithm for topological codes”, Quantum 5, 595 (2021). DOI; 1709.06218
Guillaume Duclos-Cianci and David Poulin, “Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes”. 1304.6100
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
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
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
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
G. Torlai and R. G. Melko, “Neural Decoder for Topological Codes”, Physical Review Letters 119, (2017). DOI; 1610.04238
C. Chamberland and P. Ronagh, “Deep neural decoders for near term fault-tolerant experiments”, Quantum Science and Technology 3, 044002 (2018). DOI; 1802.06441
R. Sweke et al., “Reinforcement learning decoders for fault-tolerant quantum computation”, Machine Learning: Science and Technology 2, 025005 (2020). DOI; 1810.07207
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
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
H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012). DOI; 1202.1852
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
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
A. M. Stephens, “Fault-tolerant thresholds for quantum error correction with the surface code”, Physical Review A 89, (2014). DOI; 1311.5003
M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009). DOI; 0811.0464
C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020). DOI; 1912.09410
A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021). DOI; 2006.03071
J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021). DOI; 2102.13071
Zijun Chen et al., “Exponential suppression of bit or phase flip errors with repetitive error correction”. 2102.06132
Dolev Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”. 2112.03923
Sebastian Krinner et al., “Realizing Repeated Quantum Error Correction in a Distance-Three Surface Code”. 2112.03708
Youwei Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits”. 2112.13505
K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor”, Science 374, 1237 (2021). DOI; 2104.01180
G. Semeghini et al., “Probing topological spin liquids on a programmable quantum simulator”, Science 374, 1242 (2021). DOI; 2104.04119
Anbang Wu et al., “Mapping Surface Code to Superconducting Quantum Processors”. 2111.13729
F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971). DOI
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015). DOI; 1503.02065
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
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
H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014). DOI; 1107.2707
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.