Surface code

{{Short description|Topological quantum error correcting code}}{{More citations needed|date=March 2026}} The '''surface code''' is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice.<ref>A. Y. Kitaev, Proceedings of the 3rd International Conference of Quantum Communication and Measurement, Ed. O. Hirota, A. S. Holevo, and C. M. Caves (New York, Plenum, 1997)</ref> The first type of surface code introduced by Alexei Kitaev in 1997 was the '''toric code''', which gets its name from its periodic boundary conditions, giving it the shape of a torus. These conditions give the model translational invariance, which is useful for analytic study. The toric code is the simplest and most well studied of the quantum double models.<ref>{{cite journal | last=Kitaev | first=Alexei | title=Anyons in an exactly solved model and beyond | journal=Annals of Physics | volume=321 | issue=1 | year=2006 | issn=0003-4916 | doi=10.1016/j.aop.2005.10.005 | pages=2–111| arxiv=cond-mat/0506438 | bibcode=2006AnPhy.321....2K | s2cid=118948929 }}</ref> It is also the simplest example of topological order—''Z''2 topological order (first studied in the context of ''Z''2 spin liquid in 1991).<ref>{{cite journal|last1=Read |first1=N. |last2=Sachdev |first2=Subir |title=Large-Nexpansion for frustrated quantum antiferromagnets |journal=Physical Review Letters|volume=66 |issue=13 |date=1 March 1991 |issn=0031-9007 |doi=10.1103/physrevlett.66.1773 |pages=1773–1776 |pmid=10043303 |bibcode=1991PhRvL..66.1773R}}</ref><ref>{{cite journal | last=Wen | first=X. G. |author-link=Xiao-Gang Wen| title=Mean-field theory of spin-liquid states with finite energy gap and topological orders | journal=Physical Review B| volume=44 | issue=6 | date=1 July 1991 | issn=0163-1829 | doi=10.1103/physrevb.44.2664 | pages=2664–2672 | pmid=9999836 | bibcode=1991PhRvB..44.2664W}}</ref> The toric code can also be considered to be a ''Z''2 lattice gauge theory in a particular limit.<ref>{{cite journal | last1=Fradkin | first1=Eduardo | last2=Shenker | first2=Stephen H. | title=Phase diagrams of lattice gauge theories with Higgs fields | journal=Physical Review D| volume=19 | issue=12 | date=15 June 1979 | issn=0556-2821 | doi=10.1103/physrevd.19.3682 | pages=3682–3697 | bibcode=1979PhRvD..19.3682F}}</ref>

However, on many quantum computation platforms, experimental realization of a surface code is much easier if the code can be embedded on a 2D plane. This motivated the design of another type of surface code with open boundary conditions, the '''planar code'''.<ref>{{cite journal | last1=Ghosh | first1=Joydip | last2=Fowler | first2=Austin G. | last3=Geller | first3=Michael R. | title=Surface code with decoherence: An analysis of three superconducting architectures | journal=Physical Review A | volume=86 | issue=6 | date=19 December 2012 | issn=1050-2947 | doi=10.1103/physreva.86.062318 | article-number=062318| arxiv=1210.5799 | bibcode=2012PhRvA..86f2318G | s2cid=10196488 }}</ref> As of 2025, Google Quantum AI has implemented a distance-7 planar code on their newest generation of superconducting quantum processors, Willow processor, demonstrating a below-threshold physical error rate.<ref name="willow">{{cite journal | last1=((Google Quantum AI)) | author-link1 = Google Quantum AI | last2=((Collaborators)) | title=Quantum error correction below the surface code threshold | journal=Nature | publisher=Springer Science and Business Media LLC | volume=638 | issue=8052 | date=2024-12-09 | issn=0028-0836 | doi=10.1038/s41586-024-08449-y | pages=920–926}}</ref>

== Definition ==

The surface code is defined on a two-dimensional lattice, usually chosen to be the square lattice. Below, we will first illustrate the basic concept with the toric code, where the lattice has periodic boundary conditions, i.e., the top boundary is connected to the bottom and the left boundary to the right. Topologically, this is equivalent to defining the lattice on a torus.

thumb|250px|A section of the toric code. A vertex and plaquette are highlighted, along with the qubits used in the definition of their stabilizers.

A qubit is located on each ''edge'' of the lattice. For a {{math|''d'' × ''d''}} lattice, there are {{math|''d''2}} horizontal edges and {{math|''d''2}} vertical edges, thus {{math|2''d''2}} qubits in total. Stabilizer operators are defined on the qubits around each vertex {{mvar|v}} and plaquette (face) {{mvar|p}} of the lattice as follows:

Here <math> i \in v </math> denotes the edges touching the vertex {{mvar|v}}, and <math> i \in p </math> denotes the edges surrounding the plaquette <math> p </math>. The code space of the toric code is the subspace for which all stabilizers act trivially, hence for any state <math> | \psi \rangle </math> in this space it holds that

<math> A_v | \psi \rangle = | \psi \rangle, \,\, \forall v, \,\, B_p | \psi \rangle = | \psi \rangle, \,\, \forall p. </math>

For the toric code, this space is four-dimensional, and so can be used to store two qubits of quantum information. This can be proven by considering the number of independent stabilizer operators: For a {{math|''d'' × ''d''}} lattice, there are {{math|''d''2}} vertex stabilizers and {{math|''d''2}} plaquette stabilizers, but the product of all vertex stabilizers is {{mvar|I}} and so is the product of all plaquette stabilizers. Therefore there are {{math|2''d''2 − 2}} independent stabilizers, leaving 2 qubits worth of degrees of freedom.

The occurrence of errors will usually move the state out of the code space, resulting in vertices and plaquettes for which the above condition does not hold. Specifically, a Pauli {{mvar|Z}} error on qubit {{mvar|i}} flips the two vertex stabilizers {{mvar|Av}} such that <math> i \in v </math> (the endpoints of the edge {{mvar|i}}), and a Pauli {{mvar|X}} error on qubit {{mvar|i}} flips the two plaquette stabilizers {{mvar|Bp}} such that <math> i \in p </math> (the plaquettes on either side of the edge {{mvar|i}}). The positions of these violations is the syndrome of the code, which can be used for error correction.

The unique nature of topological codes such as the surface code is that stabilizer violations can be interpreted as quasiparticles. Specifically, if the code is in a state <math> | \phi \rangle </math> such that <math> A_v | \phi \rangle = - | \phi \rangle </math>, a quasiparticle known as an {{mvar|e}} anyon can be said to exist on the vertex {{mvar|v}}. Similarly a violation of some <math>B_p</math> is associated with an {{mvar|m}} anyon on the plaquette {{mvar|p}}. The code space, with no stabilizer violation, corresponds to the anyonic vacuum. The above-mentioned fact that the product of all vertex (resp. plaquette) stabilizers is {{mvar|I}} means that the number of {{mvar|e}} (resp. {{mvar|m}}) anyons on a toric code is always even.

thumb|250px|The topologically nontrivial loops of the torus. Moving anyons along these implement logical Pauli operators on the stored qubits.

A single-qubit {{mvar|Z}} error can be associated with an edge, and creates a pair of {{mvar|e}} anyons at both endpoints of that edge. However, two {{mvar|e}} anyons at the same location will annihilate each other, so a {{mvar|Z}} error can also effectively move an {{mvar|e}} anyon along an edge, allowing {{mvar|e}} anyons to be transported on the lattice. If the initial pair of anyons end up meeting each other and annihilating, then their paths form a loop. * If the loop is topologically trivial, then it can be written as a combination of plaquettes ({{mvar|Z}} stabilizer generators) on the lattice. Therefore the loop represents a {{mvar|Z}} stabilizer of the code, and has no effect on the stored information. The annihilation of the anyons, in this case, corrects all of the errors involved in their creation and transport. * However, if the loop is topologically non-trivial, then it represents a non-trivial logical operator. Although re-annihilation of the anyons returns the state to the code space, it also implements a logical operation on the stored information. The errors, in this case, are therefore not corrected but consolidated. ** On a torus, there are two independent topologically non-trivial loops: One looping horizontally on the toric code lattice, one looping vertically. These can be identified with the {{mvar|Z}} operators of the two logical qubits encoded in the toric code.

By considering the dual graph of the lattice, it can be seen that the above paragraph also applies to {{mvar|X}} errors and {{mvar|m}} anyons. Note that the horizontal {{mvar|Z}} operator and the vertical {{mvar|X}} operator belong to the same qubit, and vice versa. This ensures the correct commutation relations between logical operators.

=== Error correction ===

Consider the noise model for which bit and phase errors occur independently on each qubit, both with probability {{mvar|p}}. When {{mvar|p}} is low, this will create sparsely distributed pairs of anyons which have not moved far from their point of creation. Correction can be achieved by identifying the pairs that the anyons were created in (up to an equivalence class), and then re-annihilating them to remove the errors. As {{mvar|p}} increases, however, it becomes more ambiguous as to how the anyons may be paired without risking the formation of topologically non-trivial loops. This gives a threshold probability, under which the error correction will almost certainly succeed. Through a mapping to the random-bond Ising model, this critical probability has been found to be around 11%.<ref>{{cite journal | last1=Dennis | first1=Eric | last2=Kitaev | first2=Alexei | last3=Landahl | first3=Andrew | last4=Preskill | first4=John |author-link4=John Preskill | title=Topological quantum memory | journal=Journal of Mathematical Physics| volume=43 | issue=9 | year=2002 | issn=0022-2488 | doi=10.1063/1.1499754 | pages=4452–4505| arxiv=quant-ph/0110143 | bibcode=2002JMP....43.4452D | s2cid=36673677 }}</ref>

Other error models may also be considered, and thresholds found. In all cases studied so far, the code has been found to saturate the Hashing bound. For some error models, such as biased errors where bit errors occur more often than phase errors or vice versa, lattices other than the square lattice must be used to achieve the optimal thresholds.<ref>{{cite journal | last1=Röthlisberger | first1=Beat | last2=Wootton | first2=James R. | last3=Heath | first3=Robert M. | last4=Pachos | first4=Jiannis K. | last5=Loss | first5=Daniel | title=Incoherent dynamics in the toric code subject to disorder | journal=Physical Review A | volume=85 | issue=2 | date=13 February 2012 | issn=1050-2947 | doi=10.1103/physreva.85.022313 | article-number=022313| arxiv=1112.1613 | bibcode=2012PhRvA..85b2313R | s2cid=118585279 }}</ref><ref>{{cite journal | last1=Bombin | first1=H. | last2=Andrist | first2=Ruben S. | last3=Ohzeki | first3=Masayuki | last4=Katzgraber | first4=Helmut G. | last5=Martin-Delgado | first5=M. A. | title=Strong Resilience of Topological Codes to Depolarization | journal=Physical Review X | volume=2 | issue=2 | date=30 April 2012 | issn=2160-3308 | doi=10.1103/physrevx.2.021004 | article-number=021004| arxiv=1202.1852 | bibcode=2012PhRvX...2b1004B |doi-access=free}}</ref>

Erasure errors have also been studied using statistical-physics mappings; Masayuki Ohzeki estimated error thresholds for surface codes with lost qubits by analyzing related spin-glass models.<ref>{{cite journal |last=Ohzeki |first=Masayuki |date=June 2012 |title=Error threshold estimates for surface code with loss of qubits |journal=Physical Review A |volume=85 |issue=6 |article-number=060301 |doi=10.1103/PhysRevA.85.060301 |arxiv=1202.2593 |bibcode=2012PhRvA..85f0301O}}</ref>

These thresholds are upper limits and are useless unless efficient algorithms are found to achieve them. The most well-used algorithm is minimum weight perfect matching.<ref>{{cite journal | last=Edmonds | first=Jack | title=Paths, Trees, and Flowers | journal=Canadian Journal of Mathematics | volume=17 | year=1965 | issn=0008-414X | doi=10.4153/cjm-1965-045-4 | pages=449–467| s2cid=247198603 | doi-access=free }}</ref> When applied to the noise model with independent bit and flip errors, a threshold of around 10.5% is achieved. This falls only a little short of the 11% maximum. However, matching does not work so well when there are correlations between the bit and phase errors, such as with depolarizing noise.

== Open boundary conditions == When adapting the surface code to open boundary conditions, special boundary behaviors arise. As a motivating example, consider defining a surface code in the same way as above, but on an {{math|''n'' × ''n''}} square grid graph. Some vertices on the boundary will have degree 3 instead of 4 (and the corner vertices will have degree 2), so there will be some weight-3 (and weight-2) {{mvar|X}} stabilizers.

The most important characteristic of such an open boundary is that a Pauli {{mvar|X}} error no longer necessarily flips two plaquette stabilizers. An edge on the boundary only has one adjacent plaquette, and thus an {{mvar|X}} error on the corresponding qubit will only flip a single <math>B_p</math>, or in the language of anyons, only create or annihilate a single {{mvar|m}} anyon. One could say that this type of code boundary (known as a ''smooth boundary'') is a source and sink for {{mvar|m}} anyons.

In a surface code with open boundaries, in addition to true loops, one needs to consider paths that start and end at a boundary, which are usually specific to anyon types. In our {{math|''n'' × ''n''}} grid graph example, an {{mvar|m}} anyon can be created at any location on the boundary, move across the grid, and be annihilated at any other location on the boundary. However, since there is only one type of boundary, all these paths are topologically trivial. For example, if an {{mvar|m}} anyon is created somewhere in the middle of the top boundary, moves one step horizontally, then is annihilated again by the top boundary, then its path corresponds to one of the weight-3 {{mvar|X}} stabilizers mentioned above. Meanwhile, {{mvar|e}} anyons can only move within the boundary, so all {{mvar|e}} anyon loops are topologically trivial too. This indicates that this code does not encode any logical qubit, which can be verified by counting qubits and stabilizers: There are {{math|2''n''(''n'' − 1)}} qubits (lattice edges), {{math|''n''2 − 1}} independent vertex stabilizers, and {{math|(''n'' − 1)2}} independent plaquette stabilizers (due to the boundary, the product of all plaquette stabilizers is no longer {{mvar|I}} and all of them are independent), and {{math|1=2''n''(''n'' − 1) − (''n''2 − 1) − (''n'' − 1)2 = 0}}.

To design an open-boundary surface code with a non-trivial codespace, it is necessary to use another type of boundary, the ''rough boundary'' which acts as the dual of the smooth boundary. To create a rough boundary, we start from the smooth boundary and remove the edges (qubits) on the boundary that only neighbor one plaquette, but keep those plaquettes as weight-3 {{mvar|Z}} stabilizers. The vertex stabilizers on the original boundary are now weight-1 and no longer properly commute with the modified plaquette stabilizers, so those vertex stabilizers are removed too, leaving some "dangling" edges on the lattice (thus the name "rough"). The result is a boundary that acts as a source and sink for {{mvar|e}} anyons.

Now consider a lattice with smooth top and bottom boundaries, and rough left and right boundaries. Such a lattice defines an '''(unrotated) planar code'''. An {{mvar|m}} anyon moving from the top boundary to the top boundary is still a topologically trivial path, but one moving from the top boundary to the bottom boundary is no longer topologically trivial, because the anyon could not have exited at the left or right boundary anymore. Similarly, the topologically non-trivial path for {{mvar|e}} anyons is one moving from the left boundary to the right boundary. If the original grid has {{mvar|d}} rows and {{math|''d'' + 1}} columns of vertices (before removing the vertex stabilizers on the left and right boundaries), then both types of topologically non-trivial paths have minimum length {{mvar|d}}, indicating that the code encodes a single logical qubit with code distance {{mvar|d}}. The number of logical qubits can again be checked by counting stabilizers: {{math|1=(''d''2 + (''d'' − 1)2) − ''d''(''d'' − 1) − ''d''(''d'' − 1) = 1}} (now the vertex stabilizers are all independent too, due to the "dangling edges" that are only part of one vertex stabilizer).

=== Rotated planar code === The '''rotated planar code''' is a variant of the planar code that removes almost half of the physical qubits without affecting the code distance. A distance-{{mvar|d}} rotated planar code has {{math|''d''2}} physical qubits, compared to {{math|''d''2 + (''d'' − 1)2}} for the unrotated code. The four corners of the unrotated planar code lattice are cut off along diagonal lines, creating new boundaries. The resulting lattice is still in the shape of a square, but rotated by 45°, thus the name.

Conceptually, a surface code can encode a logical qubit with code distance {{mvar|d}} as long as it has four boundaries with alternating types (smooth or rough), and the distance between opposing boundaries (i.e., boundaries of the same type) along lattice edges (Manhattan distance for the square lattice) is at least {{mvar|d}}. This condition holds for the rotated square lattice when these four boundaries coincide with the four sides of the rotated square. For example, the northeast and southwest boundaries may be smooth boundaries while the northwest and southeast boundaries are rough boundaries. Diagonal smooth and rough boundaries have weight-2 {{mvar|X}} and {{mvar|Z}} stabilizers respectively.

A detailed description of the layout of the rotated planar code is more easily done by rotating the coordinate system by 45°. In this orientation, the {{math|''d''2}} qubits are located on the ''vertices'' of a {{math|''d'' × ''d''}} square lattice, and both {{mvar|Z}} and {{mvar|X}} stabilizers are located on the plaquettes of this rotated lattice, with the two types in a checkerboard pattern. Furthermore, some weight-2 {{mvar|Z}} (resp. {{mvar|X}}) stabilizers exist on the left and right (resp. top and bottom) boundaries of the lattice, with one boundary stabilizer on every other boundary edge. The total number of stabilizers is {{math|1=(''d'' − 1)2 + 4(''d'' − 1) / 2 = ''d''2 − 1}}.

Compared with an unrotated planar code with the same number of physical qubits, a rotated planar code can increase the code distance {{mvar|d}} by a factor of approximately {{tmath|\sqrt{2} }}. However, there are also many more ways that an anyon can traverse from one boundary to the opposite boundary in {{mvar|d}} steps, resulting in a larger number of minimum-weight error mechanisms. This may even cause the logical error rate of the rotated code to be higher than the unrotated code with the same number of qubits, despite the larger code distance. Regardless, the rotated planar code prevails in the low-error regime where the physical error rate is significantly below the threshold.<ref name="r488">{{cite journal | last1=Beverland | first1=Michael E | last2=Brown | first2=Benjamin J | last3=Kastoryano | first3=Michael J | last4=Marolleau | first4=Quentin | title=The role of entropy in topological quantum error correction | journal=Journal of Statistical Mechanics: Theory and Experiment | volume=2019 | issue=7 | date=2019-07-18 | issn=1742-5468 | doi=10.1088/1742-5468/ab25de | page=073404 | url=https://iopscience.iop.org/article/10.1088/1742-5468/ab25de | access-date=2026-01-12| arxiv=1812.05117 }}</ref>

== Quantum computation ==

The means to perform quantum computation on logical information stored within the surface code has been considered, with the properties of the code providing fault-tolerance. It has been shown that extending the stabilizer space using 'holes', vertices or plaquettes on which stabilizers are not enforced, allows many qubits to be encoded into the code. However, a universal set of unitary gates cannot be fault-tolerantly implemented by unitary operations{{huh|date=January 2026}} and so additional techniques are required to achieve quantum computing. For example, universal quantum computing can be achieved by preparing magic states via encoded quantum stubs called tidBits used to teleport in the required additional gates when replaced as a qubit. Furthermore, preparation of magic states must be fault tolerant, which can be achieved by magic state distillation on noisy magic states. A measurement based scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two-dimensional architecture.<ref>{{cite journal | last1=Raussendorf | first1=Robert | last2=Harrington | first2=Jim | title=Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions | journal=Physical Review Letters | volume=98 | issue=19 | date=11 May 2007 | issn=0031-9007 | doi=10.1103/physrevlett.98.190504 | article-number=190504| arxiv=quant-ph/0610082 | pmid=17677613 | bibcode=2007PhRvL..98s0504R| s2cid=39504821 }}</ref><ref>{{cite journal | last1=Raussendorf | first1=R | last2=Harrington | first2=J | last3=Goyal | first3=K | title=Topological fault-tolerance in cluster state quantum computation | journal=New Journal of Physics | volume=9 | issue=6 | date=29 June 2007 | issn=1367-2630 | arxiv=quant-ph/0703143 | doi=10.1088/1367-2630/9/6/199 | page=199|doi-access=free | bibcode=2007NJPh....9..199R}}</ref>

==Hamiltonian and self-correction==

Since the stabilizer operators of the surface code are quasilocal, acting only on spins located near each other on a two-dimensional lattice, it is not unrealistic to define the following Hamiltonian,

The ground state space of this Hamiltonian is the stabilizer space of the code. Excited states correspond to those of anyons, with the energy proportional to their number. Local errors are therefore energetically suppressed by the gap, which has been shown to be stable against local perturbations.<ref>{{cite journal | last1=Bravyi | first1=Sergey | last2=Hastings | first2=Matthew B. | last3=Michalakis | first3=Spyridon | title=Topological quantum order: Stability under local perturbations | journal=Journal of Mathematical Physics| volume=51 | issue=9 | year=2010 | issn=0022-2488 | doi=10.1063/1.3490195 | page=093512| arxiv=1001.0344 | bibcode=2010JMP....51i3512B | s2cid=115166306 }}</ref> However, the dynamic effects of such perturbations can still cause problems for the code.<ref>{{cite journal |author1=F. Pastawski|author2=A. Kay|author3=N. Schuch|author4=J. I. Cirac| journal=Quantum Information and Computation |title=Limitations of passive protection of quantum information| volume=10 | issue=7&8 | year=2010 | issn=1533-7146 | doi=10.26421/qic10.7-8 | page=580|arxiv=0911.3843|s2cid=3076085}}</ref><ref>{{cite journal | last1=Freeman | first1=C. Daniel | last2=Herdman | first2=C. M. | last3=Gorman | first3=D. J. | last4=Whaley | first4=K. B. | title=Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime | journal=Physical Review B | volume=90 | issue=13 | date=7 October 2014 | issn=1098-0121 | doi=10.1103/physrevb.90.134302 | article-number=134302| arxiv=1405.2315 | bibcode=2014PhRvB..90m4302F | s2cid=118724410 }}</ref>

The gap also gives the code a certain resilience against thermal errors, allowing it to be correctable almost surely for a certain critical time. This time increases with <math> J </math>, but since arbitrary increases of this coupling are unrealistic, the protection given by the Hamiltonian still has its limits.

The means to make a surface code into a fully self-correcting quantum memory is often considered. Self-correction means that the Hamiltonian will naturally suppress errors indefinitely, leading to a lifetime that diverges in the thermodynamic limit. It has been found that this is possible in the toric code only if long range interactions are present between anyons.<ref>{{cite journal | last1=Hamma | first1=Alioscia | last2=Castelnovo | first2=Claudio | last3=Chamon | first3=Claudio | title=Toric-boson model: Toward a topological quantum memory at finite temperature | journal=Physical Review B | volume=79 | issue=24 | date=18 June 2009 | issn=1098-0121 | doi=10.1103/physrevb.79.245122 | article-number=245122| arxiv=0812.4622 | bibcode=2009PhRvB..79x5122H | hdl=1721.1/51820 | s2cid=5202832 | hdl-access=free }}</ref><ref>{{cite journal | last1=Chesi | first1=Stefano | last2=Röthlisberger | first2=Beat | last3=Loss | first3=Daniel | title=Self-correcting quantum memory in a thermal environment | journal=Physical Review A | volume=82 | issue=2 | date=6 August 2010 | issn=1050-2947 | doi=10.1103/physreva.82.022305 | article-number=022305| arxiv=0908.4264 | bibcode=2010PhRvA..82b2305C | s2cid=118400202 }}</ref> Proposals have been made for realization of these in the lab <ref>{{cite journal | last1=Pedrocchi | first1=Fabio L. | last2=Chesi | first2=Stefano | last3=Loss | first3=Daniel | title=Quantum memory coupled to cavity modes | journal=Physical Review B | volume=83 | issue=11 | date=10 March 2011 | issn=1098-0121 | doi=10.1103/physrevb.83.115415 | article-number=115415| arxiv=1011.3762 | bibcode=2011PhRvB..83k5415P | s2cid=118595257 }}</ref> Another approach is the generalization of the model to higher dimensions, with self-correction possible in 4D with only quasi-local interactions.<ref>{{cite journal | last1=Alicki | first1=R. | last2=Horodecki | first2=M. | last3=Horodecki | first3=P. | last4=Horodecki | first4=R. | title=On Thermal Stability of Topological Qubit in Kitaev's 4D Model | journal=Open Systems & Information Dynamics | volume=17 | issue=1 | year=2010 | issn=1230-1612 | doi=10.1142/s1230161210000023 | pages=1–20| arxiv=0811.0033 | s2cid=26719502 }}</ref>

==Generalizations==

It is possible to define similar codes using higher-dimensional spins. These are the quantum double models<ref>{{cite journal | last1=Bullock | first1=Stephen S | last2=Brennen | first2=Gavin K | title=Qudit surface codes and gauge theory with finite cyclic groups | journal=Journal of Physics A: Mathematical and Theoretical | volume=40 | issue=13 | date=14 March 2007 | issn=1751-8113 | doi=10.1088/1751-8113/40/13/013 | pages=3481–3505| arxiv=quant-ph/0609070 | bibcode=2007JPhA...40.3481B | s2cid=15630224 }}</ref> and string-net models,<ref>{{cite journal |author=Levin, Michael A. and Xiao-Gang Wen |title=String-net condensation: A physical mechanism for topological phases |journal=Physical Review B |volume=71 |issue=45110 |page=21 |date=12 January 2005|arxiv=cond-mat/0404617 |doi=10.1103/PhysRevB.71.045110 |bibcode=2005PhRvB..71d5110L |s2cid=51962817 }}</ref> which allow a greater richness in the behaviour of anyons, and so may be used for more advanced quantum computation and error correction proposals.<ref>{{cite journal | last1=Wootton | first1=James R. | last2=Lahtinen | first2=Ville | last3=Doucot | first3=Benoit | last4=Pachos | first4=Jiannis K. | title=Engineering complex topological memories from simple Abelian models | journal=Annals of Physics | volume=326 | issue=9 | year=2011 | issn=0003-4916 | doi=10.1016/j.aop.2011.05.008 | pages=2307–2314|arxiv=0908.0708| bibcode=2011AnPhy.326.2307W | s2cid=119288871 }}</ref> These not only include models with Abelian anyons, but also those with non-Abelian statistics.<ref>{{cite journal | last1=Aguado | first1=M. | last2=Brennen | first2=G. K. | last3=Verstraete | first3=F. | last4=Cirac | first4=J. I. | title=Creation, Manipulation, and Detection of Abelian and Non-Abelian Anyons in Optical Lattices | journal=Physical Review Letters | volume=101 | issue=26 | date=22 December 2008 | issn=0031-9007 | doi=10.1103/physrevlett.101.260501 | article-number=260501| pmid=19113760 | arxiv=0802.3163 | bibcode=2008PhRvL.101z0501A | hdl=1854/LU-8589252 | s2cid=11619038 | hdl-access=free }}</ref><ref>{{cite journal | last1=Brennen | first1=G K | last2=Aguado | first2=M | last3=Cirac | first3=J I | title=Simulations of quantum double models | journal=New Journal of Physics | volume=11 | issue=5 | date=22 May 2009 | issn=1367-2630 | doi=10.1088/1367-2630/11/5/053009 | article-number=053009| arxiv=0901.1345 | bibcode=2009NJPh...11e3009B |doi-access=free}}</ref><ref>{{Cite journal |last1=Liu |first1=Yu-Jie |last2=Shtengel |first2=Kirill |last3=Smith |first3=Adam |last4=Pollmann |first4=Frank |date=2022-11-07 |title=Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer |url=https://link.aps.org/doi/10.1103/PRXQuantum.3.040315 |journal=PRX Quantum |volume=3 |issue=4 |article-number=040315 |doi=10.1103/PRXQuantum.3.040315|arxiv=2110.02020 |bibcode=2022PRXQ....3d0315L |s2cid=238353914 }}</ref>

==Experimental progress==

The most explicit demonstration of the properties of the toric code has been in state based approaches. Rather than attempting to realize the Hamiltonian, these simply prepare the code in the stabilizer space. Using this technique, experiments have been able to demonstrate the creation, transport and statistics of the anyons<ref>{{cite journal | last1=Pachos | first1=J K | last2=Wieczorek | first2=W | last3=Schmid | first3=C | last4=Kiesel | first4=N | last5=Pohlner | first5=R | last6=Weinfurter | first6=H | title=Revealing anyonic features in a toric code quantum simulation | journal=New Journal of Physics | volume=11 | issue=8 | date=12 August 2009 | issn=1367-2630 | doi=10.1088/1367-2630/11/8/083010 | article-number=083010| bibcode=2009NJPh...11h3010P |doi-access=free| arxiv=0710.0895 }}</ref><ref>C.-Y. Lu, et al., Phys. Rev. Lett. '''102''', 030502 (2009).</ref><ref name=":0">{{cite journal|last1=Satzinger|first1=K. J.|last2=Liu|first2=Y.|last3=Smith|first3=A.|last4=Knapp|first4=C.|last5=Newman|first5=M.|last6=Jones|first6=C.|last7=Chen|first7=Z.|last8=Quintana|first8=C.|last9=Mi|first9=X.|last10=Dunsworth|first10=A.|last11=Gidney|first11=C.|date=2021-04-02|title=Realizing topologically ordered states on a quantum processor|journal=Science|volume=374|issue=6572|pages=1237–1241|doi=10.1126/science.abi8378|pmid=34855491|arxiv=2104.01180|bibcode=2021Sci...374.1237S|s2cid=233025160}}</ref> and measurement of the topological entanglement entropy.<ref name=":0" /> More recent experiments have also been able to demonstrate the error correction properties of the code.<ref>{{cite journal | last1=Yao | first1=Xing-Can | last2=Wang | first2=Tian-Xiong | last3=Chen | first3=Hao-Ze | last4=Gao | first4=Wei-Bo | last5=Fowler | first5=Austin G. | last6=Raussendorf | first6=Robert | last7=Chen | first7=Zeng-Bing | last8=Liu | first8=Nai-Le | last9=Lu | first9=Chao-Yang | last10=Deng | first10=You-Jin | last11=Chen | first11=Yu-Ao | last12=Pan | first12=Jian-Wei | title=Experimental demonstration of topological error correction | journal=Nature| volume=482 | issue=7386 | date=22 February 2012 | issn=0028-0836 | doi=10.1038/nature10770 | pages=489–494| arxiv=0905.1542 | pmid=22358838 | bibcode=2012Natur.482..489Y| s2cid=4307662 }}</ref><ref name=":0" />

For realizations of the toric code and its generalizations with a Hamiltonian, much progress has been made using Josephson junctions. The theory of how the Hamiltonians may be implemented has been developed for a wide class of topological codes.<ref>{{cite journal | last1=Douçot | first1=Benoit | last2=Ioffe | first2=Lev B. | last3=Vidal | first3=Julien | title=Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation | journal=Physical Review B | volume=69 | issue=21 | date=3 June 2004 | issn=1098-0121 | doi=10.1103/physrevb.69.214501 | article-number=214501| arxiv=cond-mat/0302104 | bibcode=2004PhRvB..69u4501D | s2cid=119407144 }}</ref> An experiment has also been performed, realizing the toric code Hamiltonian for a small lattice, and demonstrating the quantum memory provided by its degenerate ground state.<ref>{{cite journal | last1=Gladchenko | first1=Sergey | last2=Olaya | first2=David | last3=Dupont-Ferrier | first3=Eva | last4=Douçot | first4=Benoit | last5=Ioffe | first5=Lev B. | last6=Gershenson | first6=Michael E. | title=Superconducting nanocircuits for topologically protected qubits | journal=Nature Physics | volume=5 | issue=1 | year=2009 | issn=1745-2473 | doi=10.1038/nphys1151 | pages=48–53| arxiv=0802.2295 | bibcode=2009NatPh...5...48G | s2cid=118359424 }}</ref>

Other theoretical and experimental works towards realizations are based on cold atoms. A toolkit of methods that may be used to realize topological codes with optical lattices has been explored, <ref>{{cite journal | last1=Micheli | first1=A. | last2=Brennen | first2=G. K. | last3=Zoller | first3=P. | title=A toolbox for lattice-spin models with polar molecules | journal=Nature Physics | volume=2 | issue=5 | date=30 April 2006 | issn=1745-2473 | doi=10.1038/nphys287 | pages=341–347| arxiv=quant-ph/0512222 | bibcode=2006NatPh...2..341M | s2cid=108289844 }}</ref> as have experiments concerning minimal instances of topological order.<ref>{{cite journal | last1=Paredes | first1=Belén | last2=Bloch | first2=Immanuel | title=Minimum instances of topological matter in an optical plaquette | journal=Physical Review A| volume=77 | issue=2 | date=1 January 2008 | issn=1050-2947 | doi=10.1103/physreva.77.023603 | article-number=023603| arxiv=0711.3796 | bibcode=2008PhRvA..77b3603P | s2cid=46143303 }}</ref> Such minimal instances of the toric code has been realized experimentally within isolated square plaquettes.<ref>{{cite journal | last1=Dai | first1=Hanning| last2=Yang | first2=Bing | last3=Reingruber | first3=Andreas | last4=Sun | first4=Hui | last5=Xu| first5=Xiao-Fan| last6=Chen| first6=Yu-Ao| last7=Yuan| first7=Zhen-Sheng | last8=Pan | first8=Jian-Wei |title=Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian | journal=Nature Physics | volume=13 | issue=2 | date=28 August 2017 | pages=1195–1200| issn=1745-2473 | doi= 10.1038/NPHYS4243 | arxiv=1602.05709 | bibcode=2017NatPh..13.1195D| s2cid=118604118}}</ref> Progress is also being made into simulations of the toric model with Rydberg atoms, in which the Hamiltonian and the effects of dissipative noise can be demonstrated.<ref>{{cite journal | last1=Weimer | first1=Hendrik | last2=Müller | first2=Markus | last3=Lesanovsky | first3=Igor | last4=Zoller | first4=Peter | last5=Büchler | first5=Hans Peter | title=A Rydberg quantum simulator | journal=Nature Physics | volume=6 | issue=5 | date=14 March 2010 | issn=1745-2473 | doi=10.1038/nphys1614 | pages=382–388| arxiv=0907.1657 | bibcode=2010NatPh...6..382W | s2cid=54710282 }}</ref><ref>{{cite journal|last1=Semeghini|first1=Giulia|last2=Levine|first2=Harry|last3=Keesling|first3=Alexander|last4=Ebadi|first4=Sepehr|last5=Wang|first5=Tout T.|last6=Bluvstein|first6=Dolev|last7=Verresen|first7=Ruben|last8=Pichler|first8=Hannes|last9=Kalinowski|first9=Marcin|last10=Samajdar|first10=Rhine|last11=Omran|first11=Ahmed|title=Probing Topological Spin Liquids on a Programmable Quantum Simulator|journal=Science|year=2021|volume=374|issue=6572|pages=1242–1247|arxiv=2104.04119|doi=10.1126/science.abi8794|pmid=34855494|bibcode=2021Sci...374.1242S|s2cid=233204440}}</ref> Experiments in Rydberg atom arrays have also successfully realized the toric code with periodic boundary conditions in two dimensions by coherently transporting arrays of entangled atoms.<ref name="CoherentArray2">{{cite journal |last1=Bluvstein |first1=Dolev |last2=Levine |first2=Harry |last3=Semeghini |first3=Giulia |last4=Wang |first4=Tout |last5=Ebadi |first5=Sepehr |last6=Kalinowski |first6=Marcin |last7=Maskara |first7=Nishad |last8=Pichler |first8=Hannes |last9=Greiner |first9=Marcus |last10=Vuletic |first10=Vladan |last11=Lukin |first11=Misha |title=A quantum processor based on coherent transport of entangled atom arrays |journal=Nature |date=April 20, 2022 |volume=604 |issue=7906 |pages=451–456 |doi=10.1038/s41586-022-04592-6 |pmid=35444318 |pmc=9021024 |arxiv=2112.03923 |bibcode=2022Natur.604..451B |s2cid=244954259 |doi-access=free }}</ref>

As of 2025, Google Quantum AI has implemented the rotated planar code for up to code distance 7 on their newest generation of superconducting quantum processors, Willow, demonstrating a logical error suppression factor {{mvar|Λ}} slightly larger than 2 when the code distance is increased by 2, indicating below-threshold behavior.<ref name="willow" />

==References== {{reflist}}

==External links== *https://skepsisfera.blogspot.com/2010/04/kitaevs-toric-code.html

Category:Quantum information science Category:Fault-tolerant computer systems Category:Quantum phases Category:Condensed matter physics