Thesurface code is atopologicalquantum error correcting code, and an example of astabilizer code, defined on a two-dimensionalspinlattice.^[1] The first type of surface code introduced byAlexei Kitaev in 1997 was thetoric code, which gets its name from its periodic boundary conditions, giving it the shape of atorus. 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.^[2] It is also the simplest example oftopological order—Z₂ topological order(first studied in the context ofZ₂spin liquid in 1991).^[3][4] The toric code can also be considered to be aZ₂lattice gauge theory in a particular limit.^[5]

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, theplanar code.^[6] As of 2025,Google Quantum AI has implemented a distance-7 planar code on their newest generation of superconducting quantum processors, Willow, demonstrating a below-threshold physical error rate.^[7]

The surface code is defined on a two-dimensional lattice, usually chosen to be thesquare 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 atorus.

Aqubit is located on eachedge of the lattice. For ad ×d lattice, there ared² horizontal edges andd² vertical edges, thus2d² qubits in total.Stabilizer operators are defined on the qubits around each vertexv and plaquette (face)p of the lattice as follows:

${\displaystyle A_v=\prod _{i\in v}{X}_i,B_p=\prod _{i\in p}{Z}_i.}$

Here${\displaystyle i\in v}$ denotes the edges touching the vertexv, and${\displaystyle i\in p}$ denotes the edges surrounding the plaquette${\displaystyle p}$. The code space of the toric code is the subspace for which all stabilizers act trivially, hence for any state${\displaystyle |\psi ⟩}$ in this space it holds that

${\displaystyle A_v|\psi ⟩=|\psi ⟩,\mathrm{\forall }v,B_p|\psi ⟩=|\psi ⟩,\mathrm{\forall }p.}$

For the toric code, this space is four-dimensional, and so can be used to store twoqubits ofquantum information. This can be proven by considering the number of independent stabilizer operators: For ad ×d lattice, there ared² vertex stabilizers andd² plaquette stabilizers, but the product of all vertex stabilizers isI and so is the product of all plaquette stabilizers. Therefore there are2d² − 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 PauliZ error on qubiti flips the two vertex stabilizersA_v such that${\displaystyle i\in v}$ (the endpoints of the edgei), and a PauliX error on qubiti flips the two plaquette stabilizersB_p such that${\displaystyle i\in p}$ (the plaquettes on either side of the edgei). The positions of these violations is thesyndrome 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 asquasiparticles. Specifically, if the code is in a state${\displaystyle |\varphi ⟩}$ such that${\displaystyle A_v|\varphi ⟩=-|\varphi ⟩}$, a quasiparticle known as aneanyon can be said to exist on the vertexv. Similarly a violation of some${\displaystyle B_p}$ is associated with anm anyon on the plaquettep. 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 isI means that the number ofe (resp.m) anyons on a toric code is always even.

A single-qubitZ error can be associated with an edge, and creates a pair ofe anyons at both endpoints of that edge. However, twoe anyons at the same location will annihilate each other, so aZ error can also effectively move ane anyon along an edge, allowinge 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 (Z stabilizer generators) on the lattice. Therefore the loop represents aZ 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 theZ operators of the two logical qubits encoded in the toric code.

By considering thedual graph of the lattice, it can be seen that the above paragraph also applies toX errors andm anyons. Note that the horizontalZ operator and the verticalX operator belong to the same qubit, and vice versa. This ensures the correctcommutation relations between logical operators.

Consider the noise model for which bit and phase errors occur independently on each qubit, both with probabilityp. Whenp 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. Asp 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%.^[8]

Other error models may also be considered, and thresholds found. In all cases studied so far, the code has been found to saturate theHashing 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.^[9][10]

These thresholds are upper limits and are useless unless efficient algorithms are found to achieve them. The most well-used algorithm isminimum weight perfect matching.^[11] 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.

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 ann ×nsquare 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)X stabilizers.

The most important characteristic of such an open boundary is that a PauliX error no longer necessarily flips two plaquette stabilizers. An edge on the boundary only has one adjacent plaquette, and thus anX error on the corresponding qubit will only flip a single${\displaystyle B_p}$, or in the language of anyons, only create or annihilate a singlem anyon. One could say that this type of code boundary (known as asmooth boundary) is a source and sink form 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 ourn ×n grid graph example, anm 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 anm 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-3X stabilizers mentioned above. Meanwhile,e anyons can only move within the boundary, so alle 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 are2n(n − 1) qubits (lattice edges),n² − 1 independent vertex stabilizers, and(n − 1)² independent plaquette stabilizers (due to the boundary, the product of all plaquette stabilizers is no longerI and all of them are independent), and2n(n − 1) − (n² − 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, therough boundary which acts as thedual 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-3Z 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 fore 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. Anm 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 fore anyons is one moving from the left boundary to the right boundary. If the original grid hasd rows andd + 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 lengthd, indicating that the code encodes a single logical qubit with code distanced. The number of logical qubits can again be checked by counting stabilizers:(d² + (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).

Therotated planar code is a variant of the planar code that removes almost half of the physical qubits without affecting the code distance. A distance-d rotated planar code hasd² physical qubits, compared tod² + (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 distanced 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 leastd. 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-2X andZ 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, thed² qubits are located on thevertices of ad ×d square lattice, and bothZ andX stabilizers are located on the plaquettes of this rotated lattice, with the two types in a checkerboard pattern. Furthermore, some weight-2Z (resp.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(d − 1)² + 4(d − 1) / 2 =d² − 1.

Compared with an unrotated planar code with the same number of physical qubits, a rotated planar code can increase the code distanced by a factor of approximately⁠${\displaystyle \sqrt{2}}$⁠. However, there are also many more ways that an anyon can traverse from one boundary to the opposite boundary ind 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.^[12]

The means to performquantum 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 unitarygates cannot be fault-tolerantly implemented by unitary operations^{[clarification needed]} 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. Ameasurement based scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two-dimensional architecture.^[13][14]

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,

${\displaystyle H_{\mathrm{T}\mathrm{C}}=-J_e\sum _v{A}_v-J_m\sum _p{B}_p,J_e,J_m>0.}$

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.^[15] However, the dynamic effects of such perturbations can still cause problems for the code.^[16][17]

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${\displaystyle J}$, 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.^[18][19] Proposals have been made for realization of these in the lab^[20] Another approach is the generalization of the model to higher dimensions, with self-correction possible in 4D with only quasi-local interactions.^[21]

As mentioned above, so called${\displaystyle e}$ and${\displaystyle m}$ quasiparticles are associated with the vertices and plaquettes of the model, respectively. These quasiparticles can be described asanyons, due to the non-trivial effect of their braiding. Specifically, though both species of anyons are bosonic with respect to themselves, the braiding of two${\displaystyle e}$'s or${\displaystyle m}$'s having no effect, a full monodromy of an${\displaystyle e}$ and an${\displaystyle m}$ will yield a phase of${\displaystyle -1}$. Such a result is not consistent with eitherbosonic orfermionicstatistics, and hence is anyonic.

The anyonic mutual statistics of the quasiparticles demonstrate the logical operations performed by topologically non-trivial loops. Consider the creation of a pair of${\displaystyle e}$ anyons followed by the transport of one around a topologically nontrivial loop, such as that shown on the torus in blue on the figure above, before the pair are reannhilated. The state is returned to the stabilizer space, but the loop implements a logical operation on one of the stored qubits. If${\displaystyle m}$ anyons are similarly moved through the red loop above a logical operation will also result. The phase of${\displaystyle -1}$ resulting when braiding the anyons shows that these operations do not commute, but rather anticommute. They may therefore be interpreted as logical${\displaystyle Z}$ and${\displaystyle X}$ Pauli operators on one of the stored qubits. The corresponding logical Pauli's on the other qubit correspond to an${\displaystyle m}$ anyon following the blue loop and an${\displaystyle e}$ anyon following the red. No braiding occurs when${\displaystyle e}$ and${\displaystyle m}$ pass through parallel paths, the phase of${\displaystyle -1}$ therefore does not arise and the corresponding logical operations commute. This is as should be expected since these form operations acting on different qubits.

Due to the fact that both${\displaystyle e}$ and${\displaystyle m}$ anyons can be created in pairs, it is clear to see that both these quasiparticles are their own antiparticles. A composite particle composed of two${\displaystyle e}$ anyons is therefore equivalent to the vacuum, since the vacuum can yield such a pair and such a pair will annihilate to the vacuum. Accordingly, these composites have bosonic statistics, since their braiding is always completely trivial. A composite of two${\displaystyle m}$ anyons is similarly equivalent to the vacuum. The creation of such composites is known as the fusion of anyons, and the results can be written in terms of fusion rules. In this case, these take the form,

${\displaystyle e\times e=1,m\times m=1.}$

Where${\displaystyle 1}$ denotes the vacuum. A composite of an${\displaystyle e}$ and an${\displaystyle m}$ is not trivial. This therefore constitutes another quasiparticle in the model, sometimes denoted${\displaystyle \psi }$, with fusion rule,

${\displaystyle e\times m=\psi .}$

From the braiding statistics of the anyons we see that, since any single exchange of two${\displaystyle \psi }$'s will involve a full monodromy of a constituent${\displaystyle e}$ and${\displaystyle m}$, a phase of${\displaystyle -1}$ will result. This implies fermionic self-statistics for the${\displaystyle \psi }$'s.

Since the Hamiltonian is a sum of commuting projectors with eigenvalues${\displaystyle \pm 1}$, the ground state is the state${\displaystyle |{\psi }_{GS}⟩}$ that is a +1 eigenstate of every single star and plaquette operator:

${\displaystyle A_s|{\psi }_{GS}⟩=+1|{\psi }_{GS}⟩\mathrm{\forall }s}$

${\displaystyle B_p|{\psi }_{GS}⟩=+1|{\psi }_{GS}⟩\mathrm{\forall }p}$

This is the "frustration-free" ground state, where all local constraints are satisfied.

Excitations above the ground state correspond to violations of these conditions.

• Anelectric charge (ore-particle) exists at a vertexs if${\displaystyle A_s=-1}$.
• Amagnetic vortex (orm-particle) exists at a plaquettep if${\displaystyle B_p=-1}$.

These excitations are created in pairs at the ends of "string" operators. Applying a string of${\displaystyle {\sigma }^z}$ operators along a path creates e-particles at its endpoints. Applying a string of${\displaystyle {\sigma }^x}$ operators along a path on the dual lattice creates m-particles at its endpoints. The energy of an excited state is proportional to the number of such violations, leading to a gapped energy spectrum. These particles are examples ofanyons due to their non-trivial braiding statistics.

A defining feature of the surface code, and of topological order in general, is that its quasiparticle excitations exhibit non-trivialbraiding statistics. While the electric charges (e) and magnetic fluxes (m) are both individually bosons with respect to themselves, they have a non-trivialmutual statistics with respect to each other. Specifically, they aremutual semions: adiabatically moving ane particle in a full counterclockwise cycle around anm particle imparts a phase of -1 to the system's wavefunction. This is a topological analogue of theAharonov–Bohm effect, where the role of the electromagnetic vector potential is played by the non-local presence of the other quasiparticle.

This property is a direct consequence of the algebraic structure of the stabilizer operators and the string operators that create the particles. The argument can be understood as follows:

1.Creating the Quasiparticles: We begin in the ground state|Ψ₀⟩, which is annihilated by all stabilizer operators (${\displaystyle A_s=+1}$ and${\displaystyle B_p=+1}$ for alls,p).

• A stationary magnetic flux (m) is created at a plaquettep by applying astring operator of Pauli-X matrices,${\displaystyle W_m(p)}$, along a path on the dual lattice${\displaystyle C_m}$ from a faraway plaquetteq top. This flips the sign of the plaquette stabilizer${\displaystyle B_p}$ to -1.
• An electric charge (e) is moved along a path by applying a string operator of Pauli-Z matrices. A closed loop ofZ operators,${\displaystyle L_e(C)}$, corresponds to creating ane particle, moving it around the closed pathC, and then annihilating it.

2.The Braiding Process: Consider the process of moving ane charge in a closed loopC that encloses the plaquettep where the stationarym particle resides. The final state of the system is given by applying the loop operator${\displaystyle L_e(C)}$ to the state containing them particle:

${\displaystyle |{\mathrm{\Psi }}_{\text{final}}⟩=L_e(C)|m⟩=L_e(C)W_m(p)|{\mathrm{\Psi }}_0⟩}$

3.The Algebraic Origin of the Phase: The key insight comes from the commutation relation between the loop operator${\displaystyle L_e(C)}$ and the string operator${\displaystyle W_m(p)}$.

• ${\displaystyle L_e(C)}$ is a product ofZ operators on the edges forming the loopC.
• ${\displaystyle W_m(p)}$ is a product ofX operators on the edges forming the path${\displaystyle C_m}$.
• The loopC and the path${\displaystyle C_m}$ must cross at exactly one edge. Let this edge bej.
• For any edgei ≠ j, the operators${\displaystyle Z_i}$ from the loop and${\displaystyle X_i}$ from the string commute. However, at the crossing edgej, the operators anti-commute:${\displaystyle Z_j{X}_j=-X_j{Z}_j}$.

Because of this single anti-commutation, the operators as a whole anti-commute:

${\displaystyle L_e(C)W_m(p)=-W_m(p)L_e(C)}$

4.Deriving the Phase Factor: We can now substitute this back into the expression for the final state:

${\displaystyle |{\mathrm{\Psi }}_{\text{final}}⟩=L_e(C)W_m(p)|{\mathrm{\Psi }}_0⟩=-W_m(p)L_e(C)|{\mathrm{\Psi }}_0⟩}$

The operator${\displaystyle L_e(C)}$ is a closed loop ofZ operators. Any such operator can be written as a product of the vertex stabilizers${\displaystyle A_s}$ for all verticess inside the loop.^[22] Since the ground state|Ψ₀⟩ is an eigenstate of all${\displaystyle A_s}$ with eigenvalue +1, the loop operator leaves the ground state unchanged:${\displaystyle L_e(C)|{\psi }_0⟩=|{\psi }_0⟩}$. Therefore, the final state is:

${\displaystyle |{\mathrm{\Psi }}_{\text{final}}⟩=-W_m(p)|{\mathrm{\Psi }}_0⟩=-|m⟩}$

The system returns to its initial state (a singlem particle at plaquettep), but its wavefunction has acquired a phase factor of -1. This result is topological because it is independent of the exact shape of the loopC, so long as it encloses them particle. This non-trivial mutual statistics is a fundamental signature of the${\displaystyle Z_2}$ topological order present in the toric code and is the basis for proposals to use such systems for fault-tolerant quantum information processing.^[23]

On a torus, the local constraints${\displaystyle A_s=1}$ and${\displaystyle B_p=1}$ are not sufficient to uniquely define the ground state. The non-trivial topology allows for the existence of non-local operators that commute with the Hamiltonian but act non-trivially within the ground state subspace. These are thelogical operators orWilson loops.

A torus has two independent non-contractible loops (or cycles), often denoted${\displaystyle C_1}$ (e.g., "horizontal") and${\displaystyle C_2}$ (e.g., "vertical"). We can define four logical operators corresponding to strings of Pauli operators wrapping around these loops.

1.Electric Wilson Loops (${\displaystyle W^{(e)}}$): These are products of${\displaystyle {\sigma }^z}$ operators along the non-contractible loops.

   *${\displaystyle W_1^{(e)}=\prod _{i\in C_1}{\sigma }_i^z}$   *${\displaystyle W_2^{(e)}=\prod _{i\in C_2}{\sigma }_i^z}$

2.Magnetic 't Hooft Loops (${\displaystyle W^{(m)}}$): These are products of${\displaystyle {\sigma }^x}$ operators along non-contractible loops on the dual lattice, denoted${\displaystyle C_1^{\ast }}$ and${\displaystyle C_2^{\ast }}$, which intersect${\displaystyle C_2}$ and${\displaystyle C_1}$ respectively.

   *${\displaystyle W_1^{(m)}=\prod _{i\in C_1^{\ast }}{\sigma }_i^x}$   *${\displaystyle W_2^{(m)}=\prod _{i\in C_2^{\ast }}{\sigma }_i^x}$

These loop operators all commute with the Hamiltonian${\displaystyle H}$. For example, a loop operator${\displaystyle W_1^{(e)}}$ anticommutes with the two star operators at the "ends" of each link in its path. Since the path is a closed loop, it anticommutes with every neighboring star operator twice, resulting in commutation. A similar argument holds for the magnetic loops and plaquette operators.

The key to the ground state degeneracy lies in the commutation relations between these logical operators.

• Operators of the same type always commute:${\displaystyle [W_i^{(e)},W_j^{(e)}]=0}$ and${\displaystyle [W_i^{(m)},W_j^{(m)}]=0}$.
• Loops wrapping around different cycles do not intersect, and thus they commute:

${\displaystyle [W_1^{(e)},W_1^{(m)}]=0}$${\displaystyle [W_2^{(e)},W_2^{(m)}]=0}$

• Crucially, an electric loop and a magnetic loop wrapping around corresponding cycles intersect at exactly one qubit. For example, the loop${\displaystyle C_1}$ and the dual loop${\displaystyle C_2^{\ast }}$ must cross once.

Let's examine the commutation of${\displaystyle W_1^{(e)}}$ and${\displaystyle W_2^{(m)}}$.${\displaystyle W_1^{(e)}{W}_2^{(m)}=\left(\prod _{i\in C_1}{\sigma }_i^z\right)\left(\prod _{j\in C_2^{\ast }}{\sigma }_j^x\right)}$

The two strings of operators commute for every qubitexcept for the single qubit where the loops${\displaystyle C_1}$ and${\displaystyle C_2^{\ast }}$ intersect. At that intersection point, we have${\displaystyle {\sigma }^z{\sigma }^x=-{\sigma }^x{\sigma }^z}$. Because there is only one such anticommutation, the operators as a whole anticommute:

${\displaystyle W_1^{(e)}{W}_2^{(m)}=-W_2^{(m)}{W}_1^{(e)}⟹\{W_1^{(e)},W_2^{(m)}\}=0}$

Similarly, for the second cycle:

${\displaystyle \{W_2^{(e)},W_1^{(m)}\}=0}$

We have two pairs of operators,${\displaystyle (W_1^{(e)},W_2^{(m)})}$ and${\displaystyle (W_2^{(e)},W_1^{(m)})}$, that each obey the algebra of a logical qubit's Pauli operators (e.g.,${\displaystyle Z_L\equiv W^{(e)},X_L\equiv W^{(m)}}$). Since these two pairs act independently (commute with each other), they describetwo independent logical qubits.

A system of two independent qubits has a${\displaystyle 2\times 2=4}$-dimensional state space. This implies that the ground state subspace of the toric code on a torus isfour-fold degenerate.

We can explicitly construct these four states.

1. Start with one ground state,${\displaystyle |{\psi }_0⟩}$, which satisfies${\displaystyle A_s=1}$,${\displaystyle B_p=1}$ for alls, p. Let's define it as a +1 eigenstate of the electric operators:

${\displaystyle W_1^{(e)}|{\psi }_0⟩=+1|{\psi }_0⟩}$${\displaystyle W_2^{(e)}|{\psi }_0⟩=+1|{\psi }_0⟩}$

2. We can now generate the other three orthogonal ground states by acting with the magnetic 't Hooft loop operators. Since${\displaystyle W_2^{(m)}}$ anticommutes with${\displaystyle W_1^{(e)}}$, acting with it flips the eigenvalue of${\displaystyle W_1^{(e)}}$ from +1 to -1.

   *${\displaystyle |{\psi }_1⟩=W_2^{(m)}|{\psi }_0⟩}$. This state has eigenvalues${\displaystyle (-1,+1)}$ for${\displaystyle (W_1^{(e)},W_2^{(e)})}$.   *${\displaystyle |{\psi }_2⟩=W_1^{(m)}|{\psi }_0⟩}$. This state has eigenvalues${\displaystyle (+1,-1)}$ for${\displaystyle (W_1^{(e)},W_2^{(e)})}$.   *${\displaystyle |{\psi }_3⟩=W_1^{(m)}{W}_2^{(m)}|{\psi }_0⟩}$. This state has eigenvalues${\displaystyle (-1,-1)}$ for${\displaystyle (W_1^{(e)},W_2^{(e)})}$.

These four states${\displaystyle \{|{\psi }_0⟩,|{\psi }_1⟩,|{\psi }_2⟩,|{\psi }_3⟩\}}$ are all degenerate in energy (they are all +1 eigenstates of all local${\displaystyle A_s}$ and${\displaystyle B_p}$ operators), and they are mutually orthogonal. They form a basis for the 4-dimensional ground state subspace.

It is possible to define similar codes using higher-dimensional spins. These are the quantum double models^[24] andstring-net models,^[25] which allow a greater richness in the behaviour of anyons, and so may be used for more advanced quantum computation and error correction proposals.^[26] These not only include models with Abelian anyons, but also those with non-Abelian statistics.^[27][28][29]

The most explicit demonstration of the properties of the toric code has been in state based approaches. Rather than attempting to realize theHamiltonian, 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^[30][31][32] and measurement of thetopological entanglement entropy.^[32] More recent experiments have also been able to demonstrate the error correction properties of the code.^[33][32]

For realizations of the toric code and its generalizations with a Hamiltonian, much progress has been made usingJosephson junctions. The theory of how the Hamiltonians may be implemented has been developed for a wide class of topological codes.^[34] 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.^[35]

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,^[36] as have experiments concerning minimal instances of topological order.^[37] Such minimal instances of the toric code has been realized experimentally within isolated square plaquettes.^[38]Progress is also being made into simulations of the toric model withRydberg atoms, in which the Hamiltonian and the effects of dissipative noise can be demonstrated.^[39][40] 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.^[41]

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 factorΛ slightly larger than 2 when the code distance is increased by 2, indicating below-threshold behavior.^[7]

• https://skepsisfera.blogspot.com/2010/04/kitaevs-toric-code.html

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