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Toric code

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The toric code is a <a href="Topological_order" title="Topological order">topological</a> <a href="Quantum_error_correction" title="Quantum error correction">quantum error correcting code</a>, and an example of a <a href="Stabilizer_code" title="Stabilizer code">stabilizer code</a>, defined on a two-dimensional <a href="Spin_(physics)" title="Spin (physics)">spin</a> <a href="Planar_graph" title="Planar graph">lattice</a>.<a href="#cite_note-1">[1]</a> It is the simplest and most well studied of the quantum double models.<a href="#cite_note-2">[2]</a> It is also the simplest example of <a href="Topological_order" title="Topological order">topological order</a>—Z2 topological order

(first studied in the context of Z2 <a href="Spin_liquid" class="mw-redirect" title="Spin liquid">spin liquid</a> in 1991).<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> The toric code can also be considered to be a Z2 <a href="Lattice_gauge_theory" title="Lattice gauge theory">lattice gauge theory</a> in a particular limit.<a href="#cite_note-5">[5]</a> It was introduced by <a href="Alexei_Kitaev" title="Alexei Kitaev">Alexei Kitaev</a>.

The toric code gets its name from its periodic boundary conditions, giving it the shape of a <a href="Torus" title="Torus">torus</a>. These conditions give the model translational invariance, which is useful for analytic study. However, some experimental realizations require open boundary conditions, allowing the system to be embedded on a 2D surface. The resulting code is typically known as the planar code. This has identical behaviour to the toric code in most, but not all, cases.

<div class="mw-heading mw-heading2"><h2 id="Error_correction_and_computation">Error correction and computation</h2></div>

The toric code is defined on a two-dimensional lattice, usually chosen to be the <a href="Square_lattice" title="Square lattice">square lattice</a>, with a <a href="Spin-%C2%BD" class="mw-redirect" title="Spin-½">spin-½</a> degree of freedom located on each edge. They are chosen to be periodic. <a href="Stabilizer_code" title="Stabilizer code">Stabilizer</a> operators are defined on the spins around each vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> and plaquette (or face i.e. a vertex of the dual lattice) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> of the lattice as follows,

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<annotation encoding="application/x-tex">{\displaystyle A_{v}=\prod _{i\in v}\sigma _{i}^{x},\,\,B_{p}=\prod _{i\in p}\sigma _{i}^{z}.}</annotation>

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</math><img src="./872fd70bc8fb188884557fd5dc7dfee16c6b3d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.796ex; height:5.843ex;" alt="{\displaystyle A_{v}=\prod _{i\in v}\sigma _{i}^{x},\,\,B_{p}=\prod _{i\in p}\sigma _{i}^{z}.}" loading="lazy">

Where here we use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in v}">

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<annotation encoding="application/x-tex">{\displaystyle i\in v}</annotation>

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</math><img src="./51ceefdea2d52522e0b670cba6155cc252bd71de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.771ex; height:2.176ex;" alt="{\displaystyle i\in v}" loading="lazy"> to denote the edges touching the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in p}">

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<annotation encoding="application/x-tex">{\displaystyle i\in p}</annotation>

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</math><img src="./4219e50ddfbcf295d19ed53e214eec35fdc50724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.813ex; height:2.509ex;" alt="{\displaystyle i\in p}" loading="lazy"> to denote the edges surrounding the plaquette <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. The stabilizer space of the code is that for which all stabilizers act trivially, hence for any state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation>

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</math><img src="./cc27f1893b769a08cd6b296e115a29e61cab675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }" loading="lazy"> in this space it holds that

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<annotation encoding="application/x-tex">{\displaystyle A_{v}|\psi \rangle =|\psi \rangle ,\,\,\forall v,\,\,B_{p}|\psi \rangle =|\psi \rangle ,\,\,\forall p.}</annotation>

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</math><img src="./9ca18da11071b17d46ed023a77cf3ecd04184114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.005ex; height:3.009ex;" alt="{\displaystyle A_{v}|\psi \rangle =|\psi \rangle ,\,\,\forall v,\,\,B_{p}|\psi \rangle =|\psi \rangle ,\,\,\forall p.}" loading="lazy">

For the toric code, this space is four-dimensional, and so can be used to store two <a href="Qubit" title="Qubit">qubits</a> of <a href="Quantum_information" title="Quantum information">quantum information</a>. This can be proven by considering the number of independent stabilizer operators. The occurrence of errors will move the state out of the stabilizer space, resulting in vertices and plaquettes for which the above condition does not hold. The positions of these violations is the <a href="Syndrome_decoding" class="mw-redirect" title="Syndrome decoding">syndrome</a> of the code, which can be used for error correction.

The unique nature of the topological codes, such as the toric code, is that stabilizer violations can be interpreted as <a href="Quasiparticles" class="mw-redirect" title="Quasiparticles">quasiparticles</a>. Specifically, if the code is in a state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation>

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</math><img src="./312d43de853a9e6ca74888e63394fc8081f56a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }" loading="lazy"> such that,

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<annotation encoding="application/x-tex">{\displaystyle A_{v}|\phi \rangle =-|\phi \rangle }</annotation>

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</math><img src="./abb0257bc5ace49a2489ec67bbff7b8b605e36d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.553ex; height:2.843ex;" alt="{\displaystyle A_{v}|\phi \rangle =-|\phi \rangle }" loading="lazy">,

a quasiparticle known as an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> <a href="Anyon" title="Anyon">anyon</a> can be said to exist on the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">. Similarly violations of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{p}}</annotation>

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</math><img src="./18a1069ed15b2551691c1f85039842d7e7642f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.823ex; height:2.843ex;" alt="{\displaystyle B_{p}}" loading="lazy"> are associated with so called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> anyons on the plaquettes. The stabilizer space therefore corresponds to the anyonic vacuum. Single spin errors cause pairs of anyons to be created and transported around the lattice.

When errors create an anyon pair and move the anyons, one can imagine a path connecting the two composed of all links acted upon. If the anyons then meet and are annihilated, this path describes a loop. If the loop is topologically trivial, it 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, though re-annihilation of the anyons returns the state to the stabilizer space, it also implements a logical operation on the stored information. The errors, in this case, are therefore not corrected but consolidated.

Consider the noise model for which bit and phase errors occur independently on each spin, both with probability p. When 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 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%.<a href="#cite_note-6">[6]</a>

Other error models may also be considered, and thresholds found. In all cases studied so far, the code has been found to saturate the <a href="Hashing_bound" class="mw-redirect" title="Hashing bound">Hashing bound</a>. 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.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>

These thresholds are upper limits and are useless unless efficient algorithms are found to achieve them. The most well-used algorithm is <a href="Edmonds's_matching_algorithm" class="mw-redirect" title="Edmonds's matching algorithm">minimum weight perfect matching</a>.<a href="#cite_note-9">[9]</a> 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.

The means to perform <a href="Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a> on logical information stored within the toric 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 <a href="Quantum_gate" class="mw-redirect" title="Quantum gate">gates</a> cannot be fault-tolerantly implemented by unitary operations 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 <a href="One-way_quantum_computer" title="One-way quantum computer">measurement based</a> scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two-dimensional architecture.<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Hamiltonian_and_self-correction">Hamiltonian and self-correction</h2></div>

Since the stabilizer operators of the toric 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,

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<annotation encoding="application/x-tex">{\displaystyle H_{\rm {TC}}=-J_{e}\sum _{v}A_{v}-J_{m}\sum _{p}B_{p},\,\,\,J_{e},J_{m}>0.}</annotation>

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</math><img src="./b03dcbca96018cf2582c4bf895e8443abc8be793.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.782ex; height:5.676ex;" alt="{\displaystyle H_{\rm {TC}}=-J_{e}\sum _{v}A_{v}-J_{m}\sum _{p}B_{p},\,\,\,J_{e},J_{m}>0.}" loading="lazy">

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.<a href="#cite_note-12">[12]</a> However, the dynamic effects of such perturbations can still cause problems for the code.<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

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</math><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy">, but since arbitrary increases of this coupling are unrealistic, the protection given by the Hamiltonian still has its limits.

The means to make the toric code, or the planar 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.<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> Proposals have been made for realization of these in the lab <a href="#cite_note-17">[17]</a> Another approach is the generalization of the model to higher dimensions, with self-correction possible in 4D with only quasi-local interactions.<a href="#cite_note-18">[18]</a>

<div class="mw-heading mw-heading2"><h2 id="Anyon_model">Anyon model</h2></div>

As mentioned above, so called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> quasiparticles are associated with the vertices and plaquettes of the model, respectively. These quasiparticles can be described as <a href="Anyons" class="mw-redirect" title="Anyons">anyons</a>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">'s or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">'s having no effect, a full monodromy of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> will yield a phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

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<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

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</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy">. Such a result is not consistent with either <a href="Bosonic" class="mw-redirect" title="Bosonic">bosonic</a> or <a href="Fermionic" class="mw-redirect" title="Fermionic">fermionic</a> <a href="Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">statistics</a>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> anyons are similarly moved through the red loop above a logical operation will also result. The phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

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<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

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</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"> resulting when braiding the anyons shows that these operations do not commute, but rather anticommute. They may therefore be interpreted as logical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> Pauli operators on one of the stored qubits. The corresponding logical Pauli's on the other qubit correspond to an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> anyon following the blue loop and an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> anyon following the red. No braiding occurs when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> pass through parallel paths, the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> 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,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\times e=1,\,\,\,m\times m=1.}</annotation>

</semantics>

</math><img src="./a4f5420c67babdca6817817fc67701718b1a11fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.292ex; height:2.509ex;" alt="{\displaystyle e\times e=1,\,\,\,m\times m=1.}" loading="lazy">

Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> denotes the vacuum. A composite of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is not trivial. This therefore constitutes another quasiparticle in the model, sometimes denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">, with fusion rule,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\times m=\psi .}</annotation>

</semantics>

</math><img src="./e0658544961a9fd59f9ec5c6d7403c442e3200d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.223ex; height:2.509ex;" alt="{\displaystyle e\times m=\psi .}" loading="lazy">

From the braiding statistics of the anyons we see that, since any single exchange of two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">'s will involve a full monodromy of a constituent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">, a phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"> will result. This implies fermionic self-statistics for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">'s.

<div class="mw-heading mw-heading2"><h2 id="Ground_State_and_Excitations">Ground State and Excitations</h2></div>

Since the Hamiltonian is a sum of commuting projectors with eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation>

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</math><img src="./0bfeaa85da53ad1947d8000926cfea33827ef1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" loading="lazy">, the ground state is the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{GS}\rangle }">

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\psi _{GS}\rangle }</annotation>

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</math><img src="./6176df38f2b9d69dea45cc0570501b22479c19e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.649ex; height:2.843ex;" alt="{\displaystyle |\psi _{GS}\rangle }" loading="lazy"> that is a +1 eigenstate of every single star and plaquette operator:

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}|\psi _{GS}\rangle =+1|\psi _{GS}\rangle \quad \forall s}</annotation>

</semantics>

</math><img src="./32460cd221d7f42a824a30ce810676529e1e34c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.818ex; height:2.843ex;" alt="{\displaystyle A_{s}|\psi _{GS}\rangle =+1|\psi _{GS}\rangle \quad \forall s}" loading="lazy">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle B_{p}|\psi _{GS}\rangle =+1|\psi _{GS}\rangle \quad \forall p}</annotation>

</semantics>

</math><img src="./c227c88974238d3cca5043943830289d0e013cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.974ex; height:3.009ex;" alt="{\displaystyle B_{p}|\psi _{GS}\rangle =+1|\psi _{GS}\rangle \quad \forall p}" loading="lazy">

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

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

<ul><li>An electric charge (or e-particle) exists at a vertex s if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}=-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}=-1}</annotation>

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</math><img src="./0e2d4ac460864d6c8474161b2cb39e1e29d55054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.815ex; height:2.509ex;" alt="{\displaystyle A_{s}=-1}" loading="lazy">.</li>

<li>A magnetic vortex (or m-particle) exists at a plaquette p if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}=-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p}=-1}</annotation>

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</math><img src="./c4da09f1afdcc6f06d3be7a756e6597d4dfb5042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.892ex; height:2.843ex;" alt="{\displaystyle B_{p}=-1}" loading="lazy">.</li></ul>

These excitations are created in pairs at the ends of "string" operators. Applying a string of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{z}}</annotation>

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</math><img src="./652e198985e0a62a24868d93be5694afa644a123.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.332ex; height:2.343ex;" alt="{\displaystyle \sigma ^{z}}" loading="lazy"> operators along a path creates e-particles at its endpoints. Applying a string of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{x}}</annotation>

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</math><img src="./a97835fb405cf78fc95d93ad138551d37fcea399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.503ex; height:2.343ex;" alt="{\displaystyle \sigma ^{x}}" loading="lazy"> 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 of anyons due to their non-trivial braiding statistics.

<div class="mw-heading mw-heading2"><h2 id="Mutual_Statistics_of_Quasiparticles">Mutual Statistics of Quasiparticles</h2></div>

A defining feature of the Toric Code, and of topological order in general, is that its quasiparticle excitations exhibit non-trivial braiding statistics. While the electric charges (e) and magnetic fluxes (m) are both individually bosons with respect to themselves, they have a non-trivial mutual statistics with respect to each other. Specifically, they are mutual semions: adiabatically moving an e particle in a full counterclockwise cycle around an m particle imparts a phase of -1 to the system's wavefunction. This is a topological analogue of the <a href="Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a>, 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 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}=+1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}=+1}</annotation>

</semantics>

</math><img src="./bffd0e94d6fb0996801b36ec947fecb93ada2707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.815ex; height:2.509ex;" alt="{\displaystyle A_{s}=+1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}=+1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p}=+1}</annotation>

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</math><img src="./c0c0d47dfa3ff0bb9fda4a0e478f4149bf0c9ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.892ex; height:2.843ex;" alt="{\displaystyle B_{p}=+1}" loading="lazy"> for all s, p).

<ul><li>A stationary magnetic flux (m) is created at a plaquette p by applying a string operator of Pauli-X matrices, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{m}(p)}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{m}(p)}</annotation>

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</math><img src="./891c01c1e681be6a932c7dc01111eeef6616bd24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle W_{m}(p)}" loading="lazy">, along a path on the dual lattice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle C_{m}}</annotation>

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</math><img src="./957e2166e45b11d83b27710666eb7aa38f5e4b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.337ex; height:2.509ex;" alt="{\displaystyle C_{m}}" loading="lazy"> from a faraway plaquette q to p. This flips the sign of the plaquette stabilizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle B_{p}}</annotation>

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</math><img src="./18a1069ed15b2551691c1f85039842d7e7642f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.823ex; height:2.843ex;" alt="{\displaystyle B_{p}}" loading="lazy"> to -1.</li>

<li>An electric charge (e) is moved along a path by applying a string operator of Pauli-Z matrices. A closed loop of Z operators, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(C)}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)}</annotation>

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</math><img src="./7b5ef320571cc9d9750c3defafd857d64edc9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.157ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)}" loading="lazy">, corresponds to creating an e particle, moving it around the closed path C, and then annihilating it.</li></ul>

2. The Braiding Process: Consider the process of moving an e charge in a closed loop C that encloses the plaquette p where the stationary m particle resides. The final state of the system is given by applying the loop operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(C)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)}</annotation>

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</math><img src="./7b5ef320571cc9d9750c3defafd857d64edc9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.157ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)}" loading="lazy"> to the state containing the m particle:

</mrow>

<msub>

<mtext>final</mtext>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |\Psi _{\text{final}}\rangle =L_{e}(C)|m\rangle =L_{e}(C)W_{m}(p)|\Psi _{0}\rangle }</annotation>

</semantics>

</math><img src="./f670405f51bf5802ef10d89a38721155d89131e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.11ex; height:2.843ex;" alt="{\displaystyle |\Psi _{\text{final}}\rangle =L_{e}(C)|m\rangle =L_{e}(C)W_{m}(p)|\Psi _{0}\rangle }" loading="lazy">

</pre>

3. The Algebraic Origin of the Phase: The key insight comes from the commutation relation between the loop operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(C)}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)}</annotation>

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</math><img src="./7b5ef320571cc9d9750c3defafd857d64edc9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.157ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)}" loading="lazy"> and the string operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{m}(p)}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W_{m}(p)}</annotation>

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</math><img src="./891c01c1e681be6a932c7dc01111eeef6616bd24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle W_{m}(p)}" loading="lazy">.

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)}</annotation>

</semantics>

</math><img src="./7b5ef320571cc9d9750c3defafd857d64edc9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.157ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)}" loading="lazy"> is a product of Z operators on the edges forming the loop C.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{m}(p)}</annotation>

</semantics>

</math><img src="./891c01c1e681be6a932c7dc01111eeef6616bd24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle W_{m}(p)}" loading="lazy"> is a product of X operators on the edges forming the path <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{m}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{m}}</annotation>

</semantics>

</math><img src="./957e2166e45b11d83b27710666eb7aa38f5e4b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.337ex; height:2.509ex;" alt="{\displaystyle C_{m}}" loading="lazy">.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{m}}</annotation>

</semantics>

</math><img src="./957e2166e45b11d83b27710666eb7aa38f5e4b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.337ex; height:2.509ex;" alt="{\displaystyle C_{m}}" loading="lazy"> must cross at exactly one edge. Let this edge be j.</li>

<li>For any edge i ≠ j, the operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{i}}</annotation>

</semantics>

</math><img src="./b102e069bd5758b0c3d0511c68a12827a36dd761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle Z_{i}}" loading="lazy"> from the loop and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> from the string commute. However, at the crossing edge j, the operators anti-commute: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{j}X_{j}=-X_{j}Z_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{j}X_{j}=-X_{j}Z_{j}}</annotation>

</semantics>

</math><img src="./4e791a506701fd1711179254c2af9d2a7f1bdf16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.569ex; height:2.843ex;" alt="{\displaystyle Z_{j}X_{j}=-X_{j}Z_{j}}" loading="lazy">.</li></ul>

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

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)W_{m}(p)=-W_{m}(p)L_{e}(C)}</annotation>

</semantics>

</math><img src="./e211769d0a1340219f7352bd0509c582dc06532a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.915ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)W_{m}(p)=-W_{m}(p)L_{e}(C)}" loading="lazy">

</pre>

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

</mrow>

<msub>

<mtext>final</mtext>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\Psi _{\text{final}}\rangle =L_{e}(C)W_{m}(p)|\Psi _{0}\rangle =-W_{m}(p)L_{e}(C)|\Psi _{0}\rangle }</annotation>

</semantics>

</math><img src="./3251d91accf752f9e932d439c3adb7089f998c62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.587ex; height:2.843ex;" alt="{\displaystyle |\Psi _{\text{final}}\rangle =L_{e}(C)W_{m}(p)|\Psi _{0}\rangle =-W_{m}(p)L_{e}(C)|\Psi _{0}\rangle }" loading="lazy">

</pre>

The operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(C)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)}</annotation>

</semantics>

</math><img src="./7b5ef320571cc9d9750c3defafd857d64edc9711.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.157ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)}" loading="lazy"> is a closed loop of Z operators. Any such operator can be written as a product of the vertex stabilizers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}}</annotation>

</semantics>

</math><img src="./5cc9b664ef7e1dca131e7f345b4321bd3a07a7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.509ex;" alt="{\displaystyle A_{s}}" loading="lazy"> for all vertices s inside the loop.<a href="#cite_note-Kitaev2003-19">[19]</a> Since the ground state |Ψ0⟩ is an eigenstate of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}}</annotation>

</semantics>

</math><img src="./5cc9b664ef7e1dca131e7f345b4321bd3a07a7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.509ex;" alt="{\displaystyle A_{s}}" loading="lazy"> with eigenvalue +1, the loop operator leaves the ground state unchanged: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(C)|\psi _{0}\rangle =|\psi _{0}\rangle }">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{e}(C)|\psi _{0}\rangle =|\psi _{0}\rangle }</annotation>

</semantics>

</math><img src="./21a8eed52246270e30c00da58d88d3ccd92e5401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.493ex; height:2.843ex;" alt="{\displaystyle L_{e}(C)|\psi _{0}\rangle =|\psi _{0}\rangle }" loading="lazy">. Therefore, the final state is:

</mrow>

<msub>

<mtext>final</mtext>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\Psi _{\text{final}}\rangle =-W_{m}(p)|\Psi _{0}\rangle =-|m\rangle }</annotation>

</semantics>

</math><img src="./86a99ee61908731b2e2f58006be09b3bdfc050b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.412ex; height:2.843ex;" alt="{\displaystyle |\Psi _{\text{final}}\rangle =-W_{m}(p)|\Psi _{0}\rangle =-|m\rangle }" loading="lazy">

</pre>

The system returns to its initial state (a single m particle at plaquette p), but its wavefunction has acquired a phase factor of -1. This result is topological because it is independent of the exact shape of the loop C, so long as it encloses the m particle. This non-trivial mutual statistics is a fundamental signature of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{2}}</annotation>

</semantics>

</math><img src="./c98d433ae289ecb2b88f895b407538b0e4183b28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{2}}" loading="lazy"> topological order present in the Toric Code and is the basis for proposals to use such systems for fault-tolerant quantum information processing.<a href="#cite_note-Nayak2008-20">[20]</a>

<div class="mw-heading mw-heading2"><h2 id="Ground_State_Degeneracy_on_a_Torus">Ground State Degeneracy on a Torus</h2></div>

On a torus, the local constraints <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}=1}</annotation>

</semantics>

</math><img src="./cfc9341a9da138eabf812bc471a11bd448476328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.509ex;" alt="{\displaystyle A_{s}=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p}=1}</annotation>

</semantics>

</math><img src="./84752afd4af48fec48953a39ccd9df73e49684c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.084ex; height:2.843ex;" alt="{\displaystyle B_{p}=1}" loading="lazy"> 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 the logical operators or Wilson loops.

<div class="mw-heading mw-heading3"><h3 id="Logical_Operators:_Wilson_and_'t_Hooft_Loops">Logical Operators: Wilson and 't Hooft Loops</h3></div>

A torus has two independent non-contractible loops (or cycles), often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

</semantics>

</math><img src="./babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> (e.g., "horizontal") and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation>

</semantics>

</math><img src="./7ec545f7870665e1028b7492746848d149878808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}" loading="lazy"> (e.g., "vertical"). We can define four logical operators corresponding to strings of Pauli operators wrapping around these loops.

1. Electric Wilson Loops (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{(e)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W^{(e)}}</annotation>

</semantics>

</math><img src="./1bb677f174552fe74f787335b59c35c18859513a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.786ex; height:2.843ex;" alt="{\displaystyle W^{(e)}}" loading="lazy">): These are products of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{z}}</annotation>

</semantics>

</math><img src="./652e198985e0a62a24868d93be5694afa644a123.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.332ex; height:2.343ex;" alt="{\displaystyle \sigma ^{z}}" loading="lazy"> operators along the non-contractible loops.

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}=\prod _{i\in C_{1}}\sigma _{i}^{z}}</annotation>

</semantics>

</math><img src="./055e049ebcf72fecfa43920374ea643b4d472289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.274ex; height:6.009ex;" alt="{\displaystyle W_{1}^{(e)}=\prod _{i\in C_{1}}\sigma _{i}^{z}}" loading="lazy">

* <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}^{(e)}=\prod _{i\in C_{2}}\sigma _{i}^{z}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{2}^{(e)}=\prod _{i\in C_{2}}\sigma _{i}^{z}}</annotation>

</semantics>

</math><img src="./53760bf945b4396f751dc7de94d647b276a3abe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.274ex; height:6.009ex;" alt="{\displaystyle W_{2}^{(e)}=\prod _{i\in C_{2}}\sigma _{i}^{z}}" loading="lazy">

</pre>

2. Magnetic 't Hooft Loops (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{(m)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W^{(m)}}</annotation>

</semantics>

</math><img src="./ed31899b2a2eb6c5e0d0d2a73dbc7293bb98e4c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.843ex;" alt="{\displaystyle W^{(m)}}" loading="lazy">): These are products of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{x}}</annotation>

</semantics>

</math><img src="./a97835fb405cf78fc95d93ad138551d37fcea399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.503ex; height:2.343ex;" alt="{\displaystyle \sigma ^{x}}" loading="lazy"> operators along non-contractible loops on the dual lattice, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1}^{*}}</annotation>

</semantics>

</math><img src="./4bbf361be926883a6e065eef4eb40574c19c971a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{1}^{*}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}^{*}}</annotation>

</semantics>

</math><img src="./546459155d231f9b0865b4a37a6f38f1f14b351f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{2}^{*}}" loading="lazy">, which intersect <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation>

</semantics>

</math><img src="./7ec545f7870665e1028b7492746848d149878808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

</semantics>

</math><img src="./babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> respectively.

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(m)}=\prod _{i\in C_{1}^{*}}\sigma _{i}^{x}}</annotation>

</semantics>

</math><img src="./326cdb27a720c72d2b22c4068e5b606d6a91bd43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:15.217ex; height:6.343ex;" alt="{\displaystyle W_{1}^{(m)}=\prod _{i\in C_{1}^{*}}\sigma _{i}^{x}}" loading="lazy">

* <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}^{(m)}=\prod _{i\in C_{2}^{*}}\sigma _{i}^{x}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{2}^{(m)}=\prod _{i\in C_{2}^{*}}\sigma _{i}^{x}}</annotation>

</semantics>

</math><img src="./4373bfdec6e8f4166cb3dcd68621a52d97b3249c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:15.217ex; height:6.343ex;" alt="{\displaystyle W_{2}^{(m)}=\prod _{i\in C_{2}^{*}}\sigma _{i}^{x}}" loading="lazy">

</pre>

These loop operators all commute with the Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">. For example, a loop operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}^{(e)}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}}</annotation>

</semantics>

</math><img src="./342d758a14bb4d8f253f6d056828210f5aea9845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.786ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}}" loading="lazy"> 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.

<div class="mw-heading mw-heading3"><h3 id="The_Algebra_of_Logical_Operators">The Algebra of Logical Operators</h3></div>

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

<ul><li>Operators of the same type always commute: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [W_{i}^{(e)},W_{j}^{(e)}]=0}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [W_{i}^{(e)},W_{j}^{(e)}]=0}</annotation>

</semantics>

</math><img src="./f82a2cbfdd4502a1a5ff38c57734a27f3dfa4921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.16ex; height:4.009ex;" alt="{\displaystyle [W_{i}^{(e)},W_{j}^{(e)}]=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [W_{i}^{(m)},W_{j}^{(m)}]=0}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [W_{i}^{(m)},W_{j}^{(m)}]=0}</annotation>

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</math><img src="./ed5a57c6f1dee187df4787d360ceb39b7d510fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.513ex; height:4.009ex;" alt="{\displaystyle [W_{i}^{(m)},W_{j}^{(m)}]=0}" loading="lazy">.</li>

<li>Loops wrapping around different cycles do not intersect, and thus they commute:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle [W_{1}^{(e)},W_{1}^{(m)}]=0}</annotation>

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</math><img src="./3bec726493fbb69d2b2d0adf7331f283e17e2fa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.836ex; height:3.676ex;" alt="{\displaystyle [W_{1}^{(e)},W_{1}^{(m)}]=0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle [W_{2}^{(e)},W_{2}^{(m)}]=0}</annotation>

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</math><img src="./eba22f3f606f4080171b8a1e5364a90ae5910585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.836ex; height:3.676ex;" alt="{\displaystyle [W_{2}^{(e)},W_{2}^{(m)}]=0}" loading="lazy">

</pre>

<ul><li>Crucially, an electric loop and a magnetic loop wrapping around corresponding cycles intersect at exactly one qubit. For example, the loop <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

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</math><img src="./babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> and the dual loop <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{2}^{*}}</annotation>

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</math><img src="./546459155d231f9b0865b4a37a6f38f1f14b351f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{2}^{*}}" loading="lazy"> must cross once.</li></ul>

Let's examine the commutation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}^{(e)}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}}</annotation>

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</math><img src="./342d758a14bb4d8f253f6d056828210f5aea9845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.786ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}^{(m)}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{2}^{(m)}}</annotation>

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</math><img src="./6619a8c1987771ce7a5db77efdf20853a593f756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.462ex; height:3.676ex;" alt="{\displaystyle W_{2}^{(m)}}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}W_{2}^{(m)}=\left(\prod _{i\in C_{1}}\sigma _{i}^{z}\right)\left(\prod _{j\in C_{2}^{*}}\sigma _{j}^{x}\right)}</annotation>

</semantics>

</math><img src="./9b0f55718f9b1bdfc95c7b5359c6672d23f205d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:34.638ex; height:8.509ex;" alt="{\displaystyle W_{1}^{(e)}W_{2}^{(m)}=\left(\prod _{i\in C_{1}}\sigma _{i}^{z}\right)\left(\prod _{j\in C_{2}^{*}}\sigma _{j}^{x}\right)}" loading="lazy">

The two strings of operators commute for every qubit except for the single qubit where the loops <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

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</math><img src="./babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{2}^{*}}</annotation>

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</math><img src="./546459155d231f9b0865b4a37a6f38f1f14b351f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{2}^{*}}" loading="lazy"> intersect. At that intersection point, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{z}\sigma ^{x}=-\sigma ^{x}\sigma ^{z}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{z}\sigma ^{x}=-\sigma ^{x}\sigma ^{z}}</annotation>

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</math><img src="./bd6be3c645526ff1a6925c099532f0238bf404a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.577ex; height:2.509ex;" alt="{\displaystyle \sigma ^{z}\sigma ^{x}=-\sigma ^{x}\sigma ^{z}}" loading="lazy">. Because there is only one such anticommutation, the operators as a whole anticommute:

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}W_{2}^{(m)}=-W_{2}^{(m)}W_{1}^{(e)}\quad \implies \quad \{W_{1}^{(e)},W_{2}^{(m)}\}=0}</annotation>

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</math><img src="./ebc184c0f55e5100db2cde1982f06ee5d8bf74c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.301ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}W_{2}^{(m)}=-W_{2}^{(m)}W_{1}^{(e)}\quad \implies \quad \{W_{1}^{(e)},W_{2}^{(m)}\}=0}" loading="lazy">

Similarly, for the second cycle:

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<annotation encoding="application/x-tex">{\displaystyle \{W_{2}^{(e)},W_{1}^{(m)}\}=0}</annotation>

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</math><img src="./d6a697319e366f50a1d0407dc5b68fb5fea9d934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.868ex; height:3.676ex;" alt="{\displaystyle \{W_{2}^{(e)},W_{1}^{(m)}\}=0}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Construction_of_the_4-Fold_Degenerate_Ground_States">Construction of the 4-Fold Degenerate Ground States</h3></div>

We have two pairs of operators, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{1}^{(e)},W_{2}^{(m)})}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{1}^{(e)},W_{2}^{(m)})}</annotation>

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</math><img src="./3ad699c923096f729529152d64b399f36b36ae5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.091ex; height:3.676ex;" alt="{\displaystyle (W_{1}^{(e)},W_{2}^{(m)})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{2}^{(e)},W_{1}^{(m)})}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{2}^{(e)},W_{1}^{(m)})}</annotation>

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</math><img src="./6231514e09833b17b2db2c5e60b79f2218db927b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.091ex; height:3.676ex;" alt="{\displaystyle (W_{2}^{(e)},W_{1}^{(m)})}" loading="lazy">, that each obey the algebra of a logical qubit's Pauli operators (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{L}\equiv W^{(e)},X_{L}\equiv W^{(m)}}">

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<annotation encoding="application/x-tex">{\displaystyle Z_{L}\equiv W^{(e)},X_{L}\equiv W^{(m)}}</annotation>

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</math><img src="./13ef366fcba8a6ff785f58cb234430c023d8b3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.693ex; height:3.176ex;" alt="{\displaystyle Z_{L}\equiv W^{(e)},X_{L}\equiv W^{(m)}}" loading="lazy">). Since these two pairs act independently (commute with each other), they describe two independent logical qubits.

A system of two independent qubits has a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2=4}">

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<annotation encoding="application/x-tex">{\displaystyle 2\times 2=4}</annotation>

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</math><img src="./4e7a3168e66ac9bce53f8647304c799d746860d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.426ex; height:2.176ex;" alt="{\displaystyle 2\times 2=4}" loading="lazy">-dimensional state space. This implies that the ground state subspace of the toric code on a torus is four-fold degenerate.

We can explicitly construct these four states.

1. Start with one ground state, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{0}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi _{0}\rangle }</annotation>

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</math><img src="./c6c1c429af2a5f67ca03dabc80a872dfd9a3768b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.119ex; height:2.843ex;" alt="{\displaystyle |\psi _{0}\rangle }" loading="lazy">, which satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}=1}">

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<annotation encoding="application/x-tex">{\displaystyle A_{s}=1}</annotation>

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</math><img src="./cfc9341a9da138eabf812bc471a11bd448476328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.007ex; height:2.509ex;" alt="{\displaystyle A_{s}=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}=1}">

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<annotation encoding="application/x-tex">{\displaystyle B_{p}=1}</annotation>

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</math><img src="./84752afd4af48fec48953a39ccd9df73e49684c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.084ex; height:2.843ex;" alt="{\displaystyle B_{p}=1}" loading="lazy"> for all s, p. Let's define it as a +1 eigenstate of the electric operators:

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}|\psi _{0}\rangle =+1|\psi _{0}\rangle }</annotation>

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</math><img src="./c986c0e33f1e7bb539c1b1ab69cd4a2d4e0abe11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.092ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}|\psi _{0}\rangle =+1|\psi _{0}\rangle }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle W_{2}^{(e)}|\psi _{0}\rangle =+1|\psi _{0}\rangle }</annotation>

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</math><img src="./cbcc10c37ac137dabae8a3deac0d940ae33a3307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.092ex; height:3.676ex;" alt="{\displaystyle W_{2}^{(e)}|\psi _{0}\rangle =+1|\psi _{0}\rangle }" loading="lazy">

</pre>

2. We can now generate the other three orthogonal ground states by acting with the magnetic 't Hooft loop operators. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}^{(m)}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle W_{2}^{(m)}}</annotation>

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</math><img src="./6619a8c1987771ce7a5db77efdf20853a593f756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.462ex; height:3.676ex;" alt="{\displaystyle W_{2}^{(m)}}" loading="lazy"> anticommutes with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}^{(e)}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}}</annotation>

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</math><img src="./342d758a14bb4d8f253f6d056828210f5aea9845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.786ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}}" loading="lazy">, acting with it flips the eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}^{(e)}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{1}^{(e)}}</annotation>

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</math><img src="./342d758a14bb4d8f253f6d056828210f5aea9845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.786ex; height:3.676ex;" alt="{\displaystyle W_{1}^{(e)}}" loading="lazy"> from +1 to -1.

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<annotation encoding="application/x-tex">{\displaystyle |\psi _{1}\rangle =W_{2}^{(m)}|\psi _{0}\rangle }</annotation>

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</math><img src="./c294f35bcc6d2a733bb19a3afa83fb06029076f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.798ex; height:3.676ex;" alt="{\displaystyle |\psi _{1}\rangle =W_{2}^{(m)}|\psi _{0}\rangle }" loading="lazy">. This state has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,+1)}">

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<annotation encoding="application/x-tex">{\displaystyle (-1,+1)}</annotation>

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</math><img src="./833e170cbe821f4fa6b1cca3ce85db4f533f3529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle (-1,+1)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}</annotation>

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</math><img src="./a5804384bfc3d3c78737a70d105036ca000a95ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.414ex; height:3.676ex;" alt="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}" loading="lazy">.

* <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{2}\rangle =W_{1}^{(m)}|\psi _{0}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi _{2}\rangle =W_{1}^{(m)}|\psi _{0}\rangle }</annotation>

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</math><img src="./1c567005d86a8a61c33df1294612092c9df6c774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.798ex; height:3.676ex;" alt="{\displaystyle |\psi _{2}\rangle =W_{1}^{(m)}|\psi _{0}\rangle }" loading="lazy">. This state has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+1,-1)}">

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<annotation encoding="application/x-tex">{\displaystyle (+1,-1)}</annotation>

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</math><img src="./f284dc01a5b99cbe421bb918930db0831029b6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle (+1,-1)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}</annotation>

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</math><img src="./a5804384bfc3d3c78737a70d105036ca000a95ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.414ex; height:3.676ex;" alt="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}" loading="lazy">.

* <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{3}\rangle =W_{1}^{(m)}W_{2}^{(m)}|\psi _{0}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |\psi _{3}\rangle =W_{1}^{(m)}W_{2}^{(m)}|\psi _{0}\rangle }</annotation>

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</math><img src="./300ab7fbbae104d99d144938cd91087568f353e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.261ex; height:3.676ex;" alt="{\displaystyle |\psi _{3}\rangle =W_{1}^{(m)}W_{2}^{(m)}|\psi _{0}\rangle }" loading="lazy">. This state has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,-1)}">

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<annotation encoding="application/x-tex">{\displaystyle (-1,-1)}</annotation>

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</math><img src="./c96b373a30a5cd3f87bba41b65da5fb522ea58a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle (-1,-1)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}</annotation>

</semantics>

</math><img src="./a5804384bfc3d3c78737a70d105036ca000a95ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.414ex; height:3.676ex;" alt="{\displaystyle (W_{1}^{(e)},W_{2}^{(e)})}" loading="lazy">.

</pre>

These four states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|\psi _{0}\rangle ,|\psi _{1}\rangle ,|\psi _{2}\rangle ,|\psi _{3}\rangle \}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{|\psi _{0}\rangle ,|\psi _{1}\rangle ,|\psi _{2}\rangle ,|\psi _{3}\rangle \}}</annotation>

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</math><img src="./51d3565bb63db6782755190065412f41e846c261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.902ex; height:2.843ex;" alt="{\displaystyle \{|\psi _{0}\rangle ,|\psi _{1}\rangle ,|\psi _{2}\rangle ,|\psi _{3}\rangle \}}" loading="lazy"> are all degenerate in energy (they are all +1 eigenstates of all local <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{s}}</annotation>

</semantics>

</math><img src="./5cc9b664ef7e1dca131e7f345b4321bd3a07a7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.509ex;" alt="{\displaystyle A_{s}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p}}</annotation>

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</math><img src="./18a1069ed15b2551691c1f85039842d7e7642f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.823ex; height:2.843ex;" alt="{\displaystyle B_{p}}" loading="lazy"> operators), and they are mutually orthogonal. They form a basis for the 4-dimensional ground state subspace.

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

The use of a torus is not required to form an error correcting code. Other surfaces may also be used, with their topological properties determining the degeneracy of the stabilizer space. In general, quantum error correcting codes defined on two-dimensional spin lattices according to the principles above are known as surface codes.<a href="#cite_note-21">[21]</a>

It is also possible to define similar codes using higher-dimensional spins. These are the quantum double models<a href="#cite_note-22">[22]</a> and <a href="String-net_liquid" title="String-net liquid">string-net</a> models,<a href="#cite_note-23">[23]</a> which allow a greater richness in the behaviour of anyons, and so may be used for more advanced quantum computation and error correction proposals.<a href="#cite_note-24">[24]</a> These not only include models with Abelian anyons, but also those with non-Abelian statistics.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a><a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading2"><h2 id="Experimental_progress">Experimental progress</h2></div>

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<a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a><a href="#cite_note-:0-30">[30]</a> and measurement of the <a href="Topological_entanglement_entropy" class="mw-redirect" title="Topological entanglement entropy">topological entanglement entropy</a>.<a href="#cite_note-:0-30">[30]</a> More recent experiments have also been able to demonstrate the error correction properties of the code.<a href="#cite_note-31">[31]</a><a href="#cite_note-:0-30">[30]</a>

For realizations of the toric code and its generalizations with a Hamiltonian, much progress has been made using <a href="Josephson_junctions" class="mw-redirect" title="Josephson junctions">Josephson junctions</a>. The theory of how the Hamiltonians may be implemented has been developed for a wide class of topological codes.<a href="#cite_note-32">[32]</a> 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.<a href="#cite_note-33">[33]</a>

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,

<a href="#cite_note-34">[34]</a> as have experiments concerning minimal instances of topological order.<a href="#cite_note-35">[35]</a> Such minimal instances of the toric code has been realized experimentally within isolated square plaquettes.<a href="#cite_note-36">[36]</a> <a href="Quantum_spin_liquid#Toric_code_type" title="Quantum spin liquid">Progress</a> is also being made into simulations of the toric model with <a href="Rydberg_atoms" class="mw-redirect" title="Rydberg atoms">Rydberg atoms</a>, in which the Hamiltonian and the effects of dissipative noise can be demonstrated.<a href="#cite_note-37">[37]</a><a href="#cite_note-38">[38]</a> 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.<a href="#cite_note-CoherentArray2-39">[39]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> A. Y. Kitaev, Proceedings of the 3rd International Conference of Quantum Communication and Measurement, Ed. O. Hirota, <a href="Alexander_Holevo" title="Alexander Holevo">A. S. Holevo</a>, and C. M. Caves (New York, Plenum, 1997)

</li>

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</style><cite id="CITEREFKitaev2006" class="citation journal cs1">Kitaev, Alexei (2006). "Anyons in an exactly solved model and beyond". Annals of Physics. 321 (1): 2–111. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0506438">cond-mat/0506438</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006AnPhy.321....2K">2006AnPhy.321....2K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2005.10.005">10.1016/j.aop.2005.10.005</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4916">0003-4916</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118948929">118948929</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFReadSachdev1991" class="citation journal cs1">Read, N.; Sachdev, Subir (1 March 1991). "Large-Nexpansion for frustrated quantum antiferromagnets". Physical Review Letters. 66 (13): 1773–1776. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvL..66.1773R">1991PhRvL..66.1773R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.66.1773">10.1103/physrevlett.66.1773</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10043303">10043303</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFWen1991" class="citation journal cs1"><a href="Xiao-Gang_Wen" title="Xiao-Gang Wen">Wen, X. G.</a> (1 July 1991). "Mean-field theory of spin-liquid states with finite energy gap and topological orders". Physical Review B. 44 (6): 2664–2672. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvB..44.2664W">1991PhRvB..44.2664W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.44.2664">10.1103/physrevb.44.2664</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0163-1829">0163-1829</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9999836">9999836</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFFradkinShenker1979" class="citation journal cs1">Fradkin, Eduardo; Shenker, Stephen H. (15 June 1979). "Phase diagrams of lattice gauge theories with Higgs fields". Physical Review D. 19 (12): 3682–3697. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979PhRvD..19.3682F">1979PhRvD..19.3682F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.19.3682">10.1103/physrevd.19.3682</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0556-2821">0556-2821</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFDennisKitaevLandahlPreskill2002" class="citation journal cs1">Dennis, Eric; Kitaev, Alexei; Landahl, Andrew; Preskill, John (2002). "Topological quantum memory". Journal of Mathematical Physics. 43 (9): 4452–4505. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0110143">quant-ph/0110143</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JMP....43.4452D">2002JMP....43.4452D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1499754">10.1063/1.1499754</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:36673677">36673677</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFRöthlisbergerWoottonHeathPachos2012" class="citation journal cs1">Röthlisberger, Beat; Wootton, James R.; Heath, Robert M.; Pachos, Jiannis K.; Loss, Daniel (13 February 2012). "Incoherent dynamics in the toric code subject to disorder". Physical Review A. 85 (2): 022313. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.1613">1112.1613</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhRvA..85b2313R">2012PhRvA..85b2313R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.85.022313">10.1103/physreva.85.022313</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118585279">118585279</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFBombinAndristOhzekiKatzgraber2012" class="citation journal cs1">Bombin, H.; Andrist, Ruben S.; Ohzeki, Masayuki; Katzgraber, Helmut G.; Martin-Delgado, M. A. (30 April 2012). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevx.2.021004">"Strong Resilience of Topological Codes to Depolarization"</a>. Physical Review X. 2 (2): 021004. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1202.1852">1202.1852</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhRvX...2b1004B">2012PhRvX...2b1004B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevx.2.021004">10.1103/physrevx.2.021004</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2160-3308">2160-3308</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFEdmonds1965" class="citation journal cs1">Edmonds, Jack (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1965-045-4">"Paths, Trees, and Flowers"</a>. Canadian Journal of Mathematics. 17: 449–467. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1965-045-4">10.4153/cjm-1965-045-4</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-414X">0008-414X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:247198603">247198603</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFRaussendorfHarrington2007" class="citation journal cs1">Raussendorf, Robert; Harrington, Jim (11 May 2007). "Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions". Physical Review Letters. 98 (19): 190504. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0610082">quant-ph/0610082</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007PhRvL..98s0504R">2007PhRvL..98s0504R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.98.190504">10.1103/physrevlett.98.190504</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17677613">17677613</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39504821">39504821</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFRaussendorfHarringtonGoyal2007" class="citation journal cs1">Raussendorf, R; Harrington, J; Goyal, K (29 June 2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F9%2F6%2F199">"Topological fault-tolerance in cluster state quantum computation"</a>. New Journal of Physics. 9 (6): 199. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0703143">quant-ph/0703143</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007NJPh....9..199R">2007NJPh....9..199R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F9%2F6%2F199">10.1088/1367-2630/9/6/199</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFBravyiHastingsMichalakis2010" class="citation journal cs1">Bravyi, Sergey; Hastings, Matthew B.; Michalakis, Spyridon (2010). "Topological quantum order: Stability under local perturbations". Journal of Mathematical Physics. 51 (9): 093512. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1001.0344">1001.0344</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010JMP....51i3512B">2010JMP....51i3512B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3490195">10.1063/1.3490195</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-2488">0022-2488</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115166306">115166306</a>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFF._PastawskiA._KayN._SchuchJ._I._Cirac2010" class="citation journal cs1">F. Pastawski; A. Kay; N. Schuch; J. I. Cirac (2010). "Limitations of passive protection of quantum information". Quantum Information and Computation. 10 (7&8): 580. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.3843">0911.3843</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.26421%2Fqic10.7-8">10.26421/qic10.7-8</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1533-7146">1533-7146</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3076085">3076085</a>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFFreemanHerdmanGormanWhaley2014" class="citation journal cs1">Freeman, C. Daniel; Herdman, C. M.; Gorman, D. J.; Whaley, K. B. (7 October 2014). "Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime". Physical Review B. 90 (13): 134302. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1405.2315">1405.2315</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvB..90m4302F">2014PhRvB..90m4302F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.90.134302">10.1103/physrevb.90.134302</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118724410">118724410</a>.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFHammaCastelnovoChamon2009" class="citation journal cs1">Hamma, Alioscia; Castelnovo, Claudio; Chamon, Claudio (18 June 2009). "Toric-boson model: Toward a topological quantum memory at finite temperature". Physical Review B. 79 (24): 245122. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0812.4622">0812.4622</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PhRvB..79x5122H">2009PhRvB..79x5122H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.79.245122">10.1103/physrevb.79.245122</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F51820">1721.1/51820</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5202832">5202832</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFChesiRöthlisbergerLoss2010" class="citation journal cs1">Chesi, Stefano; Röthlisberger, Beat; Loss, Daniel (6 August 2010). "Self-correcting quantum memory in a thermal environment". Physical Review A. 82 (2): 022305. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.4264">0908.4264</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PhRvA..82b2305C">2010PhRvA..82b2305C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.82.022305">10.1103/physreva.82.022305</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118400202">118400202</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFPedrocchiChesiLoss2011" class="citation journal cs1">Pedrocchi, Fabio L.; Chesi, Stefano; Loss, Daniel (10 March 2011). "Quantum memory coupled to cavity modes". Physical Review B. 83 (11): 115415. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1011.3762">1011.3762</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhRvB..83k5415P">2011PhRvB..83k5415P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.83.115415">10.1103/physrevb.83.115415</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118595257">118595257</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFAlickiHorodeckiHorodeckiHorodecki2010" class="citation journal cs1">Alicki, R.; Horodecki, M.; Horodecki, P.; Horodecki, R. (2010). "On Thermal Stability of Topological Qubit in Kitaev's 4D Model". Open Systems & Information Dynamics. 17 (1): 1–20. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.0033">0811.0033</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs1230161210000023">10.1142/s1230161210000023</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1230-1612">1230-1612</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:26719502">26719502</a>.</cite>

</li>

<li id="cite_note-Kitaev2003-19"><a href="#cite_ref-Kitaev2003_19-0">^</a> <cite id="CITEREFKitaev2003" class="citation journal cs1">Kitaev, A. Y. (2003). "Fault-tolerant quantum computation by anyons". Annals of Physics. 303 (1): 2–30. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9707021">quant-ph/9707021</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003AnPhy.303....2K">2003AnPhy.303....2K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0003-4916%2802%2900018-0">10.1016/S0003-4916(02)00018-0</a>.</cite>

</li>

<li id="cite_note-Nayak2008-20"><a href="#cite_ref-Nayak2008_20-0">^</a> <cite id="CITEREFNayakSimonSternFreedman2008" class="citation journal cs1">Nayak, C.; Simon, S. H.; Stern, A.; Freedman, M.; Das Sarma, S. (2008). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics. 80 (3): 1083–1159. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0707.1889">0707.1889</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008RvMP...80.1083N">2008RvMP...80.1083N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.80.1083">10.1103/RevModPhys.80.1083</a>.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFGhoshFowlerGeller2012" class="citation journal cs1">Ghosh, Joydip; Fowler, Austin G.; Geller, Michael R. (19 December 2012). "Surface code with decoherence: An analysis of three superconducting architectures". Physical Review A. 86 (6): 062318. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1210.5799">1210.5799</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhRvA..86f2318G">2012PhRvA..86f2318G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.86.062318">10.1103/physreva.86.062318</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10196488">10196488</a>.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFBullockBrennen2007" class="citation journal cs1">Bullock, Stephen S; Brennen, Gavin K (14 March 2007). "Qudit surface codes and gauge theory with finite cyclic groups". Journal of Physics A: Mathematical and Theoretical. 40 (13): 3481–3505. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0609070">quant-ph/0609070</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JPhA...40.3481B">2007JPhA...40.3481B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F40%2F13%2F013">10.1088/1751-8113/40/13/013</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1751-8113">1751-8113</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15630224">15630224</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFLevin,_Michael_A._and_Xiao-Gang_Wen2005" class="citation journal cs1">Levin, Michael A. and Xiao-Gang Wen (12 January 2005). "String-net condensation: A physical mechanism for topological phases". Physical Review B. 71 (45110): 21. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0404617">cond-mat/0404617</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005PhRvB..71d5110L">2005PhRvB..71d5110L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.71.045110">10.1103/PhysRevB.71.045110</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51962817">51962817</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFWoottonLahtinenDoucotPachos2011" class="citation journal cs1">Wootton, James R.; Lahtinen, Ville; Doucot, Benoit; Pachos, Jiannis K. (2011). "Engineering complex topological memories from simple Abelian models". Annals of Physics. 326 (9): 2307–2314. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.0708">0908.0708</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011AnPhy.326.2307W">2011AnPhy.326.2307W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2011.05.008">10.1016/j.aop.2011.05.008</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4916">0003-4916</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119288871">119288871</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFAguadoBrennenVerstraeteCirac2008" class="citation journal cs1">Aguado, M.; Brennen, G. K.; Verstraete, F.; Cirac, J. I. (22 December 2008). "Creation, Manipulation, and Detection of Abelian and Non-Abelian Anyons in Optical Lattices". Physical Review Letters. 101 (26): 260501. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0802.3163">0802.3163</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvL.101z0501A">2008PhRvL.101z0501A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.101.260501">10.1103/physrevlett.101.260501</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1854%2FLU-8589252">1854/LU-8589252</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19113760">19113760</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11619038">11619038</a>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFBrennenAguadoCirac2009" class="citation journal cs1">Brennen, G K; Aguado, M; Cirac, J I (22 May 2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F11%2F5%2F053009">"Simulations of quantum double models"</a>. New Journal of Physics. 11 (5): 053009. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0901.1345">0901.1345</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NJPh...11e3009B">2009NJPh...11e3009B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F11%2F5%2F053009">10.1088/1367-2630/11/5/053009</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFLiuShtengelSmithPollmann2022" class="citation journal cs1">Liu, Yu-Jie; Shtengel, Kirill; Smith, Adam; Pollmann, Frank (2022-11-07). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PRXQuantum.3.040315">"Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer"</a>. PRX Quantum. 3 (4): 040315. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2110.02020">2110.02020</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022PRXQ....3d0315L">2022PRXQ....3d0315L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPRXQuantum.3.040315">10.1103/PRXQuantum.3.040315</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:238353914">238353914</a>.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFPachosWieczorekSchmidKiesel2009" class="citation journal cs1">Pachos, J K; Wieczorek, W; Schmid, C; Kiesel, N; Pohlner, R; Weinfurter, H (12 August 2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F11%2F8%2F083010">"Revealing anyonic features in a toric code quantum simulation"</a>. New Journal of Physics. 11 (8): 083010. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0710.0895">0710.0895</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NJPh...11h3010P">2009NJPh...11h3010P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F11%2F8%2F083010">10.1088/1367-2630/11/8/083010</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> C.-Y. Lu, et al., Phys. Rev. Lett. 102, 030502 (2009).

</li>

<li id="cite_note-:0-30">^ <a href="#cite_ref-:0_30-0">a</a> <a href="#cite_ref-:0_30-1">b</a> <a href="#cite_ref-:0_30-2">c</a> <cite id="CITEREFSatzingerLiuSmithKnapp2021" class="citation journal cs1">Satzinger, K. J.; Liu, Y.; Smith, A.; Knapp, C.; Newman, M.; Jones, C.; Chen, Z.; Quintana, C.; Mi, X.; Dunsworth, A.; Gidney, C. (2021-04-02). "Realizing topologically ordered states on a quantum processor". Science. 374 (6572): 1237–1241. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2104.01180">2104.01180</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021Sci...374.1237S">2021Sci...374.1237S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.abi8378">10.1126/science.abi8378</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34855491">34855491</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:233025160">233025160</a>.</cite>

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite id="CITEREFYaoWangChenGao2012" class="citation journal cs1">Yao, Xing-Can; Wang, Tian-Xiong; Chen, Hao-Ze; Gao, Wei-Bo; Fowler, Austin G.; Raussendorf, Robert; Chen, Zeng-Bing; Liu, Nai-Le; Lu, Chao-Yang; Deng, You-Jin; Chen, Yu-Ao; Pan, Jian-Wei (22 February 2012). "Experimental demonstration of topological error correction". Nature. 482 (7386): 489–494. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0905.1542">0905.1542</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012Natur.482..489Y">2012Natur.482..489Y</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature10770">10.1038/nature10770</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22358838">22358838</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4307662">4307662</a>.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite id="CITEREFDouçotIoffeVidal2004" class="citation journal cs1">Douçot, Benoit; Ioffe, Lev B.; Vidal, Julien (3 June 2004). "Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation". Physical Review B. 69 (21): 214501. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0302104">cond-mat/0302104</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004PhRvB..69u4501D">2004PhRvB..69u4501D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevb.69.214501">10.1103/physrevb.69.214501</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119407144">119407144</a>.</cite>

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> <cite id="CITEREFGladchenkoOlayaDupont-FerrierDouçot2009" class="citation journal cs1">Gladchenko, Sergey; Olaya, David; Dupont-Ferrier, Eva; Douçot, Benoit; Ioffe, Lev B.; Gershenson, Michael E. (2009). "Superconducting nanocircuits for topologically protected qubits". Nature Physics. 5 (1): 48–53. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0802.2295">0802.2295</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009NatPh...5...48G">2009NatPh...5...48G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys1151">10.1038/nphys1151</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2473">1745-2473</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118359424">118359424</a>.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFMicheliBrennenZoller2006" class="citation journal cs1">Micheli, A.; Brennen, G. K.; Zoller, P. (30 April 2006). "A toolbox for lattice-spin models with polar molecules". Nature Physics. 2 (5): 341–347. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0512222">quant-ph/0512222</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006NatPh...2..341M">2006NatPh...2..341M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys287">10.1038/nphys287</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2473">1745-2473</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:108289844">108289844</a>.</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFParedesBloch2008" class="citation journal cs1">Paredes, Belén; Bloch, Immanuel (1 January 2008). "Minimum instances of topological matter in an optical plaquette". Physical Review A. 77 (2): 023603. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0711.3796">0711.3796</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvA..77b3603P">2008PhRvA..77b3603P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.77.023603">10.1103/physreva.77.023603</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:46143303">46143303</a>.</cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFDaiYangReingruberSun2017" class="citation journal cs1">Dai, Hanning; Yang, Bing; Reingruber, Andreas; Sun, Hui; Xu, Xiao-Fan; Chen, Yu-Ao; Yuan, Zhen-Sheng; Pan, Jian-Wei (28 August 2017). "Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian". Nature Physics. 13 (2): 1195–1200. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.05709">1602.05709</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017NatPh..13.1195D">2017NatPh..13.1195D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2FNPHYS4243">10.1038/NPHYS4243</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2473">1745-2473</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118604118">118604118</a>.</cite>

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFWeimerMüllerLesanovskyZoller2010" class="citation journal cs1">Weimer, Hendrik; Müller, Markus; Lesanovsky, Igor; Zoller, Peter; Büchler, Hans Peter (14 March 2010). "A Rydberg quantum simulator". Nature Physics. 6 (5): 382–388. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.1657">0907.1657</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010NatPh...6..382W">2010NatPh...6..382W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys1614">10.1038/nphys1614</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2473">1745-2473</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54710282">54710282</a>.</cite>

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFSemeghiniLevineKeeslingEbadi2021" class="citation journal cs1">Semeghini, Giulia; Levine, Harry; Keesling, Alexander; Ebadi, Sepehr; Wang, Tout T.; Bluvstein, Dolev; Verresen, Ruben; Pichler, Hannes; Kalinowski, Marcin; Samajdar, Rhine; Omran, Ahmed (2021). "Probing Topological Spin Liquids on a Programmable Quantum Simulator". Science. 374 (6572): 1242–1247. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2104.04119">2104.04119</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021Sci...374.1242S">2021Sci...374.1242S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.abi8794">10.1126/science.abi8794</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34855494">34855494</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:233204440">233204440</a>.</cite>

</li>

<li id="cite_note-CoherentArray2-39"><a href="#cite_ref-CoherentArray2_39-0">^</a> <cite id="CITEREFBluvsteinLevineSemeghiniWang2022" class="citation journal cs1">Bluvstein, Dolev; Levine, Harry; Semeghini, Giulia; Wang, Tout; Ebadi, Sepehr; Kalinowski, Marcin; Maskara, Nishad; Pichler, Hannes; Greiner, Marcus; Vuletic, Vladan; Lukin, Misha (April 20, 2022). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9021024">"A quantum processor based on coherent transport of entangled atom arrays"</a>. Nature. 604 (7906): 451–456. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2112.03923">2112.03923</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022Natur.604..451B">2022Natur.604..451B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41586-022-04592-6">10.1038/s41586-022-04592-6</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9021024">9021024</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35444318">35444318</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:244954259">244954259</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external free" href="https://skepsisfera.blogspot.com/2010/04/kitaevs-toric-code.html">https://skepsisfera.blogspot.com/2010/04/kitaevs-toric-code.html</a></li></ul>

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</style><div id="Quantum_information_science667" style="font-size:114%;margin:0 4em"><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="DiVincenzo's_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li>

<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a>

<ul><li><a href="Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></li>

<li><a href="Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li>

<li><a href="Qubit" title="Qubit">Qubit</a>

<ul><li><a href="Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li>

<li><a href="List_of_quantum_processors" title="List of quantum processors">Quantum processors</a>

<ul><li><a href="Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li>

<li><a href="Gleason's_theorem" title="Gleason's theorem">Gleason's</a></li>

<li><a href="Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li>

<li><a href="Holevo's_theorem" title="Holevo's theorem">Holevo's</a></li>

<li><a href="No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li>

<li><a href="No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li>

<li><a href="No-communication_theorem" title="No-communication theorem">No-communication</a></li>

<li><a href="No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li>

<li><a href="No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li>

<li><a href="No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li>

<li><a href="Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li>

<li><a href="Threshold_theorem" title="Threshold theorem">Threshold</a></li>

<li><a href="Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li>

<li><a href="Schr%C3%B6dinger%E2%80%93HJW_theorem" class="mw-redirect" title="Schrödinger–HJW theorem">Schrödinger-HJW</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_capacity" title="Classical capacity">Classical capacity</a>

<ul><li><a href="Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li>

<li><a href="Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li>

<li><a href="Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li>

<li><a href="Entanglement_swapping" title="Entanglement swapping">Entanglement swapping</a></li>

<li><a href="Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li>

<li><a href="Quantum_channel" title="Quantum channel">Quantum channel</a>

<ul><li><a href="Quantum_network" title="Quantum network">quantum network</a></li></ul></li>

<li><a href="Quantum_state_purification" title="Quantum state purification">State purification</a></li>

<li><a href="Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a>

<ul><li><a href="Quantum_energy_teleportation" title="Quantum energy teleportation">quantum energy teleportation</a></li>

<li><a href="Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li>

<li><a href="Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography24" scope="row" class="navbox-group" style="width:1%"><a href="Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li>

<li><a href="Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li>

<li><a href="Quantum_money" title="Quantum money">Quantum money</a></li>

<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a>

<li><a href="List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li>

<li><a href="Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithmic_cooling" title="Algorithmic cooling">Algorithmic cooling</a></li>

<li><a href="Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li>

<li><a href="Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li>

<li><a href="Boson_sampling" title="Boson sampling">Boson sampling</a></li>

<li><a href="Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li>

<li><a href="Grover's_algorithm" title="Grover's algorithm">Grover's</a></li>

<li><a href="Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li>

<li><a href="Magic_state_distillation" title="Magic state distillation">Magic state distillation</a></li>

<li><a href="Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li>

<li><a href="Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li>

<li><a href="Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li>

<li><a href="Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li>

<li><a href="Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li>

<li><a href="Simon's_problem" title="Simon's problem">Simon's</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="PostBQP" title="PostBQP">PostBQP</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li>

<li><a href="Quantum_volume" title="Quantum volume">Quantum volume</a></li>

<li><a href="Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a>

<li><a href="Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <a href="Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li>

<li><a href="Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li>

<li><a href="One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a>

<ul><li><a href="Cluster_state" title="Cluster state">cluster state</a></li></ul></li>

<li><a href="Quantum_circuit" title="Quantum circuit">Quantum circuit</a>

<ul><li><a href="Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a>

<ul><li><a href="Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li>

<li><a href="Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li>

<li><a href="Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li>

<li><a href="Hamiltonian_quantum_computation" title="Hamiltonian quantum computation">Hamiltonian quantum computation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Codes

<ul><li><a href="Five-qubit_error_correcting_code" title="Five-qubit error correcting code">5 qubit</a></li>

<li><a href="Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li>

<li><a href="Stabilizer_code" title="Stabilizer code">stabilizer</a></li>

<li><a href="Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li>

<li><a href="Steane_code" title="Steane code">Steane</a></li>

<li><a href="Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li>

<li><a href="Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li>

<li><a href="Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li>

<li><a href="KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li>

<li><a href="Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li>

<li><a href="Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Charge_qubit" title="Charge qubit">Charge qubit</a></li>

<li><a href="Flux_qubit" title="Flux qubit">Flux qubit</a></li>

<li><a href="Phase_qubit" title="Phase qubit">Phase qubit</a></li>

<li><a href="Transmon" title="Transmon">Transmon</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="Qiskit" title="Qiskit">Qiskit</a>–<a href="IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li>

<li><a href="Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li>

<li><a href="Libquantum" title="Libquantum">libquantum</a></li>

<li><a href="Quantum_programming" title="Quantum programming">many others...</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Quantum information science</li>

<li> Quantum mechanics topics</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-25" href="https://en.wikipedia.org/wiki/?title=Toric_code&oldid=1302414067">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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