Toric code

The Toric Code is a topological quantum error correcting code defined on a two dimensional spin lattice Though many more examples of such codes exist , the toric code is the simplest and most well studied.

Error correcting and computation

The toric code is defined on a two dimensional lattice, usually chosen to be the square lattice, with a spin-½ particle located on each edge. The periodic boundary conditions of the lattice are chosen such that it is wrapped around a torus, giving the code its name. Stabilizer operators are defined on the spins around each vertex $v$ and plaquette (or face) $p$ of the lattice as follows,

$A_{v}=\prod _{i\in v}\sigma _{i}^{x},\,\,B_{p}=\prod _{i\in p}\sigma _{i}^{z}.$

Where here we use $i\in v$ to denote the edges touching the vertex $v$, and $i\in p$ to denote the edges surrounding the plaquette $p$. The stabilizer space of the code is that for which all stabilizers act trivially, hence,

$A_{v}|\psi \rangle =|\psi \rangle ,\,\,\forall v,\,\,B_{p}|\psi \rangle =|\psi \rangle ,\,\,\forall p,$

for any state $|\psi \rangle$. For the toric code, this space is four dimensional, and so can be used to store two qubits of quantum information. 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 syndrome 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 quasiparticles. Specifically, if the code is in a state $|\phi \rangle$ such that,

$A_{v}|\phi \rangle =-|\phi \rangle$,

a quasiparticle known as an $e$ anyon can be said to exist on the vertex $v$. Similarly violations of the $B_{p}$ are associated with so called $m$ 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.

If the pairs of anyons are sparsely distributed, and if the anyons are not moved far, error correction may be performed with a high success rate by first using minimum weight perfect matching to identify the pairs of anyons and then reannhilating them. However, if the movement or annihilation of anyons causes loops to form around topologically non-trivial paths of the torus, logical errors will occur and error correction will fail. This sets a bound on the number of errors the code can successfully correct. For an $L\times L$ lattice, the minimum number of spins on which errors must act to cause a logical error is then $L/2$, enough to create an anyon pair and transport one half way around the lattice. If, however, the errors occur randomly on each spin with a probability $p$, large $L$ will allow errors to be successfully corrected almost surely for $p<0.11$

The means to perform quantum computation on logical information stored within the toric code has also been considered. It has been shown that extending the stabilizer space using 'holes', vertices or plaquettes on which stabilizers are not enforced, allows a universal set of gates to be performed. A measurement based scheme for quantum computation based upon this principle has been found, whose error threshold is the highest known for a two dimensional architecture.

Hamiltonian

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,

$H_{TC}=-J\sum _{v}A_{v}-J\sum _{p}B_{p},\,\,\,J>0.$

The ground state 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. However, the dynamic effects of such perturbations can still cause problems for the code.

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 $J$, but since arbitrary increases of this coupling are unrealistic, the code still has its limits. Accordingly, many attempts are being made to make the toric code fully thermally stable .

Other uses

As a simple model of anyons, the toric code can be used in studies of anyonic behaviour and topologically ordered systems. For example, it has been used to study the effects of thermalization on the topological entropy of states. It has also been shown that, using superposition states of the Abelian toric code anyons, some properties of the more complex non-Abelian anyons can be realized.

Generalizations

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.

It is also possible to define similar codes using higher dimensional spins. These allow a greater richness in the behaviour of anyons, which may be used for quantum computation and error correction. These not only include models with Abelian anyons, but also those with non-Abelian statistics.