Toric code
The toric code gets its name from its periodic boundary conditions, giving it the shape of a torus.
However, some experimental realizations require open boundary conditions, allowing the system to be embedded on a 2D surface.
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.
Single spin errors cause pairs of anyons to be created and transported around the lattice.
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.
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.
[7][8] These thresholds are upper limits and are useless unless efficient algorithms are found to achieve them.
[9] When applied to the noise model with independent bit and flip errors, a threshold of around 10.5% is achieved.
However, matching does not work so well when there are correlations between the bit and phase errors, such as with depolarizing noise.
[10][11] 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,
[13][14] The gap also gives the code a certain resilience against thermal errors, allowing it to be correctable almost surely for a certain critical time.
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.
[15][16] Proposals have been made for realization of these in the lab [17] Another approach is the generalization of the model to higher dimensions, with self-correction possible in 4D with only quasi-local interactions.
The anyonic mutual statistics of the quasiparticles demonstrate the logical operations performed by topologically non-trivial loops.
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.
anyons are similarly moved through the red loop above a logical operation will also result.
[23][24][25] 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[26][27][28] and measurement of the topological entanglement entropy.
[29][28] For realizations of the toric code and its generalizations with a Hamiltonian, much progress has been made using Josephson junctions.
The theory of how the Hamiltonians may be implemented has been developed for a wide class of topological codes.
[30] 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.
[33] Such minimal instances of the toric code has been realized experimentally within isolated square plaquettes.
[34] Progress is also being made into simulations of the toric model with Rydberg atoms, in which the Hamiltonian and the effects of dissipative noise can be demonstrated.
