Designing a Topological Medium

When designing a topological medium, we start with a local model, a system which has localised components (spins on a lattice, dimers, josephson junctions,...) which have local interactions (that is, components that are near each other interact). This class of systems covers just about all of physics. The special thing about topologcal media is that the interactions are tailored so that the system has a number of independent ground states (states with the lowest posible energy), which are distinguished from each other by some global or topological feature. These ground states are then used as code states to store information. The idea is that no local perturbation can take you far away from one ground state and close to another which is topologically different. A simple topological picture medium is shown below

States and Pictures

States of our topological medium can be represented as superpositions of pictures of the medium. An example of such a state is shown on the right.

Ground States

The pictures in low energy states consist completely of loops. Also, in the lowest energy states, pictures which can be deformed into each other and pictures related by the addition or removal of some small loops are superimposed.

A Local Model

One possible local model for this picture medium is a system of qbits (for example spins) on the links of a square lattice. When a qbit take the "classical" value 1, we show this by coloring its link. When it takes the classical value 0, we don't color its link and when it takes a superposition of values, we show this as a superposition of pictures where the link is or is not colored. Now if we introduce an interaction which favors an even number of colored edges at each vertex, this makes sure that the colored edges in the ground states must form loops. Another interaction which favors equal superposition of each coloring of each lattice plaquette with the opposite coloring will make sure that the ground states contain all loop pictures that can be obtained from each other by "box flipping". With a bit of experimentation, you can see that this means exactly that pictures which can be deformed into each other and pictures related by the addition or removal of some small loops are superimposed.