My approximate understanding: Some problems reduce to calculating the trace of a matrix. The matrix might be really complicated, but sometimes you can approximate it by having particles braid around each other and measuring how likely the particles end up in a particular state. In theory it can be put into practice using materials called "topological insulators."

The Jones polynomial is a thing which can tell knots apart. The quantum computer can calculate the Jones polynomial by having particles trace out the knot. The Jones polynomial usually takes exponential time to calculate.

The "topological" in "topological quantum computing" refers to the braiding and the fact that they're computing topological invariants of a space.

These particles (anyons) live in two dimensions, so their world-lines trace out paths in three dimensions (two space dimensions and one time dimension.) These world-line paths get braided or knotted, just like hair does. And even though your hair gets jiggled around by the wind (quantum noise) those braids remain intact (the quantum information is preserved.)

Edit. For the very bravest of five year olds, I wrote a paper on these things here: https://arxiv.org/abs/1610.05384

Not exactly for a five year old, but see the Wikipedia article: https://en.wikipedia.org/wiki/Topological_quantum_computer#T...

Here is a fantastic video about it:https://www.youtube.com/watch?v=hKFecm9NKbM

if done, said someone, would be in line for the nobel prize.