This equation could end up being as important to Quantum Computing as the Schrödinger equation is to solid state physics. So if you understand what a CNOT gate is, what computational universality is, then this paper explains the Yang-Baxter equation in those terms.
Some recent solid-state physics: Hitachi claims to have the world's highest resolution (43 pm) electron microscope. Here is video of quantised magnetic vortices moving on the surface of a thin film of Niobium, under a strong magnetic field which is perpendicular (or at some angle) to the film. You can see them "hopping" or moving suddenly, but in a regular way so there is a bulk flow. The vortices heat the substrate such that they tend to follow the same track(s). You also get annihilation of vortices with opposite spins, later in the vid http://rdg.ext.hitachi.co.jp/rd/moviee/vortices.mpeg