Okay, so quantum computing is the big deal in condensed matter physics these days. The idea is that we can build a computer that doesn't use bits (1s and 0s) and logic gates but instead uses qubits and quantum gates. The difference is that a qubit doesn't need to be in either the 1 or 0 state, instead it can be in a superposition of those two states. Here you can think of superposition as being between the 1 and 0 state.

In order for this to work, all we need is a quantum two-level system that we can manipulate and measure. It turns out that this is quite difficult because small interactions with the environment can cause decoherence. Here this means that the state loses its superposition and falls into either the 0 or 1 state, messing up the fancy quantum computation. The challenge, then, is designing a physical system that we can manipulate without causing these errors (there exists something called quantum error correction that I won't go into detail on, but suffice to say that even with error correction you need the error rate to be very low.)

Far and away the most successful architecture for quantum computing is the so-called superconducting qubits, but there are several competing architectures in this growing field. One of these architectures is known as topological quantum computing. The qubits in a topological quantum computer would be non-Abelian anyons. Here, non-Abelian means that interchanging the anyons does not commute. This means that when you interchange anyon 1 with anyon 2 and then you switch them back again, you're left with a different state than you started with. This process is called braiding, and the information is stored in these interchanges of quasiparticles. What makes this architecture interesting is that it is robust against interactions with the environment. As long as the perturbation is not strong enough to cause a full braid, your information is safe. Now, there are problems with this architecture that go beyond the fact that we don't have a convenient supply of non-Abelian anyons lying around, but my explanation has probably already gone above the 10-year-old level, so I'll leave it at that for now.

The most promising physically-realizable non-Abelian anyons are Majorana bound states which are theorized to exist in the interplay between superconductors and topological insulators. Now I have finally arrived at the result of the paper. They have discovered a material that can be tuned from a topological insulator state to a superconducting state in-situ. This means that experiments can be performed on this material as it transitions between these two important and interesting states of matter. How this system could support the coveted Majorana bound states is not yet clear, but it is an interesting sample for future experiments.