Eventually, the universe is going to die. It's not going to be soon, but it's going to happen, and when it does, our top priority should definitely be to have a computer that can survive it. Theoretical physicists have speculated that we can do this with something called a time crystal, and they may have just figured out how to actually make one. What's A Time Crystal?
A regular crystal (like a diamond or a quartz crystal or whatever) is something made up of orderly, repeating patterns of atoms or molecules that are symmetrical in space. The reason that atoms and molecules like to form crystals is because a crystallizing puts them in their lowest energy state. For example, if you put a bunch of energy into water (by heating it), it turns into a gas, but when you take energy away (by cooling it), the water will crystallize into ice. This leaves us with two key characteristics of crystals: spatial symmetry, and a lowest energy state.
A time crystal is like a spatial crystal that adds symmetry in the fourth dimension, in time as well as in space. Something that's symmetric in time is sort of like a clock: there's a repeating pattern of motion through time, and it's this pattern of motion that's analogous to the repeating pattern of atoms in space. That's the symmetry part of a time crystal, but the tricky part is getting it down to its lowest energy state, 'cause finding something with a repeating pattern of motion usually requires energy. So how do you make one, then? How You Make One
The key to making a time crystal is finding a way to create a repeating pattern of atoms in time without adding energy to the system, and the way to do it might be with a ring of superconducting ions.
Here's the idea: if you take a bunch of ions (atoms with a charge) and cool them way, way down to their absolute lowest energy state, the ions will naturally repel each other, forming a ring-shaped spatial crystal. Add a little bit of an electric field to nudge those ions, and they should begin to rotate, at which point you can remove the electric field and the ions will keep on rotating all by themselves, forever,* since they act like superconductors. And there you go, it's a time crystal, created using technology that we (almost) have right now.
*This isn't a perpetual motion machine because you can't get any energy out of the system: it's moving, but it can't do any work. Surviving Heat Death
Once you have one of these things, you can make a computer out of it by assigning zeros and ones to the different rotational states that you'd get by using different kinds of ions. This may seem like a needlessly complicated way of creating a very simple computer, but the reason to build it is that it can continue to function after the heat death of the universe.
Local
"Heat death" is what will eventually happen when the entire universe homogenizes. A long long long (like 10^100-years-from-now long) long time from now, it's possible that the universe will all have equalized to the same temperature, with energy completely evenly distributed everywhere. This sounds soothing, I know, but it's going to screw us, since nobody will be able to get any work done.
Here's what I mean: let's say you've got a bucket of hot water and a bucket of cold water. Energy wants to flow from the hot water to the cold water, and you can suck work out of it as it does so, using something like a Sterling engine. If you mix the two buckets of water together, though, it just goes lukewarm and you can't extract energy out of it anymore. And this is what might happen to the universe.
The advantage of a time crystal computer, though, is that it can run quite happily without needing any energy to keep it going. And since it got created in its lowest energy state to begin with, it doesn't have any energy to lose to the rest of the universe either. So that all sounds good, but as the researchers from UC Berkeley who conceptualized this thing admit: "we focus on a space-time crystal that can be created in a laboratory, so you need to figure out a method to make a laboratory that can survive in the heat-death of the universe."
