I just started watching the so far excellent documentary series Through the Wormhole.
The first episode talks about whether there's a creator, and what such a creator may be like. One presented possibility is that we live in a computer simulation, that is, our entire universe is a (gigantic) computer simulation, and thus, that we are digital beings unknowingly "trapped" inside this simulation (think about the Matrix).
This actually presupposes a lot more than you may think. First, it assumes the laws of nature are possible to simulate on a computer. This is for example impossible if there is such a thing as process that requires infinite computation. Essentially, it constrains our universe to a finite one: particles must be finitely divisible, time must be finitely divisible, and the universe extends finitely much in all directions.
The first two are not controversial aspects of reality: already the ancients postulated the existence of indivisible matter, and they called it the atom (it turns out that atoms are in fact divisible, but quarks or superstrings may not be; the misnomer is however our mistake, not the ancient's).
That the universe is finite may sit uncomfortably with an intuitive sense that "if I just keep going in one direction with my space ship, I should not expect to hit a wall somewhere". Indeed, that would be very strange. But it is possible that the universe is closed, in the sense that if you travel in one direction for long enough, you'll end up at the same place you began (and no, this is not inconsistent with the measure flatness of the universe. The flatness is based on the notion of parallel lines: read about it here).
None of these postulates are controversial: it is widely believed that matter is finitely divisible, and, as for the universe, the evidence with respect to closeness is inconclusive so far (when looking out into the void, they search for repetitions as we "go around the universe one full turn", but due to the vast size of the universe, this is easier said than done).
Now that we know nothing about our current understanding for physics excludes the possibility of a computer simulation, let's look at how likely it is. The claim made by one of the speakers is that quantum physics bears a resemblance to the way a computer simulation works: from far away, it all looks good. But as you zoom in, you find that things start getting blurry. If you've ever tried to zoom in on an image on your computer, you've probably seen it: at some point, the details starts breaking down, and you start to see how the pixels are blurred out. Now, quantum physics surely look that way: as you start zooming in, particles start looking less like matter and more like waves, i.e. blurry.
I think this may be misrepresenting quantum physics. The double slit experiment does not only imply that "things blur out when nobody is looking", it also implies the existence of "virtual particles" that, at the other end of the slit, interfere with each other. More precisely, the result of the experiment heavily relies on the fact that, when you are not looking at which slit the particles go through, they seem to go through both, and interfere with each other to create a pattern that would be impossible to get if the particles only went through one slit at a time. So in order to get the final pattern we see on the wall, our laws of nature have to compute not only what happens to the particle, but also to the virtual particle that it interferes with. This makes the calculation harder, not easier, which is in sharp contrast to a computer simulation, where the blur is omission of computation for the sake of efficiency. Now, it's not too hard to change the experiment so that you would have more than one virtual particle, in which case you would get a interference between all virtual particles, implying a vastly greater computational effort.
Moreover, there's in fact no reason to believe that quantum superposition "collapses" at some point, only that it decoheres into a macroscopic state where you can't measure it anymore. Dropping the unnecessary assumption of wave collapse, and drawing the logic to its end, you'll find the many world interpretation, which briefly says that there are multiple universes, which can all interfere with ours when the quantum states resemble each other. It doesn't take much to see that have multiple universes makes data storage and computation vastly more demanding for a classic computer. Moreover, as the entropy increases over time, the number of possible states increases exponentially, meaning that any computer would run out of memory (rather quickly too). It is definitely not the way in which you would try to write simulate a universe in the cheapest possible way.
To me, it seems like quantum physics require more, not less, computation. That's also the consensus among quantum information scientists, as it takes exponential time to simulate a quantum computer on a classic computer.
Is it likely that we live in a quantum simulated reality? If the simulation is performed on a classical computer, the probability is, well, probably exponentially decreasing with time. In fact, if we have an infinity of universes in our multiverse, that would both theoretically and practically exclude the possibility of simulation on any digital and quantum computer with finite number of bits/qubits.
We could of course be simulated by a quantum computer that stores all the quantum states of our universe in memory, but my point is that the majority of arguments in favor of a digitally simulated reality do not hold, because quantum physics makes it less likely, not more. It should also be noted, that arguing for a quantum simulated reality is pretty much the same as asking: "what if all this reality is just an illusion, made to look like all this reality?". That is obviously not falsifiable, now or ever, because it presupposes that everything is the way it looks like, but at the same times suggest that it may not be real. As Morpheus would say then: what is real?