Will Quantum Computation fail if spacetime is discrete? Will Quantum Computation fail if spacetime is discrete? Basically, would a discrete spacetime impose unexpected limits on how many Qubits could be used in calculations?
Conversely, can quantum computing tell us anything about the discrete nature (or not) of spacetime?
 A: I would say no. Quantum computers will probably leverage properties of energy levels in ions. These have typical energies around $10eV$. 
By contrast the Planck scale where quantum gravity effects come into play has an energy of $10^{27}eV$! 
This illustrates why the problem of quantum gravity is so much harder than usual quantum mechanics: you need to go to energy scale which are ridiculously far removed from everyday life!
A: I will direct you to an answer on a different site, written by the notorious former user Ron Maimon.
http://www.quora.com/Is-our-universe-cellular-automata-at-a-fundamental-level

Not in any obvious way, it would conflict with Bell's theorem if the mapping were local. For nonlocal mappings, the answer isn't clear, but nobody has any clue how it would work, and the result would have to magically reproduce quantum mechanics at least for small systems.
For larger systems, for a physical size automaton, it would have to fail at quantum computation, simply because a cellular automaton the size of the universe cannot factor 10,000 digit numbers, and quantum computers can. So this model is making a prediction, it predicts that quantum computers will fail when they get to a certain size, less than 10,000 qubits.

Even this doesn't make the argument obvious, but from other things he's written I've understood it.
The limitation referred to in the quote above is holography. Draw a sphere around a quantum computer. Now we have a maximum possible entropy. That entropy is dictated by the surface area of the sphere. Presumably this sphere would be rather small. The entropy implicit within the calculation for 10,000 would be huge, and exceed the physical limitation, thus it will never happen.
This is based on the assumption that the universe is described by a non-local cellular automaton. However, I'm unclear on many of the details. Would any sphere you could draw suffice (as I suggested)? Or would the relevant sphere be the cosmic horizon, which clearly has a larger area? From my knowledge of black holes, I would hazard a guess that it is the former. The central idea is, thus, that the 10,000 qbit quantum computer would become a black hole upon doing the calculation.
We have other ways this might be refuted. Perhaps the type of problem that the quantum computer solves is a subset of information theory that allows the entanglement information to somehow not manifest as entropy as we know it. This is a vague notion, but I can somewhat imagine the problem of treating entangled states like particles or energy in the conventional sense, when we have no reason to do so.
Even Ron is awfully unsure, saying he's about 50/50. I take it that means there's a 50% chance that quantum computers do have a fundamental limitation. Perhaps we should also poll about the chance it turns into a black hole.
I think the idea is controversial either way. It's also controversial whether it's practical to ever build such a thing.
