We actually know quite a bit about quantum gravity: it must fall under a quantum mechanical framework, and it needs to result in gravity, and gravitons haven't been directly detected yet. This isn't enough to determine what the theory is, but it is enough to say some things about it. The main two things are:
1: Since it's just quantum mechanics, whatever it does, it'll just set another Hamiltonian. If it changes the ground rules, then it's not a theory of quantum gravity. It's a theory of something else gravity.
2: Gravity is weak. Ridiculously weak. Simply getting the states to not mush up into a continuum will be more difficult by a factor for which 'billions of times' would be a drastic understatement.
In order for gravity to be even noticeable, let alone the main driver of action, you either need to have really really enormous amounts of stuff, or things have to be insanely high energy and short-ranged and short-lived (unification energies).
Either of these would utterly murder coherence. In the former case your device would be big enough (and/or slow enough) that even neutrino collisions would decohere it fairly comprehensively long before the first operation could complete. In the latter case your computer is exploding at nearly the speed of light every time you turn it on and incidentally requires a particle accelerator that makes CERN look like 5V power cable,
So, everything that makes gravity different from electromagnetism makes it much much worse for computing.
Not that I actually believe most of what I wrote above (just that it hasn't yet been completely excluded), if QG introduced small nonlinearities to quantum mechanics, fun things could happen, like superluminal signaling as well as the ability to solve NP-Complete and P#-Complete problems in polynomial time (which is probably better seen as a reason to believe that QG won't have a nonlinearity).
Speculation is important for forecasting; it's also fun. Speculation is usually conveyed in two forms: in the form of an argument, or encapsulated in fiction; each has their advantages, but both tend to be time-consuming. Presenting speculation in the form of an argument involves researching relevant background and formulating logical arguments. Presenting speculation in the form of fiction requires world-building and storytelling skills, but it can quickly give the reader an impression of the "big picture" implications of the speculation; this can be more effective at establishing the "emotional plausibility" of the speculation.
I suggest a storytelling medium which can combine attributes of both arguments and fiction, but requires less work than either. That is the "wikipedia article from the future." Fiction written by inexperienced sci-fi writers tends to generate into a speculative encyclopedia anyways--why not just admit that you want to write an encyclopedia in the first place? Post your "Wikipedia articles from the future" below.