I think that the problem of qualia is introduced by taking the wrong first step in the contemporary philosophy.
A simplified sketch of how we approach things from scientific point:
We see something, let’s name it A, and then by measurements, dividing and analysis, or in general through scientific method we figure out that A is B – e.g. we see an apple and find it to be such and such configuration of molecules, or that A as a phenomenon (e.g. lightning) is such and such B regularity (electric discharge in the atmosphere), and so on. Now, it is good that philosophy takes science seriously so far, namely as telling us that A is B. But usually step further is taken, and that is what I think is a wrong step – The results of science are interpreted as A=B, and it is gone even one step further and any ontological importance to A is negated. It is found acceptable to leave out A from our basic ontology and put the description of the situation solely in terms of B – it is thought that doing so we don’t leave out anything important in our story about the world.
But it is easy in so doing to forget what one is doing. What A was, was a thing with all those things, e.g. in case of apple – redness, taste, form and so on. In reducing A to B, we are in fact abstracting from and removing the redness, taste, form and so on, and we are abstracting the world in the terms of notions which happen to be good for science. Those scientific notions like numbers, causality, space, time, thing and so on, don’t really come from outside of our thinking – they are subset of the notions we have and are on same level with the ones we are removing – namely color, taste etc… What those later notions are guilty of is that they don’t present themselves as approachable by scientific method (most importantly measurement).
But now, once we remove those notions, and limit ourselves to the notions like movement, force, position, moment, number, equation, etc… and when we find our model of the brain and the world as whole fully described in those notions, of course we won’t find those things we already removed! As much as one might search and combine those colorless, tasteless, etc… abstractions (thing, space, causality, number, etc..), they won’t get back what was removed.
So, that is where I think the problem of qualia lies. It is not problem that Philosophy of Mind can solve, because it is a problem of a wrong metaphysics. The problem is introduced at the first step.