SUMMARY -- I don't have references for you for whether a dialethic quantum computer could model, approximate, emulate, or achieve free will
What I can offer is a reasoning path that brings rationalism and analyticity into doubt, and which endorses pragmatism. And notes how pragmatism allows for accepting incompatible causation and free will models simultaneously, through tolerance for cognitive dissonance.
Whether cognitive dissonance can be coded through using features of a dialethic quantum computer -- that will have to be further research.
Pluralism breaks rationalism
The first hint I had that rationalism is a doomed project was when I explored alternative ethics thinking to utilitarianism. Utilitarians face an apparently insurmountable calculation challenge -- future effects are impossible to predict, and spin-off effects can be VERY significant, plus the challenge of weighting welfare, and summing over the future vs the present, and for communities vs individuals, lead to no unique "utilitarian calculus", and all being impossible to ever perform.
But Rights ethics presented even worse problems. And this, I realized, was because rights were plural. Multiple rights lead intrinsically to conflicts between rights, and the need to make rights less than absolute -- which produces an indeterminate problem even worse than that faced by utilitarians. By extension -- the legal code, which is constantly growing as individual judges try to reconcile conflicting laws -- is likewise in a situation of congenital indeterminacy, and always subject to logic explosion.
I believe this is intrinsic to a pluralist reference base. As you note, physical reduction itself is subject to logic problems, due to the incompatibility of QM and relativity. And not all of physics is even reducible to those two subjects, as several aspects of solid state physics appear to be emergent, not reducible. Which leads to further pluralist reference logic problems.
Note also, the SEP article on scientific reduction admits taht philosophers of science have abandoned strong reduction, and "the unity of sciences" as a program, basically at the turn of the millennia. See section 5. https://plato.stanford.edu/entries/scientific-reduction/ So -- physics is pluralist, and science is pluralist, and both can reach different conclusions in the same circumstance, when starting reasoning from different starting reference points. And sometimes one reference is wrong when they are tested against each other, and sometimes the other.
Plus, almost everyone in academia at least has abandoned scientism -- the view that science is the only way to gain valid knowledge. So there are valid NON-scientific ways to gain knowledge of our world. Pluralism, and logical incoherence, seem to be intrinsic to our world model -- and by the indirect realist inference that is central to science -- to our world.
Additionally -- logic itself is plural. My first realization of this was when I realized that methodological naturalism reasoning operates off a four state logic not the two state of classical logic (sufficiently supported to be taken as valid, sufficiently refuted to be taken as not valid, still indeterminate, and incoherent/irresolvable and therefore "not even wrong" question). Logicians have, also with the turn of the millennia, now wholesale abandoned the belief in a "one true logic", in favor of logical pluralism. https://www.cambridge.org/core/journals/think/article/abs/guide-to-logical-pluralism-for-nonlogicians/EDFDFA1C9EB65DB71848DABD6B12D877 Without "one true logic" and in the face of pluralism, rationalism as a knowledge method looks to be unsustainable.
Along the way, I also realized analyticity -- the use of careful definition-based reasoning to solve philosophic problems, was also invalid. The first key insights was in Quine's "Two Dogmas of Empiricism", where he showed that NO language can ever have sufficiently precise shared definitions to support analytic reasoning. The other was in Popper's detailing of the scientific method, in which exploration, idea development, and guided exploration precede even the first hypothesis forming, and his treatment of definitions as hypotheses. IE definitions FOLLOW the first several stages of science. Popper also locates philosophy as the parent of sciences, hence philosophy is in the stage where even subjects of inquiry are in dispute conceptually -- it is therefore even more inappropriate to start with definitions in philosophy than it is in the early stages of science.
Pragmatism as path forward
Pragmatism offers an alternative philosophic framework to rationalism, and supports pluralist reality. The key feature of pragmatism is its abandonment of "truth" as something that is singular. Sciences, and in even more so engineering, are VERY pragmatic. There are alternative methods of calculating most problems in engineering, but some are more useful in one domain than another. So good engineers work with a sute of potential starting points, and when one isn't sure which is more valid for a problem -- the solution is to design to satisfy ALL of them!
Cognitive dissonance is the human skill of maintaining incompatible reference frameworks simultaneously. I note in this PhilSE answer, how cognitive dissonance relative to free will v. determinism, is successfully maintained even among Illusionist advocates: If Free Will Is Proven Illusory, Is There a Case for Suppressing the Finding?
Your embrace of para-consistent logic is an embrace of pragmatism, and no "one truth".
Objections to your assumptions
BUT -- your continued search for "one true paraconsistent logic", and pursuit of logic itself -- are not consistent with the critiques of rationalism and universal pluralism above. Logic needs definitions, and definitions have failings noted above, plus A=A is not true of any object in our world (objects change logically over time).
Also, your example of a paraconsistent problem for free will and determinism uses two stage logic, and any science-consistent paraconsistent logic will have to be at least a 4 stage logic. Based on logical pluralism, if you want a universal paraconsistent logic, it will have to be an N-stage, universally generalizable logic. And my suspicion is that it will not be possible to make a consistent universal "one true" N-stage paraconsistent logic.
Meanwhile, you assume in your paraconsistent example problem that all macro scale events are deterministic. That this is not so is explained in this PhilSE answer: Is it the incorrect assumption of an "a priori determined universe", which creates the paradox of determinism versus free will? Our world is not determinist at the macro scale. Macro scale determinism is a useful APROXIMATION to the world, but is not logically universal.
Implicit in some of your discussion is the inference from causation, to determinism. This is a natural inference, and I consider it logically valid under the "one true" classical logic paradigm. However, I also note that causation suffers from definitional ambiguity. this was the central premise of a question I asked my philosophy cafe recently: https://www.meetup.com/philosophy-cafe-central-maryland/events/klgqtsydckbbc/ The indeterminacy of definition of free will has been a major objection to libertarian free will among rationalists, but causation suffers from the same problem, though to a lesser degree.
A final objection is that I have outlined multiple cases of logical indeterminacy in our world -- and quantum mechanics is only one of multiple such cases. There is no good reason to infer that free will and determinism are coupled with quantum indeterminacy. There are so many intrinsic logical indeterminacies to our universe, that it is far more reasonable to think that indeterminacy is intrinsic, and any coupling between indeterminacies is possible, but far from necessary.
Path Forward?
For me, I have just accepted pragmatism and pluralism. Causation is a super useful assumption, despite our definitional issues, and the indeterminacy of our world. And Free Will is also a super useful assumption, as it seems to be intrinsic to selfhood, agency, and consciousness.
Could you reconcile these two with paraconsistent logic? Possibly. It seems like an interesting, and possibly fruitful program of investigation in the logic field.
Could computers approximate, emulate, or instantiate free will if they do paraconsistent logic using dialethic quantum computing? I think the first two are likely, and the last is unlikely, but possible. I DO think that computing with paraconsistent logic has a lot of potential to be more applicable to our paraconsistent world, and that a logic system that can sustain cognitive dissonance will be essential to approximate human thinking. it seems like an interesting, and potentially fruitful project. But will it get to free willing, or to consciousness? Unlikely, but at least plausible.
