Turing machines aren't quantum mechanical, and computation is based on logic. This discussion is philosophical, so I guess it's philosophy all the way down.
Turing machines are deterministic. Quantum Mechanics is not, unless you go with a deterministic interpretation, like Many Worlds. But even then, you won't be able to compute all the branches of the universal wave equation. My guess is any deterministic interpretation of QM will have a computational bullet to bite.
As such, it doesn't look like reality can be fully simulated by a Turing machine.
Giving a Turing machine access to a quantum RNG oracle is a trivial extension that doesn't meaningfully change anything. If quantum woo is necessary to make consciousness work (there is no empirical evidence for this, BTW), such can be built into computers.
You can use a classic transistor turing machine to solve quantum problems, it's just gonna take way longer.