* - I'd note that, if quantum mechanics are really probabilistic, then the level at which mutations occur is probably close enough to the quantum to be non-deterministic and thus it's quite plausible that, in the real universe, random mutations are not determined at all.
This is most certainly the case for many mutations, if not all. One of the best examples is mutations caused by radiation, and more specifically high energy photons (e.g. UV radiation). The double slit experiment is one of the classic QM experiments, and these same photons with wave probabilities are causing mutations.
Even at the level of DNA replication there are quantum effects. For example, in
this paper the authors describe how mutations are due to a loose fit between free nucleotides and the active site of polymerases (enzymes that replicate DNA). They describe the results in probabilities, the same as you would for any quantum or thermodynamic event.
There are also larger molecular interactions that would probably not be described in terms of QM. For example, the insertion of transposons occurs between very large macromolecules. However, these events are again only describable in terms of probability. The same transposon will insert in many different places, even in genetically identical organisms in the exact same environment. I know this from my personal experience with random transposon mutagenesis in bacteria.
With evolution, we are looking at a system that is very, very susceptible to the "Butterfly Effect". Small changes in the beginning conditions can produce very large changes in the results. Something as simple as a low pressure system moving 10 miles south instead of north can result in a hurricane. This hurricane can result in flooding in a specific lake system causing large standing areas of stagnant water. Within these stagnant pools you can have algal blooms that would have otherwise not occured. This increase in algae allows for a massive increase in genetic diversity that would have otherwise not occurred, possibly resulting in an algae population that is capable of producing a new neurotoxin that would have otherwise not existed if that low pressure system had moved north 10 miles instead of south. . . just as one example.