It should be stated at the outset that this theory would only be useful if indeed there were evidence that some sort of adaptive mutation occurs - in other words, if there were experimental data showing that the classical model of random mutation is lacking, and that certain mutations are "preferred" (occur more frequently) because they confer a greater benefit to the organism. This is in and of itself a controversial subject; to date there is no such generally accepted evidence, and all evidence appears to be consistent with a process of random mutation.
However, if we decide to accept such a premise, then a mechanism which would provide for such a result is currently lacking. The mechanism proposed by quantum evolution is to imagine that the configuration of DNA in a cell is held in a quantum superposition of states, and that "mutations" occur as a result of a collapse of the superposition into the "best" configuration for the cell. The proponents of this approach liken the operation of DNA to the operation of a quantum computer, which selects one from a multitude of possible outcomes.
Several problems need to be overcome for this theory to be consistent with our current knowledge of quantum physics. Most importantly, the state of quantum superposition must last long enough to allow the DNA to do its normal job (form proteins); otherwise, there would be no way for a comparison of the various outcomes of various mutations to occur, and thus no basis for the system to "decide" which mutations are more useful. Protein formation occurs at a rate of on the order of 10,000 times a second (10-5 seconds per protein formed).
Although some have, by analogy to the technique of NMR imaging, posed state coherence times as long as half a second, (see Johnjoe McFadden's book "Quantum Evolution", [1]), this analysis has been challenged (see Mathew Donald, A Review of "Quantum Evolution", [1]) and coherence times on the order of 10-13 seconds seems to be a much more realistic outcome. This latter time would be far too short by many orders of magnitude for the protein formation required for a superposition of quantum states to affect mutations.
If the theory of quantum evolution were indeed true, one could further speculate that a similar, more robust process could explain observed phenomena such as the apparent "jumps" in the fossil record as adaptive mutations on an even larger scale; this would require even longer periods of state coherence than those described by McFadden et al.
Science fiction writer Greg Egan, in his book Teranesia, posed a similar yet more sweeping mechanism, whereby large sections of our observable universe are modeled as quantum superpositions of states, affecting not just biology, but the nature of space-time itself.
See also: irreducible complexity