Firstly to store quantum information it must pack into a quantum spin. How that spin is encoded by the organisms DNA we don't have to get involved in it will be very complex. DNA will only have the ability to encode a certain number of spins per base pair and it may not even be per base pair it may need sequences that is all biological stuff I don't study perhaps ImagingGeek knows. I think it has something to do with protein foldings or something like that.
Biological information is not stored as quantum states - and again, I'm questioning your knowledge of QM. By its very nature, spin is both unstable (indeed, we manipulate electron spin all the time, in several forms of microscopy & biomedical imaging), and generally you cannot extract the information stored in quantum states (i.e. spin) without altering other aspects of the quantum state.
Biological 'information' (whether that is a meaningful term in biology, and whether information theory can be applied to it, is hotly debated) is stored in a much more stable form than a quantum state - it is stored in chemical form, specifically in the form of triplets of DNA (or in some cases, RNA) nucleotides. The closest you get to 'quantum' interactions in the system is in read-out. Genetic information is 'read' from the DNA strand via non-covalent chemical bonding (hydrogen + ionic) of the mRNA intermediary by pair-bonding with tRNAs. Again, the readout isn't dependent on the quantum state of the electrons forming the bonds (indeed, spin, excitation state, etc, are all irrelevant to the process).
Now your argument that the organism can stay the same is the same as saying I can hold a computer byte to a set value. I can do any problems I like so long as I don't require more than 256 different values.
The problem becomes the moment I need to add in a 257th entry I need to expand the information base and on a computer you could jump to a 9th bit but for simplicity they don't they jump to 16 bits.
You're making a fatal flaw here - assuming that different or better adapted always requires more information. This is not the case - indeed, outside of eukaryota, there is a strong selective force against the accumulation of additional DNA.
For animals to evolve ergo they must encode more information or else they violate this central tennant of QM or don't evolve take your pick.
And again, you've made assumptions about evolution & biology that are false, and on those falsehoods your claims fail.
You do not need more biological 'information' to evolve; indeed, loosing unneeded information is observed as often as is the formation of new information. Information is not free - it carries a very real biological expense (notably, the energy and resources needed to replicate, maintain and process your genetic material). Nor do you need new information to increase complexity - few would argue that an amoeba or onion are more complex than humans, and yet they have more information. I'd argue a human is more complex than a mouse, and yet we have roughly the same amount of DNA with mice having slightly more functional genes.
What you've left out of your equation is the very principal of emergence. Emergence allows for the formation of hugely complex structures based on minimal information. Biology relies heavily on this. You DNA doesn't say "to make a person put protein X in position Y). Rather, it says "make proteins X, Y, Z; the interactions between them will the create a person". Very minute changes to "X, Y and Z" - rather than new proteins/information, are what creates the vast majority of the differences between you and a less 'complex' organisms such as a fly.
The standard metaphor for biological information (and a surprisingly accurate one at that) is that of a cake recipe - our DNA does not say "put a chocolate chip at 1.2,3.4,7.8cm [x,y,z coordinate]"; rather, it says "mix flour, eggs, milk, sugar, chocolate chips & yeast". Most evolution - including changes in apparent complexity - arise simply through minor tweaks to the recipe, rather than the addition of new ingredients.