I’m not a quantum physicist, but I am familiar with computational complexity theory, and have some knowledge about quantum complexity classes. My answer is no.
The issue you’re describing is a matter of throughput, not complexity. It is inherently parallelizable throughput, but still a throughput issue. A fractal is just a really, really long series of computations. Bigger resolutions mean more computations have to be done, but bigger resolutions can also be computed in parallel chunks by farming different parts of the image off to multiple cores (though some of this depends on how sequential your fractal generation technique is).
This is just a lot of computation, but nothing is really unknown. You’re never enumerating possibilities and guessing their candidacy, everything is known a priori.
What quantum computers theoretically excel at are certain variations of classes of problems known as “promise classes”. One of the classic “promise classes” is BPP or “Bounded-error Probabilistic Polynomial time”, that is, the set of problems such that, given a question, you can check very quickly whether:
- If the answer is ‘yes’ then at least 2/3 of all possible computation paths accept.
- If the answer is ‘no’ then at most 1/3 of all possible computation paths accept.
In fact, BPP is contained in BQP, which is one of the simpler quantum classes. Meaning a quantum computer can solve all BPP problems in polynomial time with error no more than a 1 in 3 chance of making an error. Or, more simply, a quantum computer can solve these problems quickly.
Technically, this means BQP can solve P problems, like fractal generation, as well. However, it won’t be any faster at this. This is because any computer that can solve a higher complexity class problem generally solves a less complex problem by merely simulating a less powerful machine. (There are a few exceptions here, I think)
Basically quantum computers aren’t inherently any “faster” it’s just that they can solve more types of problems. The reason we may describe them as “faster” is that our current computers can solve these problems, but they do it by simulating a more powerful machine. Essentially, our current computers enumerate possibilities, guess, and check whereas a more powerful machine could simply try all guesses at once (I’m being imprecise here, but that’s the gist). This is extremely, extremely slow.
In fact, almost all of these classes I’m describing are defined in terms of being able to solve a certain class of problem as fast as a current computer. BQP is effectively saying “a machine that can solve BQP can solve quantum problems as fast as a normal computer solves non-quantum problems”. It makes no claims about how fast it can solve problems that a normal computer already can solve.