I've seen that quantum computing calculations have their own complexity classes https://en.wikipedia.org/wiki/Quantum_complexity_theory, namely BQP and QMA.
I've heard that this would beat any classical computer, but I wonder if it would also beat a custom-made massively parallel computer. By massively parallel computer I imagine a huge number of little mini-computers each with a tiny memory and all of them arranged in 3-dimensional space. Each mini-computer can communicate with its local neighbors. During one step each mini-computer performs some internal computation and then sends information to its neighbors. I suppose this massively parallel setup allows for faster computation that a classical computer?
How would one define an appropriate complexity class for such a massively parallel setup? I suppose one needs to limit the number of mini-computers used? Is there an existing computation class for that?
Are the quantum complexity classes faster than this massively parallel class (for large problem instances)?
By "faster" I generally mean faster for some problem and a sufficiently large problem size. I know that quantum computers are always probabilistic, so for comparison it's ok if the parallel computers implement a probabilistic method, too. In terms of resources (qubits, mini-computers) their scaling should be comparable (otherwise one method would outperform the other due to limitation of resources).
For the physics-minded people: If the quantum complexity class is faster than the massively parallel class, does that mean that quantum computers cannot be described by a conventional field theory? In a field theory each position in space has some information - essentially my mini-computers. During each time step there are simple local rules (a differential equation) according to which the information at each position evolves. This is the next-neighbor-communication of my mini-computers.
In the end, I'm looking for a rigorous answer to the question: If quantum mechanics were described by some differential equation (a field theory, essentially the massively parallel mini-computers), then quantum computing would be impossible, because quantum computing promises to be faster than any massively parallel classical computer?