Turning to quantum mechanics … secret, secret, close the doors! we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. … It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. So that's why I like to investigate things.
The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative.
Our first result is the existence of an efficient universal quantum Turning machine in Deutsch's model. … We give the first formal evidence that quantum Turning machines violate the modern complexity theoretic formulation of the Church Turning thesis. We show the existence of a problem relative to an oracle that can be solved in polynomial time on a quantum Turning machine but require super polynomial time on a bounded error probabilistic Turning machine.
Quantum computing is a new computational model that may be physically realizable and may provide an exponential advantage over “classical” computational models such as probabilistic and deterministic Turing machines.