Simulating quantum effects with classical mechanics

Is it theoretically possible to simulate any quantum effects with classical mechanics, if there are no time and processing power constraints? In quantum mechanics, special mathematics is needed to include effects related to time, superposition etc. But can we still theoretically simulate all of this with perfect accuracy on a classical computer?

The classical computer simulation should have all the effective internal interaction and the results of the simulation should be effectively the same, just the speed of events and the underlying physical form of events would differ.

The classical computer would lack a quantum random number generator but a separate quantum random number generator could be used for that purpose. Although does the source for the random number generation make any difference if we just want to simulate possible events and end results ie. it's enough if the system just recreates everything that effectively has the same kind of chance to produce the same events and end results?

A simulation of a simple quantum effect interaction probably doesn't have any fundamental difference to a complex one, the latter would just require more processing power?