No, they can run on quantum computers, but to properly simulate a quantum computer, you may need exponential space/time. Googling for supporting data brought up http://alumni.imsa.edu/~matth/quant/433/shor-par/node19.html which discusses simulating Shor's algorithm on a classical computer.

The reason for the exponential is that the bits are all in a superposition of all possible states (so for a 14-bit register, that's a superposition of 2 to the 14 states), which operations all act on. Or something like that. I'm not really that clear on the details. :(