The power of quantum computing is constructing the solution to a problem out of an interference pattern. Classical probabilities don’t interfere, but quantum probabilities do. Loosely, quantum probabilities can be constructed to cancel, since their amplitudes can be negative.

Shor’s algorithm works on the quantum Fourier transform. The quantum Fourier transform works because you can pick a frequency out of a signal using a “test wave.” The test wave can select out the amplitude of interest because the information of the test wave constructively interferes, whereas every other frequency cancels. This is the interference effect that can only happen with complex/negative probability amplitudes.