What is meant by the term "quantum interference"?

Question

What is meant by the term "quantum interference" in the context of quantum computing and quantum algorithms?

Answer 1

The term is most often used in the context of the two-slit experiment. The interference pattern that you see, which is not simply what you'd get if you added the wavefunction ($\Psi_A$) associated with going through slit A to the wavefunction ($\Psi_B$) going through slit B, but there is a "mixed" term as follows:

$$ \tag{1} |\psi|^2 = |\psi_A + \psi_B|^2 = |\psi_A|^2 + |\psi_B|^2 + \psi_A^*\psi_B + \psi_A\psi_B^*. $$

This interference pattern occurs even when you reduce the number of photons until there's only one photon at a time! When there's only one photon at a time, what could it possibly be interfering with to cause the patter on the screen to be different from $P_A + P_B$?

We say the photon interferes with itself. This is a quantum mechanical phenomenon that doesn't exist in classical mechanics.

The two-slit experiment was used by Feynman when he explained the Feynman integral, because the Feynman integral is a sum/integral over paths which each have an amplitude. These amplitudes interfere with each other, which again is a signature of quantum mechanical effects, since classically there would only be one path in the Feynman integral: the path with the smallest Lagrangian action.