I wonder: given that a Photon travels at the speed of light, then everything happens instantaneously, from its point of view. So, is choosing the path that takes the shortest time like a standing wave?
Note that photons don't take the shortest path; they take a variationally stationary path (not necessarily the shortest one). My suspicion is that this is because all the possible photon states are coherent along and immediately nearby that path, whereas if the path is not variationally stationary then it interferes with nearby paths. I've done very little variationally physics, however, so take this with a grain of salt.
Well, am new to the subject, so not sure what this means. Am curious if there is any relationship between Fermat's principle of least time (https://en.wikipedia.org/wiki/Fermat's_principle) and Quantum FFTs (https://www.quora.com/How-does-Shors-algorithm-work-in-layma...)
I don't know enough about quantum FFTs to know if there is an analogy. However, I'm taking Aaronson's class next semester so I'll be able to tell you in a few months ;)