  In linear optics, photons are scattered in a network through passive optical
elements including beamsplitters and phase shifters, leading to many intriguing
applications in physics, such as Mach-Zehnder interferometry, Hong-Ou-Mandel
effect, and tests of fundamental quantum mechanics. Here we present a general
analytic expression governing the upper limit of the transition amplitudes in
sampling bosons, through all realizable linear optics. Apart from boson
sampling, this transition bound results in many other interesting applications,
including behaviors of Bose-Einstein Condensates (BEC) in optical networks,
counterparts of Hong-Ou-Mandel effects for multiple photons, and approximating
permanents of matrices. Also, this general bound implies the existence of a
polynomial-time randomized algorithm for estimating transition amplitudes of
bosons, which represents a solution to an open problem raised by Aaronson and
Hance in 2012.
