bosons unitarity probability distribution permanent linear optical network Fock state computational complexity algorithm computer P
The Boson Sampling Problem in an experimental configuration assumes an input of bosons (ex. photons of light) of moderate number getting randomly scattered into a large number of output modes constrained by a defined unitarity. When individual photons of light are used the problem is isomorphic to a multi-photon quantum walk. The problem is then to produce a fair sample of the probability distribution of the output which is dependent on the input arrangement of bosons and the Unitarity. Solving this problem with a classical computer algorithm requires computing the permanent of the unitary transform matrix, which may be either impossible or take a prohibitively long time. In 2014, it was proposed that existing technology and standard probabilistic methods of generating single photon states could be used as input into a suitable quantum computable linear optical network and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted the sampling problem had similar complexity for inputs other than Fock state photons and identified a transition in computational complexity from classically simulatable to just as hard as the Boson Sampling Problem, dependent on the size of coherent amplitude inputs
