  The determination of the computational complexity of the boson sampling
problem with single boson sources has opened a novel research direction in the
quantum computing field. Some research effort has also been devoted towards the
use of different input sources where the sampling from the output distribution
of a linear interferometer is still a complex task not achievable by a
classical device. In this letter, we discuss the use of multi-mode thermal
sources, based on a generalization of the boson sampling problem which relies
on time-correlation measurements. We derive, for any possible sample, two
equivalent formulations for the respective multi-boson correlation functions in
terms of matrix permanents. The properties of these permanents emerging from
the physics of multi-boson interference raise essential questions in the field
of complexity theory. Our results are a manifestation of the physics behind
correlated measurements in either photonic or atomic interferometers with
bosonic thermal sources. This fascinating physics paves the way towards
important applications not only in quantum computing but also in quantum
metrology and quantum communication, by taking advantage of different kinds of
bosonic sources.
