  We study bosonic quantum computations using the Segal-Bargmann representation
of quantum states. We argue that this holomorphic representation is a natural
one which not only gives a canonical description of bosonic quantum computing
using basic elements of complex analysis but also provides a unifying picture
which delineates the boundary between discrete- and continuous-variable quantum
information theory. Using this representation, we show that the evolution of a
single bosonic mode under a Gaussian Hamiltonian can be described as an
integrable dynamical system of classical Calogero-Moser particles corresponding
to the zeros of the holomorphic function, together with a conformal evolution
of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to
unique features of bosonic Hilbert spaces such as squeezing. We then generalize
the properties of this holomorphic representation to the multimode case,
deriving a non-Gaussian hierarchy of quantum states and relating entanglement
to factorization properties of holomorphic functions. Finally, we apply this
formalism to discrete- and continuous- variable quantum measurements and obtain
a classification of subuniversal models that are generalizations of Boson
Sampling and Gaussian quantum computing.
