  The unitary description of beam splitters (BSs) and optical parametric
amplifiers (OPAs) in terms of the dynamical Lie groups $SU(2)$ and $SU(1,1)$
has a long history. Recently, an inherent duality has been proposed that
relates the unitaries of both optical devices. At the physical level, this
duality relates the linear nature of a lossless BS to the nonlinear Parametric
Down-Conversion (PDC) process exhibited by an OPA. Here, we argue that the
duality between BS and PDC can instead be naturally interpreted by analyzing
the geometrical properties of both Lie groups, an approach that explicitly
connects the dynamical group description of the optical devices with the
aforementioned duality. Furthermore, we show that the BS-PDC duality can be
represented through tensor network diagrams, enabling the implementation of a
PDC as a circuit on a standard quantum computing platform. Thus, it is feasible
to simulate nonlinear processes by using single-qubit unitaries that can be
implemented on currently available digital quantum processors.
