  A universal programmable quantum processor uses program quantum states to
apply an arbitrary quantum channel to an input state. We generalize the concept
of a finite-dimensional programmable quantum processor to infinite dimension
assuming an energy constraint on the input and output of the target quantum
channels. By proving reductions to and from finite-dimensional processors, we
obtain upper and lower bounds on the program dimension required to
approximately implement energy-limited quantum channels. In particular, we
consider the implementation of Gaussian channels. Due to their practical
relevance, we investigate the resource requirements for gauge-covariant
Gaussian channels. Additionally, we give upper and lower bounds on the program
dimension of a processor implementing all Gaussian unitary channels. These
lower bounds rely on a direct information-theoretic argument, based on the
generalization from finite to infinite dimension of a certain replication lemma
for unitaries.
