  Circuit QED based quantum information processing relies on low noise
amplification for signal readout. In the realm of microwave superconducting
circuits, this amplification is often achieved via Josephson parametric
amplifiers (JPA). In the past, these amplifiers exhibited low power added
efficiency (PAE), which is roughly the fraction of pump power that is converted
to output signal power. This is increasingly relevant because recent attempts
to build high saturation power amplifiers achieve this at the cost of very low
PAE, which in turn puts a high heat load on the cryostat and limits the number
of these devices that a dilution refrigerator can host. Here, we numerically
investigate upper bounds on PAE. We focus on a class of parametric amplifiers
that consists of a capacitor shunted by a nonlinear inductive block. We first
set a benchmark for this class of amplifiers by considering nonlinear blocks
described by an arbitrary polynomial current-phase relation. Next, we propose
two circuit implementations of the nonlinear block. Finally, we investigate
chaining polynomial amplifiers. We find that while amplifiers with higher gain
have a lower PAE, regardless of the gain there is considerable room to improve
as compared to state of the art devices. For example, for a phase-sensitive
amplifier with a power gain of 20 dB, the PAE is ~0.1% for typical JPAs, 5.9%
for our simpler circuit JPAs, 34% for our more complex circuit JPAs, 48% for
our arbitrary polynomial amplifiers, and at least 95% for our chained
amplifiers.
