  Parameterized quantum circuits play an essential role in the performance of
many variational hybrid quantum-classical (HQC) algorithms. One challenge in
implementing such algorithms is to choose an effective circuit that well
represents the solution space while maintaining a low circuit depth and number
of parameters. To characterize and identify expressible, yet compact,
parameterized circuits, we propose several descriptors, including measures of
expressibility and entangling capability, that can be statistically estimated
from classical simulations of parameterized quantum circuits. We compute these
descriptors for different circuit structures, varying the qubit connectivity
and selection of gates. From our simulations, we identify circuit fragments
that perform well with respect to the descriptors. In particular, we quantify
the substantial improvement in performance of two-qubit gates in a ring or
all-to-all connected arrangement compared to that of those on a line.
Furthermore, we quantify the improvement in expressibility and entangling
capability achieved by sequences of controlled X-rotation gates compared to
sequences of controlled Z-rotation gates. In addition, we investigate how
expressibility "saturates" with increased circuit depth, finding that the rate
and saturated-value appear to be distinguishing features of a parameterized
quantum circuit template. While the correlation between each descriptor and
performance of an algorithm remains to be investigated, methods and results
from this study can be useful for both algorithm development and design of
experiments for general variational HQC algorithms.
