  To harness the potential of noisy intermediate-scale quantum devices, it is
paramount to find the best type of circuits to run hybrid quantum-classical
algorithms. Key candidates are parametrized quantum circuits that can be
effectively implemented on current devices. Here, we evaluate the capacity and
trainability of these circuits using the geometric structure of the parameter
space via the effective quantum dimension, which reveals the expressive power
of circuits in general as well as of particular initialization strategies. We
assess the expressive power of various popular circuit types and find striking
differences depending on the type of entangling gates used. Particular circuits
are characterized by scaling laws in their expressiveness. We identify a
transition in the quantum geometry of the parameter space, which leads to a
decay of the quantum natural gradient for deep circuits. For shallow circuits,
the quantum natural gradient can be orders of magnitude larger in value
compared to the regular gradient; however, both of them can suffer from
vanishing gradients. By tuning a fixed set of circuit parameters to randomized
ones, we find a region where the circuit is expressive, but does not suffer
from barren plateaus, hinting at a good way to initialize circuits. We show an
algorithm that prunes redundant parameters of a circuit without affecting its
effective dimension. Our results enhance the understanding of parametrized
quantum circuits and can be immediately applied to improve variational quantum
algorithms.
