  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 representation 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. Our results
enhance the understanding of parametrized quantum circuits for improving
variational quantum algorithms.
