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 . We find that the capacity exhibits scaling laws and striking differences depending on the type of entangling gates used. Based on our methods, we propose an initialization strategy where the circuit is expressive but does not suffer from barren plateaus. Further, we identify a transition in the quantum geometry when the circuit becomes overparameterized. Finally, 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.
 Tobias Haug, Kishor Bharti, and M.S. Kim PRX Quantum 2, 040309 (2021)