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To harness the potential of noisy intermediate-scale quantum devices, it isparamount to find the best type of circuits to run hybrid quantum-classicalalgorithms. Key candidates are parametrized quantum circuits that can beeffectively implemented on current devices. Here, we evaluate the capacity andtrainability of these circuits using the geometric structure of the parameterspace via the effective quantum dimension, which reveals the expressive powerof circuits in general as well as of particular initialization strategies. Weassess the expressive power of various popular circuit types and find strikingdifferences depending on the type of entangling gates used. Particular circuitsare characterized by scaling laws in their expressiveness. We identify atransition in the quantum geometry of the parameter space, which leads to adecay of the quantum natural gradient for deep circuits. For shallow circuits,the quantum natural gradient can be orders of magnitude larger in valuecompared to the regular gradient; however, both of them can suffer fromvanishing gradients. By tuning a fixed set of circuit parameters to randomizedones, we find a region where the circuit is expressive, but does not sufferfrom barren plateaus, hinting at a good way to initialize circuits. We show analgorithm that prunes redundant parameters of a circuit without affecting itseffective dimension. Our results enhance the understanding of parametrizedquantum circuits and can be immediately applied to improve variational quantumalgorithms.

Capacity and quantum geometry of parametrized quantum circuits