Capacity and quantum geometry of parametrized quantum circuits

TL;DR

This paper evaluates the capacity and geometry of parametrized quantum circuits, revealing how their expressiveness and trainability depend on circuit type, entangling gates, and initialization strategies, with implications for improving variational quantum algorithms.

Contribution

It introduces a geometric framework to analyze circuit expressiveness, identifies key scaling laws and transitions in quantum geometry, and proposes initialization and pruning methods to enhance trainability.

Findings

Different entangling gates significantly affect circuit expressiveness.

A transition in quantum geometry impacts the quantum natural gradient in deep circuits.

Proper initialization can avoid barren plateaus and improve trainability.

Abstract

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.