Optimal complexity of parameterized quantum circuits

TL;DR

This paper investigates the complexity of parameterized quantum circuits, showing they reach asymptotic complexity faster than random circuits, with topology and majorization as key factors influencing their expressibility and entanglement.

Contribution

It provides a comparative analysis of parameterized quantum circuits versus random circuits, highlighting the importance of topology and majorization in their complexity and expressibility.

Findings

Parameterized quantum circuits reach asymptotic complexity faster than random circuits.

Topology significantly influences the complexity and expressibility of quantum circuits.

Majorization is a useful tool alongside entanglement measures for analyzing circuit capabilities.

Abstract

Parameterized quantum circuits play a key role for the development of quantum variational algorithms in the realm of the NISQ era. Knowing their actual capability of performing different kinds of tasks is then of the utmost importance. By comparing them with a prototypical class of universal random circuits we have found that their approach to the asymptotic complexity defined by the Haar measure is faster, needing less gates to reach it. Topology has been revealed crucial for this. The majorization criterion has proven as a relevant complementary tool to the expressibility and the mean entanglement.