# Optimal Complexity of Parameterized Quantum Circuits

## TL;DR

This paper studies how different quantum circuits generate complex quantum states and finds that some circuits reach high complexity faster than others.

## Contribution

The paper introduces a comparison of parameterized quantum circuits with random circuits to assess their complexity and convergence rates.

## Key findings

- Parameterized circuits converge faster to asymptotic complexity than random circuits.
- Qubit connection topology strongly influences entanglement and complexity growth.
- Majorization-based measures provide new insights into random state generation.

## Abstract

Parameterized quantum circuits are central to the development of variational quantum algorithms in the NISQ era. A key feature of these circuits is their ability to generate an expressive set of quantum states, enabling the approximation of solutions to diverse problems. The expressibility of such circuits can be assessed by analyzing the ensemble of states produced when their parameters are randomly sampled, a property closely tied to quantum complexity. In this work, we compare different classes of parameterized quantum circuits with a prototypical family of universal random circuits to investigate how rapidly they approach the asymptotic complexity defined by the Haar measure. We find that parameterized circuits exhibit faster convergence in terms of the number of gates required, as quantified through expressibility and majorization-based complexity measures. Moreover, the topology of qubit connections proves crucial, significantly affecting entanglement generation and, consequently, complexity growth. The majorization criterion emerges as a valuable complementary tool, offering distinct insights into the behavior of random state generation in the considered circuit families.