Exploring Methods for Quantifying Expressibility and Entangling Capability of Parameterized Quantum Circuits

Abstract

Quantum Machine Learning (QML) is an exciting fusion of two disciplines, wherein classical machine learning techniques are augmented by quantum computation. Central to QML is the utilization of quantum bits, exploiting quantum mechanical principles such as superposition and entanglement to process vast amounts of information in parallel. This thesis focuses on parameterized quantum circuits, where gates with modifiable parameters facilitate the exploration of quantum states. Through the adjustment of these parameters, circuit behavior can be fine-tuned to execute computational tasks, including regression, classification and optimization. Nevertheless, there is a lack of comprehension regarding the factors that render a specific parameterized circuit more powerful or advantageous compared to others.

To investigate these questions, three descriptors for quantifying the expressibility and entanglement capabilities of parameterized quantum circuits are proposed. These quantities are computed using statistical properties based on sampling states from a circuit model family. For this thesis, a Python framework for quantum machine learning was developed. The provided code facilitates the sampling, training, and evaluation of quantum circuits, as well as implementation of the descriptors. Utilizing this framework, a total of 10,000 randomly selected quantum circuits were trained and scored on a classification task of the popular Iris dataset. Subsequently, a correlation analysis was conducted to explore the relationship between calculated descriptor values and the obtained scores.

This study aims to provide insights into the role of entanglement and expressibility in quantum machine learning models, offering guidance for the design of future QML algorithms.