Quantum Walks and Application to Quantum Money

Abstract

This thesis explores the foundations of quantum computation, focusing on quantum walks and their application to quantum money. Quantum walks, particularly continuous-time quantum walks based on group actions, serve as a powerful computational tool with applications in search algorithms and cryptographic protocols. We examine their mathematical structure and their advantages over classical random walks, emphasizing their efficiency in state evolution and probability distribution spreading. As a part of this work, we examine efficient implementations of transforms such as the Quantum Fourier Transform (QFT) and the Quantum Hartley Transform (QHT), analyzing their role in encoding quantum states for secure cryptographic applications. In particular, we discuss a novel instantiation of a quantum money scheme based on QHT, leveraging its unique properties for improved security and efficiency. To ensure the robustness of this quantum money scheme, we develop a verification mechanism utilizing quantum walks. Unlike previous approaches, which rely on standard quantum state measurements, our method employs continuous-time quantum walks to authenticate quantum money, preventing counterfeiting while maintaining computational feasibility. Additionally, we present a detailed discussion on the efficient implementation of this scheme, including optimized circuit designs and error mitigation strategies.