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Lecture notes in networks and systems

Graph Theory Intersects with Cryptographic Algorithm Design in the Quantum-Computing Context: A Survey

Bharati R. Kale, Rani Desurkar, Jotiba P. Kitturkar, S. L. Deshpande

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.1007/978-3-032-26908-9_39

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Quantum computing (QC) is a new computational paradigm whose foundations relate to quantum physics. Notable progress has been made, driving the birth of a series of quantum-based algorithms that take advantage of quantum computational power. In this paper, we provide a targeted survey of the development of QC for graph-related tasks. We first elaborate the correlations between quantum mechanics and graph theory to show that quantum computers are able to generate useful solutions that can not be produced by classical systems efficiently for some problems related to graphs. For its practicability and wide-applicability, we give a brief review of typical graph learning techniques designed for various tasks. Inspired by these powerful methods, we note that advanced quantum algorithms have been proposed for characterizing the graph structures. We give a snapshot of quantum graph learning where expectations serve as a catalyst for subsequent research. We further discuss the challenges of using quantum algorithms in graph learning, and future directions towards more flexible and versatile quantum graph learning solvers.
Recent progress in quantum computing shows the need to incorporate many branches of mathematics (graph theory, matrix theory, optimization, theory of orthogonal polynomials and more) into physics, computer science and chemistry. At the 2024 SIAM Quantum Intersections Convening, Bert de Jong (Lawrence Berkeley National Laboratory) gave a talk entitled 'Quantum Science Needs Mathematicians' (Report of the SIAM Quantum Intersections Convening. Integrating Mathematical Scientists into Quantum Research, 7-9 October 2024, Tysons, Virginia (doi:10.11337/25M1741017)), since despite the growing demand for research in these domains, the mathematical sciences community has remained largely disengaged from quantum research, with only a few isolated areas of active involvement. This issue brings together researchers from different areas of mathematics to show the relation between spectral graph theory, the theory of orthogonal polynomials and numerical analysis. This interconnectedness highlights the versatility and importance of these areas of mathematics in the context of quantum computing.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
Quantum computing has emerged as a transformative computing paradigm capable of solving problems that remain computationally infeasible for classical systems; however, its rapid advancement also introduces significant security, privacy, and reliability concerns. In this context, this survey presents a comprehensive review of security challenges and mitigation strategies associated with quantum computing, focusing on security issues in quantum computers, security threats caused by quantum computers, and security mechanisms developed for quantum systems. The paper examines vulnerabilities in quantum hardware and software, the impact of quantum computing on existing cryptographic infrastructures and cybersecurity mechanisms, and the development of quantum-resilient solutions such as post-quantum cryptography, quantum-safe communication protocols, quantum intrusion detection systems, and quantum-aware software engineering techniques. In addition, the survey discusses emerging applications of quantum technologies in cybersecurity domains, including malware detection, network intrusion detection, Internet of Things (IoT) security, and secure communication systems. Furthermore, the paper analyzes existing quantum error mitigation and fault-tolerance approaches designed to improve the robustness and trustworthiness of quantum computation under realistic noisy conditions. By consolidating recent advances, open research challenges, and future directions, this survey provides a structured overview of the evolving intersection between quantum computing and cybersecurity, while serving as a reference for researchers and practitioners working toward secure, resilient, and quantum-ready computing infrastructures.