Quantum algorithms in group theory

M Batty, SL Braunstein, Andrew Duncan +1 more

Jincheng Zhang

This paper investigates the synergistic relationship between high-dimensional group theory and quantum computing. Traditional quantum computing approaches often lack the structural advantages offered by group theory, leading to limitations in algorithm efficiency and stability. We propose a novel framework based on the principles of group theory to design and optimize quantum algorithms, aiming to significantly enhance their performance. This work explores how the inherent properties of groups can be leveraged to improve the generation and execution of quantum states, thereby reducing computational complexity. The research highlights the potential for a new generation of quantum algorithms that are intrinsically more robust and efficient.

M Batty, SL Braunstein, Andrew J. Duncan +1 more

We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.

Stephen Fenner, Yong Zhang

Akshay Kishanlal Jaiswal

Quantum computing represents a transformative paradigm that leverages the principles of quantum mechanics, such as superposition and entanglement, to perform computations beyond the capabilities of classical systems. The development of efficient quantum algorithms and architectures relies heavily on robust mathematical frameworks, among which group theory plays a fundamental role. Group-theoretic structures provide a systematic way to describe symmetries, unitary transformations, and state evolution in quantum systems. In particular, unitary and special unitary groups govern the behavior of quantum gates, while representation theory enables the analysis of multi-qubit operations and composite systems. Moreover, group theory underpins several key aspects of quantum computing, including the design of quantum algorithms, error correction schemes, and cryptographic protocols. Notable algorithms such as Shor’s Algorithm and Grover’s Algorithm exploit algebraic structures to achieve computational advantages. This review aims to provide a comprehensive overview of the interplay between group theory and quantum computing, highlighting fundamental concepts, major applications, and recent developments. The paper also identifies current challenges and outlines future research directions, thereby offering a consolidated resource for researchers in mathematics, physics, and quantum information science.