Linear Algebra and Quantum Mechanics

Yinshan Jiang

The theoretical breakthroughs and practical applications of quantum mechanics have always been deeply intertwined with the innovative applications of linear algebra [1]. As a core mathematical tool for describing the motion laws of microscopic particles, linear algebra not only provides a rigorous framework for fundamental concepts such as quantum states and quantum operators [12], but also achieves methodological innovation, scenario innovation, and theoretical integration innovation in the evolution of quantum technologies [19]. This paper, based on the core connection between linear algebra and quantum mechanics, focuses on analyzing the innovative application directions of linear algebra in the field of quantum mechanics [2]. These include linear algebra simplification methods for high-dimensional quantum systems [6], innovative linear algebra representations of quantum entanglement [10], algorithmic innovations in linear algebra for quantum computing [20], and optimization of linear algebra in quantum measurement [7]. By integrating cutting-edge practical cases, the paper demonstrates the value of these innovative applications, explores the bottlenecks and optimization paths encountered during the innovation process, and looks forward to the future development trends of their integrated innovation [6]. The findings aim to provide references for innovative research and technological implementation in quantum mechanics-related fields [2].

BongJu Kim

In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The formulation uses functional analysis, linear algebra and probability theory. The knowledge of the mathematical formulation of QM is enough quantum mechanical knowledge for approaching to quantum algorithm and it might be efficient way for mathematicians that starting with mathematical formulations of QM. We explain the mathematical formulations of quantum mechanics briefly, quantum bits, quantum gates, quantum discrete Fourier transformation, Deutsch's algorithm and Shor's algorithm.

John David Jackson, Robert C. Whitten

Steven Herbert

Abstract Quantum mechanics is mathematically formulated in terms of a Hilbert space, which is a vector space with a defined inner product. Linear algebra is the study of vector spaces and the (linear) operations thereon, and so it follows that having a solid grasp of the basics of linear algebra is essential for studying quantum mechanics and, therefore, quantum computation. All the linear algebra prerequisites necessary to understand the theoretical results presented in the remainder of the book are detailed in this chapter. The chapter also acts as a nomenclature reference for the book, by providing all the notations that are used thereafter.