Linear algebra and quantum algorithm

TL;DR

This paper explores the mathematical foundations of quantum algorithms, emphasizing linear algebra and quantum mechanics formulations, and discusses key algorithms like Deutsch's and Shor's within this framework.

Contribution

It provides a mathematical perspective on quantum algorithms based on linear algebra and quantum mechanics formulations, connecting foundational theory with specific algorithms.

Findings

Quantum algorithms are expressed using linear algebra on complex vector spaces.

The paper explains key quantum algorithms such as Deutsch's and Shor's algorithms.

Mathematical formulations of quantum mechanics underpin quantum computing methods.

Abstract

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.