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Journal of information engineering and applications

Analysis and Review of Mathematical Concepts in Quantum Computation

M. A. Bashar, Mahmudul Hasan, M. Syduzzaman M. A. Chowdhury

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.Mathematics lies at the heart of quantum computation and it is needed in every aspect of the research and development of quantum computation.In this paper we have presented mathematical concepts that are closely related to quantum computation and their relation to quantum mechanics in a way which is innovative and very easy to comprehend.Most of the mathematical concepts presented in this paper are deduced from the postulates of quantum mechanics and linear algebra.

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The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.Mathematics lies at the heart of quantum computation and it is needed in every aspect of the research and development of quantum computation.In this paper we have presented mathematical concepts that are closely related to quantum computation and their relation to quantum mechanics in a way which is innovative and very easy to comprehend.Most of the mathematical concepts presented in this paper are deduced from the postulates of quantum mechanics and linear algebra.
Quantum computation, the ultimate goal of future computing, is an interesting field for researchers.The concept of quantum computation is based on basics of quantum mechanics.A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena such as superposition and entanglement, to perform operations on data.The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.A quantum computer operates by manipulating the qubits with a fixed sequence of quantum logic gates.The sequence of gates to be applied is called a quantum algorithm.The field of quantum computation algorithm is fast moving and the scope is vast.Major quantum algorithms are summarized in this paper along with their applications.
It is not in the scope of this elementary textbook to introduce quantum mechanics in detail or to provide an elaborate mathematical background required for quantum computing. However, we will provide some basic ideas that are essential for us to proceed further. To be precise, here we will briefly describe vector space, different types of operators, Gram-Schmidt procedure, postulates of quantum mechanics, POVM, density operator, entanglement and its measures, partial trace, Schmidt decomposition, Bloch sphere, nocloning theorem, etc. Let us start our journey toward the world of quantum computation by reviewing some simple ideas of algebra.
Quantum mechanics is more fundamental than classical mechanics, and it works at both the microscopic and macroscopic levels. However, the manifestation of quantum mechanics becomes more significant for particles and systems in the microscopic domain, which moves with high velocity. There are still questions about the interpretational aspects of quantum mechanics. However, at an operational level, it works over a wide range of phenomena with a high level of precision. The mathematics of quantum mechanics is much simpler than classical mechanics, and as a calculation device, quantum mechanics has been hugely successful. In this chapter, we will go through some of the topics in linear algebra and then move to the postulates of quantum mechanics.
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].
The aim of this paper is to present some basic notions of the theory of quantum computing and to compare them with the basic notions of the classical theory of computation. I am convinced, that the results of quantum computation theory (QCT) are not only interesting in themselves, but also should be taken into account in discussions concerning the nature of mathematical knowledge. The philosophical discussion will however be postponed to another paper. QCT seems not to be well-known among philosophers (at least not to the degree it deserves), so the aim of this paper is to provide the necessary technical preliminaries presented in a way accessible to the general philosophical audience.