Mathematical Tools

Ramona Wolf

Riccardo Manenti, Mário Motta

Abstract In this chapter, we introduce some mathematical tools used throughout the rest of the textbook, to provide readers from different backgrounds with a good understanding of our notation and terminology. These mathematical tools include the binary system, Boolean functions, and sets of logic gates universal for classical computation. We then introduce the notion of a decision problem, a mathematical problem whose solution is either yes or no and discuss how a computation solves it by exploiting two physical resources: time and space. We conclude by presenting several examples of decision problems, including satisfiability, MAXCUT, traveling salesman, and the hidden subgroup problem

Walter Greiner

Benjamin H. Feintzeig

This Element provides an entry point for philosophical engagement with quantization and the classical limit. It introduces the mathematical tools of C*-algebras as they are used to compare classical and quantum physics. It then employs those tools to investigate philosophical issues surrounding theory change in physics. It discusses examples in which quantization bears on the topics of reduction, structural continuity, analogical reasoning, and theory construction. In doing so, it demonstrates that the precise mathematical tools of algebraic quantum theory can aid philosophers of science and philosophers of physics.

Yousra Mahmoudi, Nadjet Zioui, Hacène Belbachir +2 more

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PDF

Walter Greiner