QUANTUM MODELING AND LINEAR ALGEBRA

Published by Marcos Elias on

Abstract

This paper presents a concise yet comprehensive introduction to the mathematical foundations of quantum algorithms. Without assuming extensive prior knowledge, the work maintains full mathematical rigour while avoiding the interpretational and philosophical challenges commonly associated with quantum mechanics.

The central objetive is to highlight the fundamental role of linear algebra in the design, formulation and operation of quantum algorithms. The paper provides a structured overview of the most essential and widely studied quantum algorithms, offering a clear conceptual bridge between abstract mathematical formalism and practical algorithmic implementation.

Additionally, this work summarises key insights from an upcoming book by the author, extending and contextualising the theoretical framework that underpins modern quantum computation.

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Marcos Elias

I am the founder of the Rāmānujan Institute for the Development of Prodigious Young Mathematicians, which aims to foster the education and growth of children and young people aged 2 to 16 who demonstrate exceptional mathematical abilities. Our institute is dedicated to providing an advanced and challenging curriculum that not only prepares students for prestigious competitions, such as the International Mathematical Olympiad (IMO), but also supports their overall development. We focus on nurturing mathematical talent while ensuring that our students develop the skills necessary for success in academic and professional environments. Through mentorship programs and a commitment to inclusivity, we strive to create a community where young mathematicians can thrive and make significant contributions to the field. I am also Marcos Eduardo Elias, an engineer and mathematician. My academic journey began with a Bachelor’s degree in Mechatronic Engineering from the Polytechnic School of the University of São Paulo (USP). My education culminated in a Doctorate in Mathematics from St. Petersburg State University, specializing in Real, Complex, and Functional Analysis. I have held teaching positions at institutions such as Ibmec, Insper, and FGV-SP. I am also a member of the Brazilian Mathematical Society. In the financial sector, I founded several institutions, including GAS Investimentos (later Vinci Partners), Empiricus Research, Turing High Frequency Trading, Modena Capital and Guiar Investments. In the financial markets, I am known as the Volatility Hunter, and - in the last 30 years, I have been relentlessly searching for highly asymmetric upside risk (I am on the right side of the barbell). I have been contributing to the MIT Sloan Management Review, primarily on topics related to risk management. My current research interests include real, complex, and functional analysis, linear algebra for quantum algorithmics, Feigenbaum-Coullet-Tresser universality, stochastic calculus, and Grothendieck's Standard Conjectures on Algebraic Cycles.

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