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Logical structures underlying quantum computing

Federico Holik, Giuseppe Sergioli, Héctor Freytes, Angel Ricardo Plastino

Published Jan 1, 2019• 6 citations
OpenAlex

Abstract

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.

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The report deals with the axiomatic approach to the phenomena of quantum computation and algorithms. It gives an overview of the classical approach (due to von Neumann) using the structures of Hilbert spaces, as well as mentions some major results in the field and discusses the flaws of the model and its current dominance. This leads to the introduction of a new axiomatization, first proposed by S. Abramsky and B. Coecke in 2007, which relies on the modern concepts of category theory. It is a field of mathematics originally created in the 1940s by specialists in algebraic topology but currently gaining popularity in other branches because of its abstract nature and emphasis on operations rather than objects. Most importantly, category theory is a powerful tool in functional analysis allowing mathematicians to reformulate the statements of Hilbert space theory in a more intuitive way, without matrix calculus. Finally, the logic behind quantum operations is discussed: it is revealed that the ideas of quantum algorithms have a strong connection to Jean-Yves Girard’s linear logic.
This paper explores a novel computational framework based on quantum entanglement, aiming to revolutionize logical reasoning and computation. Traditional logical computation relies on classical logic gates, but this research leverages the unique properties of quantum entanglement to establish a fundamentally different approach. We propose a system that utilizes entangled states to perform quantum logic operations, offering a potential paradigm shift for quantum computing. The core mechanism involves designing a quantum logic algorithm predicated on entanglement, exploring its implications for the advancement of quantum computation. The paper presents a conceptual outline, detailing the design process and potential applications. We argue that this approach presents a significant advancement in computational modeling, potentially unlocking new capabilities within the field of quantum information processing.