Quantum Computing Algebra (QCA), the theory and implementation

• URL: http://arxiv.org/abs/2606.17621v1
• Date: Tue, 16 Jun 2026 07:27:40 GMT
• Title: Quantum Computing Algebra (QCA), the theory and implementation
• Abstract summary: We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations.<n>We show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally.
• Score: 0.2249916681499244
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a real geometric algebra framework designed for the direct translation of the Dirac formalism into geometric algebra representations. Unlike previous approaches based on positive-definite signatures, QCA employs a split-signature construction that enables a natural realization of quantum states and operators while simplifying computational implementation. We further present an implementation of QCA using the \textit{GAALOP} software and show how quantum gates and multi-qubit systems can be efficiently represented and generated computationally. As an application, we demonstrate the use of QCA in quantum game theory, where the real-algebraic formulation provides computational advantages for modeling entangled strategies and quantum interactions. The proposed framework establishes a practical bridge between the abstract formalism of quantum computation and efficient geometric algebra implementations.

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