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Representation of $\mathfrak{su}(8)$ in Pauli Basis

Quantum computation started to become significant field of studies as it hold great promising towards the upgrade of our current computational power. Studying the evolution of quantum states serves as a good fundamental in understanding quantum information which lead to quantum computation. This was assisted with the respective mathematical tools such as Lie group and Lie algebra. In this study, the Lie algebra of $\mathfrak{su}(8)$ is represented in tensor product between three Pauli matrices. This is done by constructing the generalized Gell-Mann matrices and compared to the Pauli basis. This study will explicitly shows the one-to-one correlation of Gell-Mann matrices with the Pauli basis resembled change of coordinates. This is particularly useful when dealing with quantum circuit problems.