This thesis explores the classical Sewing Lemma and introduces Young’s integration theory. The key concept is the non-commutative extension of the Sewing Lemma, which provides a method to solve differential equations involving matrix-valued functions. These equations lead to a new concept, called "integral product," which plays a crucial role in describing the time evolution of quantum systems.

The thesis then generalizes the fourth postulate of quantum mechanics, focusing on the time evolution operator that governs quantum systems. Finally, the work explores the potential applications of this generalization in quantum computing, particularly in the context of quantum machine learning.

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