<![CDATA[A comprehensive mathematical model formulation is presented, encompassing gate fidelity optimization, coherence time extension, stabilizer code evolution, and surface code implementation. The research demonstrates significant advancements in qubit stability, with a 7% increase in gate fidelity and a remarkable 50% extension in coherence time achieved through optimized gate operations and material improvements. Quantum error correction techniques, guided by the Lindblad master equation and the surface code, result in a 25% reduction in error rates, contributing to the overall stability of the quantum processor. The outcomes not only bring practical quantum computing closer to realization but also provide a foundation for future innovations. The research identifies avenues for continued optimization, including advanced gate designs, exploration of emerging qubit technologies, and the development of sophisticated error correction codes. Further interdisciplinary collaborations and investigations into scalable quantum architectures, materials science, and cryogenic engineering are essential for overcoming remaining challenges. The insights gained contribute to the advancement of fault-tolerant quantum computing systems, offering transformative capabilities for computation and technology.]]>
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