In this study, we propose an explicit quantum circuit design for the Hamiltonian simulation of nonlinear reaction-diffusion equations, aiming to bridge the gap between theoretical quantum algorithms and practical implementation. Specifically, we utilize the Carleman Linearization + Schrödingerization (CLS) algorithm, focusing on the explicit construction of its time evolution operator into a concrete quantum circuit. Our strategy exploits the specific sparse, block-tridiagonal structure of the Hamiltonian derived from the linearization process to effectively apply Trotter decomposition. Each resulting term in the decomposed operator is then explicitly synthesized via diagonalization, which involves a sequence of basis transformation, phase rotation, and inverse transformation using standard quantum gates. This concrete construction advances the theoretical frameworks established in prior research. While previous studies mainly focused on oracle-based algorithmic efficiency, our work provides the detailed gate implementations necessary for precise resource estimation. We validated our design through numerical simulations, confirming that it operates with tolerance precision. By enabling precise gate-based evaluation, our work helps overcome the fundamental barrier of mapping nonlinear, dissipative dynamics onto linear, unitary quantum systems. Ultimately, this contributes to the realization of Quantum Computer-Aided Engineering, offering a potential solution to the curse of dimensionality that hinders high-fidelity classical simulations in modern manufacturing.