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For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\\varepsilon) \\le \\left( \\frac{2d+3}{8} \\right) \\|L\\|_\\infty^2 \\left( \\frac{t^2}{\\varepsilon} \\right)$, holding for simulation time $t$ and error $\\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. 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