How many quantum samples do you actually need to simulate noise?

Simulating how quantum systems lose coherence (Lindbladian dynamics) requires copying quantum states as 'samples' — but how many? This work tightens the count to roughly (2d+3)/8 × ‖L‖²(t²/ε), cutting the previous d² dimension dependence nearly in half. Better yet, for random noise operators the dimensional overhead vanishes entirely, while a constructed worst-case example proves the algorithm can't escape an Ω(d) penalty — a clean separation with real implications for quantum simulation resource estimates.