Explore the challenges and recent advancements in numerical simulations for open quantum system dynamics in this conference talk by Jianfeng Lu from Duke University Mathematics. Delve into diagrammatic algorithms, focusing on the interplay between the dynamical sign problem and error amplification in numerical integration. Discover how partial resummation techniques can balance these errors and learn about the inchworm Monte Carlo method's success in suppressing the numerical sign problem. Gain insights into various approaches, mathematical frameworks, and numerical examples presented throughout the talk, including the Brick Theorem, propagator analysis, and integral differential interpretations.