In this 37-minute lecture, Bobak Toussi Kiani from Harvard University explores the challenges of quantum Gibbs sampling at IPAM's Optimal Transport for Density Operators workshop. Discover a novel method based on quantum bottleneck lemma that generalizes classical techniques for establishing lower bounds on mixing times in quantum systems. Learn how this approach focuses on quantum analogs of distance, including Bohr spectrum jumps and operator locality, enabling the establishment of exponential lower bounds on mixing times across various Hamiltonian families. Examine specific applications to random K-SAT, spin glasses, stabilizer codes, and models with transverse fields, including the ferromagnetic 2D transverse field Ising model. Recorded April 2, 2025, at the Institute for Pure & Applied Mathematics (IPAM) at UCLA.