Syllabus (subjected to changes):

1. Overview of sampling & applications. Draft notes for Lecture 1

+  Approximate sampling and approximate counting

+ Total variation distance, coupling for distributions, data processing inequality.

+ Some example algorithms/stochastic processes for sampling: the Glauber dynamics, auto-regression, masked diffusion.

2. Markov chains and mixing time: an introduction Lecture 2

+ Stationary distribution, ergodicity, reversibility, mixing time

+ Example: random walk on graphs

3. Spectral gap, Poincaré inequality, (modified) Log-Sobolev inequality

4. Conductance, Cheeger's inequality Lecture 4

5. Equivalence between approximate sampling and counting

6. Equivalence between approximate sampling and counting (continue) Concentration inequalities: Lecture 6 Concentration Inequalities

7. Path coupling + Dobrushin condition Lecture 7

8. Overview of sampling from Ising models (4/22)

9. & 10. Hardness of sampling from low-temperatured Ising models (4/29-5/4) Lecture 9-10

11. Measure decomposition via stochastic localization

12. Spectral independence, entropic independence, and bounded covariance

13. Bounding covariance via trickle-down

14. Sampling from continuous domains I: Langevin dynamics

15. Sampling from continuous domains II: Continuous and discrete diffusion models

16-17. Quantum Markov chains Slides on quantum to classical spectral gap comparison for Davies generator

18. Quantum Monte-Carlo: computing the partition function of transverse-field Ising model (TFIM) Slides on partition function of TFIM