The Local Hamiltonian problem (LHP) is the quantum analogy of MAX-SAT, and plays a fundamental role in both quantum complexity and condense matter physics. In this talk, we give an overview of results and techniques for studying a subclass of LHP, where all terms pairwise commute (Commuting LHP).  Commuting local Hamiltonian is interesting since it not only has a simpler form, but  also captures highly quantum phenomemon like long-range entanglement in Toric code. Commuting local Hamiltonians (CLHs) provide a way to study the role of non-commutativity in the complexity of quantum systems, and provides a test ground for attacking difficult problems such as the quantum-PCP conjecture [AE11a, Has12b, Has13], NLTS conjecture [ABN22], Gibbs states preparation [KB16] and fast thermalization [BCG+23, CRF20].


Video Recording