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This course explores the complexity of the local Hamiltonian problem, covering topics such as problem definition, NP containment, proof, ground state, locality, adiabatic computation, propagation, clock state, energy density, Wang tiles, common models, interaction graph, and open quantum systems. The course aims to teach students about these concepts and their applications in quantum computing. The teaching method involves lectures and discussions, making it suitable for individuals interested in quantum computing and theoretical computer science.