Complexity phase diagrams

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In this talk, I argue that the question of whether a physical system can be simulated on a computer is important not just from a practical perspective but also a fundamental one. We consider the complexity of simulating Hamiltonians with respect to both dynamics and equilibrium properties. This gives us a classification and a phase diagram of the complexity.  I will cover recent results in this topic, such as a dynamical complexity phase diagram for a long-range bosonic Hamiltonian and a complexity classification of the local Hamiltonian problem in the presence of a spectral gap. I will talk about the physical implications of these results and cover some of the basic proof ideas if time permits.