The Hubbard model (see also the wikipedea article on the Bose-Hubbard model) is a basic quantum model of solid-state physics. Question: What is the computational complexity of approximating the ground state of a 2-D Hubbard model? Remarks:  1) This question is inspired by lecture-series by Frank Verstraete in Jerusalem. (See here for the videotaped lectures.) 2) The question is a tiny input question, (see this question:Counting Isomorphism Types of Graphs  ) so hardness result for questions of this kind is largely an uncharted territory.We can ask: 1) is there a polynomial time algorithm? 2) Is there a polynomial time quantum algorithm? Frank suggested that algorithms based on adiabatic evolution (starting from a product state) would work, since gaps at phase transitions typically only scale as 1/poly in the system size, (which keeps it in BQP). My question is if this is known/proven.3) Is the problem in NP?
How hard it is to approximate the ground state of the (2-D) Hubbard model
quantum computing