The Bose-Hubbard Model is QMA-complete

摘要

The Bose-Hubbard model is a system of interacting bosons that liveon the vertices of a graph. The particles can move between adjacentvertices and experience a repulsive on-site interaction. TheHamiltonian is determined by a choice of graph that specifies thegeometry in which the particles move and interact. We prove thatapproximating the ground energy of the Bose-Hubbard model on a graphat fixed particle number is QMA-complete. In our QMA-hardness proof,we encode the history of an $n$-qubit computation in the subspace withat most one particle per site (i.e., hard-core bosons). This feature,along with the well-known mapping between hard-core bosons and spinsystems, lets us prove a related result for a class of $2$-localHamiltonians defined by graphs that generalizes the XY model. Byavoiding the use of perturbation theory in our analysis, we circumventthe need to multiply terms in the Hamiltonian by large coefficients.