Rapid Adiabatic Preparation of Injective Projected Entangled Pair States and Gibbs States

We propose a quantum algorithm for many-body state preparation. It is especially suited for injective projected entangled pair states and thermal states of local commuting Hamiltonians on a lattice. We show that for a uniform gap and sufficiently smooth paths, an adiabatic runtime and circuit depth of O(polylogN) can be achieved for O(N) spins. This is an almost exponential improvement over previous bounds. The total number of elementary gates scales as O(NpolylogN). This is also faster than the best known upper bound of O(N^{2}) on the mixing times of Monte Carlo Markov chain algorithms for sampling classical systems in thermal equilibrium.