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 $O(\mathrm{polylog}N)$ can be achieved for $O(N)$ spins. This is an almost exponential improvement over previous bounds. The total number of elementary gates scales as $O(N\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}N)$. 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.

• Received 14 August 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.080503