  We give efficient quantum algorithms to estimate the partition function of
(i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the
Ising model with magnetic fields on a planar graph, (iii) the Potts model on a
quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a
three-dimensional square lattice. Moreover, we prove that these problems are
BQP-complete, that is, that estimating these partition functions is as hard as
simulating arbitrary quantum computation. The results are proven for a complex
parameter regime of the models. The proofs are based on a mapping relating
partition functions to quantum circuits introduced in [Van den Nest et al.,
Phys. Rev. A 80, 052334 (2009)] and extended here.
