  We show that the formalism of tensor-network states, such as the matrix
product states (MPS), can be used as a basis for variational quantum Monte
Carlo simulations. Using a stochastic optimization method, we demonstrate the
potential of this approach by explicit MPS calculations for the transverse
Ising chain with up to N=256 spins at criticality, using periodic boundary
conditions and D*D matrices with D up to 48. The computational cost of our
scheme formally scales as ND^3, whereas standard MPS approaches and the related
density matrix renromalization group method scale as ND^5 and ND^6,
respectively, for periodic systems.
