  The generalization of matrix product states (MPS) to continuous systems, as
proposed in the breakthrough paper [F. Verstraete, J.I. Cirac, Phys. Rev. Lett.
104, 190405(2010)], provides a powerful variational ansatz for the ground state
of strongly interacting quantum field theories in one spatial dimension. A
continuous MPS (cMPS) approximation to the ground state can be obtained by
simulating an Euclidean time evolution. In this Letter we propose a cMPS
optimization algorithm based instead on energy minimization by gradient
methods, and demonstrate its performance by applying it to the Lieb Liniger
model (an integrable model of an interacting bosonic field) directly in the
thermodynamic limit. We observe a very significant computational speed-up, of
more than two orders of magnitude, with respect to simulating an Euclidean time
evolution. As a result, much larger cMPS bond dimension D can be reached (e.g.
D = 256 with moderate computational resources) thus helping unlock the full
potential of the cMPS representation for ground state studies.
