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.
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