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

Continuous Matrix Product States for Quantum Fields: an Energy Minimization Algorithm