An approach to the simulation of locally interacting systems is demonstrated and assayed. The proposal is built upon the concept of folding of bosonic modes previously introduced in the context of linear dynamics and can be seen as an alternative to Trotter–Susuki expansion in studies of quantum propagation based on tensor states. It is shown that evolution as well as ground state computations can be implemented and that test simulations deliver comparatively accurate results. The whole analysis provides insight into the way well-known quantum structures affect mean values and fluctuations in realistic setups. The proposed method can be used to study a variety of systems and phenomena arising in optical lattice experiments such as superfluid or Mott insulator regimes, quantum phase transitions, soliton propagation in Bose–Einstein condensates or Bose mixtures and elementary vortex excitations.