This is the fourth part of the series of papers devoted to some relationships between the computer science and the quantum gravity. We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with the quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of the quantum computers. We consider simpler models for which one can obtain exact mathemati- cal results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice Z, defined by a renormalized Hamiltonian.