Schmidt eigenvalues in decreasing order and to choose the first and the appropriate Schmidt eigenvectors. Mind this would imply diagonalizing somewhat generous reduced density matrices, which, depending on the system one has to simulate, might be a task beyond our reach and patience. Instead, one can try to do the following: i) construct the decomposition for a simple initial state, let us say, some product state , for which the decomposition is straightforward. ii) relate to the ground state of a Hamiltonian by a sufficiently local transformation Q (one that can be expressed as a product of TQGs, for example) iii) make an imaginary-time evolution towards the ground state of the Hamiltonian , according to: or, alternatively, simulate an isentropic evolution using a time-dependent Hamiltonian, which interpolates between the Hamiltonian , which has the product state as its ground state, and the Hamiltonian ; the evolution must be done slow enough, such that the system is always in the ground state or, at least, very close to it. iv)finally, make the time-evolution of the state towards using the Hamiltonian : Error sources The errors in the simulation are resulting from the Suzuki–Trotter approximation and the involved truncation of the Hilbert space. Errors coming from the Suzuki–Trotter expansion In the case of a Trotter approximation of order, the error is of order . Taking into account steps, the error after the time T is: The unapproximated state is: where is the state kept after the Trotter expansion and accounts for the part that is neglected when doing the expansion. The total error scales with time as: The Trotter error is independent of the dimension of the chain. Errors coming from the truncation of the Hilbert space Considering the errors arising from the truncation of the Hilbert space comprised in the decomposition D, they are twofold. First, as we have seen above, the smallest contributions to the Schmidt spectrum are left away, the state being faithfully represented up to: where is the sum of all the discarded eigenvalues of the reduced density matrix, at the bond . The state is, at a given bond , described by the Schmidt decomposition: where is the state kept after the truncation and is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected. Now, because they are spanned by vectors corresponding to orthogonal spaces. Using the same argument as