within reach of eventual numerical simulations. The update of the decomposition One can proceed now to investigate the behaviour of the decomposition D when acted upon with one-qubit gates (OQG) and two-qubit gates (TQG) acting on neighbouring qubits. Instead of updating all the coefficients , we will restrict ourselves to a number of operations that increase in as a polynomial of low degree, thus saving computational time. One-qubit gates acting on qubit k The OQGs are affecting only the qubit they are acting upon, the update of the state after a unitary operator at qubit k does not modify the Schmidt eigenvalues or vectors on the left, consequently the 's, or on the right, hence the 's. The only 's that will be updated are the 's (requiring only at most operations), as Two-qubit gates acting on qubits k, k+1 The changes required to update the 's and the 's, following a unitary operation V on qubits k, k+1, concern only , and . They consist of a number of basic operations. Following Vidal's original approach, can be regarded as belonging to only four subsystems: The subspace J is spanned by the eigenvectors of the reduced density matrix : In a similar way, the subspace K is spanned by the eigenvectors of the reduced density matrix: The subspaces and belong to the qubits k and k + 1. Using this basis and the decomposition D, can be written as: Using the same reasoning as for the OQG, the applying the TQG V to qubits k, k + 1 one needs only to update , and We can write as: where To find out the new decomposition, the new 's at the bond k and their corresponding Schmidt eigenvectors must be computed and expressed in terms of the 's of the decomposition D. The reduced density matrix is therefore diagonalized: The square roots of its eigenvalues are the new 's. Expressing the eigenvectors of the diagonalized matrix in the basis : the 's are obtained as well: From the left-hand eigenvectors, after expressing them in the basis , the 's are: The computational cost The dimension of the largest tensors in D is of the order ; when constructing the one makes the summation over , and for each , adding up to a total of operations. The same holds for the formation of the elements , or for computing the