left-hand eigenvectors , a maximum of , respectively basic operations. In the case of qubits, , hence its role is not very relevant for the order of magnitude of the number of basic operations, but in the case when the on-site dimension is higher than two it has a rather decisive contribution. The numerical simulation The numerical simulation is targeting (possibly time-dependent) Hamiltonians of a system of particles arranged in a line, which are composed of arbitrary OQGs and TQGs: It is useful to decompose as a sum of two possibly non-commuting terms, , where Any two-body terms commute: , This is done to make the Suzuki–Trotter expansion (ST) of the exponential operator, named after Masuo Suzuki and Hale Trotter. The Suzuki–Trotter expansion The Suzuki–Trotter expansion of the first order (ST1) represents a general way of writing exponential operators: or, equivalently The correction term vanishes in the limit For simulations of quantum dynamics it is useful to use operators that are unitary, conserving the norm (unlike power series expansions), and there's where the Trotter-Suzuki expansion comes in. In problems of quantum dynamics the unitarity of the operators in the ST expansion proves quite practical, since the error tends to concentrate in the overall phase, thus allowing us to faithfully compute expectation values and conserved quantities. Because the ST conserves the phase-space volume, it is also called a symplectic integrator. The trick of the ST2 is to write the unitary operators as: where . The number is called the Trotter number. Simulation of the time-evolution The operators , are easy to express, as: since any two operators , (respectively, ,) commute for and an ST expansion of the first order keeps only the product of the exponentials, the approximation becoming, in this case, exact. The time-evolution can be made according to For each "time-step" , are applied successively to all odd sites, then to the even ones, and again to the odd ones; this is basically a sequence of TQG's, and it has been explained above how to update the decomposition when applying them. Our goal is to make the time evolution of a state for a time T, towards the state using the n-particle Hamiltonian . It is rather troublesome, if at all possible, to construct the decomposition for an arbitrary n-particle state, since this would mean one has to compute the Schmidt decomposition at each bond, to arrange the