Abstract: The complexity of the simulating the out-of-equilibrium evolution of local operators in the Heisenberg picture is dictated by the operator entanglement, which grows linearly in time for generic non-integrable systems; this translates into an exponential increase of the required computational resources. Recent works on spin chains proposed to project the evolved operators onto the subspace of strings with few Pauli matrices, also termed as low Pauli weight strings. Here we explore whether this strategy can be applied to quenches starting from Products States, that can equilibrate to very different temperatures. These initial states allow us to define a single-spin-basis in which we can split the operators into parallel and orthogonal superpositions: at a given time, only the parallel part contributes to the expectation value, while the orthogonal one will affect the later times through backflow. We characterize these two effects, finding that the former dominates over the later. In order to connect the leading direct contributions with the Pauli weight content and the hardness of simulating local observables, we introduce a new measure of complexity, the Operator Length Entropy. This entropy allows us to identify sets of states whose simulation is harder. As a by-product, we detect a dip in the complexity and Pauli weight needed to simulate states that equilibrate a high temperature states, opposed to those close to the ground state. This study allowed us to propose an extension of the classical TEBD method by truncating orthogonal non-contributing components on the evolving operators; such algorithm is able to break the operator entanglement barrier for a wide range of initial states, reaching long-term simulations of local dynamics.