Pauli weight requirement of the matrix elements in time-evolved local operators: Dependence beyond the equilibration temperature

The complexity of simulating the out-of-equilibrium evolution of local operators in the Heisenberg picture is governed by the operator entanglement, which grows linearly in time for generic nonintegrable systems, leading to an exponential increase in computational resources. A promising approach to simplify this challenge involves discarding parts of the operator and focusing on a subspace formed by ``light'' Pauli strings---strings with few Pauli matrices---as proposed by Rakovszki et al. [Phys. Rev. B 105, 075131 (2022)]. In this work, we investigate whether this strategy can be applied to quenches starting from homogeneous product states. For ergodic dynamics, these initial states grant access to a wide range of equilibration temperatures. By concentrating on the desired matrix elements and retaining only the portion of the operator that contains Pauli strings parallel to the initial state, we uncover a complex scenario. In some cases, the light Pauli strings suffice to describe the dynamics, enabling efficient simulation with current algorithms. However, in other cases, heavier strings become necessary, pushing computational demands beyond our current capabilities. We analyze this behavior by introducing a measure of complexity, the operator weight entropy, which we compute for different operators across most points on the Bloch sphere.