This is the dataset used for generating the plots of our work (accepted in PRA) https://doi.org/10.48550/arXiv.2407.03421 The dataset is generated by a simulation protocol implemented in Python. The file CORR_NN_spin1_XXZ_quench_N_10lam_0.2budg1500.csv contains the data depicted in FIG. 2. The columns contain each time point of evalutation, as well as the analytical value of the dynamical commutator and anti-commutator, their sampled values according to the Hadamard test and the linear response protocols, as well as the numerical value of the estimated variance of both. The files FINAL_spin1_XXZ_quench_N_10... contain the same information for ten different strengths of the linear response parameter $\lambda$. This data is used for calculating the relative error and the time average of the standard deviation plotted in FIG. 3.