We assess the convergence of the Carleman linearization of advection–diffusion–reaction (ADR) equations with a logistic nonlinearity. It is shown that five Carleman iterates provide a satisfactory approximation of the original ADR across a broad range of parameters and strength of nonlinearity. To assess the feasibility of a quantum algorithm based on this linearization, we analyze the projection of the Carleman ADR matrix onto the tensor Pauli basis.