Abstract Ultracold atoms offer a unique opportunity to study many-body physics in a clean and well-controlled environment. However, the isolated nature of quantum gases makes it difficult to study transport properties of the system, which are among the key observables in condensed matter physics. In this work, we employ Markovian feedback control to synthesize two effective thermal baths that couple to the boundaries of a one-dimensional Bose–Hubbard chain. This allows for the realization of a heat-current-carrying state. We investigate the steady-state heat current, including its scaling with system size and its response to disorder. In order to study large systems, we use semi-classical Monte-Carlo simulation and kinetic theory. The numerical results from both approaches show, as expected, that for non- and weakly interacting systems with and without disorder one finds the same scaling of the heat current with respect to the system size as it is found for systems coupled to thermal baths. Finally, we propose and test a scheme for measuring the energy flow. Thus, we provide a route for the quantum simulation of heat-current-carrying steady states of matter in atomic quantum gases.