We consider reset of a qubit by means of a controlled interaction with a finite-dimensional reservoir within the framework of Landauer’s principle. If the two systems are initially uncorrelated, we give a protocol for finding the optimal unitary operator acting on the composite system that minimises heat dissipation conditional on maximising the probability of bit reset. A specific model is presented where the qubit is initially in a maximally mixed state, and the Hamiltonian of the reservoir has an even-gapped, non-degenerate spectrum. We show that the presence of initial thermal correlations, so as to be consistent with the framework of Landauer’s principle, are not sufficient to constitute a resource for bit erasure.