[en] Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state $|{\mathrm{\phi <!--\; ??\; -->}}_1⟩<!--\; ???\; -->$ be no smaller than $p_{{\mathrm{\phi <!--\; ??\; -->}}_1}^{\mathrm{m}\mathrm{a}\mathrm{x}}-<!--\; ???\; -->\mathrm{\delta <!--\; ??\; -->}$. Here $p_{{\mathrm{\phi <!--\; ??\; -->}}_1}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum probability of information erasure that is permissible by the physical context, and $\mathrm{\delta <!--\; ??\; -->}⩾<!--\; ???\; -->0$ the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework. (paper)