We determine the unitary evolution of a system, composed of an object to be erased and a thermal reservoir, so as to minimise the heat dissipation to the latter conditional on maximising the probability of purifying the former: of maximally erasing its information. A model is considered where a qubit is erased using a harmonic oscillator, with a log(2) reduction in entropy accompanied by heat dissipation Q > k_B T, which exceeds Landauer's bound. Moreover in the presence of energy conserving dephasing, the optimal cases are when only the first 2^n levels of the reservoir are used.