We propose and analyze Maxwell's demon based on a single qubit with avoided level crossing. Its operation cycle consists of adiabatic drive to the point of minimum energy separation, measurement of the qubit state, and conditional feedback. We show that the heat extracted from the bath at temperature T can ideally approach the Landauer limit of k(B)T ln 2 per cycle even in the quantum regime. Practical demon efficiency is limited by the interplay of Landau-Zener transitions and coupling to the bath. We suggest that an experimental demonstration of the demon is fully feasible using one of the standard superconducting qubits.