Traditional simulated annealing uses thermal fluctuations for convergence in optimization problems. Quantum tunneling provides a different mechanism for moving between states, with the potential for reduced time scales and different outcomes. Thermal and quantum annealing are compared in two concentration regimes of a model disordered magnet, where the effects of quantum mechanics can be tuned both by varying an applied magnetic field and by controlling the strength of thermal coupling between the magnet and an external heat bath. The results indicate that quantum annealing hastens convergence to the final state, and that the quantum character of the final state can be engineered thermodynamically.