It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima. Since thermal transition probabilities (proportional to formula_1, with formula_2 the temperature and formula_3 the Boltzmann constant) depend only on the height formula_4 of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier in 1989 by Ray, Chakrabarti & Chakrabarti, the quantum tunneling probability through the same barrier (considered in isolation) depends not only on the height formula_4 of the barrier, but also on its width formula_6 and is approximately given by formula_7, where formula_8 is the tunneling field. This additional handle through the width formula_6, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e. formula_10), quantum fluctuations can surely bring the system out of the shallow local minima. For an formula_11-spin glass, the barrier height formula_4 becomes of order formula_11. For constant value of formula_6 one gets formula_15 proportional to formula_16 for the annealing time (instead of formula_15 proportional to formula_18 for thermal annealing), while formula_15 can even become formula_11-independent for cases where formula_21 decreases as formula_22.