  We study the effects of diffusion on a $\Lambda$-gradient echo memory, which
is a coherent optical quantum memory using thermal gases. The efficiency of
this memory is high for short storage time, but decreases exponentially due to
decoherence as the storage time is increased. We study the effects of both
longitudinal and transverse diffusion in this memory system, and give both
analytical and numerical results that are in good agreement. Our results show
that diffusion has a significant effect on the efficiency. Further, we suggest
ways to reduce these effects to improve storage efficiency.
