A locally compact group G is said to have Reiter’s property (Pp) with p ∈ [1, ∞) if there is a net (mα)α of non-negative norm one functions in L p (G) such that kLxmα −mαkp → 0 uniformly in x on compact subsets of G. It is well known that property (Pp) for any p is equivalent to G being amenable. The fact that an amenable locally compact group has property (P2) can be used to give a proof of Leptin’s theorem that avoids Følner type conditions. We formulate Reiter’s property (P1) for locally compact quantum groups and show that it is equivalent to amenability amenability.