  Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz
rank of a permutation polynomial is a important measure of complexity of the
polynomial. In this paper we find the sharp lower bound for the weight of any
permutation polynomial with Carlitz rank $2$, improving the bound found by
G\'omez-P\'erez, Ostafe and Topuzo\u{g}lu in that case.
