Let $��=(G, ��)$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(��)$ be its adjacency matrix, where $G$ is called the underlying graph of $��$. The rank of $��$, denoted by $r(��)$, is the rank of $A(��)$. Denote by $��(G)=|E(G)|-|V(G)|+��(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $��(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(��)$ in terms of $r(G)$, that is, $r(G)-2��(G)\leq r(��)\leq r(G)+2��(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-��(G)\leq\frac{r(��)}{r(G)}\leq1+��(G)$. All corresponding extremal graphs are characterized.

17 pages. arXiv admin note: text overlap with arXiv:1612.05043