  In this paper, I consider a fine-grained dichotomy of Boolean counting
constraint satisfaction problem (#CSP), under the exponential time hypothesis
of counting version (#ETH). Suppose $\mathscr{F}$ is a finite set of algebraic
complex-valued functions defined on Boolean domain. When $\mathscr{F}$ is a
subset of either two special function sets, I prove that #CSP($\mathscr{F}$) is
polynomial-time solvable, otherwise it can not be computed in sub-exponential
time unless #ETH fails. I also improve the result by proving the same dichotomy
holds for #CSP with bounded degree (every variable appears at most constant
constraints), even for #R$_3$-CSP.
  An important preparation before proving the result is to argue that pinning
(two special unary functions $[1,0]$ and $[0,1]$ are used to reduce arity) can
also keep the sub-exponential lower bound of a Boolean #CSP problem. I discuss
this issue by utilizing some common methods in proving #P-hardness of counting
problems. The proof illustrates the internal correlation among these commonly
used methods.

Computational Complexity