  The $q$-fractional differential equation usually describe the physics process
imposed on the time scale set $T_q$. In this paper, we first propose a
difference formula for discretizing the fractional $q$-derivative $^cD_q^\alpha
x(t)$ on the time scale set $T_q$ with order $0<\alpha<1$ and scale index
$0<q<1$. We establish a rigours truncation error boundness and prove that this
difference formula is unconditionally stable. Then, we consider the difference
method for solving the initial problem of $q$-fractional differential equation:
$^cD_q^\alpha x(t)=f(t,x(t))$ on the time scale set. We prove the unique
existence and stability of the difference solution and give the convergence
analysis. Numerical experiments show the effectiveness and high accuracy of the
proposed difference method.
