On the least almost-prime in arithmetic progression

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In this paper, it is proved for sufficiently large $q$, there holds \begin{equation*} \mathcal{P}_2(a,q)\ll q^{1.82193}. \end{equation*} This result constitutes improvement upon of Iwaniec, who obtained same conclusion, but range $1.845$ in place $1.82193$.