  The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful
framework to study invariants of smooth varieties via invariants of pairs
$(X,D)$ of a variety $X$ and a divisor $D$. We develop a generalization of this
theory where $D$ can be a $\mathbb{Q}$-divisor. As an application, we provide a
motivic construction of the de Rham-Witt complex, which is analogous to the
motivic construction of the Milnor $K$-theory due to Suslin-Voevodsky.
