Given $n\in \mathbb{Z}_{\geq 0}$ denote by $B_n$ the set of binary sequences of length $n$. Denote $B=\bigcup_{n\geq 0} B_n$.

Let $P, Q:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ be two computable functions. We have a map $f_{P, Q}:B\to \mathbb{Z}_{\geq 0}$ given by applying $P$ and $Q$ to $0$ in the order specified by the sequence.

Are there $P$ and $Q$ such that $f_{P, Q}$ is bijective? In particular $P$ and $Q$ cannot have fixed points.