  A $1$-parameter family of invertible states gives a topological transport
phenomenon, similar to the Thouless pumping. As a natural generalization of
this, we can consider a family of invertible states parametrized by some
topological space $X$. This is called a higher pump. It is conjectured that
$(1+1)$-dimensional bosonic invertible state parametrized by $X$ is classified
by $\mathrm{H}^{3}(X;\mathbb{Z})$. In this paper, we construct two higher
pumping models parametrized by $X=\mathbb{R}P^{2}\times S^1$ and
$X=\mathrm{L}(3,1)\times S^1$ that corresponds to the torsion part of
$\mathrm{H}^{3}(X;\mathbb{Z})$. As a consequence of the nontriviality as a
family, we find that a quantum mechanical system with a nontrivial discrete
Berry phase is pumped to the boundary of the $(1+1)$-dimensional system. We
also study higher pump phenomena by using matrix product states (MPS), and
construct a higher pump invariant which takes value in a torsion part of
$\mathrm{H}^{3}(X;\mathbb{Z})$. This is a higher analog of the ordinary
discrete Berry phase that takes value in the torsion part of
$\mathrm{H}^{2}(X;\mathbb{Z})$. In order to define the higher pump invariant,
we utilize the smooth Deligne cohomology and its integration theory. We confirm
that the higher pump invariant of the model has a nontrivial value.
