  High-order topological phases of matter refer to the systems of
$n$-dimensional bulk with the topology of $m$-th order, exhibiting
$(n-m)$-dimensional boundary modes and can be characterized by topological
pumping. Here, we experimentally demonstrate two types of second-order
topological pumps, forming four 0-dimensional corner localized states on a
4$\times$4 square lattice array of 16 superconducting qubits. The initial
ground state of the system for half-filling, as a product of four identical
entangled 4-qubit states, is prepared using an adiabatic scheme. During the
pumping procedure, we adiabatically modulate the superlattice Bose-Hubbard
Hamiltonian by precisely controlling both the hopping strengths and on-site
potentials. At the half pumping period, the system evolves to a
corner-localized state in a quadrupole configuration. The robustness of the
second-order topological pump is also investigated by introducing different
on-site disorder. Our work studies the topological properties of high-order
topological phases from the dynamical transport picture using superconducting
qubits, which would inspire further research on high-order topological phases.
