  How to characterize topological quantum phases is a fundamental issue in the
broad field of topological matter. From a dimension reduction approach, we
propose the concept of high-order band inversion surfaces (BISs) which enable
the optimal schemes to characterize equilibrium topological phases by
far-from-equilibrium quantum dynamics, and further report the experimental
simulation. We show that characterization of a d-dimensional (dD) topological
phase can be reduced to lower-dimensional topological invariants in the
high-order BISs, of which the nth-order BIS is a (d-n)D interface in momentum
space. In quenching the system from trivial phase to topological regime, we
unveil a high-order dynamical bulk-surface correspondence that the quantum
dynamics exhibits nontrivial topological pattern in arbitrary nth-order BISs,
which universally corresponds to and so characterizes the equilibrium
topological phase of the post-quench Hamiltonian. This high-order dynamical
bulk-surface correspondence provides new and optimal dynamical schemes with
fundamental advantages to simulate and detect topological states, in which
through the highest-order BISs that are of zero dimension, the detection of
topological phase relies on only minimal measurements. We experimentally build
up a quantum simulator with spin qubits to investigate a 3D chiral topological
insulator through emulating each momentum one by one and measure the high-order
dynamical bulk-surface correspondence, with the advantages of topological
characterization via highest-order BISs being demonstrated.
