  Originating from the Hamiltonian of a single qubit system, the phenomenon of
the avoided level crossing is ubiquitous in multiple branches of physics,
including the Landau-Zener transition in atomic, molecular and optical physics,
the band structure of condensed matter physics and the dispersion relation of
relativistic quantum physics. We revisit this fundamental phenomenon in the
simple example of a spinless relativistic quantum particle traveling in
(1+1)-dimensional space-time and establish its relation to a spin-1/2 system
evolving under a $\mathcal{PT}$-symmetric Hamiltonian. This relation allows us
to simulate 1-dimensional eigenvalue problems with a single qubit. Generalizing
this relation to the eigenenergy problem of a bulk system with $N$ spatial
dimensions reveals that its eigenvalue problem can be mapped onto the time
evolution of the edge state with $(N-1)$ spatial dimensions governed by a
non-Hermitian Hamiltonian. In other words, the bulk eigenenergy state is
encoded in the edge state as a hologram, which can be decoded by the
propagation of the edge state in the temporal dimension. We argue that the
evolution will be $\mathcal{PT}$-symmetric as long as the bulk system admits
parity symmetry. Our work finds the application of $\mathcal{PT}$-symmetric and
non-Hermitian physics in quantum simulation and provides insights into the
fundamental symmetries.
