  While the Hamiltonians used in standard quantum mechanics are Hermitian, it
is also possible to extend the theory to non-Hermitian Hamiltonians.
Particularly interesting are non-Hermitian Hamiltonians satisfying parity-time
(PT) symmetry, or more generally pseudo-Hermiticity, since such non-Hermitian
Hamiltonians can still exhibit real eigenvalues. In this work, we present a
quantum simulation of the time-dependent non-Hermitian non-PT-symmetric
Hamiltonian used in a pseudo-Hermitian extension of the
Landau-Zener-St\"uckelberg-Majorana (LZSM) model. The simulation is implemented
on a superconducting processor by using Naimark dilation to transform a
non-Hermitian Hamiltonian for one qubit into a Hermitian Hamiltonian for a
qubit and an ancilla; postselection on the ancilla state ensures that the qubit
undergoes nonunitary time-evolution corresponding to the original non-Hermitian
Hamiltonian. We observe properties such as the dependence of transition rates
on time and the replacement of conservation of total probability by other
dynamical invariants in agreement with predictions based on a theoretical
treatment of the pseudo-Hermitian LZSM system.
