  Non-Hermitian quantum systems have recently attracted considerable attention
due to their exotic properties. Though many experimental realizations of
non-Hermitian systems have been reported, the non-Hermiticity usually resorts
to the hard-to-control environments and cannot last for too long times. An
alternative approach is to use quantum simulation with the closed system,
whereas how to simulate non-Hermitian Hamiltonian dynamics remains a great
challenge. To tackle this problem, we propose a protocol which combines a
dilation method with the variational quantum algorithm. The dilation method is
used to transform a non-Hermitian Hamiltonian into a Hermitian one through an
exquisite quantum circuit, while the variational quantum algorithm is for
efficiently approximating the complex entangled gates in this circuit. As a
demonstration, we apply our protocol to simulate the dynamics of an Ising chain
with nonlocal non-Hermitian perturbations, which is an important model to study
quantum phase transition at nonzero temperatures. The numerical simulation
results are highly consistent with the theoretical predictions, revealing the
effectiveness of our protocol. The presented protocol paves the way for
practically simulating small-scale non-Hermitian dynamics.
