  Solving non-Hermitian quantum many-body systems on a quantum computer by
minimizing the variational energy is challenging as the energy can be complex.
Here, based on energy variance, we propose a variational method for solving the
non-Hermitian Hamiltonian, as zero variance can naturally determine the
eigenvalues and the associated left and right eigenstates. Moreover, the energy
is set as a parameter in the cost function and can be tuned to obtain the whole
spectrum, where each eigenstate can be efficiently obtained using a two-step
optimization scheme. Through numerical simulations, we demonstrate the
algorithm for preparing the left and right eigenstates, verifying the
biorthogonal relations, as well as evaluating the observables. We also
investigate the impact of quantum noise on our algorithm and show that its
performance can be largely improved using error mitigation techniques.
Therefore, our work suggests an avenue for solving non-Hermitian quantum
many-body systems with variational quantum algorithms on near-term noisy
quantum computers.
