  Ground-state preparation for a given Hamiltonian is a common
quantum-computing task of great importance and has relevant applications in
quantum chemistry, computational material modeling, and combinatorial
optimization. We consider an approach to simulate dissipative non-Hermitian
Hamiltonian quantum dynamics using Hamiltonian simulation techniques to
efficiently recover the ground state of a target Hamiltonian. The proposed
method facilitates the energy transfer by repeatedly projecting ancilla qubits
to the desired state, rendering the effective non-Hermitian Hamiltonian
evolution on the system qubits. To make the method more resource friendly in
the noisy intermediate-scale quantum (NISQ) and early fault-tolerant era, we
combine the non-Hermitian projection algorithm with multiple variational
gadgets, including variational module enhancement and variational state
recording, to reduce the required circuit depth and avoid the exponentially
vanishing success probability for post-selections. We compare our method, the
non-Hermitian-variational algorithm, with a pure variational method -- QAOA for
solving the 3-SAT problem and preparing the ground state for the
transverse-field Ising model. As demonstrated by numerical evidence, the
non-Hermitian-variational algorithm outperforms QAOA in convergence speed with
improved quantum resource efficiency.
