Variational-quantum-eigensolver solution for Lindblad-driven nonequilibrium electron-transport problems

We propose a quantum algorithm that solves the electron transport problem over a molecular junction via the variational quantum eigensolver (VQE) method. We regard the molecular junction as an impurity model coupled with discrete Lindblad dissipaters. The problem of solving for the nonequilibrium steady state (NESS) of the electron transport is converted to a variational optimization problem for finding the zero eigenstates of non-Hermitian Liouville superoperators (Liouvillian). With a modified Jordan-Wigner transformation, the variational kernel of the Hermitian product of fermionic Liouville superoperators would be mapped as the cost function on the quantum circuit, and the corresponding density matrix of the NESS can be found by the VQE process. We demonstrate this approach on a single-level molecular model and estimate the $I\text{\ensuremath{-}}V$ curve, verifying the accuracy of this algorithm.