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Physical Review A

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

Yi Liu, Zixuan Hu, Xiao Zheng, Zhenyu Li

Published Dec 4, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1103/b8tq-169m

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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\ensuremath-VI\text{\ensuremath{-}}VI\ensuremath-V curve, verifying the accuracy of this algorithm.
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We propose a variational quantum algorithm named variational open quantum eigensolver (VOQE) for solving steady states of open quantum systems described by either Lindblad master equations or non-Hermitian Hamiltonians. In VOQE, density matrices of mixed states are represented by pure states in doubled Hilbert space. We give a framework for building a circuit ansatz which we call the Hermitian-preserving ansatz to restrict the searching space. We also give a method to efficiently measure the operators' expectation values by postselection measurements. We show the workflow of VOQE on solving steady states of the Lindblad master equations of the driven XXZ model and implement VOQE to solve the spectrum of the non-Hermitian Hamiltonians of the Ising spin chain in an imaginary field.
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The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
Non-Hermitian operators naturally arise in the description of open quantum systems, which exhibit features such as resonances and decay processes, where the associated eigenvalues are complex. Standard quantum algorithms, including the Variational Quantum Eigensolver (VQE), are designed for Hermitian operators and are ineffective in recovering correct eigenvalues for non-Hermitian matrices. We present a systematic formulation based on a Real Variance-based Variational Quantum Eigensolver (RVVQE) for non-Hermitian operators. A correct cost function that guarantees convergence to the true eigenstates is identified. Our implementation utilizes Hermitian measurements only, rendering the algorithm easily deliverable. The performance and scalability of the proposed algorithm on a hierarchy of dense non-Hermitian matrices of increasing dimension are demonstrated with numerical results and computational metrics.