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.
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.
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, we propose a variational quantum algorithm for solving the non-Hermitian Hamiltonian by minimizing a type of energy variance, where 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 scan the whole spectrum efficiently by 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.
Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices-Inspired by Schur's triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver-With additional [Formula: see text] quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm's feasibility. We also undertake a comprehensive parametric study to validate VQUE's scalability, generality, and performance in realistic applications.
Abstract Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The variational quantum eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix ρ . We introduce the variational quantum state eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of ρ as well as a gate sequence V that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function C=Tr(ρ~H) C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when ρ~=VρV† ρ ̃ = V ρ V † is diagonal in the eigenbasis of H . VQSE only requires a single copy of ρ (only n qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.
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.
Abstract The advancement of quantum computation paves a novel way for addressing the issue of eigenstates. In this paper, two full quantum eigenvalue solvers based on quantum gradient descent are put forward. Compared to the existing classical-quantum hybrid approaches such as the variance-variational quantum eigenvalue solver, our method enables faster convergent computations on quantum computers without the participation of classical algorithms. As any eigenstate of a Hamiltonian has zero variance, this paper takes the variance as the objective function and utilizes the quantum gradient descent method to optimize it, demonstrating the optimization of the objective function on the quantum simulator. With the swift progress of quantum computing hardware, the two variance full quantum eigensolvers proposed in this paper are anticipated to be implemented on quantum computers, thereby offering an efficient and potent calculation approach for solving eigenstate problems. Employing this algorithm, we showcase 2 qubits of deuterium and hydrogen molecule. Furthermore, we numerically investigate the energy and variance of the Ising model in larger systems, including 3, 4, 5, 6, and 10 qubits.
While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.