Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Scientific Reports

A universal variational quantum eigensolver for non-Hermitian systems

Huanfeng Zhao, Peng Zhang, Tzu-Chieh Wei

Published Dec 15, 2023• 7 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-023-49662-5

Related papers

Real Variance-Based Variational Quantum Eigensolver for Non-Hermitian Matrices

Durgesh Pandey, Ankit Kumar Das, P. Arumugam

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.

quant-pharXiv:2603.28892•IEEE Transactions on Quantum Engineering•Apr 1, 2026•0 citations

Photonic variational quantum eigensolver for NISQ-compatible quantum technology

Kang-Min Hu, Min Namkung, Hyang-Tag Lim

Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.

quant-pharXiv:2512.18952•Nano Convergence•Dec 23, 2025•2 citations

Variational quantum algorithm for generalized eigenvalue problems of non-Hermitian systems

Jiaxin Li, Zhaobing Fan, Hongmei Yao +5 more

Non-Hermitian generalized eigenvalue problems (GEPs) play a significant role in many practical applications, such as mechanical engineering. Based on the generalized Schur decomposition, we propose a variational quantum algorithm for solving the GEPs in non-Hermitian systems. The algorithm transforms the generalized eigenvalue problem into a process of searching for unitary transformation matrices. We demonstrate a method for evaluating both the loss function and its gradients on near-term quantum devices. We validate numerically the algorithm's performance through simulations, and demonstrate its application to GEPs in ocean acoustics. The algorithm's robustness is further confirmed through noise simulations.

arXiv:2507.04783•Physical Review A•Dec 8, 2025•0 citations

Variational Quantum Shot-Based Simulations for Waveguide Modes

Emanuel Colella, Spencer Beloin, Luca Bastianelli +3 more

Current quantum computers (QCs) belong to the noisy intermediate-scale quantum (NISQ) class, characterized by noisy qubits, limited qubit capabilities, and limited circuit depth. These limitations have led to the development of hybrid quantum-classical algorithms that split the computational cost between classical and quantum hardware. Among the hybrid algorithms, the variational quantum eigensolver (VQE) is mentioned. The VQE is a variational quantum algorithm designed to estimate the eigenvalues and eigenvectors of a system on universal-gate quantum architectures. A canonical problem in electromagnetics is the computation of eigenmodes within waveguides. Following the finite difference method, the wave equation can be recast as an eigenvalue problem. This work exploits the quantum superposition and entanglement in quantum computing to solve the square waveguide mode problem. This algorithm is expected to demonstrate exponentially efficiency over classical computational techniques as the qubit count increases. The simulations were performed on IBM’s three-qubit quantum simulator, Qasm IBM Simulator. A shot-based simulation was performed considering computationally based measurements of the quantum hardware. The results of the probabilistic read-out, reported in terms of 2-D eigenmode field distributions, are close to ideal values with a few number of qubits, confirming the possibility to exploit the quantum advantage to formulate innovative eigensolvers.

IEEE Transactions on Microwave Theory and Techniques•Dec 19, 2023•15 citations

Variational quantum state eigensolver

Marco Cerezo, Kunal Sharma, Andrew Arrasmith +1 more

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 $$C={{{\rm{Tr}}}}(\tilde{\rho }H)$$ C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when $$\tilde{\rho }=V\rho {V}^{{\dagger} }$$ 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.

arXiv:2004.01372•npj Quantum Information•Sep 21, 2022•138 citations

Estimation of Waveguide Eigenmodes Based on Subspace-Search Variational Quantum Algorithm

Zhuo‐Wei Miao, Fanxu Meng

The variational quantum eigensolver (VQE), a variational algorithm to approximate the ground state of the given Hamiltonian, is the leading candidate for receiving quantum advantage on noisy intermediate-scale quantum (NISQ) devices. The eigenmode analysis within waveguides, a canonical problem in electromagnetics, can be reformulated as an eigenvalue problem adopting the finite difference method. Therefore, in this work, a subspace-search-based VQE is applied for the computation of the eigenmodes. The proposed algorithm has the promise to show exponential efficiency outperforming over the classical algorithms. Compared with the existing counterparts, our work makes the additional orthogonality constrain unnecessary in the cost function, meantime, avoids the costly inner product evaluation in the cost function, where a few number of ancillary qubits and deeper quantum circuits are indispensable. Comprehensive experimental results show that the proposed framework significantly provides more accurate eigenmode estimation with fewer iterations and shows more favorable resource efficiency.

IEEE Microwave and Wireless Technology Letters•Aug 27, 2024•1 citation
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.
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.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
Non-Hermitian generalized eigenvalue problems (GEPs) play a significant role in many practical applications, such as mechanical engineering. Based on the generalized Schur decomposition, we propose a variational quantum algorithm for solving the GEPs in non-Hermitian systems. The algorithm transforms the generalized eigenvalue problem into a process of searching for unitary transformation matrices. We demonstrate a method for evaluating both the loss function and its gradients on near-term quantum devices. We validate numerically the algorithm's performance through simulations, and demonstrate its application to GEPs in ocean acoustics. The algorithm's robustness is further confirmed through noise simulations.
Current quantum computers (QCs) belong to the noisy intermediate-scale quantum (NISQ) class, characterized by noisy qubits, limited qubit capabilities, and limited circuit depth. These limitations have led to the development of hybrid quantum-classical algorithms that split the computational cost between classical and quantum hardware. Among the hybrid algorithms, the variational quantum eigensolver (VQE) is mentioned. The VQE is a variational quantum algorithm designed to estimate the eigenvalues and eigenvectors of a system on universal-gate quantum architectures. A canonical problem in electromagnetics is the computation of eigenmodes within waveguides. Following the finite difference method, the wave equation can be recast as an eigenvalue problem. This work exploits the quantum superposition and entanglement in quantum computing to solve the square waveguide mode problem. This algorithm is expected to demonstrate exponentially efficiency over classical computational techniques as the qubit count increases. The simulations were performed on IBM’s three-qubit quantum simulator, Qasm IBM Simulator. A shot-based simulation was performed considering computationally based measurements of the quantum hardware. The results of the probabilistic read-out, reported in terms of 2-D eigenmode field distributions, are close to ideal values with a few number of qubits, confirming the possibility to exploit the quantum advantage to formulate innovative eigensolvers.
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={{{\rm{Tr}}}}(\tilde{\rho }H)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†\tilde{\rho }=V\rho {V}^{{\dagger} }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.
The variational quantum eigensolver (VQE), a variational algorithm to approximate the ground state of the given Hamiltonian, is the leading candidate for receiving quantum advantage on noisy intermediate-scale quantum (NISQ) devices. The eigenmode analysis within waveguides, a canonical problem in electromagnetics, can be reformulated as an eigenvalue problem adopting the finite difference method. Therefore, in this work, a subspace-search-based VQE is applied for the computation of the eigenmodes. The proposed algorithm has the promise to show exponential efficiency outperforming over the classical algorithms. Compared with the existing counterparts, our work makes the additional orthogonality constrain unnecessary in the cost function, meantime, avoids the costly inner product evaluation in the cost function, where a few number of ancillary qubits and deeper quantum circuits are indispensable. Comprehensive experimental results show that the proposed framework significantly provides more accurate eigenmode estimation with fewer iterations and shows more favorable resource efficiency.