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

Jiaxin Li, Zhaobing Fan, Hongmei Yao, Chunlin Yang, Shao-Ming Fei, Zi-Tong Zhou, Meng-Han Dou, Teng-Yang Ma

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 15 sections, 38 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical framework and loss function
  3. Computation of loss function and its gradient
  4. Quantum implementation of loss function
  5. Gradient estimation with the parameter shift rule
  6. Variational quantum generalized eigensolver
  7. Numerical results
  8. Two-qubit system
  9. Application in ocean acoustic fields
  10. Noise-incorporated simulation
  11. Conclusion
  12. Quantum process snapshot
  13. Data input model based on LCU
  14. Derivation of the gradient formulas
  15. Design of parameterized circuits

Figures (11)

  • Figure 1: Quantum circuit for computation of the loss function in (\ref{['loss']}).
  • Figure 2: The iterative process for computing the generalized eigenvalues in two-qubit system.
  • Figure 3: The iterative process for computing the generalized eigenvalues in ocean acoustic fields.
  • Figure 4: Comparison of exact and experimental generalized eigenvalues in ocean acoustic fields.
  • Figure 5: The iterative process under noisy and noiseless conditions.
  • ...and 6 more figures