Quantum algorithm for evaluating Lyapunov stability of non-Hermitian systems
- 1South China Normal University
Abstract
Stability is a fundamental property of dynamical systems, and Lyapunov theory provides a rigorous link between stability and eigenvalue distribution. By computing eigenvalues, quantum algorithms may offer potential advantages for evaluating the stability of high-dimensional systems under appropriate assumptions regarding data access and quantum resource requirements. In this work, we introduce a quantum algorithm based on the quantum singular value transformation (QSVT) to efficiently resolve the complex eigenspectrum of non-Hermitian matrices and determine asymptotic stability. By employing block encoding and QSVT techniques, we implement a polynomial transformation to approximate the singular value threshold problem, enabling the localization of eigenvalues with positive real parts under appropriate requirements regarding block encoding efficiency, matrix conditioning, state overlap, and precision. We validate the method numerically on a one-dimensional non-Hermitian spin-1/2 chain with local gain and loss, demonstrating accurate detection of unstable and stable cases, which is consistent with the results obtained by exact diagonalization. These results show that our quantum algorithm is a reliable and powerful tool for stability analysis and spectral characterization of high-dimensional dynamical systems.
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BibTeX
@article{he2026quantum,
title = {{Quantum algorithm for evaluating Lyapunov stability of non-Hermitian systems}},
author = {Mingwu He and Xiao-Ming Zhang},
journal = {New Journal of Physics},
year = {2026},
month = oct,
doi = {10.1088/1367-2630/aeb8b1},
url = {https://iopscience.iop.org/article/10.1088/1367-2630/aeb8b1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Quantum algorithm for evaluating Lyapunov stability of non-Hermitian systems
AU - Mingwu He
AU - Xiao-Ming Zhang
PY - 2026
DA - 2026/10/23
JO - New Journal of Physics
DO - 10.1088/1367-2630/aeb8b1
UR - https://iopscience.iop.org/article/10.1088/1367-2630/aeb8b1
AB - Stability is a fundamental property of dynamical systems, and Lyapunov theory provides a rigorous link between stability and eigenvalue distribution. By computing eigenvalues, quantum algorithms may offer potential advantages for evaluating the stability of high-dimensional systems under appropriate assumptions regarding data access and quantum resource requirements. In this work, we introduce a quantum algorithm based on the quantum singular value transformation (QSVT) to efficiently resolve the complex eigenspectrum of non-Hermitian matrices and determine asymptotic stability. By employing block encoding and QSVT techniques, we implement a polynomial transformation to approximate the singular value threshold problem, enabling the localization of eigenvalues with positive real parts under appropriate requirements regarding block encoding efficiency, matrix conditioning, state overlap, and precision. We validate the method numerically on a one-dimensional non-Hermitian spin-1/2 chain with local gain and loss, demonstrating accurate detection of unstable and stable cases, which is consistent with the results obtained by exact diagonalization. These results show that our quantum algorithm is a reliable and powerful tool for stability analysis and spectral characterization of high-dimensional dynamical systems.
SN - 1367-2630
ER -