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

arXiv:1707.07372 · IEEE Conference on Decision and Control (CDC)

Lyapunov stability analysis for invariant states of quantum systems

Muhammad F. Emzir, Ian R. Petersen, Matthew J. Woolley

Published Dec 1, 2017• 5 citations
arXivHTMLPDF

Abstract

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In contrast to many previously studied convergence analysis methods for invariant density operators which use weak convergence, in this article we analyze the convergence of density operators by considering the set of density operators as a Banach space. We show that the set of invariant density operators is both closed and convex, which implies the impossibility of having multiple isolated invariant density operators. We then show how to analyze the stability of this set via a candidate Lyapunov operator.

DOI
10.1109/cdc.2017.8264475

Related papers

Lyapunov-Based Stabilization and Control of Closed Quantum Systems

Elham Jamalinia, Peyman Azodi, Alireza Khayatian +1 more

A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.

arXiv:2102.00344•Jan 31, 2021•0 citations

Analysis of Lyapunov Method for Control of Quantum States

Xiaoting Wang, S. G. Schirmer

The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.

arXiv:0801.0702•IEEE Transactions on Automatic Control•Feb 17, 2010•152 citations

Lyapunov stabilization strategy of mixed-state quantum systems with ideal conditions

Shuang Cong

Under the ideal conditions that the controlled systems are non-degenerate,transitionally non-degenerate,and fully connected,the stabilization problem of mixed-state quantum systems is studied by using a Lyapunov function with free degrees. Based on the LaSalle's principle,the largest invariant set of the closed-loop systems and the convergent state set associated with any initial state are deduced,and the construction principles of free degrees that ensure the system can be asymptotically stabilized in any equilibrium state contained in the largest invariant set are given. A numerical simulation experiment on a two-level system shows the rationality of the obtained theoretical results.

Kongzhi yu juece•Jan 1, 2010•4 citations

Analysis of Lyapunov Control for Hamiltonian Quantum Systems

Xiaoting Wang, Sonia G. Schirmer

We present detailed analysis of the convergence properties and effectiveness of Lyapunov control design for bilinear Hamiltonian quantum systems based on the application of LaSalle's invariance principle and stability analysis from dynamical systems and control theory. For a certain class of Hamiltonians, strong convergence results can be obtained for both pure and mixed state systems. The control Hamiltonians for realistic physical systems, however, generally do not fall in this class. It is shown that the effectiveness of Lyapunov control design in this case is significantly diminished.

arXiv:0805.2882•May 19, 2008•9 citations

Analysis of Lyapunov Method for Control of Quantum States: non-generic case

Xiaoting Wang, Sonia G. Schirmer

Abstract—We analyze the state/trajectory tracking problem using Lyapunov control for finite-dimensional quantum systems when the target state is not a generic density operator and the Lyapunov function is therefore not a Morse function. For ideal systems general stability results are derived for stationary target states based on the linearization method. For the special class of pseudo-pure states a full analysis of the trajectory tracking problem for non-stationary states is given. Despite differences from the generic case such as the emergence of critical manifolds, it is shown that the method is effective for ideal systems but fails if ideal conditions are not met. I.

•Jan 28, 2009•4 citations

An implicit Lyapunov control for finite‐dimensional closed quantum systems

Shouwei Zhao, Hai Lin, Jitao Sun +1 more

SUMMARY In this paper, the state convergence problem for closed quantum systems is investigated. We consider two degenerate cases, where the internal Hamiltonian of the system is not strongly regular or the linearized system around the target state is not controllable. Both the cases are closely related to practical systems such as one‐dimensional oscillators and coupled two spin systems. An implicit Lyapunov‐based control strategy is adopted for the convergence analysis. In particular, two kinds of Lyapunov functions are defined by implicit functions and their existences are guaranteed by a fixed point theorem. The convergence analysis is investigated by the LaSalle invariance principle for both cases. Moreover, the two Lyapunov functions are unified in a general form, and the characterization of the largest invariant set is presented. Finally, simulation studies are included to show the effectiveness and advantage of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.

International Journal of Robust and Nonlinear Control•May 4, 2011•27 citations
In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In contrast to many previously studied convergence analysis methods for invariant density operators which use weak convergence, in this article we analyze the convergence of density operators by considering the set of density operators as a Banach space. We show that the set of invariant density operators is both closed and convex, which implies the impossibility of having multiple isolated invariant density operators. We then show how to analyze the stability of this set via a candidate Lyapunov operator.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.
Under the ideal conditions that the controlled systems are non-degenerate,transitionally non-degenerate,and fully connected,the stabilization problem of mixed-state quantum systems is studied by using a Lyapunov function with free degrees. Based on the LaSalle's principle,the largest invariant set of the closed-loop systems and the convergent state set associated with any initial state are deduced,and the construction principles of free degrees that ensure the system can be asymptotically stabilized in any equilibrium state contained in the largest invariant set are given. A numerical simulation experiment on a two-level system shows the rationality of the obtained theoretical results.
We present detailed analysis of the convergence properties and effectiveness of Lyapunov control design for bilinear Hamiltonian quantum systems based on the application of LaSalle's invariance principle and stability analysis from dynamical systems and control theory. For a certain class of Hamiltonians, strong convergence results can be obtained for both pure and mixed state systems. The control Hamiltonians for realistic physical systems, however, generally do not fall in this class. It is shown that the effectiveness of Lyapunov control design in this case is significantly diminished.
Abstract—We analyze the state/trajectory tracking problem using Lyapunov control for finite-dimensional quantum systems when the target state is not a generic density operator and the Lyapunov function is therefore not a Morse function. For ideal systems general stability results are derived for stationary target states based on the linearization method. For the special class of pseudo-pure states a full analysis of the trajectory tracking problem for non-stationary states is given. Despite differences from the generic case such as the emergence of critical manifolds, it is shown that the method is effective for ideal systems but fails if ideal conditions are not met. I.
SUMMARY In this paper, the state convergence problem for closed quantum systems is investigated. We consider two degenerate cases, where the internal Hamiltonian of the system is not strongly regular or the linearized system around the target state is not controllable. Both the cases are closely related to practical systems such as one‐dimensional oscillators and coupled two spin systems. An implicit Lyapunov‐based control strategy is adopted for the convergence analysis. In particular, two kinds of Lyapunov functions are defined by implicit functions and their existences are guaranteed by a fixed point theorem. The convergence analysis is investigated by the LaSalle invariance principle for both cases. Moreover, the two Lyapunov functions are unified in a general form, and the characterization of the largest invariant set is presented. Finally, simulation studies are included to show the effectiveness and advantage of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.