Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation $A^T X + X A = -Q$ . Using the Kronecker product formulation, the matrix equation is transformed into the linear system $H z = b$ , where $H = I_n \otimes A^T + A^T \otimes I_n$ and $z = \mathrm{vec}(X)$ . VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to $25 \times 25$ demonstrate residual norms as low as $10^{-9}$ , with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation $A^T X + X A = -Q$. Using the Kronecker product formulation, the matrix equation is transformed into the linear system $H z = b$, where $H = I_n \otimes A^T + A^T \otimes I_n$ and $z = \mathrm{vec}(X)$. VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to $25 \times 25$ demonstrate residual norms as low as $10^{-9}$, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.

Jun 12, 20260 citations

Analysis of Lyapunov Method for Control of Quantum

S. G. Schirmer, Xiaoting Wang

Jan 1, 20101 citation

Coherent feedback $H^\infty$ control of quantum linear systemsCoherent feedback $H^\infty$ control of quantum linear systems

Guofeng Zhang, Ian R. Petersen

The purpose of this paper is to investigate the coherent feedback $H^\infty$ control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.The purpose of this paper is to investigate the coherent feedback $H^\infty$ control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.

quant-pharXiv:2604.06574Apr 9, 20260 citations

Estimation of the Attraction Domain for the Quantum Systems Based on the Schrödinger Equation

Hongli Yang, Guohui Yu, Ivan Ganchev Ivanov

This paper investigates a quantum system described by the Schrödinger equation, utilizing the concept of the quantum Lyapunov function. The Lyapunov function is chosen based on the mean value of a virtual mechanical quantity, where different values of P, the mean value of the virtual mechanical quantity in the Lyapunov function, have an impact on the attractive domain of the quantum system. The selected primary optimization algorithms approximating matrix P are the particle swarm optimization (PSO) algorithm and the simulated annealing (SA) algorithm. This study examines the characteristics of the system’s attraction domain under these two distinct algorithms and establishes stability conditions for the nonlinear quantum system. We introduce a method to estimate the size of the attractive domain using the Lyapunov function approach, converting the attractive domain issue into an optimization challenge. Numerical simulations are conducted in various two-dimensional test systems and spin 1/2 particle systems.This paper investigates a quantum system described by the Schrödinger equation, utilizing the concept of the quantum Lyapunov function. The Lyapunov function is chosen based on the mean value of a virtual mechanical quantity, where different values of P, the mean value of the virtual mechanical quantity in the Lyapunov function, have an impact on the attractive domain of the quantum system. The selected primary optimization algorithms approximating matrix P are the particle swarm optimization (PSO) algorithm and the simulated annealing (SA) algorithm. This study examines the characteristics of the system’s attraction domain under these two distinct algorithms and establishes stability conditions for the nonlinear quantum system. We introduce a method to estimate the size of the attractive domain using the Lyapunov function approach, converting the attractive domain issue into an optimization challenge. Numerical simulations are conducted in various two-dimensional test systems and spin 1/2 particle systems.

AxiomsAug 9, 20240 citations

Quantum Lyapunov control with machine learning

S. C. Hou, X. X. Yi

arXiv:1808.02516Quantum Information ProcessingNov 26, 201916 citations

Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation

A^T X + X A = -Q

. Using the Kronecker product formulation, the matrix equation is transformed into the linear system

H z = b

, where

H = I_n \otimes A^T + A^T \otimes I_n

and

z = \mathrm{vec}(X)

. VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to

25 \times 25

demonstrate residual norms as low as

10^{-9}

, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.

Coherent feedback

H^\infty

control of quantum linear systems

The purpose of this paper is to investigate the coherent feedback

H^\infty

control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.