coherent-feedback-h-infinity-quantum-control

name	coherent-feedback-h-infinity-quantum-control
description	Simplified coherent feedback H∞ control design for linear quantum systems using Lyapunov equations instead of coupled algebraic Riccati equations — computationally efficient robust control for quantum optical systems.
category	quantum
version	1.0
created	2026-07-09
trigger_words	["coherent feedback","H-infinity quantum","quantum linear system control","Lyapunov quantum control","quantum robust control","quantum optical control"]
source_paper	arXiv:2604.06574

Coherent Feedback H∞ Control of Quantum Linear Systems

Overview

This methodology provides a simplified design approach for coherent feedback H∞ control of linear quantum systems. Instead of solving two coupled algebraic Riccati equations (standard approach), a physically realizable quantum controller is obtained by solving at most four Lyapunov equations, providing significant computational efficiency.

Core Methodology

General Case

1. Formulate the linear quantum system in state-space form
2. Solve at most four Lyapunov equations to obtain the controller
3. Verify physical realizability conditions
4. Guarantee closed-loop stability + prescribed disturbance attenuation level

Passive Case (Simplified)

1. Solve two uncoupled pairs of Lyapunov equations
2. This provides a necessary and sufficient condition for passive coherent H∞ control
3. Significantly simpler than the standard Riccati-based approach

Mathematical Framework

For a linear quantum system:

dx = A x dt + B1 dw + B2 du
dy = C1 x dt + D12 dw
du = Ck xk dt + Dk dy  (controller)

The H∞ controller design problem: find Ck, Dk such that the closed-loop system is physically realizable and achieves γ-disturbance attenuation.

Traditional: Solve 2 coupled algebraic Riccati equations (ARE) This method: Solve ≤4 Lyapunov equations

Advantages

• Computational efficiency: Lyapunov equations are simpler than coupled AREs
• Numerical stability: Lyapunov solvers are more robust than ARE solvers
• Scalability: Better suited for larger quantum systems
• Same guarantees: Closed-loop stability + prescribed H∞ performance

Demonstrated Applications

1. Empty optical cavity — standard quantum optical benchmark
2. Degenerate parametric amplifier — nonlinear quantum optical device (linearized)

When to Use

• Linear quantum systems (or linearized around operating point)
• Quantum optical systems (cavities, amplifiers, optomechanical systems)
• When computational efficiency is important
• When robust disturbance attenuation is required

Pitfalls

• Only applies to linear quantum systems
• Physical realizability constraints must still be verified
• For highly nonlinear systems, linearization may be insufficient
• The four Lyapunov equations must all have solutions (feasibility check needed)
• Does not handle measurement-based feedback (this is coherent/coherent-only)

Verification

1. Verify all Lyapunov equations have positive-definite solutions
2. Check physical realizability conditions on the resulting controller
3. Simulate closed-loop response to verify H∞ performance bound
4. Validate on standard benchmarks (optical cavity, parametric amplifier)