[Paper Review] Fault-Tolerant Control of Linear Quantum Stochastic Systems

Shi Wang, Daoyi Dong|arXiv (Cornell University)|Jun 7, 2018

Quantum Information and Cryptography19 references3 citations

TL;DR

This paper proposes an estimator-based fault-tolerant control framework for linear quantum stochastic systems subject to classical fault signals. By estimating fault signals and commutative system observables via a reduced-order dynamic estimator, the method designs a compensatory controller that stabilizes the system and maintains performance despite faults, demonstrated through semi-definite programming and a quantum optical example with a 0.001 estimation error bound.

ABSTRACT

In quantum engineering, faults may occur in a quantum control system, which will cause the quantum control system unstable or deteriorate other relevant performance of the system. This note presents an estimator-based fault-tolerant control design approach for a class of linear quantum stochastic systems subject to fault signals. In this approach, the fault signals and some commutative components of the quantum system observables are estimated, and a fault-tolerant controller is designed to compensate the effect of the fault signals. Numerical procedures are developed for controller design and an example is presented to demonstrate the proposed design approach.

Motivation & Objective

• To address instability and performance degradation in quantum control systems caused by fault signals such as environmental fluctuations or voltage variations.
• To develop a fault-tolerant control strategy tailored for linear quantum stochastic systems, accounting for non-commutative observables and measurement collapse.
• To design a reduced-order dynamic estimator that estimates fault signals and commutative components of system observables for feedback compensation.
• To provide a numerically tractable design procedure using semi-definite programming for controller synthesis.
• To validate the approach through a quantum optical plant example with time-varying fault signals and a specified estimation error bound.

Proposed method

• Uses a measurement-based feedback architecture where a homodyne detector measures the system output, feeding data to a classical estimator.
• Employs a reduced-order dynamic estimator to estimate fault signals and commutative components of system observables, modeled via linear stochastic differential equations.
• Designs a fault-tolerant controller using a separation principle, where the controller gain is derived from the estimator's output.
• Applies semi-definite programming (SDP) to solve the feasibility problem for controller and estimator design, subject to linear matrix inequalities (LMIs).
• Imposes constraints on the estimation error covariance matrix to ensure boundedness, with a target trace of the error covariance matrix ≤ γ.
• Uses physical realizability conditions to ensure the resulting controller and estimator are compatible with quantum system dynamics.

Experimental results

Research questions

• RQ1 How can fault signals in linear quantum stochastic systems be estimated and compensated in a way that preserves system stability?
• RQ2 What is the role of a reduced-order dynamic estimator in enabling fault-tolerant control for quantum systems with non-commutative observables?
• RQ3 How can semi-definite programming be used to design a fault-tolerant controller that guarantees bounded estimation error and system stability?
• RQ4 What are the necessary and sufficient conditions for the physical realizability of the proposed estimator-based controller in a quantum feedback system?
• RQ5 How does the proposed method compare to classical fault-tolerant control in terms of applicability to quantum systems with measurement collapse and non-commutativity?

Key findings

• The proposed estimator-based fault-tolerant controller successfully stabilizes the quantum system under time-varying fault signals, as verified through numerical simulation.
• For the example system, the estimation error bound was achieved at Tr(Y₂) = 0.001, indicating a high-precision estimation of fault signals and system components.
• The controller gains were computed as L = [[-4.07, 1.03], [9.22, -30.21]] and K = [[-0.1500, -2.6643], [0.3000, 2.6857]], satisfying all required LMIs and physical realizability conditions.
• The positive definiteness of matrix P was confirmed, ensuring stability of the closed-loop system under the designed controller.
• The method successfully handles fault signals independent of quantum noise, distinguishing it from uncertainty-based robust control approaches.
• The SDP-based design procedure is feasible and can be iteratively refined by adjusting the estimation error bound γ to improve performance.