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Stabilization of Time-Varying Perturbed Quantum Systems via Reduced Filters

Weichao Liang, Daoyi Dong

TL;DR

This work tackles robust measurement-based quantum feedback control for open systems subject to time-varying perturbations and uncertain initial states. It introduces a reduced-order quantum filter that tracks only the diagonal density-matrix elements in a quantum non-demolition basis, achieving $O(N)$ computational complexity instead of the full $O(N^2)$ SME while preserving convergence and robustness. The authors prove existence and well-posedness of the reduced filter, establish global exponential stabilization to a target subspace under time-varying perturbations, and demonstrate a recurrence property and an explicit Lyapunov bound. A Kalman-type rank condition on the control Hamiltonian ensures the non-invariance of undesired subspaces, enabling almost-sure convergence of the system through the reduced estimator. Numerical results on a three-level system illustrate robust exponential stabilization under slow and fast perturbations, highlighting the approach’s scalability and practicality for high-dimensional perturbed open quantum systems.

Abstract

In practical applications, quantum systems are inevitably subject to significant uncertainties, including unknown initial states, imprecise physical parameters, and unmodeled environmental noise, all of which pose major challenges to robust quantum feedback control. This paper proposes a feedback stabilization strategy based on a reduced quantum filter that achieves robustness against time-varying Hamiltonian perturbations and additional dissipative effects, without requiring prior knowledge of the initial state or exact system parameters. The proposed filter estimates only O(N) real variables corresponding to the diagonal elements of the system density matrix in a quantum non-demolition basis in contrast to the O(N^2) variables required by a full stochastic master equation, where N is the Hilbert space dimension. This dimensionality reduction substantially simplifies real-time computation and feedback implementation while preserving both convergence and robustness guarantees. Rigorous analysis further establishes global exponential stability of the target subspace. The results provide a scalable framework for robust and efficient measurement-based feedback control applicable to high-dimensional perturbed open quantum systems.

Stabilization of Time-Varying Perturbed Quantum Systems via Reduced Filters

TL;DR

This work tackles robust measurement-based quantum feedback control for open systems subject to time-varying perturbations and uncertain initial states. It introduces a reduced-order quantum filter that tracks only the diagonal density-matrix elements in a quantum non-demolition basis, achieving computational complexity instead of the full SME while preserving convergence and robustness. The authors prove existence and well-posedness of the reduced filter, establish global exponential stabilization to a target subspace under time-varying perturbations, and demonstrate a recurrence property and an explicit Lyapunov bound. A Kalman-type rank condition on the control Hamiltonian ensures the non-invariance of undesired subspaces, enabling almost-sure convergence of the system through the reduced estimator. Numerical results on a three-level system illustrate robust exponential stabilization under slow and fast perturbations, highlighting the approach’s scalability and practicality for high-dimensional perturbed open quantum systems.

Abstract

In practical applications, quantum systems are inevitably subject to significant uncertainties, including unknown initial states, imprecise physical parameters, and unmodeled environmental noise, all of which pose major challenges to robust quantum feedback control. This paper proposes a feedback stabilization strategy based on a reduced quantum filter that achieves robustness against time-varying Hamiltonian perturbations and additional dissipative effects, without requiring prior knowledge of the initial state or exact system parameters. The proposed filter estimates only O(N) real variables corresponding to the diagonal elements of the system density matrix in a quantum non-demolition basis in contrast to the O(N^2) variables required by a full stochastic master equation, where N is the Hilbert space dimension. This dimensionality reduction substantially simplifies real-time computation and feedback implementation while preserving both convergence and robustness guarantees. Rigorous analysis further establishes global exponential stability of the target subspace. The results provide a scalable framework for robust and efficient measurement-based feedback control applicable to high-dimensional perturbed open quantum systems.

Paper Structure

This paper contains 14 sections, 12 theorems, 101 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Stochastic dynamical model
  3. Stochastic Evolution of States in QND basis
  4. Stabilization by the reduced filter
  5. Motivation for a reduced filter
  6. Stochastic model for the reduced filter
  7. Feedback stabilization to target subspaces
  8. Instability of undesired invariant subsets
  9. Recurrence property
  10. Exponential stability
  11. Numerical example: Three-level systems
  12. CONCLUSIONS AND FUTURE WORKS
  13. Invariant properties of quantum trajectories
  14. Proof of Inequality \ref{['Eq:Ineq_LV_QSR']}

Key Result

Theorem 3.1

Assume that $u_t\equiv 0$, and A1, A3 and A-qsr hold. For any $\rho(0)\in\mathcal{S}(\mathcal{H})$, the system Eq:SME converges towards $\mathcal{I}(\mathcal{H}):=\bigcup^{d}_{j=0}\mathcal{I}(\mathcal{H}_j)$ in mean and almost surely with Lyapunov exponent less than or equal to $-\mathfrak{E}_{l}/2$

Figures (2)

  • Figure 1: Three-level system under slow modulation ($T=100\tau$). Left: no feedback ($u\equiv 0$). Right: reduced-filter feedback $u(\hat{q})$. Initial conditions: $\rho(0)=\mathrm{diag}(0,0,1)$, $\hat{q}(0)=[1,1,1]^\top/3$. Black curve: mean over $100$ realizations.
  • Figure 2: Three-level system under fast modulation ($T=2\tau$). Left: no feedback ($u\equiv 0$). Right: reduced-filter feedback $u(\hat{q})$. Initial conditions: $\rho(0)=\mathrm{diag}(0,0,1)$, $\hat{q}(0)=[1,1,1]^\top/3$. Black curve: mean over $100$ realizations.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1: Exponential quantum state reduction
  • Theorem 4.1
  • Lemma 4.2
  • Corollary 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Remark 4.7
  • ...and 6 more