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Adaptive Fidelity-Based Density Tracking for Open Quantum Systems

Jhon Manuel Portella Delgado, Ankit Goel

Abstract

This paper presents an online learning-based adaptive control framework for density-matrix tracking in a two-level Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) quantum system, in which the feedback control law does not require prior knowledge of the system Hamiltonian or dissipative operators. The adaptive controller is based on a continuous-time formulation of retrospective cost adaptive control (RCAC). To preserve the geometric structure of the quantum-state evolution, an adaptive PID controller driven by Uhlmann's fidelity is employed. The proposed approach is validated in numerical simulations for both low-entropy and high-entropy density-tracking tasks, and robustness to measurement noise in the feedback path is investigated.

Adaptive Fidelity-Based Density Tracking for Open Quantum Systems

Abstract

This paper presents an online learning-based adaptive control framework for density-matrix tracking in a two-level Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) quantum system, in which the feedback control law does not require prior knowledge of the system Hamiltonian or dissipative operators. The adaptive controller is based on a continuous-time formulation of retrospective cost adaptive control (RCAC). To preserve the geometric structure of the quantum-state evolution, an adaptive PID controller driven by Uhlmann's fidelity is employed. The proposed approach is validated in numerical simulations for both low-entropy and high-entropy density-tracking tasks, and robustness to measurement noise in the feedback path is investigated.
Paper Structure (12 sections, 2 theorems, 38 equations, 11 figures)

This paper contains 12 sections, 2 theorems, 38 equations, 11 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Density Error
  4. Adaptive PID Control
  5. Continuous Time Retrospective Cost Adaptive Control
  6. Numerical Simulation
  7. Hyperparameter Tuning
  8. Low-Entropy Density Tracking
  9. High-Entropy Density Tracking
  10. Shot Noise
  11. Conclusions
  12. Some Useful Facts

Key Result

Proposition III.1

Consider the cost function $J(t, \hat{\theta})$ given by eq:J_RCAC. For all $t \ge 0,$ define the minimizer of $J(t, \hat{\theta})$ by Then, for all $t \ge 0,$ the minimizer satisfies where $P(0) = R_\theta^{-1}$ and $\theta(0) = 0.$

Figures (11)

  • Figure 1: Adaptive feedback control to track desired density matrix $\rho_{\rm d}.$ The estimator, shown in red, consisting of a homodyne detection and a state observer, provides the density matrix $\rho$ for feedback.
  • Figure 2: Contour plot of $J_h$ for various values of $(P_0, \beta).$ The pair $(P_0, \beta)$ that minimizes $J_h$ is used in all numerical simulations.
  • Figure 3: Low-entropy density tracking. The first subplot shows the absolute value of the density error $e(t)$ between the system state and the desired state on a logarithmic scale, the next three subplots show the adaptive gains $k_{\rm p},$$k_{\rm i},$ and $k_{\rm d},$ updated by RCAC, and the last subplot shows the control $u$ generated by the adaptive controller.
  • Figure 4: Low-entropy density tracking. The first two subplots show the real and imaginary parts of the quantum state $\rho(t).$ The desired state components are shown with dashed lines, while the system response is shown with solid lines of the corresponding color. The third subplot shows the entropy of the quantum state.
  • Figure 5: Low-entropy density tracking. Trajectory of the quantum state of the LGKS system \ref{['eq:Lindblad_equation']} on the Bloch sphere. Note that the trajectory remains inside the Bloch ball, consistent with mixed-state evolution.
  • ...and 6 more figures

Theorems & Definitions (14)

  • Remark II.1
  • Remark III.1
  • Remark III.2
  • Proposition III.1
  • proof
  • Remark IV.1
  • Definition A.1
  • Definition A.2
  • proof
  • proof
  • ...and 4 more