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QI-MPC: A Hybrid Quantum-Inspired Model Predictive Control for Learning Optimal Policies

Muhammad Al-Zafar Khan, Jamal Al-Karaki

TL;DR

QI-MPC advances a hybrid quantum-classical approach that uses Variational Quantum Circuits to learn control policies within Model Predictive Control, aiming to alleviate combinatorial and high-dimensional optimization challenges. The framework encodes classical states into quantum states, processes them with a parameterized unitary, and extracts control signals via measurements, all within a receding-horizon MPC loop, while enforcing bounds and stability through formal safety guarantees. Across five numerical experiments, QI-MPC demonstrates opportunities for improved performance in non-oscillatory, well-behaved systems, but reveals significant challenges for oscillatory and chaotic dynamics, including instability and high variance due to quantum noise and topological constraints. The authors articulate two key theoretical propositions delineating when HQC-based MPC can outperform classical MPC and when it cannot, offering guidance on when to deploy QI-MPC and how to refine quantum encodings, learning schedules, and circuit designs for improved stability and efficiency.

Abstract

In this paper, we present Quantum-Inspired Model Predictive Control (QIMPC), an approach that uses Variational Quantum Circuits (VQCs) to learn control polices in MPC problems. The viability of the approach is tested in five experiments: A target-tracking control strategy, energy-efficient building climate control, autonomous vehicular dynamics, the simple pendulum, and the compound pendulum. Three safety guarantees were established for the approach, and the experiments gave the motivation for two important theoretical results that, in essence, identify systems for which the approach works best.

QI-MPC: A Hybrid Quantum-Inspired Model Predictive Control for Learning Optimal Policies

TL;DR

QI-MPC advances a hybrid quantum-classical approach that uses Variational Quantum Circuits to learn control policies within Model Predictive Control, aiming to alleviate combinatorial and high-dimensional optimization challenges. The framework encodes classical states into quantum states, processes them with a parameterized unitary, and extracts control signals via measurements, all within a receding-horizon MPC loop, while enforcing bounds and stability through formal safety guarantees. Across five numerical experiments, QI-MPC demonstrates opportunities for improved performance in non-oscillatory, well-behaved systems, but reveals significant challenges for oscillatory and chaotic dynamics, including instability and high variance due to quantum noise and topological constraints. The authors articulate two key theoretical propositions delineating when HQC-based MPC can outperform classical MPC and when it cannot, offering guidance on when to deploy QI-MPC and how to refine quantum encodings, learning schedules, and circuit designs for improved stability and efficiency.

Abstract

In this paper, we present Quantum-Inspired Model Predictive Control (QIMPC), an approach that uses Variational Quantum Circuits (VQCs) to learn control polices in MPC problems. The viability of the approach is tested in five experiments: A target-tracking control strategy, energy-efficient building climate control, autonomous vehicular dynamics, the simple pendulum, and the compound pendulum. Three safety guarantees were established for the approach, and the experiments gave the motivation for two important theoretical results that, in essence, identify systems for which the approach works best.

Paper Structure

This paper contains 15 sections, 11 theorems, 30 equations, 22 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Quantum-Informed Model Predictive Control
  4. The QI-MPC Algorithm
  5. Safety Guarantees of QI-MPC
  6. Experiments
  7. Experiment 1: A Target-Tracking Proportional Control Strategy
  8. Experiment 2: Energy Efficient Building Climate Control
  9. Experiment 3: Autonomous Vehicular Dynamics on a Curvilinear Road
  10. Experiment 4: The Simple Pendulum
  11. Experiment 5: The Compound Pendulum
  12. Physical Results
  13. Nonfeasibility of hybrid QI-MPC Approaches to Control Oscillatory Systems
  14. Outperformance of Classical MPC by QI-MPC
  15. Conclusion

Key Result

Theorem 1

(Nyquist-Shannon Sampling Theorem). A band-limited signal, with the highest frequency component $f_{\max}$, can be perfectly reconstructed from its samples if it is sampled at a rate at least twice the highest frequency present in the signal. Mathematically, for a sampling rate frequency $f_{s}$,

Figures (22)

  • Figure 1: The quantum circuit used to determine the optimal control actions for the state dynamics $\mathbf{x}_{k}(t+1)=\mathbf{x}_{k}(t)+\alpha\left[\mathbf{u}_{k}(t)-\mathbf{x}_{k}(t)\right]$.
  • Figure 2: Control signals for the controls $\mathbf{u}=\left(u_{1},u_{2},u_{3}\right)\in\mathbb{R}^{3}$.
  • Figure 3: State trajectories $\mathbf{x}=\left(x_{1},x_{2},x_{3}\right)\in\mathbb{R}^{3}$.
  • Figure 4: Loss function convergence.
  • Figure 5: Quantum circuit simulating the optimal choice of controls for the energy efficient climate control model.
  • ...and 17 more figures

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • ...and 20 more