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Quantum-Assisted Barrier Sequential Quadratic Programming for Nonlinear Optimal Control

Nahid Binandeh Dehaghani, Rafal Wisniewski, A. Pedro Aguiar

TL;DR

This work introduces a quantum-assisted barrier-SQP framework for finite-horizon nonlinear OCPs, integrating a quantum subroutine based on block-encoding and QSVT to accelerate the Schur-complement solve within the SQP iterations. It establishes local input-to-state stability and explicit error bounds in the presence of quantum solver inaccuracies, showing that overall computational effort scales polylogarithmically with problem size under reasonable conditioning. The approach preserves classical outer-loop convergence guarantees while transferring the heavy linear-algebraic work to quantum subroutines, demonstrated on a HIV-1 model to validate comparable performance with potential scalability advantages. The combination of barrier methods, quantum linear algebra, and Gauss–Newton structure offers a promising path toward real-time or large-scale nonlinear control where classical SQP becomes cost-prohibitive.

Abstract

We propose a quantum-assisted framework for solving constrained finite-horizon nonlinear optimal control problems using a barrier Sequential Quadratic Programming (SQP) approach. Within this framework, a quantum subroutine is incorporated to efficiently solve the Schur complement step using block-encoding and Quantum Singular Value Transformation (QSVT) techniques. We formally analyze the time complexity and convergence behavior under the cumulative effect of quantum errors, establishing local input-to-state stability and convergence to a neighborhood of the stationary point, with explicit error bounds in terms of the barrier parameter and quantum solver accuracy. The proposed framework enables computational complexity to scale polylogarithmically with the system dimension demonstrating the potential of quantum algorithms to enhance classical optimization routines in nonlinear control applications.

Quantum-Assisted Barrier Sequential Quadratic Programming for Nonlinear Optimal Control

TL;DR

This work introduces a quantum-assisted barrier-SQP framework for finite-horizon nonlinear OCPs, integrating a quantum subroutine based on block-encoding and QSVT to accelerate the Schur-complement solve within the SQP iterations. It establishes local input-to-state stability and explicit error bounds in the presence of quantum solver inaccuracies, showing that overall computational effort scales polylogarithmically with problem size under reasonable conditioning. The approach preserves classical outer-loop convergence guarantees while transferring the heavy linear-algebraic work to quantum subroutines, demonstrated on a HIV-1 model to validate comparable performance with potential scalability advantages. The combination of barrier methods, quantum linear algebra, and Gauss–Newton structure offers a promising path toward real-time or large-scale nonlinear control where classical SQP becomes cost-prohibitive.

Abstract

We propose a quantum-assisted framework for solving constrained finite-horizon nonlinear optimal control problems using a barrier Sequential Quadratic Programming (SQP) approach. Within this framework, a quantum subroutine is incorporated to efficiently solve the Schur complement step using block-encoding and Quantum Singular Value Transformation (QSVT) techniques. We formally analyze the time complexity and convergence behavior under the cumulative effect of quantum errors, establishing local input-to-state stability and convergence to a neighborhood of the stationary point, with explicit error bounds in terms of the barrier parameter and quantum solver accuracy. The proposed framework enables computational complexity to scale polylogarithmically with the system dimension demonstrating the potential of quantum algorithms to enhance classical optimization routines in nonlinear control applications.
Paper Structure (7 sections, 8 theorems, 10 equations, 2 figures, 2 algorithms)

This paper contains 7 sections, 8 theorems, 10 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Finite-Horizon Nonlinear OCP
  3. Hybrid Quantum--Classical SQP
  4. Quantum Assited framework for SQP
  5. Convergence and Complexity Analysis
  6. Simulation Results
  7. Conclusions

Key Result

Proposition 1

In Algorithm Qschur, the block encodings propagate such that the final block encoding of the step direction $\Delta z$ has normalization factor $\alpha_{\Delta z} \;=\; \tfrac{\kappa_Q \beta_Q}{\alpha_Q} \left( \alpha_g + \alpha_{\mathcal{A}} \cdot \tfrac{\kappa_S \beta_S}{\alpha_{\mathcal{A}}^2 } \

Figures (2)

  • Figure 1: High level quantum circuit for calculating $\Delta z$.
  • Figure 2: Comparison of classical SQP and hybrid quantum-classical SQP for the HIV-1 model.

Theorems & Definitions (26)

  • Definition 1: Block Encoding
  • Remark 1: Matrix Padding for Block Encoding
  • Remark 2: Addition of Block Encodings
  • Remark 3: Multiplication of Block Encodings
  • Remark 4: Matrix Inversion via QSVT
  • Proposition 1: Normalization Factor
  • proof
  • Remark 5: Success Probability
  • Proposition 2: Accuracy
  • proof
  • ...and 16 more