Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Using quantum computers in control: interval matrix properties

Jan Schneider, Julian Berberich

TL;DR

The paper tackles the problem of verifying robust interval-matrix properties, specifically robust non-singularity and robust stability, which are NP-hard in classical computation. It develops a QAOA-based quantum algorithm that encodes the radius of non-singularity $d(A,\Delta)$ for rank-1 $\Delta$ into a binary optimization problem via a problem Hamiltonian $H_P$, enabling quantum approximation of the worst-case singularity distance. Two numerical simulations implemented in Pennylane demonstrate the method can recover the correct radius values (e.g., $d=\tfrac{1}{3}$ in a 2D test and $d=\tfrac{1}{4.0833}$ for a symmetric RL circuit), validating the approach for robust verification and, by extension, stability certification under symmetry. The work demonstrates a path toward quantum-accelerated verification in robust control and suggests future directions to handle general interval matrices and real-device implementations.

Abstract

Quantum computing provides a powerful framework for tackling computational problems that are classically intractable. The goal of this paper is to explore the use of quantum computers for solving relevant problems in systems and control theory. In the recent literature, different quantum algorithms have been developed to tackle binary optimization, which plays an important role in various control-theoretic problems. As a prototypical example, we consider the verification of interval matrix properties such as non-singularity and stability on a quantum computer. We present a quantum algorithm solving these problems and we study its performance in simulation. Our results demonstrate that quantum computers provide a promising tool for control whose applicability to further computationally complex problems remains to be explored.

Using quantum computers in control: interval matrix properties

TL;DR

The paper tackles the problem of verifying robust interval-matrix properties, specifically robust non-singularity and robust stability, which are NP-hard in classical computation. It develops a QAOA-based quantum algorithm that encodes the radius of non-singularity for rank-1 into a binary optimization problem via a problem Hamiltonian , enabling quantum approximation of the worst-case singularity distance. Two numerical simulations implemented in Pennylane demonstrate the method can recover the correct radius values (e.g., in a 2D test and for a symmetric RL circuit), validating the approach for robust verification and, by extension, stability certification under symmetry. The work demonstrates a path toward quantum-accelerated verification in robust control and suggests future directions to handle general interval matrices and real-device implementations.

Abstract

Quantum computing provides a powerful framework for tackling computational problems that are classically intractable. The goal of this paper is to explore the use of quantum computers for solving relevant problems in systems and control theory. In the recent literature, different quantum algorithms have been developed to tackle binary optimization, which plays an important role in various control-theoretic problems. As a prototypical example, we consider the verification of interval matrix properties such as non-singularity and stability on a quantum computer. We present a quantum algorithm solving these problems and we study its performance in simulation. Our results demonstrate that quantum computers provide a promising tool for control whose applicability to further computationally complex problems remains to be explored.
Paper Structure (11 sections, 42 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 42 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Interval Matrices
  3. Non-singularity of interval matrices
  4. Stability of interval matrices
  5. Quantum Algorithm
  6. Quantum Basics
  7. Quantum approximate optimization algorithm
  8. Formulating the problem Hamiltonian
  9. The Quantum Algorithm
  10. Implementation
  11. Conclusion

Figures (4)

  • Figure 1: Quantum circuit implementing the operation \ref{['eq:X01']} followed by a measurement.
  • Figure 2: Circuit representation of QAOA for $n=2$ qubits and $p=2$ layers.
  • Figure 3: Basic scheme of QAOA as an iterative optimization containing classical parameter updates and executions of parametrized quantum circuits.
  • Figure 4: Counts per string for the output of the proposed algorithm when applied to the first example.

Theorems & Definitions (1)

  • proof