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i-QLS: Quantum-supported Algorithm for Least Squares Optimization in Non-Linear Regression

Supreeth Mysore Venkatesh, Antonio Macaluso, Diego Arenas, Matthias Klusch, Andreas Dengel

TL;DR

i-QLS tackles the scalability and precision limits of quantum-annealing-based least squares by introducing an iterative, qubit-efficient QUBO refinement that dynamically narrows the weight search region, achieving exponential convergence toward the LS solution. The method generalizes to non-linear regression through spline-based modeling, enabling quantum-assisted regression beyond linear models. Empirical results on near-term hardware show scalability to high-dimensional problems (e.g., up to 175 features) with competitive accuracy and clear advantages over prior quantum approaches, while highlighting practical constraints such as latency and noise. This work advances practical quantum-assisted machine learning by combining iterative refinement with QUBO-based LS and extending it to non-linear function approximation.

Abstract

We propose an iterative quantum-assisted least squares (i-QLS) optimization method that leverages quantum annealing to overcome the scalability and precision limitations of prior quantum least squares approaches. Unlike traditional QUBO-based formulations, which suffer from a qubit overhead due to fixed discretization, our approach refines the solution space iteratively, enabling exponential convergence while maintaining a constant qubit requirement per iteration. This iterative refinement transforms the problem into an anytime algorithm, allowing for flexible computational trade-offs. Furthermore, we extend our framework beyond linear regression to non-linear function approximation via spline-based modeling, demonstrating its adaptability to complex regression tasks. We empirically validate i-QLS on the D-Wave quantum annealer, showing that our method efficiently scales to high-dimensional problems, achieving competitive accuracy with classical solvers while outperforming prior quantum approaches. Experiments confirm that i-QLS enables near-term quantum hardware to perform regression tasks with improved precision and scalability, paving the way for practical quantum-assisted machine learning applications.

i-QLS: Quantum-supported Algorithm for Least Squares Optimization in Non-Linear Regression

TL;DR

i-QLS tackles the scalability and precision limits of quantum-annealing-based least squares by introducing an iterative, qubit-efficient QUBO refinement that dynamically narrows the weight search region, achieving exponential convergence toward the LS solution. The method generalizes to non-linear regression through spline-based modeling, enabling quantum-assisted regression beyond linear models. Empirical results on near-term hardware show scalability to high-dimensional problems (e.g., up to 175 features) with competitive accuracy and clear advantages over prior quantum approaches, while highlighting practical constraints such as latency and noise. This work advances practical quantum-assisted machine learning by combining iterative refinement with QUBO-based LS and extending it to non-linear function approximation.

Abstract

We propose an iterative quantum-assisted least squares (i-QLS) optimization method that leverages quantum annealing to overcome the scalability and precision limitations of prior quantum least squares approaches. Unlike traditional QUBO-based formulations, which suffer from a qubit overhead due to fixed discretization, our approach refines the solution space iteratively, enabling exponential convergence while maintaining a constant qubit requirement per iteration. This iterative refinement transforms the problem into an anytime algorithm, allowing for flexible computational trade-offs. Furthermore, we extend our framework beyond linear regression to non-linear function approximation via spline-based modeling, demonstrating its adaptability to complex regression tasks. We empirically validate i-QLS on the D-Wave quantum annealer, showing that our method efficiently scales to high-dimensional problems, achieving competitive accuracy with classical solvers while outperforming prior quantum approaches. Experiments confirm that i-QLS enables near-term quantum hardware to perform regression tasks with improved precision and scalability, paving the way for practical quantum-assisted machine learning applications.
Paper Structure (12 sections, 1 theorem, 27 equations, 2 figures, 1 algorithm)

This paper contains 12 sections, 1 theorem, 27 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Background and Related Works
  2. Methods
  3. Problem Formulation
  4. Discretization and QUBO Construction
  5. Quadratic Cost Function
  6. i-QLS Algorithm
  7. Experiments
  8. Convergence and Search Space Refinement
  9. Non-Linear Function Approximation
  10. Discussion and Conclusion
  11. Code Availability.
  12. Acknowledgments.

Key Result

lemma thmcounterlemma

If the underlying least squares problem is well-posed (i.e., there exists a unique optimal weight vector $w^*$ such that $y=Xw^*$) and $w_i^* \in [l_i^{(0)}, u_i^{(0)}] \forall i$, then the mean squared error evaluated from the weights estimated at each iteration exponentially converges to $0$ as $k

Figures (2)

  • Figure 1: Assessment of convergence rate and accuracy of i-QLS
  • Figure 2: Spline-Based Approximation of Non-Linear Functions. The green curves illustrate the iterative refinement of the regression fit obtained using our quantum-assisted least squares approach with linear splines (20 knots), one qubit per parameter, and up to 10 iterations. The lighter green lines correspond to intermediate fits across iterations, while the darker green curve represents the final approximation after 10 iterations.

Theorems & Definitions (2)

  • lemma thmcounterlemma
  • proof