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H-DES: a Quantum-Classical Hybrid Differential Equation Solver

Hamza Jaffali, Jonas Bastos de Araujo, Nadia Milazzo, Marta Reina, Henri de Boutray, Karla Baumann, Frédéric Holweck, Youcef Mohdeb, Roland Katz

TL;DR

H-DES presents a quantum-classical hybrid framework for solving differential equations by encoding the solution in a spectral basis using a variational quantum circuit. The method leverages Chebyshev polynomials, with two encoding schemes (basis-state and Pauli-monomial) to evaluate the function and its derivatives via diagonal observables, enabling an exact derivative calculation within a single circuit. It extends naturally to multivariate PDEs, analyzes scalability, and demonstrates promising results on emulators and real hardware across several ODE examples. The work indicates that shallow, few-qubit circuits can model diverse DEs with high accuracy on NISQ devices, suggesting practical potential for industrial simulations and multi-physics problems.

Abstract

In this article, we introduce an original hybrid quantum-classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions that are encoded directly in the quantum states generated by different parametrized circuits, and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we elaborate preliminary elements for a complexity analysis to highlight some of its scaling properties. We apply our algorithm to a set of examples, running on emulators and real hardware showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement.

H-DES: a Quantum-Classical Hybrid Differential Equation Solver

TL;DR

H-DES presents a quantum-classical hybrid framework for solving differential equations by encoding the solution in a spectral basis using a variational quantum circuit. The method leverages Chebyshev polynomials, with two encoding schemes (basis-state and Pauli-monomial) to evaluate the function and its derivatives via diagonal observables, enabling an exact derivative calculation within a single circuit. It extends naturally to multivariate PDEs, analyzes scalability, and demonstrates promising results on emulators and real hardware across several ODE examples. The work indicates that shallow, few-qubit circuits can model diverse DEs with high accuracy on NISQ devices, suggesting practical potential for industrial simulations and multi-physics problems.

Abstract

In this article, we introduce an original hybrid quantum-classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions that are encoded directly in the quantum states generated by different parametrized circuits, and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we elaborate preliminary elements for a complexity analysis to highlight some of its scaling properties. We apply our algorithm to a set of examples, running on emulators and real hardware showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement.
Paper Structure (39 sections, 80 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 39 sections, 80 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Description of the algorithm
  3. Variational Quantum Circuit and Ansatz
  4. Spectral decomposition and Chebyshev polynomials
  5. Encoding and evaluation of the function
  6. Basis-state encoding
  7. Pauli-monomial encoding
  8. Evaluation of the derivatives
  9. Loss function
  10. Validation of the results
  11. Multivariate functions
  12. Computing the derivatives efficiently
  13. Implementation and related complexity
  14. Pseudo-code of the algorithm
  15. Line-by-line analysis
  16. ...and 24 more sections

Figures (8)

  • Figure 1: An illustration of the workflow of our approach.
  • Figure 2: A representation of the hardware-efficient Ansatz used in our variational quantum algorithm.
  • Figure 3: The exact (blue solid) versus attempted solutions (orange dashed) for $f(x)$ (left) and $g(x)$ (right) mediated over 100 attempts after 150 iterations of the BFGS optimizer. The fit was obtained by minimizing the loss function over 20 points equally spaced in the $x = [0, 0.95]$ domain. The area at the bottom shows the standard deviation (between the attempted and target solution) associated with each point.
  • Figure 4: A comparison between the target function (blue solid) obtained analytically and the attempted function of the algorithm (orange dashed). The area at the bottom shows the standard deviation (between the two functions) associated with each point.
  • Figure 5: The exact (blue solid) versus attempted solutions (orange dashed) for the displacement (left) and stress (right) average over 100 attempts and after 400 iterations of the BFGS optimizer. The fit was obtained by minimizing the loss function over 20 points equally spaced in the $x = [0, 0.95]$ domain. The area at the bottom shows the standard deviation associated with each point.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Remark 1