Structure preserving Quantum Algorithms for Linear Ordinary Differential Equations

Many scientific and engineering problems are governed by differential equations whose analytical solutions are unavailable, making numerical discretization indispensable. Classical numerical schemes generally fail to preserve the geometric and symmetry properties of the underlying differential equations, which can adversely affect numerical accuracy. In this work, we combine Lie symmetry-preserving numerical discretization with Quantum Linear Systems Algorithms (QLSAs) to develop a quantum framework for solving first-order linear ordinary differential equations. Using the moving-frame invariantization method, we construct a Lie symmetry-preserving variant of the classical Euler method, referred to as the Lie–Euler scheme. Both the classical Euler and Lie–Euler discretizations are formulated as sparse linear systems and solved using the Harrow–Hassidim–Lloyd algorithm. The Lie–Euler discretization preserves the underlying Lie symmetry and improves the global discretization error from \(O(h)\) to \(O(h^2)\), allowing a prescribed accuracy to be achieved using fewer discretization points and hence smaller linear systems with reduced qubit requirements. Furthermore, for an important class of dissipative problems, we prove that the associated Lie–Euler matrices possess uniformly bounded spectral condition numbers, yielding stronger asymptotic complexity guarantees for modern QLSAs. Numerical examples demonstrate the effectiveness of the proposed framework.

Recommended citation: M. Rohida, A. Shukla, and P. Vedula, Structure preserving Quantum Algorithms for Linear Ordinary Differential Equations, 2026. hal- 05680703.
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