A Quantum Linear Systems Pathway for Solving Differential Equations

We present a systematic pathway for solving differential equations within the quantum linear systems framework by combining block encoding with Quantum Singular Value Transformation (QSVT). The approach is demonstrated on a complex tridiagonal linear system and extended to problems in computational fluid dynamics: the heat equation with mixed boundary conditions and the nonlinear Burgers'equation. Our scaling analysis of the heat equation shows how discretization influences the minimum singular value and the polynomial degree required for QSVT, identifying circuit-depth overhead as a key bottleneck. For Burgers'equation, we illustrate how Carleman-linearized nonlinear dynamics can be efficiently block encoded and solved within the QSVT framework. These results highlight both the potential and limitations of current methods, underscoring the need for efficient estimation of minimum singular value, depth-reduction techniques, and benchmarks against classical reachability. This pathway lays a foundation for advancing quantum linear system methods toward large-scale applications.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-10-08

• Fields of study

  Physics, Computer Science

• Identifiers
  arXiv 2510.06837
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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