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Quantum HHL Algorithm for Solving 1D Linear Advection Equation Using Explicit and Implicit Finite Difference Schemes — arXiv:2611.01167

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.01167
License (as archived): CC0-1.0
Archived: 2026-11-03T11:49:48.455Z
SHA-256 of the archived text: 01cf977b8d31c5fb99de553de91fa6a27bf67795894fb9e366e719e3be0bd125
Stable URL of this page: https://metraipolis.com/documento/dc_28b8efbf63be

How to cite

Quantum HHL Algorithm for Solving 1D Linear Advection Equation Using Explicit and Implicit Finite Difference Schemes — arXiv:2611.01167 — arXiv (en), https://arxiv.org/abs/2611.01167 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_28b8efbf63be (Metraipolis archive dc_28b8efbf63be, sha256 01cf977b8d31c5fb99de553de91fa6a27bf67795894fb9e366e719e3be0bd125)

BibTeX

@misc{dc28b8efbf63be,
  title = {Quantum HHL Algorithm for Solving 1D Linear Advection Equation Using Explicit and Implicit Finite Difference Schemes — arXiv:2611.01167},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.01167},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_28b8efbf63be; sha256 01cf977b8d31c5fb99de553de91fa6a27bf67795894fb9e366e719e3be0bd125},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.01167. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This study develops and implements a Harrow-Hassidim-Lloyd (HHL) quantum algorithm for solving a one-dimensional linear advection equation with periodic boundary conditions. The governing equation is first discretized using a classical finite difference (FD) scheme to obtain a linear system suitable for quantum solution. We present a newly designed, circuit-level implementation of HHL assembled from Qiskit circuit components and exploit the circulant structure of the periodic discretization matrices through Fourier-based matrix evolution. The HHL algorithm is benchmarked with three explicit schemes and one implicit scheme. The implicit scheme requires a nontrivial linear solve arising from the discretized PDE on progressively finer grids, whereas the explicit schemes provide different input fields for testing state preparation and reconstruction with identity coefficient matrices. HHL solution states are validated against direct classical solves in noiseless simulation. For hardware execution, the periodic matrix structure permits a Fourier implementation that applies reciprocal eigenvalues without phase estimation. Results on the IBM Rensselaer Nighthawk processor are compared at a shared final time. Noiseless reconstructed fields approach the transported profile on resolved grids, while substantial hardware errors remain for both explicit and implicit schemes. Crank-Nicolson remains exceptionally well-conditioned and utilizes similar circuit re sources to the explicit controls, proving that hardware degradation in our tests was not driven by poor matrix conditioning. Ultimately, this study shows new means for HHL algorithms to study quantum CFD problems involving advection equations with implicit time integrations, while demonstrating how state preparation and measurement affect the practical performance of quantum solvers on current devices.

Tema degli esploratori: fisica e astronomia (elenco arXiv physics.comp-ph, nuovi lavori del giorno).
Categorie arXiv: Computational Physics (physics.comp-ph) ; Fluid Dynamics (physics.flu-dyn).
Pagina del lavoro: https://arxiv.org/abs/2611.01167 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T11:10:22.273Z dalla pagina https://arxiv.org/list/physics.comp-ph/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).