Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS
Abstract
This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method.By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. The Robin boundary elimination is encoded by newly constructed endpoint-projector circuits, and all physical and boundary-induced inhomogeneous terms are incorporated through an outer linear combination of unitaries (LCU). The error analysis adopts an LCU-weighted form, preserving the decay of the LCHS kernel in the product-formula bounds. Complementing this analysis, a comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases.The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.
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BibTeX
@misc{chen2026quantum,
title = {{Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS}},
author = {Leyu Chen and Tiegang Liu and Liang Xu and Kun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.17542},
eprint = {2605.17542},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17542},
}
RIS
TY - UNPB
TI - Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS
AU - Leyu Chen
AU - Tiegang Liu
AU - Liang Xu
AU - Kun Wang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2605.17542
UR - https://arxiv.org/abs/2605.17542
AB - This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method.By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. The Robin boundary elimination is encoded by newly constructed endpoint-projector circuits, and all physical and boundary-induced inhomogeneous terms are incorporated through an outer linear combination of unitaries (LCU). The error analysis adopts an LCU-weighted form, preserving the decay of the LCHS kernel in the product-formula bounds. Complementing this analysis, a comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases.The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.
ER -