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Quantum algorithms for linear and non-linear fractional reaction-diffusion equations

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Dong and Trivisa developed quantum algorithms that solve high-dimensional fractional reaction-diffusion equations with polynomial complexity, outperforming classical methods that scale exponentially with spatial dimensions. The study introduces a novel algorithm combining linear Hamiltonian simulation with interaction picture formalism, achieving optimal spatial scaling for linear equations with periodic boundary conditions. For nonlinear equations, the team adapted Carleman linearization into a block-encoding framework, enabling efficient handling of dense matrices from spatial discretization. The research compares existing methods like second-order Trotter, time-marching, and truncated Dyson series, identifying trade-offs in accuracy and computational cost for linear systems. Published in January 2026, the work advances quantum simulation of complex physical, chemical, and biological systems where traditional numerical methods fail due to high dimensionality.

$Quantum algorithms for linear and non-linear fractional reaction-diffusion equations$

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AbstractHigh-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the spatial dimension, a quantum computer can produce a quantum state that encodes the solution with only polynomial complexity, provided that suitable input access is available. In this work, we investigate efficient quantum algorithms for linear and nonlinear fractional reaction-diffusion equations with periodic boundary conditions. For linear equations, we analyze and compare the complexity of various methods, including the second-order Trotter formula, time-marching method, and truncated Dyson series method. We also present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism, resulting in optimal scaling in the spatial dimension. For nonlinear equations, we employ the Carleman linearization method and propose a block-encoding version that is appropriate for the dense matrices that arise from the spatial discretization of fractional reaction-diffusion equations.► BibTeX data@article{An2026quantumalgorithms, doi = {10.22331/q-2026-01-19-1969}, url = {https://doi.org/10.22331/q-2026-01-19-1969}, title = {Quantum algorithms for linear and non-linear fractional reaction-diffusion equations}, author = {An, Dong and Trivisa, Konstantina}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1969}, month = jan, year = {2026} }► References [1] Nicholas F. Britton. ``Reaction–diffusion equations and their applications to biology''. Academic Press. (1986). url: https://www.cabidigitallibrary.org/doi/full/10.5555/19870540782. https://www.cabidigitallibrary.org/doi/full/10.5555/19870540782 [2] Robert Stephen Cantrell and Chris Cosner. ``Spatial ecology via reaction–diffusion equations''. Wiley. (2003). https://doi.org/10.1002/0470871296 [3] Peter Grindrod. ``The theory and applications of reaction-diffusion equations: Patterns and waves''. Clarendon Press. (1996). [4] Franz Rothe. ``Global solutions of reaction-diffusion systems''. Springer-Verlag, Berlin. (1984). https://doi.org/10.1007/BFb0099278 [5] Joel Smoller. ``Shock waves and reaction—diffusion equations''.

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Quantum algorithms for linear and non-linear fractional reaction-diffusion equations

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