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High-dimensional fractional reaction-diffusion equations have numerousapplications in the fields of biology, chemistry, and physics, and exhibit arange of rich phenomena. While classical algorithms have an exponentialcomplexity in the spatial dimension, a quantum computer can produce a quantumstate that encodes the solution with only polynomial complexity, provided thatsuitable input access is available. In this work, we investigate efficientquantum algorithms for linear and nonlinear fractional reaction-diffusionequations with periodic boundary conditions. For linear equations, we analyzeand compare the complexity of various methods, including the second-orderTrotter formula, time-marching method, and truncated Dyson series method. Wealso present a novel algorithm that combines the linear combination ofHamiltonian 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-encodingversion that is appropriate for the dense matrices that arise from the spatialdiscretization of fractional reaction-diffusion equations.

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