  High-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.
