  In this note we present a multigrid preconditioning method for solving
quadratic optimization problems constrained by a fractional diffusion equation.
Multigrid methods within the all-at-once approach to solve the first
order-order optimality Karush-Kuhn-Tucker (KKT) systems are widely popular, but
their development have relied on the underlying systems being sparse. On the
other hand, for most discretizations, the matrix representation of fractional
operators is expected to be dense. We develop a preconditioning strategy for
our problem based on a reduced approach, namely we eliminate the state
constraint using the control-to-state map. Our multigrid preconditioning
approach shows a dramatic reduction in the number of CG iterations. We assess
the quality of preconditioner in terms of the spectral distance. Finally, we
provide a partial theoretical analysis for this preconditioner, and we
formulate a conjecture which is clearly supported by our numerical experiments.
