  In this work, we study the iteration complexity of gradient methods
minimizing the class of uniformly convex regularized quadratic functions. We
prove lower bounds on the functional residual of the form
$\Omega(N^{-2p/(p-2)})$, where $p > 2$ is the power of the regularization term,
and $N$ is the number of calls to a first-order oracle. A special case of our
problem class is $p=3$, which is the minimization of cubically regularized
convex quadratic functions. It naturally appears as a subproblem at each
iteration of the cubic Newton method. The corresponding lower bound for $p = 3$
becomes $\Omega(N^{-6})$. Our result matches the best-known upper bounds on
this problem class, rendering a sharp analysis of the minimization of uniformly
convex regularized quadratic functions. We also establish new lower bounds on
minimizing the gradient norm within our framework.
