  We study a variant of the sparse PCA (principal component analysis) problem
in the "hard" regime, where the inference task is possible yet no
polynomial-time algorithm is known to exist. Prior work, based on the
low-degree likelihood ratio, has conjectured a precise expression for the best
possible (sub-exponential) runtime throughout the hard regime. Following
instead a statistical physics inspired point of view, we show bounds on the
depth of free energy wells for various Gibbs measures naturally associated to
the problem. These free energy wells imply hitting time lower bounds that
corroborate the low-degree conjecture: we show that a class of natural MCMC
(Markov chain Monte Carlo) methods (with worst-case initialization) cannot
solve sparse PCA with less than the conjectured runtime. These lower bounds
apply to a wide range of values for two tuning parameters: temperature and
sparsity misparametrization. Finally, we prove that the Overlap Gap Property
(OGP), a structural property that implies failure of certain local search
algorithms, holds in a significant part of the hard regime.
