  Classical models with complex energy landscapes represent a perspective
avenue for the near-term application of quantum simulators. Until now, many
theoretical works studied the performance of quantum algorithms for models with
a unique ground state. However, when the classical problem is in a so-called
clustering phase, the ground state manifold is highly degenerate. As an
example, we consider a 3-XORSAT model defined on simple hypergraphs. The
degeneracy of classical ground state manifold translates into the emergence of
an extensive number of $Z_2$ symmetries, which remain intact even in the
presence of a quantum transverse magnetic field. We establish a general duality
approach that restricts the quantum problem to a given sector of conserved
$Z_2$ charges and use it to study how the outcome of the quantum adiabatic
algorithm depends on the hypergraph geometry. We show that the tree hypergraph
which corresponds to a classically solvable instance of the 3-XORSAT problem
features a constant gap, whereas the closed hypergraph encounters a
second-order phase transition with a gap vanishing as a power-law in the
problem size. The duality developed in this work provides a practical tool for
studies of quantum models with classically degenerate energy manifold and
reveals potential connections between glasses and gauge theories.
