  Designing and optimizing cost functions and energy landscapes is a problem
encountered in many fields of science and engineering. These landscapes and
cost functions can be embedded and annealed in experimentally controllable spin
Hamiltonians. Using an approach based on group theory and symmetries, we
examine the embedding of Boolean logic gates into the ground state subspace of
such spin systems. We describe parameterized families of diagonal Hamiltonians
and symmetry operations which preserve the ground state subspace encoding the
truth tables of Boolean formulas. The ground state embeddings of adder circuits
are used to illustrate how gates are combined and simplified using symmetry.
Our work is relevant for experimental demonstrations of ground state embeddings
found in both classical optimization as well as adiabatic quantum optimization.
