  Physics-based Ising machines (IM) have risen to the challenge of solving hard
combinatorial optimization problems with higher speed and better energy
efficiency. Generally, such dedicated systems employ local search heuristics to
traverse energy landscapes in searching for optimal solutions. Extending
landscape geometry visualization tools, disconnectivity graphs, we quantify and
address some of the major challenges met by IMs in the field of combinatorial
optimization. Using efficient sampling methods, we visually capture landscapes
of problems having diverse structure and hardness and featuring strong
degeneracies, which act as entropic barriers for IMs. Furthermore, we
investigate energy barriers, local minima, and configuration space clustering
effects caused by locality reduction methods when embedding combinatorial
problems to the Ising hardware. For this purpose, we sample disconnectivity
graphs of PUBO energy landscapes and their different QUBO mappings accounting
for both local minima and saddle regions. We demonstrate that QUBO energy
landscape properties lead to the subpar performance of quadratic IMs and
suggest directions for their improvement.
