  A promising approach to achieve computational supremacy over the classical
von Neumann architecture explores classical and quantum hardware as Ising
machines. The minimisation of the Ising Hamiltonian is known to be NP-hard
problem for certain interaction matrix classes, yet not all problem instances
are equivalently hard to optimise. We propose to identify computationally
simple instances with an `optimisation simplicity criterion'. Such optimisation
simplicity can be found for a wide range of models from spin glasses to
k-regular maximum cut problems. Many optical, photonic, and electronic systems
are neuromorphic architectures that can naturally operate to optimise problems
satisfying this criterion and, therefore, such problems are often chosen to
illustrate the computational advantages of new Ising machines. We further probe
an intermediate complexity for sparse and dense models by analysing circulant
coupling matrices, that can be `rewired' to introduce greater complexity. A
compelling approach for distinguishing easy and hard instances within the same
NP-hard class of problems can be a starting point in developing a standardised
procedure for the performance evaluation of emerging physical simulators and
physics-inspired algorithms.
