  By exploiting the link between time-independent Hamiltonians and
thermalisation, heuristic predictions on the performance of continuous-time
quantum walks for MAX-CUT are made. The resulting predictions depend on the
number of triangles in the underlying MAX-CUT graph. We extend these results to
the time-dependent setting with multi-stage quantum walks and Floquet systems.
The approach followed here provides a novel way of understanding the role of
unitary dynamics in tackling combinatorial optimisation problems with
continuous-time quantum algorithms.
