  We present a theoretical framework for the analysis of amplitude transfer in
Quantum Variational Algorithms (QVAs) for combinatorial optimisation with
mixing unitaries defined by vertex-transitive graphs, based on their
continuous-time quantum walk (CTQW) representation and the theory of graph
automorphism groups. This framework leads to a heuristic for designing
efficient problem-specific QVAs. Using this heuristic, we develop novel
algorithms for unconstrained and constrained optimisation. We outline their
implementation with polynomial gate complexity and simulate their application
to the parallel machine scheduling and portfolio rebalancing combinatorial
optimisation problems, showing significantly improved convergence over
preexisting QVAs. Based on our analysis, we derive metrics for evaluating the
suitability of graph structures for specific problem instances, and for
establishing bounds on the convergence supported by different graph structures.
For mixing unitaries characterised by a CTQW over a Hamming graph on $m$-tuples
of length $n$, our results indicate that the amplification upper bound
increases with problem size like $\mathcal{O}(e^{n \log m})$.
