  We present methods for constructing any target coupling graph using limited
global controls in an Ising-like quantum spin system. Our approach is motivated
by implementing the quantum approximate optimization algorithm (QAOA) on
trapped ion quantum hardware to find approximate solutions to Max-Cut. We
present a mathematical description of the problem and provide approximately
optimal algorithmic constructions that generate arbitrary unweighted coupling
graphs with $n$ nodes in $O(n)$ global entangling operations and weighted
graphs with $m$ edges in $O(m)$ operations. These upper bounds are not tight in
general, and we formulate a mixed-integer program to solve the graph coupling
problem to optimality. We perform numeric experiments on small graphs with
$n\le8$ and show that optimal sequences, which use fewer operations, can be
found using mixed-integer programs. Noisy simulations of Max-Cut QAOA show that
our implementation is less susceptible to noise than the standard gate-based
compilation.
