  Recently, there has been considerable interest in solving optimization
problems by mapping these onto a binary representation, sparked mostly by the
use of quantum annealing machines. Such binary representation is reminiscent of
a discrete physical two-state system, such as the Ising model. As such,
physics-inspired techniques -- commonly used in fundamental physics studies --
are ideally suited to solve optimization problems in a binary format. While
binary representations can be often found for paradigmatic optimization
problems, these typically result in k-local higher-order unconstrained binary
optimization cost functions. In this work, we discuss the effects of locality
reduction needed for the majority of the currently available quantum and
quantum-inspired solvers that can only accommodate 2-local (quadratic) cost
functions. General locality reduction approaches require the introduction of
ancillary variables which cause an overhead over the native problem. Using a
parallel tempering Monte Carlo solver on Microsoft Azure Quantum, as well as
k-local binary problems with planted solutions, we show that post reduction to
a corresponding 2-local representation the problems become considerably harder
to solve. We further quantify the increase in computational hardness introduced
by the reduction algorithm by measuring the variation of number of variables,
statistics of the coefficient values, and the population annealing entropic
family size. Our results demonstrate the importance of avoiding locality
reduction when solving optimization problems.
