  We propose an annealing scheme usable on modern Ising machines for crystal
structures prediction (CSP) by taking into account the general n-body atomic
interactions, and in particular three-body interactions which are necessary to
simulate covalent bonds. The crystal structure is represented by discretizing a
unit cell and placing binary variables which express the existence or
non-existence of an atom on every grid point. The resulting quadratic
unconstrained binary optimization (QUBO) or higher-order unconstrained binary
optimization (HUBO) problems implement the CSP problem and is solved using
simulated and quantum annealing. Using the example of Lennard-Jones clusters we
show that it is not necessary to include the target atom number in the
formulation allowing for simultaneous optimization of both the particle density
and the configuration and argue that this is advantageous for use on annealing
machines as it reduces the total amount of interactions. We further provide a
scheme that allows for reduction of higher-order interaction terms that is
inspired by the underlying physics. We show for a covalently bonded monolayer
MoS2 crystal that we can simultaneously optimize for the particle density as
well as the crystal structure using simulated annealing. We also show that we
reproduce ground states of the interatomic potential with high probability that
are not represented on the initial discretization of the unit cell.
