  Programmable arrays of optical traps enable the assembly of configurations of
single atoms to perform controlled experiments on quantum many-body systems.
Finding the sequence of control operations to transform an arbitrary
configuration of atoms into a predetermined one requires solving an atom
reconfiguration problem quickly and efficiently. A typical approach to solve
atom reconfiguration problems is to use an assignment algorithm to determine
which atoms to move to which traps. This approach results in control protocols
that exactly minimize the number of displacement operations; however, this
approach does not optimize for the number of displaced atoms nor the number of
times each atom is displaced, resulting in unnecessary control operations that
increase the execution time and failure rate of the control protocol. In this
work, we propose the assignment-rerouting-ordering (aro) algorithm to improve
the performance of assignment-based algorithms in solving atom reconfiguration
problems. The aro algorithm uses an assignment subroutine to minimize the total
distance traveled by all atoms, a rerouting subroutine to reduce the number of
displaced atoms, and an ordering subroutine to guarantee that each atom is
displaced at most once. The ordering subroutine relies on the existence of a
partial ordering of moves that can be obtained using a polynomial-time
algorithm that we introduce within the formal framework of graph theory. We
numerically quantify the performance of the aro algorithm in the presence and
in the absence of loss, and show that it outperforms the exact, approximation,
and heuristic algorithms that we use as benchmarks. Our results are useful for
assembling large configurations of atoms with high success probability and fast
preparation time, as well as for designing and benchmarking novel atom
reconfiguration algorithms.
