  Our work is motivated by the challenges presented in preparing arrays of
atoms for use in quantum simulation. The recently-developed process of loading
atoms into traps results in approximately half of the traps being filled. To
consolidate the atoms so that they form a dense and regular arrangement, such
as all locations in a grid, atoms are rearranged using moving optical tweezers.
Time is of the essence, as the longer that the process takes and the more that
atoms are moved, the higher the chance that atoms will be lost in the process.
  Viewed as a problem on graphs, we wish to solve the problem of reconfiguring
one arrangement of tokens (representing atoms) to another using as few moves as
possible. Because the problem is NP-complete on general graphs as well as on
grids, we focus on the parameterized complexity for various parameters,
considering both undirected and directed graphs, and tokens with and without
labels. For unlabelled tokens, the problem is in FPT when parameterizing by the
number of tokens, the number of moves, or the number of moves plus the number
of vertices without tokens in either the source or target configuration, but
intractable when parameterizing by the difference between the number of moves
and the number of differences in the placement of tokens in the source and
target configurations. When labels are added to tokens, however, most of the
tractability results are replaced by hardness results.
