# Rearrangement of Single Atoms by Solving Assignment Problems via Convolutional Neural Network
## 1 Introduction  
Over the past decade, substantial progress has been made in the neutral atom platform, which is one of the platforms used to construct a quantum system whose applications exist in many quantum branches, for instance, quantum information processing in [PERSON] [1], quantum metrology in [PERSON] [2], and quantum simulation in [PERSON] [3]. Arrays of cold neutral atoms are necessary to gain control of single qubits. After trapping atoms, rearrangement of neutral atoms is therefore crucial, and there have been several recent impressive results. Most research on the problems of rearranging atoms from any initial array to any target array can be categorized into physical methods and mathematical algorithms. One frequently used physical technique for atom rearrangement is the optical tweezer. In the most recent work of 2024, by [PERSON] et al. [4], after the trapping of single atoms up to a 2000 site, atom-by-atom rearrangement is demonstrated with very few defects up to an 828-atom target array using this optical tweezer technique. While the task of trapping single atoms is crucial [5; 6; 7], creating defect-free atomic arrays is just as important as they have been demonstrated to be a platform for quantum simulations and quantum computations. There are methods in physics that focus on physically moving the atoms to defect-free target arrays [8] and more theoretical methods that try to find the best way to move the atoms under some specified conditions. The development of theoretical solutions will benefit those who try to physically move the atoms to the target arrays. The following are instances of work using mathematical algorithms to choose the path of atom. In 2016, [PERSON] et al. [9] proposed and demonstrated 3D rearrangements of single atoms. They considered 3D atom arrays as layers of 2D images and then presented a method for moving atoms in 2D into the right positions. The same year, [PERSON] et al. [10] relocated atoms to user-defined target arrays using a developed shortest-move heuristic. From the heuristic, the number of moves obtained is approximately \(\frac{N}{2}\), where \(N\) is the number of atoms in the initial array. However, this heuristic does not necessarily minimize the total distance of atoms' movement. Moreover, to prevent collision of atoms, their method of atom moving depends on nearest-neighbor distance in the target array. Later that year, [PERSON] et al. [11] showed moving atoms in 1D, which is a defect-free condition. Using this method, atoms can be moved simultaneously in one single movement and clusters of atoms can be generated in any form. While atoms are being moved, some of them are lost from the target array. These empty traps are then filled using the reservoir atoms. In 2017, [PERSON] et al. [12] constructed an optimization problem from an atom rearrangement problem. The Hungarian matching algorithm is used to plan collision-free paths through vacancy filling. They were concerned about the collision of atoms in multiple ways. In the end, they concluded that, if the cost function is appropriately defined, then all atom collisions can be avoided. In 2019, [PERSON] et al. [13] rearranged atoms by applying a shortest-move heuristic. In order to reduce the number of moves, the heuristic attempts to choose the paths with the fewest obstructing atoms to fill all vacancies in a pre-defined target array. The heuristic does not necessarily minimize the total distance of moving atoms but instead focuses on time efficiency. Later in the year of 2019, [PERSON] et al. [14] presented 3D atom relocation. First, they choose atoms that they want to move to the target array and then rearrange these atoms in a single movement by switching rows and columns in 2D. This work does not focus on either the number of moves or the total distance of moving atoms. The central point of this work is the rearrangement of neutral atoms using mathematical algorithms under the following conditions.