Cluster States Generation with a Quantum Metasurface

We investigate the implementation of photonic cluster state generation protocols using quantum metasurfaces comprising sub-wavelength atomic arrays which enables quantum-controlled reflectivity. These cluster states are generated using fundamental quantum logic gates and enable wide-ranging applications in quantum computation and communication. In the past few years, certain protocols have been developed, but their physical realizations induces natural losses on the system mainly originated from coupling the photonic structures, setting a limit on the efficiency and maximal qubit number. In this paper, we examine a physical implementation of two specific protocols for generating distinct cluster states: a two-dimensional cluster state and a tree cluster state. Our approach leverages the unique properties of a quantum metasurface and its free space settings to implement two-qubit quantum-logic gates, namely CNOT, CZ, and E gates, with practical fidelities exceeding 0.9, and potential speed-up due to parallelism. In addition, we analyze these protocols fidelities for practical conditions of potential implementation experiments, such as thermal fluctuation of trapped atoms.