Efficient preparation of high-dimensional hybrid entangled states in circuit quantum electrodynamics

High-dimensional quantum systems, known as qudits with dimension $d>2$, possess the capability to encode and process more information than traditional two-dimensional qubits. Entangled states serve as crucial resources for both quantum computation and quantum communication. On the basis of circuit quantum electrodynamics (QED) systems, we introduce an efficient scheme to deterministically generate high-dimensional multipartite entangled states of hybrid qutrits ($d=3$). Assuming a simple initial state has already been prepared with the use of local operations, our method requires only a single step to produce hybrid Greenberger-Horne-Zeilinger (GHZ) entangled states among circuit QED systems consisting of superconducting qutrits and a superconducting cavity. Remarkably, the number of steps and the operational time remain constant as the number of qutrits increases, highlighting the efficiency and scalability of our approach. Using numerical simulations for a three-dimensional hybrid GHZ state of seven qutrits, we demonstrate the feasibility of our scheme and study the effects of various parameters on the fidelity. The importance of this research is in directing the experimental creation of multipartite, high-dimensional, and hybrid GHZ states, which can then be used to enhance quantum communication protocols and computational processes.