Pulsed generation of continuous-variable cluster states in a phononic quantum network

Cluster states are multipartite entangled states that are maximally connected and resilient to decoherence, making them valuable resources for quantum information processing. Continuous-variable (CV) cluster states have been extensively investigated for such applications. Here we present a pulsed protocol for generating CV cluster states in a phononic quantum network composed of phonon waveguides, mechanical resonators, and optical cavities. A key feature of this architecture is its modular design, where pairs of mechanical modes serve as building blocks with only local, tunable interactions between mechanical and cavity modes. We characterize the generated cluster states through the average nullifier of the CV modes. Our numerical results show that high-quality CV cluster states can be generated with moderate squeezing parameters, whereas strong squeezing and the resulting large phonon occupations can degrade the cluster states in the presence of finite dissipation. We further show that, under moderate squeezing and dissipation, the average nullifier can decrease with increasing system size $N$, demonstrating the scalability of the proposed scheme. As a direct application, we demonstrate that distant mechanical modes can be entangled through local measurements.