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 the modular design, where pairs of mechanical modes act as building blocks with only local, tunable interactions between mechanical and cavity modes. The scheme is scalable, requiring just $4N$ driving tones for $N$ mechanical resonators. We characterize the resulting cluster states by evaluating the nullifiers of the CV modes. We also study the effects of dissipation, showing that strong squeezing with large phonon occupations can degrade the generated cluster states under finite mechanical and optical losses. As a direct application, we demonstrate that distant mechanical modes can be entangled via local measurements.