This paper presents a significant advancement in the field of circuit quantum acoustodynamics (cQAD), focusing on the engineering of multimode interactions between mechanical modes. The authors demonstrate a tunable beamsplitter-type interaction between multiple mechanical modes of a high-overtone bulk acoustic-wave resonator (HBAR) mediated by a parametrically driven superconducting transmon qubit. This interaction allows for the coupling of pairs or triplets of phononic modes and enables the realization of quantum gates, such as the iSWAP gate, which is essential for quantum computing. The study also explores the Hong–Ou–Mandel effect between phonons, a phenomenon that has been observed in photonic systems but is now demonstrated in the phononic domain. The results lay the foundation for using phononic systems as quantum memories and platforms for quantum simulations, highlighting the potential of cQAD devices in realizing quantum random-access memory and fault-tolerant quantum computing architectures. The experimental setup and theoretical framework are detailed, providing a comprehensive understanding of the system's behavior and its applications in quantum information processing.This paper presents a significant advancement in the field of circuit quantum acoustodynamics (cQAD), focusing on the engineering of multimode interactions between mechanical modes. The authors demonstrate a tunable beamsplitter-type interaction between multiple mechanical modes of a high-overtone bulk acoustic-wave resonator (HBAR) mediated by a parametrically driven superconducting transmon qubit. This interaction allows for the coupling of pairs or triplets of phononic modes and enables the realization of quantum gates, such as the iSWAP gate, which is essential for quantum computing. The study also explores the Hong–Ou–Mandel effect between phonons, a phenomenon that has been observed in photonic systems but is now demonstrated in the phononic domain. The results lay the foundation for using phononic systems as quantum memories and platforms for quantum simulations, highlighting the potential of cQAD devices in realizing quantum random-access memory and fault-tolerant quantum computing architectures. The experimental setup and theoretical framework are detailed, providing a comprehensive understanding of the system's behavior and its applications in quantum information processing.