Deterministic phonon phase gates with number-resolving phonon detection

We implement frequency-dependent scattering of itinerant phonon qubits from superconducting transmon qubits that yields phonon phase control in an acoustic Mach-Zehnder interferometer. The experiments demonstrate phonon indistinguishability and interference visibility exceeding 99%, more than an order-of-magnitude improvement over our previous demonstration [H. Qiao et al., Science (2023)]. Additionally, we propose and implement a multi-phonon detection scheme that enables coherent conversion between itinerant two-phonon Fock states and transmon qutrit states, partially transferring the Hong-Ou-Mandel two-phonon entangled output state into an entangled state of two qutrit transmons. The performance of this phononic qubit hardware holds promise for the future development of a linear mechanical quantum computing architecture.