  Massive mechanical resonators operating at the quantum scale can enable a
large variety of applications in quantum technologies, as well as fundamental
tests of quantum theory. Of crucial importance in that direction, is both their
integrability into state-of-the-art quantum platforms as well as the ability to
prepare them in generic quantum states using well-controlled high-fidelity
operations. Here, we propose a scheme for controlling a radio-frequency
mechanical resonator at the quantum scale using two superconducting transmon
qubits that can be integrated on the same chip. Specifically, we consider two
qubits coupled via a capacitor in parallel to a superconducting quantum
interference device (SQUID), which has a suspended mechanical beam embedded in
one of its arms. Following a theoretical analysis of the quantum system, we
find that this configuration, in combination with an in-plane magnetic field,
can give rise to a tuneable three-body interaction in the single-photon
strong-coupling regime, while enabling suppression of the stray qubit-qubit
coupling. Using state-of-the-art parameters and qubit operations at
single-excitation levels, we numerically demonstrate the possibility of
ground-state cooling as well as high-fidelity preparation of mechanical quantum
states and qubit-phonon entanglement, i.e. states having negative Wigner
functions and obeying non-classical correlations. Our work significantly
extends the quantum control toolbox of radio-frequency mechanical resonators
and may serve as a promising architecture for integrating such mechanical
elements with transmon-based quantum processors.
