  We report the design of a diamond-based honeycomb phononic network, in which
a mechanical resonator couples to three distinct phononic crystal waveguides.
This two-dimensional (2D) phononic network extends an earlier study on
one-dimensional (1D) phononic networks with closed mechanical subsystems. With
a special design for the phononic band structures of the waveguides, any two
neighboring resonators in the 2D network and the waveguide between them can
form a closed mechanical subsystem, which enables nearest neighbor coupling and
at the same time circumvents the scaling problems inherent in typical large
mechanical systems. In addition, the 2D network can be attached to a square
phononic crystal lattice and be protected by the large band gap of the phononic
crystal shield. Honeycomb phononic networks of spin qubits with nearest
neighbor coupling can serve as an experimental platform for quantum computing
and especially topological quantum error corrections.
