  Localized electronic and nuclear spin qubits in the solid state constitute a
promising platform for storage and manipulation of quantum information, even at
room temperature. However, the development of scalable systems requires the
ability to entangle distant spins, which remains a challenge today. We propose
and analyze an efficient, heralded scheme that employs a parity measurement in
a decoherence free subspace to enable fast and robust entanglement generation
between distant spin qubits mediated by a hot mechanical oscillator. We find
that high-fidelity entanglement at cryogenic and even ambient temperatures is
feasible with realistic parameters, and show that the entangled pair can be
subsequently leveraged for deterministic controlled-NOT operations between
nuclear spins. Our results open the door for novel quantum processing
architectures for a wide variety of solid-state spin qubits.
