  The implementation of quantum entangling gates between qubits is essential to
achieve scalable quantum computation. Here, we propose a robust scheme to
realize an entangling gate for distant solid-state spins via a mechanical
oscillator in its thermal equilibrium state. By appropriate Hamiltonian
engineering and usage of a protected subspace, we show that the proposed scheme
is able to significantly reduce the thermal effect of the mechanical oscillator
on the spins. In particular, we demonstrate that a high entangling gate
fidelity can be achieved even for a relatively high thermal occupation. Our
scheme can thus relax the requirement for ground-state cooling of the
mechanical oscillator, and may find applications in scalable quantum
information processing in hybrid solid-state architectures.
