  We show that mode-mixing quantum gates can be produced by non-uniform
relativistic acceleration. Periodic motion in cavities exhibits a series of
resonant conditions producing entangling quantum gates between different
frequency modes. The resonant condition associated with particle creation is
the main feature of the dynamical Casimir effect which has been recently
demonstrated in superconducting circuits. We show that a second resonance,
which has attracted less attention since it implies negligible particle
production, produces a beam splitting quantum gate leading to a resonant
enhancement of entanglement which can be used as the first evidence of
acceleration effects in mechanical oscillators. We propose a desktop experiment
where the frequencies associated with this second resonance can be produced
mechanically.
