We propose to measure Newton's gravitational constant G by using two
nanoparticles suspended in a trap. By having both particles evolve
without external driving, a mechanical Mach-Zehnder-type
interferometer is realized in which a breathing mode picks up a phase
proportional to G and the interaction time. We analyse the sensitivity
of the setup by evaluating the Quantum Cramér-Rao bound for several
input states, including coherent, squeezed, and thermal states. The
bound can be saturated periodically by homodyne measurements in the
original uncoupled modes. The effects of damping and finite
temperatures are evaluated. The proposal should allow a reduction of
the uncertainty with which G is known by more than an order of
magnitude compared to the current CODATA uncertainty under realistic
conditions, and to reach a regime of much smaller masses than
previously accessible.