We present an experimental procedure, based on Meissner effect levitation of neodymium ferromagnets, as a method of measuring the gravitational interactions between milligram masses. The scheme consists of two superconducting lead traps, with a magnet levitating in each trap. The levitating magnets behave as harmonic oscillators and, by carefully driving the motion of one magnet on resonance with the other, we find that it should easily be possible to measure the gravitational field produced by a 4 mg sphere, with the gravitational attraction from masses as small as $30\mu \mathrm{g}$ predicted to be measurable within a realistic measurement time frame. We apply this acceleration sensitivity to one concrete example and show the abilities of testing models of modified Newtonian dynamics.

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• Experimental studies of gravity
• Gravitation
• Quantum harmonic oscillator
• SQUID