Some years ago I thought I had found a way to build electrical logic gates through magnetoresistance using 19th-century materials science. The phenomenon in question is the altered resistance of ferromagnetic materials when magnetized, by about 5% depending on the orientation of the field, as demonstrated by Kelvin in 1856. I concluded that, by using a Wheatstone bridge, you could get large amplification and inversion from this effect. What I didn’t realize at the time was that, as my calculations later showed, the extremely high self-inductance of the ferromagnetic conductors would limit it to sub-hertz speeds.
A Hall-effect alternative might solve the problem. The Hall effect, demonstrated by Edwin Hall in 1879 in gold leaf, produces a few millivolts of voltage from one side to the other of a conducting ribbon with a magnetic field at right angles to it. The voltage is proportional to the current through the ribbon and the magnetic field and inversely proportional to the thickness of the ribbon and the charge carrier density.
The energy of whatever current is driven by the Hall-effect voltage, importantly, does not come from the applied magnetic field; it comes from the sensing current, which does not diminish that field. In theory, this means that you could use that voltage to drive another coil controlling another set of Hall-effect ribbons. In the absence of better means of amplification, you could drive the sensing current through many parallel layers of gold leaf with insulators between them from many isolated voltage sources, such as separate windings of a transformer, and put their Hall voltages in series, thus controlling an arbitrarily large amount of energy with an arbitrarily small magnetic field. This series Hall voltage could then drive a load whose resistance was not too high compared to the (ordinary, not Hall) resistance of the sensing ribbons. (Although, wait, aren’t those sensing ribbons in series? Is that in fact a limit on the amplification you can achieve without superconducting ribbons?)
Such a device could, in theory, operate even at high frequencies. In practice, though, I think it might require an unreasonably large amount of apparatus for even a single logic gate.