Massively parallel AND gate using relays
Does such a relay exist at a nanometer scale?


No.

MEMS relays have been demonstrated in R&amp;D labs, but their size scale is (AFAIK) in the 10's of microns, not nanometers. Further, as far as I know, there is no such device available commercially.

Relays are generally made at millimeter scales in order to provide sufficient clearance and creepage distances to provide the galvanic isolation that most relay applications require.

Transistors can be made at nanometer scale and can provide essentially the functionality you're looking for. Google pass-transistor logic for further information. Rather than chaining 1000 pass-gate AND gates, you might need to include buffering of some kind every few (10? 50?) gates to ensure good noise margins.

But this is almost certainly not the most efficient way (in terms of area, power, and speed) to implement a 1000-input AND gate.