I want to be sure, with arbitrary precision (let say, for example, with 10^-100 certainty), that I can check the following statement (in binary):
x = 49
That is, I want to check that a number that is (correctly) coded in a binary electrical signal is the same as a second number (also correctly coded in binary).
I'm a novice at electrical circuits, but I've worked out that I can perform this operation using 6 exNors and 5 And gates. I'm assuming there is some small uncertainty in these logic gates, and that there's a super small chance that these gates can misfire (maybe even so small it's associated with the quantum electrical noise).
If I wanted to make this system have arbitary precision, what could I do to prevent these errors from /ever/ occuring?
One option I thought of is to perform the same operation multiple times and AND-gate the results together. (I am perfectly fine losing "efficiency," that is, I don't mind if the circuit has "false negatives" where the number is correct but since there was a misfire in one of the circuits it does not return a 1.)
Is there a way that I can analyze the probabilities of the circuit being correct? And what types of electronics would be ideal for this type of system (my guess would be that higher voltages would have less fundamental noise, so they would be the best to construct the system.) It seems as though there is some work on simulating the propagation of these errors using some advance techniques like monte-carlo simulations and baysian analysis. But is it not possible to get a crude, order-of-magnitude estimate for the amount of uncertainty/error per gate?
*Moved information in comments to the question where it belongs*
The purpose of this device I think is outside the scope of electrical engineering, but it's sort of related to metrology and quantum mechanics. I agree it's a very unconventional thing to be concerned with. I am first trying to get some kind of grounding on the topic because of how unusual it is. Any implementation (or discussion of errors, no matter how small) would be helpful (not concerned with hardware, frequencies, as long as it's digital).
Maybe rephrasing the question. If I performed this comparison operation on a microcontroller over and over, how much time (or how many repetitions) would it take until it misfired and gave an incorrect comparison? (Minutes? Days? Millennia?) And does this error loosely scale linearly/polynomially with the number of gates? So, for example, if I have 600exNors now instead of 6, does my operation misfire 100 times faster? (implying the errors scale linearly)