Confirming that efficiently finding a small circuit (you don’t actually need further restrictions than size) based on its values on a fixed collection of test inputs is known to imply NP⊆BPP --- see this paper.
Confirming that efficiently finding a small circuit (you don’t actually need further restrictions than size) based on its values on a fixed collection of test inputs is known to imply NP⊆BPP --- see this paper.