Imagine you have a noise-free NOT gate, and an AND gate with the usual truth table
00 0
01 0
10 0 (*)
11 1
but such that the case (*) is wrong 1/3 of the time, i.e. it gives 1 with probability 1/3, and 0 with probability 2/3.
Is this gate family universal, in the sense that one can write a logic formula with these gates only, and that the probability of obtaining the right outcome is >1/2? (You also have noise-free SWAP and FANOUT, in case that helps).
Thanks!
Let $\land_p$ be a gate with error $p$ only when the inputs are $1$ and $0$. What can we say about $$ (x \land_p y) \land_p (x \land_p y)? $$ If $x=y=1$ then we always get $1$. If $x = 0$ then we always get $0$. When $x = 1$ and $y = 0$, we get the wrong answer $1$ with probability $$ p \cdot p + p \cdot (1-p) \cdot p = p^2(2-p). $$ Call that function $f(p)$. We conclude that this construction results in an $\land_{f(p)}$ gate.
The function $f(p)$ is monotone increasing over $[0,1]$, and satisfies $f(p) \leq 2p^2 = (2p)^2/2$. Therefore $f(f(p)) = (2f(p))^2/2 = (2p)^4/2$. More generally, $f^{(t)}(p) = (2p)^{2^t}/2$. Consequently, if we apply this construction recursively $O(\log\log(1/\epsilon))$ times to your $\land_{1/3}$ gate, we get an $\land_q$ gate with error $q \leq \epsilon$. This requires a gadget of size $2^{O(\log\log(1/\epsilon))} = \operatorname{polylog}(1/\epsilon)$.
To handle a circuit of size $S$, you need to choose $\epsilon = 1/(2S)$, which results in a blowup of $\operatorname{polylog}(S)$.
