On reliable computation over larger alphabets

Abstract

We present two new positive results for reliable computation using formulas over physical alphabets of size $q > 2$. First, we show that for logical alphabets of size $\ell = q$ the threshold for denoising using gates subject to $q$-ary symmetric noise with error probability $\epsilon$ is strictly larger that possible for Boolean computation and we demonstrate a clone of $q$-ary functions that can be reliably computed up to this threshold. Secondly, we provide an example where $\ell < q$, showing that reliable Boolean computation can be performed using $2$-input ternary logic gates subject to symmetric ternary noise of strength $\epsilon < 1/6$ by using the additional alphabet element for error signalling.Comment: 14 pages, 2 figure