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arXiv · 2306.13262

Reliable computation by large-alphabet formulas in the presence of noise

Andrew K. Tan·Matthew Ho·Isaac L. Chuang
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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 $\varepsilon$ is strictly larger than that for Boolean computation, and is possible as long as signals remain distinguishable, i.e. $\epsilon < (q - 1) / q$, in the limit of large fan-in $k \rightarrow \infty$. We also determine the point at which generalized majority gates with bounded fan-in fail, and show in particular that reliable computation is possible for $\epsilon < (q - 1) / (q (q + 1))$ in the case of $q$ prime and fan-in $k = 3$. 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 $\varepsilon < 1/6$ by using the additional alphabet element for error signaling.

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Andrew K. Tan, Matthew Ho, Isaac L. Chuang. 2023-06-23. Reliable computation by large-alphabet formulas in the presence of noise. https://doi.org/10.1109/tit.2024.3486278

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