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

TL;DR

The threshold for denoising using gates subject to q-ary symmetric noise with error probability is strictly larger than that for Boolean computation, and reliable computation is possible as long as signals remain distinguishable.

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. $ε< (q - 1) / q$, 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 $ε< (q - 1) / (q (q + 1))$ 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.

Figures (4)

• Figure 1: Plot of denoising threshold, $\beta$, using the maj-$[q, k]$ gate as a function of fan-in for different alphabet sizes. The plotted threshold values are achieved for symmetrically noisy inputs and are known to be tight for $q = 2$evans2003on-the-maximum. In the large-$k$ limit, denoising thresholds approach $(q-1)/q$.
• Figure 2: Diagram showing stable (solid lines) and unstable (dashed lines) fixed-points of the $\varepsilon$-noisy maj-$[3, 3]$ gate for symmetrically $a$-noisy inputs. Note that at the denoising lower-bound of $\varepsilon = 1 / 6$ (\ref{['lem:lower-bound-on-majqk-denoising-threshold']}) corresponds to a transcritical bifurcation, at which point the qualitative structure of the fixed-points changes, with the central $(2/3)$-noisy fixed-point becoming stable. Two stable fixed-points persist until the ultimate saddle-node bifurcation at $\varepsilon = 2 / 11$, corresponding to a discontinuous phase transition.
• Figure 3: Streamlines of the vector field $\bar{\mathcal{M}}_\varepsilon(x, y) - (x, y)$ for four values of $\varepsilon$ in the region $\bar{\mathcal{R}}^{(2)}$ along with stable fixed-points (square markers). The picture in regions $\bar{\mathcal{R}}^{(0)}$ and $\bar{\mathcal{R}}^{(1)}$ can be obtained by symmetry. Note that the fixed-point at the transcritical bifurcation (\ref{['lem:lower-bound-on-majqk-denoising-threshold']}) $\varepsilon = 1/6$ is $(1/3)$-noisy (third plot). Between the transcritical bifurcation and saddle-node bifurcation (i.e. $1 / 6 < \varepsilon < 2 / 11$), denoising is still possible for a strict subset of the original region $\bar{\mathcal{R}}^{(2)}$ (fourth plot) owing to the emergence of a fourth stable fixed-point at the center of the simplex.
• Figure 4: Streamlines of the vector field $\bar{\mathcal{D}}_\varepsilon(x, y) - (x, y)$ for three values of $\xi$ showing the $y$ nullcline (dashed black line), stable fixed-points $(x_\pm, y_\pm)$ (square markers), and line separating logical states (dashed grey line).

Theorems & Definitions (30)

• Definition 1: Superpositions and Clones
• Definition 2: Universal $q$-ary computation
• Definition 3: $\delta$-reliability
• Definition 4: $q$-ary symmetric noise
• Definition 5: $q$-ary symmetric $a$-noisy encoding
• Remark 6: Properties of $c^{[q, k]}_\ell$
• Remark 7: Uniform distribution fixed-point
• Lemma 8: Lower-bound on maj-$[q, k]$ denoising threshold
• proof
• Lemma 9: All points in $\mathcal{R}^{(i)}$ approach $\vec{\chi}^{(i)}$ upon repeated iteration of $\mathcal{M}$