Energy-reliability limits in nanoscale circuits

Abstract

In the nanoscale regime, the behavior of both extant and emerging semiconductor devices are often unreliable. Reliability of such devices often trades-off with their energy consumption, speed, and/or chip area. We study the energy-reliability limits for circuits designed using such devices. Using the mutual information propagation in logic circuits technique developed by Pippenger, together with optimization, we obtain lower bounds on the energy consumption for computing n-input boolean functions. Most extant technologies require all gates to have the same electrical operating point and in circuits of such uniform gates, the minimum energy required to achieve any nontrivial reliability scales superlinearly with the number of inputs. On the other hand, in some emerging technologies such as spin electronics, where the gates in a circuit can have different operating points, energy scaling can be linear in the number of inputs. As part of our development we find a simple procedure for energy allocation across gates in a boolean circuit with different operating points.

Energy-Reliability Limits in Nanoscale Feedforward Neural Networks and Formulas

“…Note that earlier presentations of this work [1], [2] were focused only on circuits with homogeneous electrical operating points for devices, whereas the new synthesis in the current paper emphasizes the value of heterogeneous operation of gates and neural network layers. Results on energy allocation are therefore novel to this paper.…”

“…is maximized. This is because ln 1 1−h(δ) ≥ 0, for δ < 1 2 . Thus when max i |P i | is minimized the other term is also minimized.…”

“…Theorem 1: The minimum energy required to realize any n-input boolean function using a δ-reliable (δ < 1 2 ) formula of gates with degree no more than k (with k < n), and each with the common physical energy-failure function χ is…”

“…This continues until we have only one node at the top level. Thus the total number of non-leaf nodes are lower bounded by 1 k n + 1…”

“…To understand the implications of Theorem 1, let us consider the simple case where we fix δ ∈ (0, 1 2 ) and k = O (1). Then, the theorem implies that to realize δ-reliability using gates of at most k inputs, the minimum energy requirement scales with number of inputs n as…”