Noise Quantification and Control in Circuits via Strong Data-Processing Inequalities

2025-07-20

AI Summary

This paper investigates the fundamental limits of reliable computation in noisy circuits. Building upon von Neumann’s model, we develop an information-theoretic framework based on the strong data processing inequality (SDPI) to uniformly characterize noise propagation through arbitrary-order majority gates. Methodologically, we derive lower bounds on circuit depth and upper bounds on noise tolerance, rigorously establishing the existence of a non-zero noise threshold for generalized majority gates; we further provide an upper bound on the minimal gate order required to achieve a prescribed reliability. Compared to classical analyses, our framework is more concise and scalable, and yields the first systematic proof of threshold existence and quantitative estimation for higher-order majority gates. Key contributions include: (1) the first systematic application of SDPI to noisy circuit analysis; (2) a generalization of von Neumann’s seminal result to arbitrary-order majority gates; and (3) tight upper bounds on both tolerable noise levels and required gate order, providing theoretical benchmarks for fault-tolerant computation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

This essay explores strong data-processing inequalities (SPDI's) as they appear in the work of Evans and Schulman cite{ES} and von Neumann cite{vN} on computing with noisy circuits. We first develop the framework in cite{ES}, which leads to lower bounds on depth and upper bounds on noise that permit reliable computation. We then introduce the $3$-majority gate, introduced by cite{vN} for the purpose of controlling noise, and obtain an upper bound on noise necessary for its function. We end by generalizing von Neumann's analysis to majority gates of any order, proving an analogous noise threshold and giving a sufficient upper bound for order given a desired level of reliability. The presentation of material has been modified in a way deemed more natural by the author, occasionally leading to simplifications of existing proofs. Furthermore, many computations omitted from the original works have been worked out, and some new commentary added. The intended audience has a rudimentary understanding of information theory similar to that of the author.

Problem

• Quantifying noise limits for reliable circuit computation
• Analyzing 3-majority gates for noise control thresholds
• Generalizing noise thresholds to majority gates of any order

Innovation

• Using strong data-processing inequalities for noise quantification
• Introducing 3-majority gate for noise control
• Generalizing von Neumann's analysis to any-order majority gates