Modular Debiasing: A Robust Method for Quantum Randomness Extraction

We propose a novel modular debiasing technique applicable to any discrete random source, addressing the fundamental challenge of reliably extracting high-quality randomness from inherently imperfect physical processes. The method involves summing the outcomes of multiple independent trials from a biased source and reducing the sum modulo the number of possible outcomes, $m$. We provide a rigorous theoretical framework, utilizing probability generating functions and roots of unity, demonstrating that this simple operation guarantees the exponential convergence of the output distribution to the ideal uniform distribution over $\{0, 1, \dots, m-1\}$. A key theoretical result is the method's remarkable robustness: convergence is proven for any initial bias (provided all outcomes have non-zero probability) and, crucially, is maintained even under non-stationary conditions or time-dependent noise, which are common in physical systems. Analytical bounds quantify this exponential rate of convergence, and are empirically validated by numerical simulations. This technique's simplicity, strong theoretical guarantees, robustness, and data efficiency make it particularly well-suited for practical implementation in quantum settings, such as spatial photon-detection-based Quantum Random Number Generators (QRNGs), offering an efficient method for extracting high-quality randomness resilient to experimental imperfections. This work contributes a valuable tool to the field of Quantum Information Science.