Modular Debiasing: A Robust Method for Quantum Randomness Extraction

TL;DR

The paper tackles the challenge of obtaining high-quality randomness from biased discrete sources, with a focus on quantum randomness generation. It introduces a modular debiasing technique that sums N outcomes and reduces the total modulo m, and analyzes it using probability generating functions and roots of unity. The authors prove exponential convergence of the output distribution to uniform over {0, ..., m−1}, with a convergence rate governed by ρ = max_{j≥1} |A_j| where A_j = ∑ p_l ω^{jl}, and show robustness to non-stationary noise as long as all outcomes remain possible. They validate the theory with numerical simulations across diverse bias and noise scenarios and discuss practical QRNG applications, notably spatial photon-detection-based schemes, highlighting simplicity, robustness, and data efficiency as key advantages.

Abstract

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.

Figures (7)

• Figure 1: Entropy Convergence vs N
• Figure 2: TVD Convergence vs N (Linear Scale)
• Figure 3: TVD Convergence vs N (Log Scale)
• Figure 4: Proportion Evolution for IID Strong Bias (m=10, p0=0.8)
• Figure 5: Proportion Evolution for Non-Stationary Random Noise (m=3)