Advanced Statistical Testing of Quantum Random Number Generators

Abstract

Pseudo-random number generators are widely used in many branches of science, mainly in applications related to Monte Carlo methods, although they are deterministic in design and, therefore, unsuitable for tackling fundamental problems in security and cryptography. The natural laws of the microscopic realm provide a fairly simple method to generate non-deterministic sequences of random numbers, based on measurements of quantum states. In practice, however, the experimental devices on which quantum random number generators are based are often unable to pass some tests of randomness. In this review, we briefly discuss two such tests, point out the challenges that we have encountered in experimental implementations and finally present a fairly simple method that successfully generates non-deterministic maximally random sequences.

Parity-based, bias-free optical quantum random number generation with min-entropy estimation

“…Since statistical tests such as those of the NIST suite were originally developed to test PRNGs, they are not necessarily sensitive to the particular issues and properties that characterize QRNGs. QRNGs that pass the NIST tests may struggle with tests that probe aspects of incomputability and algorithmic randomness, or show at least no marked advantage over PRNGs in terms of their algorithmic properties [28][29][30]. Here one particularly relevant and commonly used test is the Borel normality [24,34], which is a necessary (but not sufficient, e.g., Champernowne's constant [50] is normal but computable) condition for algorithmic randomness and thus incomputability.…”

“…While these are striking advantages over PRNGs, practical realizations of QRNGs exhibit problems of their own [6]. In particular, they tend to be susceptible to substantial bias and correlation effects that may detrimentally affect the quality of the output from the point of view of statistical and algorithmic randomness [28][29][30].…”

“…In the photonic scenario just described, bias may arise from sources such as imperfect state preparation with nonequalized amplitudes, a polarizing beam splitter that does not reliably separate H and V polarizations, and imbalanced detectors. Correlations may be caused, for example, by dead-time and afterpulsing effects in the detectors [29,31,32]. In practical implementations, these imperfections often have a significant influence and can be difficult to avoid [6].…”

“…In practical implementations, these imperfections often have a significant influence and can be difficult to avoid [6]. Mitigation of bias effects generally requires the application of an unbiasing algorithm during postprocessing of the output [6], and some optical QRNGs also include explicit calibration [8] and feedback [29] stages to fine-tune and balance detector efficiencies, coupling ratios of beam splitters, and other parameters.…”

“…Borel normality is an important criterion as it taps the realm of algorithmic (rather than statistical) randomness, an area of particular interest for QRNGs. Yet there exist only a few studies to date that have used Borel normality for testing QRNGs and probing the kind of incomputability and algorithmic randomness that in principle sets them apart from PRNGs [28][29][30].…”