Evaluating Quantumness, Efficiency and Cost of Quantum Random Number Generators via Photon Statistics

Despite the availability of commercial QRNG devices, distinguishing between PRNG and QRNG outputs computationally remains challenging. This paper presents two significant contributions from the perspectives of QRNG manufacturers and users. For manufacturers, the conventional method of assessing the quantumness of single-photon-based QRNGs through mean and variance comparisons of photon counts is statistically unreliable due to finite sample sizes. Given the sub-Poissonian statistics of single photons, confirming the underlying distribution is crucial for validating a QRNG's quantumness. We propose a more efficient two-fold statistical approach to ensure the quantumness of optical sources with the desired confidence level. Additionally, we demonstrate that the output of QRNGs from exponential and uniform distributions exhibit similarity under device noise, deriving corresponding photon statistics and conditions for $\epsilon$-randomness. From the user's perspective, the fundamental parameters of a QRNG are quantumness (security), efficiency (randomness and random number generation rate), and cost. Our analysis reveals that these parameters depend on three factors, expected photon count per unit time, external reference cycle duration, and detection efficiency. A lower expected photon count enhances security but increases cost and decreases the generation rate. A shorter external reference cycle boosts security but must exceed a minimum threshold to minimize timing errors, with minor impacts on cost and rate. Lower detection efficiency enhances security and lowers cost but reduces the generation rate. Finally, to validate our results, we perform statistical tests like NIST, Dieharder, AIS-31, ENT etc. over the data simulated with different values of the above parameters. Our findings can empower manufacturers to customize QRNGs to meet user needs effectively.