From Classical to Quantum: Statistical Testing of Randomness

This thesis investigates the use of statistical tests to evaluate randomness in data produced by different sources, with particular attention to quantum computing systems. The study evaluates established test suites (NIST SP 800-22, BoolTest, StringENT, and SAC) on both classical and quantum-generated sequences, assessing their performance and limitations.Results on classical algorithms show a clear dichotomy: modern generators like MT19937 consistently pass all test batteries (p − values > 0.01), whereas PCG32, despite its reputation, exhibited specific failures in the NIST Spectral and Universal tests. Low-quality generators (e.g., cong) exhibit systemic failures, yielding p-values < 10−5 in strict tests such as the NIST tests and BoolTest, while the standard rand() implementation demonstrated unexpected resilience by passing all applied suites. In addition, the thesis examines the sensitivity of these suites using artificially biased sources. The analysis demonstrates that modern testing frameworks can detect deviations as small as 0.003 from the ideal probability of 0.5, rejecting such sequences with overwhelming statistical confidence (p − values < 10−5 across all suites). Regarding quantum random number generators (QRNGs), the work analyses raw data from single-qubit and GHZ-state circuits executed on the ‘Quantum Blue” superconducting processor. While the single-qubit circuit passed the SAC test, it exhibited statistically significant deviations in the BoolTest (p − value ≈ 0.0004) and StringENT. The GHZ-based implementation, heavily influenced by hardware noise and decoherence, failed all applied test suites (p − values < 10−5), highlighting the critical necessity of randomness extraction in raw quantum streams. The results provide quantitative insight into how statistical methods can be applied to quantum data and support the development of validation tools relevant to applications such as quantum cryptography and secure communications.