Quantum Random Number Generation

• Quantum Random Number Generation is a method that leverages quantum phenomena, such as photon splitting and vacuum fluctuations, to produce true, unpredictable random numbers.
• QRNG implementations employ both discrete and continuous variable sources along with digital post-processing to mitigate hardware imperfections and bias.
• Performance metrics range from kbps to Gbps and adhere to rigorous statistical tests, ensuring security for cryptographic and high-performance computing applications.

Quantum random number generation (QRNG) leverages the intrinsic indeterminacy of quantum measurements to deliver unpredictability and information-theoretic security fundamentally inaccessible to classical sources or algorithmic pseudo-random number generators. QRNGs are realized using diverse quantum physical mechanisms, each governed by the core operational principle that—by quantum theory—no classical or quantum adversary can, even in principle, predict the outcome better than the statistical bounds defined by the system's design and measured performance.

1. Physical Principles and Quantum Randomness Sources

QRNGs rely on the irreducible statistical nature of quantum events, encoded in the Born rule. Measurement of a superposed quantum state—such as a photon split at a beam splitter, the quadrature amplitude of vacuum fluctuations, or the phase diffusion of a laser—yields outcomes with a probability distribution that cannot be decomposed into hidden variables accessible to any observer (Ma et al., 2015, Herrero-Collantes et al., 2016).

The main families of physical entropy sources are:

• Discrete-variable (DV) sources: Single-photon splitting at a balanced beam splitter (Stipčević et al., 2014), on-chip spatial multiplexing (White et al., 2020), or multiport quantum walks (Meng et al., 5 Mar 2024).
• Continuous-variable (CV) sources: Measurement of vacuum amplitude quadratures via balanced homodyne detection (Zhao et al., 5 Jun 2025, Haider et al., 3 Jun 2025, Haylock et al., 2018); phase diffusion in lasers (Álvarez et al., 2019, Li et al., 16 Jan 2024); and amplified spontaneous emission.
• Solid-state and electronic processes: Quantum tunneling in diodes (Esaki/tunnel diodes) (Aungskunsiri et al., 2020), spontaneous emission in LEDs (Moeini et al., 2023).
• Time-of-arrival based entropy: Extraction of randomness from Poisson emission statistics (Strydom et al., 2023, 0807.4111).
• Atomic, nuclear, and other non-optical sources: Radioactive decay, atomic spin noise, and quantum dot emission (Herrero-Collantes et al., 2016).

Each mechanism is subject to careful physical and statistical modeling to quantify how classical imperfections and technical noise dilute the accessible quantum entropy per sample.

2. Implementation Architectures and Methodologies

Contemporary QRNG realizations operationalize their quantum source by a chain of preparation, quantum interaction, measurement, digitization, and randomness extraction:

• Photonics-based QRNGs: Employ laser diodes or single-photon emitters. For example, pulsed LDs/LEDs paired with single-photon avalanche photodiodes (APDs) in Geiger mode (Stipčević et al., 2014), or CV systems with dual-quadrature homodyne detection at GHz bandwidth and high-bit-depth ADCs (Zhao et al., 5 Jun 2025).
• Solid-state sources: Tunnel diodes as practical, integrable quantum entropy sources, where the tunneling transition timing under swept bias is fundamentally quantum-random (Aungskunsiri et al., 2020).
• On-chip and integrated platforms: Use nanophotonic waveguides for spatial multiplexing, multiport walks, ultra-compact footprints, and scalability (Strydom et al., 2023, Meng et al., 5 Mar 2024).
• Cloud and quantum CPU QRNGs: Qubit initialization, Hadamard gate application, and measurement on contemporary superconducting and NISQ-class hardware; randomness certified empirically (but not always composably) (Tamura et al., 2019, Nath et al., 5 Feb 2025).

Digital post-processing methods (e.g., Toeplitz-hash extractors, FIR filters, Wallace mixers) are applied to compensate for hardware-induced bias, serial correlation, or residual classical noise. Architectures differ crucially in whether their output requires such extraction—e.g., intrinsically unbiased self-differencing photodiode QRNGs (0807.4111) and phase-reconstruction schemes (Li et al., 16 Jan 2024) can dispense with post-processing under correct parameterization.

3. Entropy Quantification, Statistical Validation, and Security

The quantum entropy per raw bit is stringently quantified using both information-theoretic arguments and conservative, sometimes adversarial models:

• Min-entropy: $H_{\min}(X) = -\log_2 \max_x \Pr[X=x]$ yields the extractable ε-uniform bits via strong extractors (Ma et al., 2015, Zhao et al., 5 Jun 2025).
• Conditional min-entropy: For adversarial models (allowing side information), $H_{\min}(X|E)$ is bounded, for instance, by rigorous quantum information-theoretic analysis in both device-dependent and -independent scenarios (Lin et al., 2023, Cao et al., 2015).
• Statistical testing: NIST SP 800-22, Dieharder, and TestU01 batteries provide empirical evidence of flatness, absence of correlations, and non-reproducibility in the output. Raw output in systems such as the on-demand optical QRNG (Stipčević et al., 2014) and plasmonic time-of-arrival QRNG (Strydom et al., 2023) passes these tests without post-processing.

Composability—the property that the output can safely serve as a cryptographic seed or key—is ensured when the statistical distance from uniform, given all side information, is bounded by a small ε in the trace-norm.

4. Device Trust Models: From Trusted to Device-independent QRNG

• Practical (device-dependent) QRNGs: Both source and measurement are trusted and well-characterized. The output randomness is calculated by explicit physical modeling and classical noise subtraction (Ma et al., 2015, Stipčević et al., 2014, Haider et al., 3 Jun 2025).
• Source-independent QRNGs: The entropy source is completely untrusted (e.g., sunlight or an uncharacterized laser), but measurement is trusted. Security is established via protocol-level countermeasures (e.g., basis switching and squashing models) and bounded only by measurement calibration (Li et al., 2021, Cao et al., 2015).
• Measurement-device-independent and semi–device-independent QRNGs: Source is characterized or trusted only in limited dimensions, allowing detection modules (potentially with adversarial memory/all side-channels) to be uncharacterized (Lin et al., 2023, Ma et al., 2015).
• Self-testing/device-independent QRNGs: Trust is removed from all devices; randomness is certified from observed violation of Bell (CHSH) or steering inequalities, with composable security proofs securing even against quantum side information (Joch et al., 2021, Ma et al., 2015, Nath et al., 5 Feb 2025). Throughput is limited—typically ≲1 bit/s due to efficiency and statistical requirements.
• Certified quantum computer randomness: Quantum computer circuits leveraging temporal (Leggett–Garg inequality + NSIT) or spatial (Bell–CHSH) quantum correlations enable semi-device-independent randomness expansion without full spatial separation (Nath et al., 5 Feb 2025).

This hierarchy offers a trade-off between security assumptions and generation rates, with practical QRNGs offering high-throughput and device-independent schemes maximizing adversarial resistance at the cost of bit-rate.

5. Performance Metrics, Scalability, and Integration

Key metrics in QRNG evaluation include:

• Raw and secure bit-rate: Throughput from $\sim$10 kbps (tunneling diodes (Aungskunsiri et al., 2020)) to 40 Gbps (dual-homodyne vacuum quadrature (Zhao et al., 5 Jun 2025)). Extraction efficiency (ratio of min-entropy to raw bitstring) determines the final rate.
• Entropy per sample: Homodyne and phase-reconstruction systems can provide multi-bit entropy per ADC sample (Haider et al., 3 Jun 2025, Li et al., 16 Jan 2024). Well-calibrated setups achieve $H_{\min}\to$ 1 per bit for uniform phase/q-walk QRNGs (Meng et al., 5 Mar 2024, Li et al., 16 Jan 2024).
• Latency and on-demand capabilities: Some designs offer sub-10-ns bit latency and on-demand response (no “timeout” events), critical for cryptographic protocols requiring unpredictability in “future light cone” timing (Stipčević et al., 2014).
• Temperature and time stability: Designs such as the LED QRNG exhibit entropy invariance over temperature sweeps and long run-time (Moeini et al., 2023).

Scalability is addressed through modular optoelectronic architectures (e.g., parallel LED + PD units (Moeini et al., 2023), waveguide-integrated multiplexed CV systems (Haylock et al., 2018, Zhao et al., 5 Jun 2025)), facilitating multi-Gbps rates and integration with CMOS platforms.

6. Applications, Limitations, and Future Directions

QRNGs are integral in cryptography (QKD, random key generation), Monte Carlo simulations, high-performance computing, device security, and stochastic modeling. Ultra-fast applications (high-dimensional QKD, neural-network weight initialization) benefit from QRNGs with multi-bit, tunable distributions per sample as realized in quantum-walk architectures (Meng et al., 5 Mar 2024), and systems with software-controlled selection between uniform, Gaussian, and Rayleigh distributions (Zhao et al., 5 Jun 2025).

Limitations arise from hardware imperfections (e.g., dead time, afterpulsing, finite detection efficiency, unmodeled classical/electronic noise (Stipčević et al., 2014, Strydom et al., 2023)) and side-channel vulnerabilities. Fully device-independent and certified steering-based QRNGs remain challenging beyond laboratory scales due to rate limitations, calibration overheads, and computationally intensive certification (semidefinite program solving) (Joch et al., 2021, Nath et al., 5 Feb 2025).

Future work addresses integration (on-chip nanophotonics (Strydom et al., 2023, White et al., 2020)), protocol-level randomness amplification from weak sources, hardware-seeded extractors, and adaptive security models minimizing trust while preserving high generation speed and entropy-per-bit.

7. Comparative Summary of Key Implementations

QRNG Principle	Throughput	Security Model	Requires Extraction
Poissonian photon detection	10 Mbps–100 Mbps	Trusted source & detectors	Minimal/None (Stipčević et al., 2014)
Phase-noise (coherent detection)	1 Mbps–10+ Gbps	Trusted (source/measurement)	Sometimes unnecessary
Vacuum quadrature (homodyne)	10 Mbps–40 Gbps	Trusted measurement/CV SI	Extraction for security (Zhao et al., 5 Jun 2025)
Tunnel diode fluctuations	10–100 kbps	Trusted or semi-trusted	Toeplitz hash (Aungskunsiri et al., 2020)
Quantum walk (on-demand distribution)	~0.4 Mbps (exp.)	Trusted photonic platform	Direct multi-bit output (Meng et al., 5 Mar 2024)
Device-independent (Bell/steering)	0.01–1 bps	Device-independent	Quantum-proof extractor
Cloud quantum computer (NSIT/LGI)	1–10 Mbps (theory)	Semi-device-independent	Analytical entropy bound (Nath et al., 5 Feb 2025)

This diversity in platforms, performance, and adversarial models allows practitioners to select QRNG technologies best aligned to application-specific throughput and security requirements. The foundational research and evolving standards continue to push both extractable randomness and assurance of unpredictability toward quantum-theoretic optima (Ma et al., 2015, Herrero-Collantes et al., 2016, Zhao et al., 5 Jun 2025).