Analytical framework for uncertainty propagation in beam-splitter-based quantum random number generators (QRNGs) and its impact on extraction efficiency, computational cost, throughput, and Gaussian random number generation.
Research-style numerical and mathematical analysis connecting physical quantum source bias to practical computation cost.
Quantum random number generators are only useful if their raw physical measurements can be converted into high-quality random bits efficiently. This project asks a practical question:
If the physical source is slightly biased or uncertain, how much extra computation does the system need to produce reliable random numbers?
The analysis connects beam-splitter uncertainty to extraction efficiency, raw-bit cost, Gaussian random number generation cost, and throughput.
| Area | Details |
|---|---|
| Topic | Quantum random number generation |
| Model | Beam-splitter source modeled as a Bernoulli process |
| Core question | How uncertainty affects extraction and computation |
| Methods | Entropy analysis, Taylor approximation, Fisher information, throughput modeling |
| Output | Analytical framework, plots, and technical report |
This research project investigates how uncertainty in beam-splitter-based quantum random number generators propagates through randomness extraction and downstream computational stages.
The work focuses on:
- Quantum randomness generation
- Bernoulli source modeling
- Randomness extraction efficiency
- Uncertainty propagation
- Uniform random bit generation cost
- Gaussian random number generation
- Throughput degradation
- Latency analysis
The project develops a unified analytical framework connecting physical source imperfections directly to computational performance overhead.
The QRNG source is modeled as a Bernoulli process:
where:
- ( p ) represents the beam-splitter transmissivity
- ( X ) represents the measured quantum outcome
Under ideal conditions:
which corresponds to maximum entropy and unbiased randomness generation.
The extraction efficiency is defined as:
where:
- ( \eta(p) ) is the output efficiency
- ( p ) is the Bernoulli source parameter
The function reaches maximum efficiency at:
meaning that even ideal sources lose a significant portion of raw measurements during extraction.
The graph illustrates:
- maximum efficiency at ( p=0.5 )
- nonlinear degradation under bias
- increasing sensitivity near the optimal operating point
Near the optimal operating point:
the extraction efficiency becomes:
This demonstrates:
- quadratic efficiency degradation
- strong sensitivity to uncertainty
- nonlinear computational overhead growth
The expected raw-bit cost for generating one unbiased output bit is:
At the optimal operating point:
meaning four raw quantum measurements are required on average for one unbiased random bit.
The Gaussian generation cost is modeled as:
At ideal conditions:
showing that Gaussian random generation amplifies uncertainty-related computational cost.
The analysis quantifies:
- extraction efficiency degradation
- raw-bit generation cost
- computational overhead
- uncertainty amplification
under varying QRNG source conditions.
Parameter estimation uncertainty is bounded by the Cramér–Rao inequality:
where:
- ( \hat{p} ) is the estimated source parameter
- ( N ) is the number of collected samples
This establishes the throughput–precision trade-off central to the project.
The project evaluates how uncertainty affects:
- randomness throughput
- extraction efficiency
- Gaussian generation latency
- computational stability
- system-level performance
The framework directly links physical QRNG imperfections to measurable computational cost.
- Unified framework for QRNG uncertainty propagation
- Closed-form extraction efficiency approximations
- Uniform and Gaussian random generation cost models
- Throughput–precision trade-off analysis
- System-level interpretation of uncertainty effects
- Quantitative performance degradation analysis
- Bernoulli modeling
- Shannon entropy analysis
- Fisher information
- Cramér–Rao bounds
- Taylor approximation
- Uncertainty propagation
- Throughput modeling
- Python
- Numerical evaluation
- Performance analysis
- Cost modeling
| Metric | Observation |
|---|---|
| Extraction Efficiency | Governed by ( p(1-p) ) |
| Sensitivity | Quadratic near ( p=0.5 ) |
| Uniform Bit Cost | Increases under source bias |
| Gaussian Generation Cost | Strongly affected by uncertainty |
| Throughput | Degrades with parameter uncertainty |
| Latency | Increases as uncertainty grows |
Even a small deviation from an ideal 50/50 quantum source can increase the raw measurements needed to produce usable random numbers. The cost becomes more visible when the random bits are used for Gaussian sampling, where uncertainty is amplified by the downstream generation process.
This makes QRNG design a full system problem: the physics, estimator precision, extraction algorithm, and application-level throughput all affect the final cost.
- Quantum random number generation
- Randomness extraction
- Entropy analysis
- Statistical estimation
- Throughput–precision trade-offs
- Computational cost modeling
- Uncertainty propagation
src/
├── efficiency_analysis.py
├── uncertainty_propagation.py
├── throughput_model.py
└── README.md
- Hasan Al Hussein
- Omar Yousef
- Ahmad Alhawamdeh
Khalifa University