A Gaussian-Distributed Quantum Random Number Generator Using Vacuum Shot Noise



1. Introduction

2. Analysis of Gaussian Distribution QRNG Scheme

2.1. Gaussian Random Source and Entropy Estimation

2.1.1. Vacuum Fluctuation

2.1.2. Entropy Estimation

2.2. Impact of Sampling Device

2.2.1. Sampling Range

  • If k is too small, () will occur too often, making the random variable more predictable, and reducing entropy . Furthermore, the worse profile of Gaussian distribution has a higher possibility to fail the GoF test, which does not match our requirement in post-processing and applications;
  • If k is too large, most signals will locate in a small range of sample bins, making the most significant bits (MSB) of samples more predictable, and also reducing entropy . On the other hand, many sampling bins are unoccupied, wasting the ability of devices and substantially reduce the sampling precision.

2.2.2. Sampling Resolution

2.2.3. Sampling Depth

3. Post-Processing

  • Elements in the matrix, which are the weights in Equation (10), is not fixed, as long as they obey fundamental rules. For matrix, each row/column should have 3 (1) positive and 1 (3) negative elements, and the position should not be the same; the absolute value of each row and column should not be the same either. Thus there is a group of with hundreds of possible matrices;
  • The size of the matrix can be designed, which indicates how many raw numbers will be used to generate a final number. We take the matrix as the simplest example for a demonstration. However, when the precision after m-MSB pre-processing is inadequate, and a larger matrix should be made. For instance, in the following section of implementation, we generate 12-bit Gaussian distribution numbers from 5-bit pre-processed data, by utilizing an matrix. If the matrix size is larger, it has a potential for even higher precision, such as five-bit pre-processed data with a matrix will generate 20-bit Gaussian distribution random numbers for high multiple-sigma applications.
  • The values of matrix elements can also be designed, which indicate shifted bits of the pre-processed data. In the discussion above, weights of adjacent numbers always follow the power of , which means that adjacent numbers in should shift one bit in the summation operation. However, if we change to , it means that adjacent numbers in should shift two bits. Remember that according to Equation (17), a normalized coefficient should be carefully calculated to match the designation, making sure that the input and output share the same variance.

4. Implementation and Results

4.1. Experimental Setup

4.2. Test Results

Normality Tests

5. Conclusions

Author Contributions


Conflicts of Interest


QRNG Quantum Random Number Generator
QKD Quantum Key Distribution
PDF/CDF Probability/Cumulative Density Function
ADC Analog-to-Digital Converter
QCNR Quantum-to-Classical Noise Ratio
GoF Goodness of Fit
MSB/LSB Most/Least Significant Bit

Appendix A. Goodness of Fit Tests

Appendix B. PDF Conversion between Uniform and Gaussian Distribution

  • Box-Muller [63]: uniform and Gaussian distribution can be easily converted between rectangular basis and polar basis. Assuming that are uniform variables, and are Gaussian variables, there exist:
  • CDF method [54]: uniform and Gaussian distribution can be converted by cumulative density function (CDF) and its inverse function, ICDF. Assuming U an uniform variable, and X a Gaussian variable, there exist:is denoted as:


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Normal t-Dist. Uniform Rayleigh
QCNR(dB) Before After Before After Before After Before After
3 1.2225 1.1653 1.2966 1.3338 64.036 4.6896 179.40 1.4993
6 1.2582 1.2320 1.4416 1.4348 9.0556 1.4507 39.991 1.2510
10 1.1920 1.2031 1.2478 1.2741 1.4064 1.3917 4.5185 1.1799
20 1.2717 1.2455 1.2132 1.2510 1.1764 1.1996 1.2150 1.1964
Function Mean AD Test JB Test t-Test
Calculated result p = 0.4788 p = 0.3678 p = 0.2023
Confidence Interval [, 0.0036] NULL NULL NULL
Hypothesis value
Status Pass Pass Pass Pass
Test Name p-Value Proportion Status
Frequency 0.811993 394 Success
Block Frequency 0.719747 396 Success
Cumulative Sums 0.785103(KS) 395.5(avg) Success
Runs 0.270275 396 Success
Longest Run 0.788728 397 Success
Rank 0.375313 396 Success
FFT 0.272297 395 Success
Non-overlapping 0.647530(KS) 394(avg) Success
Overlapping 0.830808 396 Success
Universal 0.451234 393 Success
Approx. Entropy 0.739918 397 Success
Excursions 0.726852(KS) 392(avg) Success
Excursions Var. 0.670396(KS) 395(avg) Success
Serial 0.589359(KS) 392.5(avg) Success
Complexity 0.124115 392 Success

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Huang, M.; Chen, Z.; Zhang, Y.; Guo, H. A Gaussian-Distributed Quantum Random Number Generator Using Vacuum Shot Noise. Entropy 2020, 22, 618. https://doi.org/10.3390/e22060618

Huang M, Chen Z, Zhang Y, Guo H. A Gaussian-Distributed Quantum Random Number Generator Using Vacuum Shot Noise. Entropy. 2020; 22(6):618. https://doi.org/10.3390/e22060618

Chicago/Turabian Style

Huang, Min, Ziyang Chen, Yichen Zhang, and Hong Guo. 2020. “A Gaussian-Distributed Quantum Random Number Generator Using Vacuum Shot Noise” Entropy 22, no. 6: 618. https://doi.org/10.3390/e22060618

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