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

:

## 2. Analysis of Gaussian Distribution QRNG Scheme

#### 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.

## 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.

## Abbreviations

 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 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