Skip to main content

On protocols for increasing the uniformity of random bits generated with noisy quantum computers


Generating random numbers is important for many real-world applications, including cryptography, statistical sampling and Monte Carlo simulations. Quantum systems subject to a measurement produce random results via Born’s rule, and thus it is natural to study the possibility of using such systems in order to generate high-quality random numbers. However, current quantum devices are subject to errors and noise, which can make the output bits deviate from the uniform distribution. In this work, we propose and analyse two protocols that can be used to increase the uniformity of the bits obtained when running a circuit with a Hadamard gate and a measurement in a noisy quantum computer. These protocols may be used prior to other standard processes, such as randomness amplification. We conduct experiments on both a quantum simulator and a real quantum computer, obtaining results that suggest that these protocols are useful to improve the probability of the generated bits passing statistical tests for uniformity.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Aleksandrowicz G, Alexander T, Barkoutsos P, Bello L, Ben-Haim Y, Bucher D, Cabrera-Hernández FJ, Carballo-Franquis J, Chen A, Chen C-F, Chow JM, Córcoles-Gonzales AD, Cross AJ, Cross A, Cruz-Benito J, Culver C, De La Puente González S, De La Torre E, Ding D, Dumitrescu E, Duran I, Eendebak P, Everitt M, Sertage IF, Frisch A, Fuhrer A, Gambetta J, Gago BG, Gomez-Mosquera J, Greenberg D, Hamamura I, Havlicek V, Hellmers J, Herok Ł, Horii H, Hu S, Imamichi T, Itoko T, Javadi-Abhari A, Kanazawa N, Karazeev A, Krsulich K, Liu P, Luh Y, Maeng Y, Marques M, Martín-Fernández FJ, McClure DT, McKay D, Meesala S, Mezzacapo A, Moll N, Rodríguez DM, Nannicini G, Nation P, Ollitrault P, O'Riordan LJ, Paik H, Pérez J, Phan A, Pistoia M, Prutyanov V, Reuter M, Rice J, Davila AR, Rudy RHP, Ryu M, Sathaye N, Schnabel C, Schoute E, Setia K, Shi Y, Silva A, Siraichi Y, Sivarajah S, Smolin JA, Soeken M, Takahashi H, Tavernelli I, Taylor C, Taylour P, Trabing K, Treinish M, Turner W, Vogt-Lee D, Vuillot C, Wildstrom JA, Wilson J, Winston E, Wood C, Wood S, Wörner S, Akhalwaya IY, Zoufal C (2019) Qiskit: An open-source framework for quantum computing. Zenodo, 0.7.2.

  2. 2.

    Acín A, Masanes L (2016) Certified randomness in quantum physics. Nature 540:213–219

    Article  Google Scholar 

  3. 3.

    Asmussen S, Glynn P (2007) Stochastic simulation: algorithms and analysis. Springer, Berlin

    Book  Google Scholar 

  4. 4.

    Bakiri M, Guyeux C, Couchot JF, Oudjida AK (2018) Survey on hardware implementation of random number generators on fpga: theory and experimental analyses. Comput Sci Rev 27:135–153

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bassham LE, Rukhin AL, Soto J et al (2010) A statistical test suite for random and pseudorandom number generators for cryptographic applications. Technical Report, National Institute of Standards and Technology, Gaithersburg, MD, USA

  6. 6.

    Bera MN, Acín A, Kuś M, Mitchell MW, Lewenstein M (2017) Randomness in quantum mechanics: philosophy, physics and technology. Reports Progress Phys 80(12):124,001

    MathSciNet  Article  Google Scholar 

  7. 7.

    Born M (1955) Statistical interpretation of quantum mechanics. Science 122(3172):675–679

    Article  Google Scholar 

  8. 8.

    Brown RG (2020) Dieharder: a random number test suite. Last accessed December 2nd

  9. 9.

    Combarro E, Carminati F, Vallecorsa S, Ranilla J, Rúa I (2020) Quantum random numbers generated by the cloud superconducting quantum computer. Bristol Quantum Information Technologies Workshop (BQIT:20)

  10. 10.

    Combarro E, Carminati F, Vallecorsa S, Ranilla J, Rúa I (2020) Two simple protocols for improving the uniformity of quantum random bits in the presence of noise. In 20th Computational and Mathematical Methods in Science and Engineering Conference

  11. 11.

    Combarro EF, Ranilla J, Rúa I (2019) A quantum algorithm for the commutativity of finite dimensional algebras. IEEE Access 7:45554–45562

    Article  Google Scholar 

  12. 12.

    Gentle JE (2004) Random number generation and monte carlo methods. Springer, Berlin

    MATH  Google Scholar 

  13. 13.

    Grover LK (1996) A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC ’96, pp. 212–219. ACM, New York, NY, USA

  14. 14.

    Herrero-Collantes M, Garcia-Escartin JC (2017) Quantum random number generators. Rev Mod Phys 89:015,004

    MathSciNet  Article  Google Scholar 

  15. 15.

    IBM: IBM Q Experience. Last accessed December 2nd, 2020

  16. 16.

    James F, Moneta L (2020) Review of high-quality random number generators. Comput Softw Big Sci 4:2

    Article  Google Scholar 

  17. 17.

    Karp RM (1991) An introduction to randomized algorithms. Discrete Appl Math 34(1):165–201

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kollmitzer C, Schauer S, Rass S, Rainer B (2020) Quantum random number generation theory and practice: theory and practice. Springer, Berlin

    Book  Google Scholar 

  19. 19.

    L’Ecuyer P (2012) Random number generation. In: Gentle JE, Härdle WK, Mori Y (eds) Handbook of computational statistics: concepts and methods. Springer, Berlin Heidelberg, Berlin, Heidelberg

    Google Scholar 

  20. 20.

    Li L, Yu F, Tang Q, Song Y, Xu Q, Cai S (2019) A survey on true random number generators based on Chaos. Discrete Dyn Nature Soc 2019:1–10

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Liu Y, Zhao Q, Li MH et al (2018) Device-independent quantum random-number generation. Nature 562(7728):548–551

    Article  Google Scholar 

  22. 22.

    Luby M (1996) Pseudorandomness and cryptographic applications. Princeton University Press, Princeton, NJ

    Book  Google Scholar 

  23. 23.

    L’Ecuyer P, Simard R (2007) Testu01: A c library for empirical testing of random number generators. ACM Trans Math Softw 33(4):1–40

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ma X, Yuan X, Cao Z, Qi B, Zhang Z (2016) Quantum random number generation. npj Quantum Inf 2(1):16,021

    Article  Google Scholar 

  25. 25.

    Nielsen MA, Chuang IL (2011) Quantum computation and quantum information, 10th edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  26. 26.

    Schlosshauer M, Kofler J, Zeilinger A (2013) A snapshot of foundational attitudes toward quantum mechanics. Stud History Philos Sci Part B: Stud History Philos Modern Phys 44(3):222–230

    Article  Google Scholar 

  27. 27.

    Shaltiel R (2011) An introduction to randomness extractors. In: Aceto L, Henzinger M, Sgall J (eds) Automata, languages and programming. Springer, Berlin Heidelberg, Berlin, Heidelberg, pp 21–41

    Chapter  Google Scholar 

  28. 28.

    Shor P (1994) Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings of FOCS pp. 124–134

  29. 29.

    Stipčević M, Koç ÇK (2014) True random number generators. In: Koç ÇK (ed) Open problems in mathematics and computational science. Springer, Cham

    Google Scholar 

  30. 30.

    Tamura K, Shikano Y (2019) Quantum random numbers generated by the cloud superconducting quantum computer. In International Symposium on Mathematics, Quantum Theory, and Cryptography: Proceedings of MQC 2019

Download references


This work was supported in part by the Ministry of Economy, Industry and Competitiveness from Spain/FEDER under grant MTM-2017-83506-C2-2-P, by Ministry of Science and Innovation from Spain/FEDER under grant RTI2018-098085-B-C44, and by he Regional Ministry of the Principality of Asturias under grants FC-GRUPIN-IDI/2018/000193 and FC-GRUPIN-IDI/2018/000226.

Author information



Corresponding author

Correspondence to José Ranilla.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Combarro, E.F., Carminati, F., Vallecorsa, S. et al. On protocols for increasing the uniformity of random bits generated with noisy quantum computers. J Supercomput 77, 8063–8081 (2021).

Download citation


  • Random bits
  • Quantum computers
  • Monobit test
  • Uniform distribution