Factoring larger integers with fewer qubits via quantum annealing with optimized parameters

  • WangChun Peng
  • BaoNan Wang
  • Feng Hu
  • YunJiang Wang
  • XianJin Fang
  • XingYuan Chen
  • Chao WangEmail author


RSA cryptography is based on the difficulty of factoring large integers, which is an NP-hard (and hence intractable) problem for a classical computer. However, Shor’s algorithm shows that its complexity is polynomial for a quantum computer, although technical difficulties mean that practical quantum computers that can tackle integer factorizations of meaningful size are still a long way away. Recently, Jiang et al. proposed a transformation that maps the integer factorization problem onto the quadratic unconstrained binary optimization (QUBO) model. They tested their algorithm on a D-Wave 2000Q quantum annealing machine, raising the record for a quantum factorized integer to 376289 with only 94 qubits. In this study, we optimize the problem Hamiltonian to reduce the number of qubits involved in the final Hamiltonian while maintaining the QUBO coefficients in a reasonable range, enabling the improved algorithm to factorize larger integers with fewer qubits. Tests of our improved algorithm using D-Wave’s hybrid quantum/classical simulator qbsolv confirmed that performance was improved, and we were able to factorize 1005973, a new record for quantum factorized integers, with only 89 qubits. In addition, our improved algorithm can tolerate more errors than the original one. Factoring 1005973 using Shor’s algorithm would require about 41 universal qubits, which current universal quantum computers cannot reach with acceptable accuracy. In theory, the latest IBM Q System OneTM (Jan. 2019) can only factor up to 10-bit integers, while the D-Wave have a thousand-fold advantage on the factoring scale. This shows that quantum annealing machines, such as those by D-Wave, may be close to cracking practical RSA codes, while universal quantum-circuit-based computers may be many years away from attacking RSA.


integer factorization quantum annealing QUBO D-Wave 


  1. 1.
    S. S. Wagstaff, Am. Math. Soc. 68, 293 (2013).Google Scholar
  2. 2.
    A. K. Lenstra, H. W. Lenstra Jr, M. S. Manasse, and J. M. Pollard, in Theory of Computing 1990: Proceedings of the Twenty–Second Annual ACM Symposium on Theory of Computing, edited by H. Ortiz (ACM, New York, 1990), pp. 564–572.Google Scholar
  3. 3.
    M. A. Nielsen, I. Chuang, and L. K. Grover, Am. J. Phys. 70, 558 (2002).ADSCrossRefGoogle Scholar
  4. 4.
    S. J. Wei, T. Xin, and G. L. Long, Sci. China–Phys. Mech. Astron. 61, 070311 (2018), arXiv: 1706.08080.ADSCrossRefGoogle Scholar
  5. 5.
    H. L. Huang, Y. W. Zhao, T. Li, F. G. Li, Y. T. Du, X. Q. Fu, S. Zhang, X. Wang, and W. S. Bao, Front. Phys. 12, 120305 (2017), arXiv: 1612.02886.CrossRefGoogle Scholar
  6. 6.
    T. Xin, S. Huang, S. Lu, K. Li, Z. Luo, Z. Yin, J. Li, D. Lu, G. Long, and B. Zeng, Sci. Bull. 63, 17 (2018).CrossRefGoogle Scholar
  7. 7.
    J. Zhou, B. J. Liu, Z. P. Hong, and Z. Y. Xue, Sci. China–Phys. Mech. Astron. 61, 010312 (2018), arXiv: 1705.08852.ADSCrossRefGoogle Scholar
  8. 8.
    P. W. Shor, SIAM Rev. 41, 303 (1999).ADSGoogle Scholar
  9. 9.
    L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Nature 414, 883 (2001).ADSCrossRefGoogle Scholar
  10. 10.
    E. Martín–López, A. Laing, T. Lawson, R. Alvarez, X. Q. Zhou, and J. L. O’Brien, Nat. Photon 6, 773 (2012), arXiv: 1111.4147.ADSCrossRefGoogle Scholar
  11. 11.
    J. A. Smolin, G. Smith, and A. Vargo, Nature 499, 163 (2013), arXiv: 1301.7007.ADSCrossRefGoogle Scholar
  12. 12.
    M. R. Geller, and Z. Zhou, Sci. Rep. 3, 3023 (2013), arXiv: 1304.0128.ADSCrossRefGoogle Scholar
  13. 13.
    C. Gidney, arXiv: 1706.07884v2Google Scholar
  14. 14.
    A. Cho, Science 359, 141 (2018).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001).ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    T. Albash, and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018), arXiv: 1611.04471.ADSCrossRefGoogle Scholar
  17. 17.
    N. Xu, X. Peng, M. Shi, and J. F. Du, arXiv: 65.062310Google Scholar
  18. 18.
    T. Wang, Z. Zhang, L. Xiang, Z. Gong, J. Wu, and Y. Yin, Sci. China–Phys. Mech. Astron. 61, 047411 (2018), arXiv: 1712.10089.ADSCrossRefGoogle Scholar
  19. 19.
    W. H. Wang, H. X. Cao, Z. L. Chen, and L. Wang, Sci. China–Phys. Mech. Astron. 61, 070312 (2018).CrossRefGoogle Scholar
  20. 20.
    C. J. C. Burges, Factoring As Optimization, Technical Report (Microsoft Research, 2002).Google Scholar
  21. 21.
    X. Peng, Z. Liao, N. Xu, G. Qin, X. Zhou, D. Suter, and J. Du, Phys. Rev. Lett. 101, 220405 (2008), arXiv: 0808.1935.ADSCrossRefGoogle Scholar
  22. 22.
    N. Xu, J. Zhu, D. Lu, X. Zhou, X. Peng, and J. Du, Phys. Rev. Lett. 108, 130501 (2012).ADSCrossRefGoogle Scholar
  23. 23.
    S. Pal, S. Moitra, V. S. Anjusha, A. Kumar, and T. S. Mahesh, arXiv: 1611.00998Google Scholar
  24. 24.
    R. Dridi, and H. Alghassi, Sci. Rep. 7, 43048 (2017), arXiv: 1604.05796.ADSCrossRefGoogle Scholar
  25. 25.
    Z. Yin, and Z. Wei, Sci. Bull. 62, 741 (2017).CrossRefGoogle Scholar
  26. 26.
    T. Xin, B. X. Wang, K. R. Li, X. Y. Kong, S. J. Wei, T. Wang, D. Ruan, and G. L. Long, Chin. Phys. B 27, 020308 (2018).ADSCrossRefGoogle Scholar
  27. 27.
    I. Hen, arXiv: 1612.06012Google Scholar
  28. 28.
    H. Li, Y. Liu, and G. L. Long, Sci. China–Phys. Mech. Astron. 60, 080311 (2017), arXiv: 1703.10348.ADSCrossRefGoogle Scholar
  29. 29.
    B. X. Wang, T. Xin, X. Y. Kong, S. J. Wei, D. Ruan, and G. L. Long, Phys. Rev. A 97, 042345 (2018), arXiv: 1802.01420.ADSCrossRefGoogle Scholar
  30. 30.
    C. Wang, and H. G. Zhang, Inf. Secur. Commun. Priv. 2, 31 (2012).Google Scholar
  31. 31.
    C. Wang, Y. J. Wang, and F. Hu, Chin. J. Netw. Inf. Secur. 2, 17 (2016).Google Scholar
  32. 32.
    Z. K. Li, N. S. Dattani, X. Chen, X. M. Liu, H. Y. Wang, R. Tanburn, H. W. Chen, X. H. Peng, and J. F. Du, arXiv: 1706.08061Google Scholar
  33. 33.
    S. X. Jiang, K. A. Britt, A. J. McCaskey, T. S. Humble, and S. Kais, arXiv: 1804.02733Google Scholar
  34. 34.
    E. Boros, and P. L. Hammer, Discret Appl. Math. 123, 155 (2001).CrossRefGoogle Scholar
  35. 35.
    P. A. Parrilo, and B. Sturmfels, arXiv: math/0103170Google Scholar
  36. 36.
    R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Nature 508, 500 (2014), arXiv: 1402.4848.ADSCrossRefGoogle Scholar
  37. 37.
    Z. J. Chen, Metrology of Quantum Control and Measurement in Superconducting Qubits, Dissertation for the Doctoral Degree (University of California Santa Barbara, Santa Barbara, 2018), pp. 200–201.Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • WangChun Peng
    • 1
  • BaoNan Wang
    • 1
  • Feng Hu
    • 1
  • YunJiang Wang
    • 2
  • XianJin Fang
    • 3
  • XingYuan Chen
    • 4
  • Chao Wang
    • 1
    Email author
  1. 1.Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Joint International Research Laboratory of Specialty Fiber Optics and Advanced Communication, Shanghai Institute for Advanced Communication and Data ScienceShanghai UniversityShanghaiChina
  2. 2.State Key Laboratory of Integrated Services Networks (ISN)Xidian UniversityXi’anChina
  3. 3.School of Computer Science and Technology EngineeringAnhui University of Science and TechnologyHuainanChina
  4. 4.State Key Laboratory of CryptologyBeijingChina

Personalised recommendations