Skip to main content
SpringerLink
Log in
Book cover

Quantum Computational Number Theory pp 229–245Cite as

Miscellaneous Quantum Algorithms

  • Chapter
  • First Online:

  • 1736 Accesses

Abstract

So far, we have discussed classical and particularly quantum algorithms for integer factoring, discrete logarithms and elliptic curve discrete logarithms. This does not mean quantum algorithms can only be used to solve integer factorization problem, discrete logarithm problem and elliptic curve discrete logarithm problem. In fact, quantum algorithms and quantum computers in general can solve other problems with either superpolynomially (exponentially) speedup or polynomially speedup. In this last and short chapter, we shall discuss some various other quantum algorithms and methods for more number-theoretic problems. Unlike the previous chapters, we will not emphasize on the introduction of the details quantum algorithms for number-theoretic problems, rather we shall concentrated on new ideas and new developments in quantum algorithms for number-theoretic problems.

Any noun can be verbed. Alan Perlis (1922–1990) The First (1966) Turing Award Recipient

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A typical superpolynomial complexity is \(\mathcal{O}((\log n)^{\log \log \log n})\), as \(n \rightarrow \infty\), where \(\log n\) is the input length. Note that superpolynomial is exponential not polynomial. Thus, e.g., \(\mathcal{O}((\log n)^{11})\) is polynomial, whereas \(\mathcal{O}(n^{0.1})\) is exponential.

References

  1. D. Aharonov, V. Jones, Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, in Proceedings of the 38th Annual ACM symposium on Theory of Computing, Seattle, 21–23 May 2006, pp. 427–436

    MathSciNet  MATH  Google Scholar 

  2. A. Ambainis, Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Ambainis, New developments in quantum algorithms, in Proceedings of Mathematical Foundations of Computer Science 2010. Lecture Notes in Computer Science, vol. 6281 (Springer, New York, 2010), pp. 1–11

    Google Scholar 

  4. E. Bombieri, The Riemann hypothesis, in The Millennium Prize Problems, ed. by J. Carlson, A. Jaffe, A. Wiles (Clay Mathematics Institute/American Mathematical Society, Providence, 2006), pp. 107–152 [8]: The Millennium Prize Problem (Clay Mathematical Institute and American Mathematical Society, Cambridge, 2006), pp. 105–124

    Google Scholar 

  5. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over \(\mathbb{Q}\): wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. M.L. Brown, Heegner Modules and Elliptic Curves. Lecture Notes in Mathematics, vol. 1849 (Springer, New York, 2004)

    Google Scholar 

  7. J.A. Buchmann, H.C. Williams, A key-exchange system based on real quadratic fields (extended abstract), in Advances in Cryptology–CRYPTO 1989. Lecture Notes in Computer Science, vol. 435 (Springer, New York, 1990), pp. 335–343

    Google Scholar 

  8. J. Carlson, A. Jaffe, A. Wiles (eds.), The Millennium Prize Problems (Clay Mathematics Institute and American Mathematical Society, Cambridge, 2006)

    MATH  Google Scholar 

  9. J.R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes. Scientia Sinica XVI(2), 157–176 (1973)

    Google Scholar 

  10. R. Cleve, A. Ekert, C. Macchiavello, M. Mosca, Quantum algorithms revisited. Proc. R. Soc. Lond. A 454(1969), 339–354 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Coates, A. Wiles, On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39(3), 223–251 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138 (Springer, New York, 1993)

    Google Scholar 

  13. D. Deutsch, R. Jozsa, Rapid solutions of problems by quantum computation. Proc. R. Soc. Lond. A 439(1907), 553–558 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Eisenträger, S. Hallgren, Computing the unit group, class group, and compact representations in algebraic function fields, in The Open Book Series: Tenth Algorithmic Number Theory Symposium, vol. 1 (2013), pp. 335–358

    Google Scholar 

  15. K. Eisenträger, A. Kitaev, F. Song, A quantum algorithm for computing the unit group of an arbitrary degree number field, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, New York, 31 May–4 June 2014, pp. 293–302

    Google Scholar 

  16. R. Feynman, Simulating physics with quantum computers. Int. J. Theor. Phys. 21(6–7), 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  17. B.H. Gross, D.B. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series, II. Math. Ann. 278(1–4), 497–562 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. L.K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of 28th Annual ACM Symposium on the Theory of Computing, Philadelphia, 22–24 May 1996, pp. 212–219

    MathSciNet  Google Scholar 

  20. L.K. Grover, Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325–328 (1997)

    Article  Google Scholar 

  21. L.K. Grover, From Schrödinger’s equation to quantum search algorithm. Am. J. Phys. 69(7), 769–777 (2001)

    Article  Google Scholar 

  22. L.K. Grover, A.M. Sengupta, From coupled pendulum to quantum search, in Mathematics of Quantum Computing (Chapman & Hall/CRC, Boca Raton, 2002), pp. 119–134

    MATH  Google Scholar 

  23. S. Hallgren, Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montreal, 19–21 May 2002, pp. 653–658, ed. by R.K. Brylinski, G. Chen

    Google Scholar 

  24. S. Hallgren, Fast quantum algorithms for computing the unit group and class group of a number field, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, 21–24 May 2005, pp. 468–474

    MathSciNet  MATH  Google Scholar 

  25. S. Hallgren, Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. J. ACM, 54(1), Article 4, 19 (2007)

    Google Scholar 

  26. S. Jordan, Quantum algorithm zoo (2015). http://math.nist.gov/quantum/zoo/

  27. R. Jozsa, Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation. Ann. Phys. 306(2), 241–279 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. A.Y. Kitaev, Quantum measurements and the Abelian stabilizer problem (1995). arXiv:quant-ph/95110226v1

    Google Scholar 

  29. V. Kolyvagin, Finiteness of \(E(\mathbb{Q})\) amd \(\mathrm{III}(E, \mathbb{Q})\) for a class of weil curves. Math. USSR Izvestija, 32, 523–541 (1989)

    Article  MathSciNet  Google Scholar 

  30. H.W. Lenstra, Jr., Solving the Pell equation. Not. Am. Math. Soc. 49(2), 182–192 (2002)

    MathSciNet  MATH  Google Scholar 

  31. J. Latorre, G. Sierra, Quantum computing of prime number functions (2013). arXiv:1302.6245v3 [quant-ph]

    Google Scholar 

  32. J. Latorre, G. Sierra, There is Entanglement in the primes (2014). arXiv:1403.4765v2 [quant-ph]

    Google Scholar 

  33. C. Lomont, The hidden subgroup problem – review and open problems (2004). arXiv:quant-ph/0411037v1

    Google Scholar 

  34. A.P. Lund, A. Laing, S. Rahimi-Keshari, et al., Boson sampling from gaussian states. Phys. Rev. Lett. 113(10), 100502, 1–5 (2014)

    Google Scholar 

  35. F. Magniez, A. Nayak, Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Magniez, M. Santha, M. Szegedy, Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. I. Pak, Testing commutativity of a group and the power of randomization. LMS J. Comput. Math. 15, 38–43 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. T.C. Ralph, Boson sampling on a chip. Nat. Photonics 7(7), 514–515 (2013)

    Article  Google Scholar 

  39. D. Shanks, Class number, a theory of factorization, and genera, in Proceedings of Symposia in Pure Mathematics, vol. 20 (American Mathematical Society, Providence, 1971), pp. 415–440

    Google Scholar 

  40. A. Schmidt, U. Völlmer, Polynomial-time quantum algorithm for the computation of the unit group of a number field, in Proceedings of 37th Annual ACM Symposium on Theory of Computing, New York (2005), pp. 475–480

    Google Scholar 

  41. A. Wiles, The Birch and Swinnerton-Dyer conjecture, in The Millennium Prize Problems, ed. by J. Carlson, A. Jaffe, A. Wiles (Clay Mathematics Institute/American Mathematical Society, Providence, 2006), pp. 31–44

    Google Scholar 

  42. P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. P. Shor, Why haven’t more quantum algorithms been found? J. ACM, 50(1), 87–90 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. D.R. Simon, On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. R.D. Silverman, A perspective on computational number theory. Not. Am. Math. Soc. 38(6), 562–568 (1991)

    Google Scholar 

  46. C. Thiel, On the Complexity of Some Problems in Algorithmic Algebraic Number Theory. Ph.D. Thesis, Universität des Saarlandes, Saarbrücken, 1995

    Google Scholar 

  47. W. van Dam, Quantum computing of zeroes of zeta functions (2004). arXiv:quant-ph/0405081v1

    Google Scholar 

  48. W. van Dam, G. Seroussi, Efficient quantum algorithms for estimating gauss sums (2002). arXiv:quant-ph/0207131v

    Google Scholar 

  49. P. Vitányi, The quantum computing challenge, in Informatics: 10 Years Back, 10 Years Ahead, Lecture Notes in Computer Science, vol. 2000 (Springer, New York, 2001), pp. 219–233

    Google Scholar 

  50. Wikipedia, Quantum algorithms. Wikipedia, the free encyclopedia (2015). https://en.wikipedia.org/wiki/Quantum_algorithm

  51. A. Wiles, The Birch and Swinnerton-Dyer conjecture, in The Millennium Prize Problem ed. by J. Carlson, A. Jaffe, A. Wiles (Clay Mathematics Institute/American Mathematical Society, Providence, 2006), Cambridge, 2006), pp. 31–44

    MATH  Google Scholar 

  52. H.C. Williams, Solving the pell equation, in Surveys in Number Theory: Papers from the Millennial Conference on Number Theory, ed. by M.A. Bennett, B.C. Berndt, N. Boston, et al. (AK Peters, Natick, 2002), pp. 325–363

    Google Scholar 

  53. S.Y. Yan, Number Theory for Computing, 2nd edn. (Springer, New York, 2002)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wuhan University, Wuhan, China

    Song Y. Yan

Authors
  1. Song Y. Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Yan, S.Y. (2015). Miscellaneous Quantum Algorithms. In: Quantum Computational Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-25823-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25823-2_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25821-8

  • Online ISBN: 978-3-319-25823-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation