Skip to main content
SpringerLink
Log in

New Developments in Quantum Algorithms

  • Conference paper
Mathematical Foundations of Computer Science 2010 (MFCS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6281))

Abstract

In this talk, we describe two recent developments in quantum algorithms.

The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time \(O(\sqrt{N})\). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model.

The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N 2.37...) where N is the size of the system, the quantum algorithm runs in time O(logc N). It outputs a quantum state describing the solution of the system.

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

Access this chapter

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 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

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aaronson, S.: D-Wave Easter Spectacular. A blog post (April 7, 2007), http://scottaaronson.com/blog/?p=225

  2. Altshuler, B., Krovi, H., Roland, J.: Anderson localization casts clouds over adiabatic quantum optimization, arxiv:0912.0746

    Google Scholar 

  3. Ambainis, A.: Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences 64, 750–767 (2002), quant-ph/0002066

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambainis, A.: Quantum walks and their algorithmic applications. International Journal of Quantum Information 1, 507–518 (2003), quant-ph/0403120

    Article  MATH  Google Scholar 

  5. Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM Journal on Computing 37(1), 210–239 (2007), FOCS 2004 and quant-ph/0311001

    Article  MATH  MathSciNet  Google Scholar 

  6. Ambainis, A.: Quantum search algorithms (a survey). SIGACT News 35(2), 22–35 (2004), quant-ph/0504012

    Article  Google Scholar 

  7. Ambainis, A.: Polynomial degree vs. quantum query complexity. Journal of Computer and System Sciences 72(2), 220–238 (2006), quant-ph/0305028

    Article  MATH  MathSciNet  Google Scholar 

  8. Ambainis, A.: A nearly optimal discrete query quantum algorithm for evaluating NAND formulas, arxiv:0704.3628

    Google Scholar 

  9. Ambainis, A.: Quantum random walks - New method for designing quantum algorithms. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 1–4. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  10. Ambainis, A.: Quantum algorithms for formula evaluation. In: Proceedings of the NATO Advanced Research Workshop Quantum Cryptography and Computing: Theory and Implementations (to appear)

    Google Scholar 

  11. Ambainis, A.: Variable time amplitude amplification and a faster quantum algorithm for systems of linear equations (in preparation, 2010)

    Google Scholar 

  12. Ambainis, A., Childs, A., Reichardt, B., Spalek, R., Zhang, S.: Any AND-OR formula of size N can be evaluated in time N 1/2 + o(1) on a quantum computer. In: Proceedings of FOCS 2007, pp. 363–372 (2007)

    Google Scholar 

  13. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of SODA 2005, pp. 1099–1108 (2005), quant-ph/0402107

    Google Scholar 

  14. Barnum, H., Saks, M.: A lower bound on the quantum complexity of read once functions. Journal of Computer and System Sciences 69, 244–258 (2004), quant-ph/0201007

    Article  MATH  MathSciNet  Google Scholar 

  15. Barnum, H., Saks, M.E., Szegedy, M.: Quantum query complexity and semi-definite programming. In: IEEE Conference on Computational Complexity 2003, pp. 179–193 (2003)

    Google Scholar 

  16. Berry, D.W., Ahokas, G., Cleve, R., Sanders, B.C.: Efficient quantum algorithms for simulating sparse Hamiltonians. Communications in Mathematical Physics 270(2), 359–371 (2007), quant-ph/0508139

    Article  MATH  MathSciNet  Google Scholar 

  17. Berry, D.W., Childs, A.: The quantum query complexity of implementing black-box unitary transformations, arxiv:0910.4157

    Google Scholar 

  18. Boneh, D., Lipton, R.: Quantum cryptanalysis of hidden linear functions (extended abstract). In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 424–437. Springer, Heidelberg (1995)

    Google Scholar 

  19. Buhrman, H., Špalek, R.: Quantum verification of matrix products. In: Proceedings of SODA 2006, pp. 880–889 (2006), quant-ph/0409035

    Google Scholar 

  20. Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288, 21–43 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information Science. AMS Contemporary Mathematics Series, vol. 305, pp. 53–74 (2002), quant-ph/0005055

    Google Scholar 

  22. Brassard, G., Høyer, P., Tapp, A.: Quantum cryptanalysis of hash and claw-free functions. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 163–169. Springer, Heidelberg (1998), quant-ph/9705002

    Chapter  Google Scholar 

  23. Brassard, G., Høyer, P., Tapp, A.: Quantum counting. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 820–831. Springer, Heidelberg (1998), quant-ph/9805082

    Chapter  Google Scholar 

  24. Bunch, J.R., Hopcroft, J.E.: Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation 28, 231–236 (1974)

    MATH  MathSciNet  Google Scholar 

  25. Childs, A.: On the relationship between continuous- and discrete-time quantum walk. Communications in Mathematical Physics 294, 581–603 (2010), arXiv:0810.0312

    Article  MathSciNet  Google Scholar 

  26. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by quantum walk. In: Proceedings of STOC 2003, pp. 59–68 (2003), quant-ph/0209131

    Google Scholar 

  27. Childs, A., Reichardt, B., Špalek, R., Zhang, S.: Every NAND formula on N variables can be evaluated in time O(N 1/2 + ε), quant-ph/0703015, version v1 (March 2, 2007)

    Google Scholar 

  28. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  29. van Dam, W., Mosca, M., Vazirani, U.: How Powerful is Adiabatic Quantum Computation? In: Proceedings of FOCS 2001, pp. 279–287 (2001)

    Google Scholar 

  30. D-Wave Systems, http://www.dwavesys.com

  31. Farhi, E., Goldstone, J., Gutman, S., Sipser, M.: A quantum adiabatic algorithm applied to random instances of an NP-complete problem. Science 292, 472–476 (2001), quant-ph/0104129

    Article  MathSciNet  Google Scholar 

  32. Farhi, E., Goldstone, J., Gutman, S.: A Quantum Algorithm for the Hamiltonian NAND Tree. Theory of Computing 4, 169–190 (2008), quant-ph/0702144

    Article  MathSciNet  Google Scholar 

  33. Golub, G., van Loan, C.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  34. Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of STOC 1996, pp. 212–219 (1996), quant-ph/9605043

    Google Scholar 

  35. Hallgren, S.: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. Journal of the ACM 54(1) (2007)

    Google Scholar 

  36. Harrow, A., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Physical Review Letters 103, 150502 (2008), arXiv:0907.3920

    Article  MathSciNet  Google Scholar 

  37. Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  38. Høyer, P., Lee, T., Špalek, R.: Negative weights make adversaries stronger. In: Proceedings of STOC 2007, pp. 526–535 (2007), quant-ph/0611054

    Google Scholar 

  39. Jozsa, R.: Quantum factoring, discrete logarithms, and the hidden subgroup problem. Computing in Science and Engineering 3, 34–43 (2001), quant-ph/0012084

    Article  Google Scholar 

  40. Karchmer, M., Wigderson, A.: On Span Programs. In: Structure in Complexity Theory Conference 1993, pp. 102–111 (1993)

    Google Scholar 

  41. Kempe, J.: Quantum random walks - an introductory overview. Contemporary Physics 44(4), 307–327 (2003), quant-ph/0303081

    Article  MathSciNet  Google Scholar 

  42. Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Proceedings of ICALP (to appear, 2010), arxiv:1002.2419

    Google Scholar 

  43. Laplante, S., Magniez, F.: Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments. SIAM Journal on Computing 38(1), 46–62 (2008), Also CCC 2004 and quant-ph/0311189

    Article  MATH  MathSciNet  Google Scholar 

  44. Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of STOC 2007, pp. 575–584 (2007), quant-ph/0608026

    Google Scholar 

  45. Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM Journal on Computing 37(2), 413–424 (2007), quant-ph/0310134

    Article  MATH  MathSciNet  Google Scholar 

  46. Mosca, M., Ekert, A.: The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer. In: Williams, C.P. (ed.) QCQC 1998. LNCS, vol. 1509, pp. 174–188. Springer, Heidelberg (1999), quant-ph/9903071

    Chapter  Google Scholar 

  47. Nayak, A., Wu, F.: The quantum query complexity of approximating the median and related statistics. In: Proceedings of STOC 1999, pp. 384–393 (1999), quant-ph/9804066

    Google Scholar 

  48. Reichardt, B.: Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function. In: Proceedings of FOCS (2009), arXiv:0904.2759

    Google Scholar 

  49. Reichardt, B.: Span-program-based quantum algorithm for evaluating unbalanced formulas, arXiv:09071622

    Google Scholar 

  50. Reichardt, B.: Faster quantum algorithm for evaluating game trees, arXiv:0907.1623

    Google Scholar 

  51. Reichardt, B.: Reflections for quantum query algorithms, arxiv:1005.1601

    Google Scholar 

  52. Reichardt, B., Špalek, R.: Span-program-based quantum algorithm for evaluating formulas. In: Proceedings of STOC 2008, pp. 103–112 (2008), arXiv:0710.2630

    Google Scholar 

  53. Saks, M., Wigderson, A.: Probabilistic Boolean decision trees and the complexity of evaluating game trees. In: Proceedings of FOCS 1986, pp. 29–38 (1986)

    Google Scholar 

  54. Santha, M.: Quantum walk based search algorithms. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 31–46. Springer, Heidelberg (2008), arXiv:0808.0059

    Chapter  Google Scholar 

  55. Shewchuk, J.: An introduction to the conjugate gradient method without the agonizing pain. Technical Report CMU-CS-94-125, School of Computer Science, Carnegie Mellon University (1994)

    Google Scholar 

  56. Shor, P.: Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In: Proceedings of FOCS 1994, pp. 124–134 (1994), quant-ph/9508027

    Google Scholar 

  57. Snir, M.: Lower bounds on probabilistic linear decision trees. Theoretical Computer Science 38, 69–82 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  58. Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of FOCS 2004, pp. 32–41 (2004)

    Google Scholar 

  59. Tulsi, T.A.: Faster quantum walk algorithm for the two dimensional spatial search. Physical Review A 78, 012310 (2008), arXiv:0801.0497

    Article  Google Scholar 

  60. Venegas-Andrade, S.E.: Quantum Walks for Computer Scientists. Morgan and Claypool (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computing, University of Latvia, Raina bulv. 19, Riga, LV-1586, Latvia

    Andris Ambainis

Authors
  1. Andris Ambainis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Falculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Petr Hliněný

  2. Faculty of Informatics, Masaryk University, Botanicka 68a, 602 00, Brno, Czech Republic

    Antonín Kučera

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ambainis, A. (2010). New Developments in Quantum Algorithms. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15155-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15154-5

  • Online ISBN: 978-3-642-15155-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation