Skip to main content
Log in

Measurement-Based Quantum Computation and Undecidable Logic

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs—describing the quantum correlations of the states—are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Gödel’s incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Feynman, R.P.: Opt. News 11, 11 (1985)

    Article  Google Scholar 

  2. Feynman, R.P.: Found. Phys. 16, 507 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  3. Feynman, R.P.: Int. J. Theor. Phys. 21, 467 (1982)

    Article  MathSciNet  Google Scholar 

  4. Shor, P.W.: SIAM J. Sci. Stat. Comput. 26, 1484 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Jozsa, R., Linden, N.: Proc. R. Soc. Lond. Ser. A 459, 2011 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Vidal, G.: Phys. Rev. Lett. 91, 147902 (2003)

    Article  ADS  Google Scholar 

  7. Markov, I., Shi, Y.: quant-ph/0511069

  8. Shi, Y., Duan, L., Vidal, G.: Phys. Rev. A 74, 022320 (2006)

    Article  ADS  Google Scholar 

  9. Jozsa, R.: quant-ph/0603163

  10. Yoran, N., Short, A.J.: Phys. Rev. Lett. 96, 170503 (2006)

    Article  ADS  Google Scholar 

  11. Gross, D., Eisert, J.: quant-ph/0609149

  12. Van den Nest, M., Miyake, A., Dür, W., Briegel, H.J.: Phys. Rev. Lett. 97, 150504 (2006)

    Article  Google Scholar 

  13. Van den Nest, M., Dür, W., Vidal, G., Briegel, H.J.: Phys. Rev. A 75, 012337 (2007)

    Article  ADS  Google Scholar 

  14. Raussendorf, R., Briegel, H.J.: Phys. Rev. Lett. 86, 5188 (2001)

    Article  ADS  Google Scholar 

  15. Raussendorf, R., Browne, D.E., Briegel, H.J.: Phys. Rev. A 68, 022312 (2003)

    Article  ADS  Google Scholar 

  16. Briegel, H.J., Raussendorf, R.: Phys. Rev. Lett. 86, 910 (2001)

    Article  ADS  Google Scholar 

  17. Hein, M., et al.: In: Proceedings of the International School of Physics Enrico Fermi on Quantum Computers, Algorithms and Chaos, Varenna, Italy, July 2005. quant-ph/0602096

  18. Gödel, K.: Mon. Math. Phys. 38, 173–198 (1931)

    Article  Google Scholar 

  19. Courcelle, B., Oum, S.-I.: J. Comb. Theory Ser. B 97(1), 91–126 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Courcelle, B.: In: Handbook of Graph Grammars and Computing by Graph Transformation 1, pp. 313–400. World Scientific, River Edge (1997)

    Google Scholar 

  21. Oum, S.-I.: PhD thesis, Princeton University (2005)

  22. Rabin, M.O.: Trans. Am. Math. Soc. 141, 1–35 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bravyi, S., Raussendorf, R.: Phys. Rev. A 76, 022304 (2007)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maarten Van den Nest.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van den Nest, M., Briegel, H.J. Measurement-Based Quantum Computation and Undecidable Logic. Found Phys 38, 448–457 (2008). https://doi.org/10.1007/s10701-008-9212-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-008-9212-6

Keywords

Navigation