Computational Depth Complexity of Measurement-Based Quantum Computation

  • Dan Browne
  • Elham Kashefi
  • Simon Perdrix
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6519)

Abstract

In this paper, we mainly prove that the “depth of computations” in the one-way model is equivalent, up to a classical side-processing of logarithmic depth, to the quantum circuit model augmented with unbounded fanout gates. It demonstrates that the one-way model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out, for some specific problems, a depth separation between the one-way model and the quantum circuit model. Since one-way model has the same parallel power as unbounded quantum fan-out circuits, the quantum Fourier transform can be approximated in constant depth in the one-way model, and thus the factorisation can be done by a polytime probabilistic classical algorithm which has access to a constant-depth one-way quantum computer. The extra power of the one-way model, comparing with the quantum circuit model, comes from its classical-quantum hybrid nature. We show that this extra power is reduced to the capability to perform unbounded classical parity gates in constant depth.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AB09]
    Anders, J., Browne, D.E.: Computational Power of Correlations. Physical Review Letters 102, 050502 (2009)MathSciNetCrossRefGoogle Scholar
  2. [ADH97]
    Adleman, L., DeMarrais, J., Huang, M.: Quantum computability. SIAM Journal on Computing 26, 1524–1540 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. [BK07]
    Broadbent, A., Kashefi, E.: Parallelizing quantum circuits. To appear in Theoretical Computer Science (2007) (arXiv.org preprint 0704.1736)Google Scholar
  4. [DKP07]
    Danos, V., Kashefi, E., Panangaden, P.: The measurement calculus. J. ACM 54(2) (2007)Google Scholar
  5. [DKPP09]
    Danos, V., Kashefi, E., Panangaden, P., Perdrix, S.: Semantic Techniques in quantum Computation. In: Extended measurement calculus. Cambridge University Press, Cambridge (2010)Google Scholar
  6. [FFGHZ03]
    Fang, M., Fenner, S., Green, F., Homer, S., Zhang, Y.: Quantum lower bounds for fanout. Quantum Information and Computation 6, 46–57 (2003)MathSciNetMATHGoogle Scholar
  7. [GE07]
    Gross, D., Eisert, J.: Novel schemes for measurement-based quantum computation. Physical Review Letters 98, 220503 (2007)CrossRefGoogle Scholar
  8. [GHMP02]
    Green, F., Homer, S., Moore, C., Pollett, C.: Counting, fanout, and the complexity of quantum ACC. Quantum Information and Computation 2(1), 35–65 (2002)MathSciNetMATHGoogle Scholar
  9. [GN96]
    Griffiths, R.B., Niu, C.-s.: Semiclassical Fourier transform for quantum computation. Physical Review Letters 76, 3228–3231 (1996)CrossRefGoogle Scholar
  10. [HJ85]
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  11. [HS05]
    Høyer, P., Špalek, R.: Quantum fan-out is powerful. Theory of Computing 1(1), 81–103 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. [J05]
    Jozsa, R.: An introduction to measurement based quantum computation (2005), arXiv pre-print: quant-ph/0508124Google Scholar
  13. [KOBAA09]
    Kashefi, E., Oi, D.K.L., Browne, D., Anders, J., Andersson, E.: Twisted Graph States for Ancilla-driven Universal Quantum Computation. ENTCS 249, 307–331 (2009)MATHGoogle Scholar
  14. [MN98]
    Moore, C., Nilsson, M.: Parallel Quantum Computation and Quantum Codes (1998), arXiv pre-print:quant-ph/9808027v1 Google Scholar
  15. [MN02]
    Moore, C., Nilsson, M.: Parallel quantum computation and quantum codes. SIAM Journal on Computing 31(3), 799–815 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. [Nie03]
    Nielsen, M.A.: Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state. Phys. Rev. A 308, 96–100 (2003)CrossRefGoogle Scholar
  17. [Per05]
    Perdrix, S.: State transfer instead of teleportation in measurement-based quantum computation. International Journal of Quantum Information 3(1), 219–223 (2005)CrossRefGoogle Scholar
  18. [RB00]
    Raussendorf, R., Briegel, H.J.: Quantum computing via measurements only. Physical Review Letters 86, 5188–5191 (2001)CrossRefGoogle Scholar
  19. [RBB03]
    Raussendorf, R., Browne, D.E., Briegel, H.J.: Measurement-based quantum computation on cluster states. Physical Review A 68 (2003)Google Scholar
  20. [SB08]
    Shepherd, D., Bremner, M.J.: Instantaneous Quantum Computation (2008), arXiv pre-print:0809.0847v1 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dan Browne
    • 1
  • Elham Kashefi
    • 2
  • Simon Perdrix
    • 3
  1. 1.Department of Physics and AstronomyUniversity College LondonUK
  2. 2.Laboratory for Foundations of Computer ScienceUniversity of EdinburghUK
  3. 3.CNRS, Laboratoire d’Informatique de GrenobleGrenoble UniversityFrance

Personalised recommendations