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Topological features of good resources for measurement-based quantum computation

Published online by Cambridge University Press:  28 February 2013

DAMIAM MARKHAM,
JANET ANDERS,
MICHAL HAJDUŠEK and
VLATKO VEDRAL
DAMIAM MARKHAM
Affiliation:
CNRS, LTCI, Telecom ParisTech, 37/39 rue Dareau, 75014 Paris, France Email: markham@telecom-paristech.fr
JANET ANDERS
Affiliation:
Department of Physics & Astronomy, University College London, London WC1E 6BT, United Kingdom Email: janet@qipc.org
MICHAL HAJDUŠEK
Affiliation:
The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom, and Department of Physics, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan, 113-0033 Email: michal@eve.phys.s.u-tokyo.ac.jp
VLATKO VEDRAL
Affiliation:
Centre for Quantum Technologies, National University of Singapore, Singapore, and Department of Physics, National University of Singapore, Singapore, and Clarendon Laboratory, University of Oxford, Oxford, United Kingdom Email: vlatko.vedral@qubit.org
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Abstract

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We study how graph states on fractal lattices can be used to perform measurement-based quantum computation, and investigate which topological features allow this application. We find fractal lattices of arbitrary dimension greater than one that all act as good resources for measurement-based quantum computation, and sets of fractal lattices with dimension greater than one that do not. The difference is put down to other topological factors such as ramification and connectivity. This is in direct analogy to the tendency of lattices to observe criticality in spin systems. We also discuss the analogy between thermodynamics and one-way computation in this context. This work adds confidence to the analogy and highlights new features of what we require for universal resources for measurement-based quantum computation. This paper is an extended version of Markham et al. (2010), which appeared in the proceedings of DCM 2010.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Special Issue 2: Developments In Computational Models 2010 , April 2013, pp. 441 - 453
Copyright
Copyright © Cambridge University Press 2013

References

Anders, J., Hajdušek, M., Markham, D. and Vedral, V. (2008) How much of one-way computation is just thermodynamics? Foundations of Physics 38 (6)506–522.CrossRefGoogle Scholar
Browne, D. E., Kashefi, E., Mhalla, M. and Perdrix, S. (2007) Generalized Flow and Determinism in Measurement-based Quantum Computation. New Journal of Physics 9 (250).CrossRefGoogle Scholar
Browne, D. E., Elliott, M. B., Flammia, S. T., Merkel, S. T., Miyake, A. and Short, A. J. (2008) Phase transition of computational power in the resource states for one-way quantum computation. New Journal of Physics 10 (023010).CrossRefGoogle Scholar
Danos, V. and Kashefi, E. (2006) Determinism in the one-way model. Physical Review A 74 (052310).CrossRefGoogle Scholar
van den Nest, M., Miyake, A., Dür, W. and Briegel, H. J. (2006) Universal resources for measurement-based quantum computation. Physical Review Letters 97 (150504).CrossRefGoogle ScholarPubMed
van den Nest, M., Dür, W., Miyake, A. and Briegel, H. J. (2007) Fundamentals of universality in one-way quantum computation. New Journal of Physics 9 (204).CrossRefGoogle Scholar
Falconer, K. (1990) Fractal Geometry, Mathematical Foundations and Applications, John Wiley and Sons.Google Scholar
Gefen, Y., Mandlebrot, B. B. and Aharony, A. (1980) Critical Phenomena on Fractal Lattices. Physical Review Letters 45 (855).CrossRefGoogle Scholar
Griffiths, R. B. (1964) Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet. Physical Review 136 (2A)437–439.CrossRefGoogle Scholar
Gross, D., Flammia, S. and Eisert, J. (2009) Most quantum states are too entangled to be useful as computational resources. Physical Review Letters 102 (190501).CrossRefGoogle ScholarPubMed
Hein, M., Dür, W., Raussendorf, J. E. R., van den Nest, M. and Briegel, H. J. (2005) Entanglement in Graph States and its Applications. Available at quant-ph/0602096.Google Scholar
Horodecki, R., Horodecki, P., Horodecki, M. and Horodecki, K. (2009) Quantum Entanglement. Reviews of Modern Physics 81 (2)865–942.CrossRefGoogle Scholar
Markham, D., Miyake, A. and Virmani, S. (2007) Entanglement and local information access for graph states. New Journal of Physics 9 (194).CrossRefGoogle Scholar
Markham, D., Anders, J., Hajdušek, M. and Vedral, V. (2010) Measurement Based Quantum Comutation on Fractal Lattices. Electronic Proceedings in Theoretical Computer Science 26 109–115.CrossRefGoogle Scholar
Nielsen, M. A. and Chuang, I. L. (2000) Quantum Computation and Quantum Information, Cambridge University Press.Google Scholar
Peierls, R. (1936) On the Ising model of ferromagnetism. Proceedings of the Cambridge Philosophical Society 32 477–481.CrossRefGoogle Scholar
Raussendorf, R. and Briegel, H. J. (2001) A One-Way Quantum Computer. Physical Review Letters 86 5188–5191.CrossRefGoogle ScholarPubMed
Teplyaev, A. (2005) Harmonic coordinates on fractals with finitely ramified cell structure. Available at arXiv:math/0506261.Google Scholar
Toffoli, T. (1998) How much of physics is just computation? Superlattices and Microstructures 23 381–406.CrossRefGoogle Scholar
Vedral, V. and Plenio, M. B. (1998) Entanglement measures and purification procedures. Physical Review A 57 1619–1633.CrossRefGoogle Scholar
Virmani, S. and Plenio, M. (2005) An introduction to entanglement measures. Available at quant-ph/0504163.Google Scholar