Michael Mc Gettrick
580 California St., Suite 400
San Francisco, CA, 94104
Alternate two-dimensional quantum walk with a single-qubit coin
2011, Physical Review A
https://doi.org/10.1103/PHYSREVA.84.042337
Sign up for access to the world's latest research
Abstract
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the non-localized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states, in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of x-y spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.
References (28)
- Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48, 1687 (1993).
- J. Kempe, Contemp. Phys. 44, 307327 (2003).
- M. Barber and B. W. Ninham, Random and Restricted Walks: Theory and Applications (Gordon and Breach, New York, 1970);
- B. Malkiel, A Random Walk Down Wall Street (W. W. Norton, New York, 1973);
- H. C. Berg, Random Walks in Biology (Princeton University Press, Princeton, NJ, 1993).
- A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
- R. P. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
- M. Karski et al., Science 325, 174 (2009);
- H. Schmitz et al., Phys. Rev. Lett. 103, 090504 (2009);
- A. Schreiber et al., Phys. Rev. Lett. 104, 050502 (2010);
- H. Schmitz et al., Phys. Rev. Lett. 103, 090504 (2009);
- F. Zähringer et al., Phys. Rev. Lett. 104, 100503 (2010);
- M. A. Broome et al., Phys. Rev. Lett. 104, 153602 (2010).
- S. E. Venegas-Andraca and S. Bose, arXiv:0901.3946 (2009);
- S. K. Goya and C. M. Chandrashekar, J. Phys. A 43, 235303 (2010).
- R. Horodecki et al., Rev. Mod. Phys. 81, 865 (2009).
- N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A 67, 052307 (2003);
- A. Ambainis, J. Kempe, and A. Rivosh, in Proc. 16th ACM-SIAM SODA, Vancouver (SIAM, Philadelphia, USA, 2005), p. 1099; A. Tulsi, Phys. Rev. A 78, 012310 (2008).
- C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011).
- N. Konno, Quantum Inf. Process. 1, 345 (2002);
- G. Grim- mett, S. Janson, and P. F. Scudo, Phys. Rev. E 69, 026119 (2004);
- N. Konno, J. Math. Soc. Jpn. 57, 1179 (2005).
- K. Watabe et al., Phys. Rev. A 77, 062331 (2008).
- Y. Ide, N. Konno, and T. Machida, arXiv:1012.4164 (2010).
- N. Inui, Y. Konishi, and N. Konno, Phys. Rev. A 69, 052323 (2004);
- M. Štefaňák, T. Kiss, and I. Jex, Phys. Rev. A 78, 032306 (2008).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, United Kingdom, 2000).
- S. Lee et al., Phys. Rev. A 68, 062304 (2003).