Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T07:48:24.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatial quantum search in a triangular network†

Published online by Cambridge University Press:  09 February 2012

G. ABAL ,
R. DONANGELO ,
M. FORETS  and
R. PORTUGAL
G. ABAL
Affiliation:
Instituto de Física, Facultad de Ingeniería, UdelaR, C.C. 30, C.P. 11300, Montevideo, Uruguay Email: abal@fing.edu.uy, donangel@fing.edu.uy, mforets@fing.edu.uy
R. DONANGELO
Affiliation:
Instituto de Física, Facultad de Ingeniería, UdelaR, C.C. 30, C.P. 11300, Montevideo, Uruguay Email: abal@fing.edu.uy, donangel@fing.edu.uy, mforets@fing.edu.uy
M. FORETS
Affiliation:
Instituto de Física, Facultad de Ingeniería, UdelaR, C.C. 30, C.P. 11300, Montevideo, Uruguay Email: abal@fing.edu.uy, donangel@fing.edu.uy, mforets@fing.edu.uy
R. PORTUGAL
Affiliation:
Laboratório Nacional de Computação Científica - LNCC, Av. Getúlio Vargas 333, Petrópolis, RJ, 25651-075, Brazil Email: portugal@lncc.br
Get access Rights & Permissions [Opens in a new window]

Abstract

The spatial search problem consists of minimising the number of steps required to find a given site in a network, with the restriction that only an oracle query or a translation to a neighbouring site is allowed at each step. We propose a quantum algorithm for the spatial search problem on a triangular lattice with N sites and torus-like boundary conditions. The proposed algorithm is a special case of the general framework for abstract search proposed by Ambainis, Kempe and Rivosh (AKR) in Ambainis et al. (2005) and Tulsi in Tulsi (2008) applied to a triangular network. The AKR–Tulsi formalism was employed to show that the time complexity of the quantum search on the triangular lattice is .

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 3 , June 2012 , pp. 521 - 531
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abal, G., Donangelo, R., Marquezino, F. and Portugal, R. (2010) Spatial search in a honeycomb network. Mathematical Structures in Computer Science 20 (6)999–1009.CrossRefGoogle Scholar
Ambainis, A. and Aaronson, S. (2003) Quantum search of spatial regions. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS) 200–209.Google Scholar
Ambainis, A., Kempe, J. and Rivosh, A. (2005) Coins make quantum walks faster. SODA '05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms 78 (2)1099–1108.Google Scholar
Benioff, P. (2002) Space searches with a quantum robot. In: Lomonaco, S. J. and Brandt, H. E. (eds.) Quantum Computation and Information, Contemporary Mathematics Series, AMS.Google Scholar
Grover, L. (1996) A fast quantum mechanical algorithm for database search. Proceedings 28th Annual ACM Symposium on the Theory of Computation, ACM Press 212–219.Google Scholar
Grover, L. (1997) Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters 78 (2)325–328.CrossRefGoogle Scholar
Kittel, C. (1995) Introduction to Solid State Physics, 7th edition, Wiley.Google Scholar
Krovi, H., Magniez, F., Ozols, M. and Roland, J. (2010) Finding is as easy as detecting for quantum walks. In: Abramsky, S., Gavoille, C., Kirchner, C., auf der Heide, F. M. and Spirakis, P. G. (eds.) Proceedings of the 37th international colloquium conference on Automata, languages and programming (ICALP'10). Springer-Verlag Lecture Notes in Computer Science 6198 540–551.CrossRefGoogle Scholar
Magniez, F., Nayak, A., Richter, P. and Santha, M. (2009) On the hitting times of quantum versus random walks. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09) 86–95.CrossRefGoogle Scholar
Shenvi, N., Kempe, J. and BirgittaWhaley, F. Whaley, F. (2003) A quantum random walk search algorithm. Physical Review A 67 052307.CrossRefGoogle Scholar
Tulsi, A. (2008) Faster quantum walk algorithm for the two dimensional spatial search. Physical Review A 78 (1) 012310-1–012310-6.CrossRefGoogle Scholar