Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T02:51:09.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum walks and elliptic integrals

Published online by Cambridge University Press:  08 November 2010

NORIO KONNO
NORIO KONNO*
Affiliation:
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan Email: konno@ynu.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 6: Quantum Algorithms , December 2010 , pp. 1091 - 1098
Copyright
Copyright © Cambridge University Press 2010

References

Ambainis, A., Bach, E., Nayak, A., Vishwanath, A. and Watrous, J. (2001) One-dimensional quantum walks. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing 37–49.CrossRefGoogle Scholar
Andrews, G. E., Askey, R. and Roy, R. (1999) Special Functions, Cambridge University Press.CrossRefGoogle Scholar
Cantero, M. J., Grünbaum, F. A., Moral, L. and Velázquez, L. (2010) Matrix valued Szegő polynomials and quantum random walks. Comm. Pure Appl. Math. 63 464–507.CrossRefGoogle Scholar
Cantero, M. J., Moral, L. and Velázquez, L. (2003) Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362 29–56.CrossRefGoogle Scholar
Childs, A. M. (2010) On the relationship between continuous- and discrete-time quantum walk. Commun. Math. Phys. 294 581–603.CrossRefGoogle Scholar
Durrett, R. (2004) Probability: Theory and Examples, third edition, Brooks-Cole.Google Scholar
Kempe, J. (2003) Quantum random walks – an introductory overview. Contemporary Physics 44 307–327.CrossRefGoogle Scholar
Kendon, V. (2007) Decoherence in quantum walks – a review. Mathematical Structures in Computer Science 17 1169–1220.CrossRefGoogle Scholar
Konno, N. (2008) Quantum Walks. In: Franz, U. and Schürmann, M. (eds.) Quantum Potential Theory. Springer-Verlag Lecture Notes in Mathematics 1954 309–452.CrossRefGoogle Scholar
Konno, N. (2002) Quantum random walks in one dimension. Quantum Inf. Proc. 1 345–354.CrossRefGoogle Scholar
Konno, N. (2005) A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57 1179–1195.CrossRefGoogle Scholar
Pólya, G. (1921) Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt im straßennetz. Math. Ann. 84 149–160.CrossRefGoogle Scholar
Simon, B. (2007) CMV matrices: Five years after. J. Comput. Appl. Math. 208 120–154.CrossRefGoogle Scholar
Spitzer, F. (1976) Principles of Random Walks, second edition, Springer-Verlag.CrossRefGoogle Scholar
Strauch, F. W. (2006) Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A 73 030301.CrossRefGoogle Scholar
Venegas-Andraca, S. E. (2008) Quantum Walks for Computer Scientists, Morgan and Claypool.CrossRefGoogle Scholar
Watson, G. N. (1939) Three triple integrals. Oxford Qu. J. of Math. 10 266–276.CrossRefGoogle Scholar