Weak limits for quantum random walks

Geoffrey Grimmett; Svante Janson; Petra F. Scudo

doi:10.1103/PhysRevE.69.026119

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Weak limits for quantum random walks

Geoffrey Grimmett^1,*, Svante Janson^2,†, and Petra F. Scudo^1,3,‡

¹Statistical Laboratory, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
²Department of Mathematics, Uppsala University, P.O. Box 480, S-751 06 Uppsala, Sweden
³Department of Physics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

^*Electronic address: g.r.grimmett@statslab.cam.ac.uk
^†Electronic address: svante.janson@math.uu.se
^‡Electronic address: p.scudo@statslab.cam.ac.uk

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Phys. Rev. E 69, 026119 – Published 27 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.026119

Abstract

We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With $X_{n}$ denoting position at time n, we show that $X_{n} / n$ converges weakly as $\vec{n} \infty$ to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.

References (7)

J. Kempe, Contemp. Phys. 44, 307 (2003).
N. Konno, Quantum Inf. Process. 1, 345 (2002).
N. Konno, e-print quant-ph/0206103.
A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 37–49 (unpublished).
G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, Oxford, 2001).
P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).
A.D. Gottlieb, e-print quant-ph/0310026.

[c1] J. Kempe, Contemp. Phys. 44, 307 (2003).

[c2] N. Konno, Quantum Inf. Process. 1, 345 (2002).

[c3] N. Konno, e-print quant-ph/0206103.

[c4] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 37–49 (unpublished).

[c5] G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University Press, Oxford, 2001).

[c6] P. Billingsley, Probability and Measure, 3rd ed. (Wiley, New York, 1995).

[c7] A.D. Gottlieb, e-print quant-ph/0310026.

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Weak limits for quantum random walks

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References (7)

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