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Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm

  • Mathematical Structures in Statistical and Condensed Matter Physics
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Abstract.

In papers [Jafarizadehn and Salimi, Ann. Phys. 322, 1005 (2007) and J. Phys. A: Math. Gen. 39, 13295 (2006)], the amplitudes of continuous-time quantum walk (CTQW) on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated with their adjacency matrix. Here in this paper, it is shown that the CTQW on any arbitrary graph can be investigated by spectral analysis method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition (GQD) have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm. Moreover, it is shown that probability amplitude of observing the walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. The capability of Lanczos-based algorithm for evaluation of CTQW on graphs (GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at the limit of the large number of vertices, are in agreement with those of central limit theorem of [Phys. Rev. E 72, 026113 (2005)]. At the end, some applications of the method such as implementation of quantum search algorithms, calculating the resistance between two nodes in regular networks and applications in solid state and condensed matter physics, have been discussed, where in all of them, the Lanczos algorithm, reduces the Hilbert space to some smaller subspaces and the problem is investigated in the subspace with maximal dimension.

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References

  • M.A. Jafarizadehn, S. Salimi, Ann. Phys. 322, 1005 (2007)

    Article  ADS  Google Scholar 

  • M.A. Jafarizadeh, S. Salimi, J. Phys. A: Math. Gen. 39, 13295 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • N. Konno, Phys. Rev. E 72, 026113 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  • E. Farhi, M. Childs, S. Gutmann, Quant. Inf. Process. 1, 35 (2002)

    Article  MathSciNet  Google Scholar 

  • A. Ambainis, E. Bach, A. Nayak, A. Viswanath, J. Watrous, One-Dimensional Quantum Walks, in Proceedings of the 33rd ACM Annual Symposium on Theory Computing (ACM Press, 2001), p. 60

  • D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, in Proceedings of the 33rd ACM Annual Symposium on Theory Computing (ACM Press, New York, 2001)

  • C. Moore, A. Russell, in Proceedings of the 6th Int. Workshop on Randomization and Approximation in Computer Science (RANDOM'02, 2002)

  • J. Kempe, Discrete Quantum Random Walks Hit Exponentially Faster, Proceedings of 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM'03, 2003), p. 354–69

  • W. Dur, R. Raussendorf, V. Kendon, H. Briegel, Phys. Rev. A 66, 052319 (2002)

    Article  ADS  Google Scholar 

  • B. Sanders, S. Bartlett, B. Tregenna, P. Knight, Phys. Rev. A 67, 042305 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  • J. Du, X. Xu, M. Shi, J. Wu, X. Zhou, R. Han, Phys. Rev. A 67, 042316 (2003)

    Article  ADS  Google Scholar 

  • G.P. Berman, D.I. Kamenev, R.B. Kassman, C. Pineda, V.I. Tsifrinovich, Int. J. Quant. Inf. 1, 51 (2003)

    Article  MATH  Google Scholar 

  • E.B. Feldman, R. Bruschweiler, R.R. Ernst, Chem. Phys. Lett. 249, 297 (1998); H.M. Pawstawski, G. Usaj, P.R. Levstein, Chem. Phys. Lett. 261, 329 (1996)

    Article  ADS  Google Scholar 

  • Z.L. Madi, B. Brutscher, T. Schulte-Herbuggen, R. Bruschweiler, R.R. Ernst, Chem. Phys. Lett. 268, 300 (1997)

    Article  ADS  Google Scholar 

  • P.L. Knight, E. Roldan, E. Sipe, Phys. Rev. A 68 020301(R) (2003); P.L. Knight, E. Roldan, E. Sipe, Optics Comm. 227, 147 (2003)

    Article  ADS  Google Scholar 

  • H. Jeong, M. Paternostro, M.S. Kim, Phys. Rev. A 69, 012320 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  • R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1965), Vol. 3

  • Y. Aharonov, L. Davidovich, N. Zagury, Phy. Rev. A 48, 1687 (1993)

    Article  ADS  Google Scholar 

  • A. Childs, E. Deotto, R. Cleve, E. Farhi, S. Gutmann, D. Spielman, in Proc. 35th Ann. Symp. Theory of Computing (ACM Press, 2003), p. 59

  • N. Obata, Interdiscipl. Inf. Sci. 10, 41 (2004)

    MATH  MathSciNet  Google Scholar 

  • A. Hora, N. Obata, Quantum Information V, (World Scientific, Singapore, 2002)

  • B. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall Inc., Englewood Cliffs, N.J., 1980)

  • J. Wilkkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1997)

  • L. Trefethen, D. Bau, Numerical Linear Algebra (Society for Industrial and Applied Mathematics (SIMA), Philadelphia, PA., 1997)

  • J. Cullum, R. Willoughby, L \(\acute{a}\) nczos Algorithems for Large Symmetric Eigenvalue Computations (Birkhäuser Boston Inc, Boston, MA, 1985, Theory), Vol. I

  • R.A. Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics (Cambridge University Press, Cambridge, 2004)

  • L.D. Zusman, Chem. Phys. 49, 295, (1980)

    Article  MathSciNet  ADS  Google Scholar 

  • Y. Jung, R.J. Silbey, J. Cao, e-print arXiv:physics/0008164

  • J. Cao, Y. Jung, J. Chem. Phys. (in press, 2000)

  • C.E. Porter, Statistical theories of spectra: fluctuations (Academic Press, New York, 1965)

  • M.L. Mehta, Random matrices, 2nd edn. (Academic Press, New York, 1991)

  • O. Bohigas, In Chaos and Quantum Physics, Proceedings of the Les Houges Summer School of Theoretical Physics, edited by M.J. Giannoni, A. Voros, J. Zinn-Justin (Elsevier, New York, 1991)

  • T. Guhr, A. Muller-Groeling, H.A. Weidenmuller, Phys. Rep. 299, 190 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  • D.V. Voiculescu, Invent. Math. 104, 201 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • H. Cycon, R. Forese, W. Kirsch, B. Simon, Schrödinger operators (Springer-Verlag, 1987)

  • P.D. Hislop, I.M. Sigal, Introduction to Spectral Theory With Applications to Schrödinger Operators, Applied Mathematical Sciences, Vol. 113 (Springer, 1996)

  • Y.S. Kim, Phase space picture of quantum mechanics:group theoretical approach (Science, 1991)

  • H.W. Lee, Phys. Rep. 259, 147 (1995)

    Article  MathSciNet  Google Scholar 

  • J.A. Shohat, J.D. Tamarkin, The Problem of Moments, American Mathematical Society (Providence, RI, 1943)

  • T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, Science Publishers Inc., 1978)

  • A. Hora, N. Obata, Fundamental Problems in Quantum Physics (World Scientific, 2003), Vol. 284

  • G.H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994)

  • O. Mulken, H. van Beijeren, Phys. Rev. E 69, 046203 (2004)

    Article  ADS  Google Scholar 

  • K.A. Eriksen, I. Simonsen, S. Maslov, K. Sneppen, Phys. Rev. Lett. 90, 148701 (2003)

    Article  ADS  Google Scholar 

  • I. Simonsen, K.A. Eriksen, S. Maslov, K. Sneppen, Physica A 336, (2003)

  • N. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1990)

  • L.K. Grover, Phys. Rev. Lett. 79, 325 (1997)

    Article  ADS  Google Scholar 

  • E. Farhi, S. Gutmann, Phys. Rev. A 57, 2403 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  • A.M. Childs, J. Goldstone, Phys. Rev. A 70, 022314 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  • J. Cserti, Am. J. Phys. 68, 896 (2000); Preprint arXiv:cond-mat/9909120

    Article  ADS  Google Scholar 

  • S. Kakutani, Proc. Jap. Acad. 21, 227 (1945)

    Article  MathSciNet  Google Scholar 

  • J.G. Kemeny, J.L. Snell, A.W. Knapp, Denumerable Markov Chains (Springer, New York, 1976)

  • F. Kelly, Reversibility and Stochastic Networks (Wiley, New-York, 1979)

  • P.G. Doyle, J.L. Snell, e-print arXiv:math-pr/0001057

  • B. Tadic, V. Priezzhev, e-print arXiv:cond-mat/0207100

  • F.Y. Wu, J. Phys. A: Math. Gen. 37, 6653 (2004)

    Article  MATH  ADS  Google Scholar 

  • N.W. Ashcroft, N.D. Mermin, Solid State Physics (Holt, Reihart and Winston, New York, 1976)

  • P.F. Buonsante, R. Burioni, D. Cassi, I. Meccoli, S. Regina, A. Vezzani, Physica A 280, 131 (2000)

    Article  ADS  Google Scholar 

  • E.N. Economou, Greens Functions in Quantum Physics (Springer-Verlag, Berlin, 1979)

  • J.P. Keating, N. Linden, J.C.F. Matthews, A. Winter, e-print arXiv:quant-ph/0606205

  • B. Georgeot, D.L. Shepelyansky, e-print arXiv:quant-ph/9909074

  • B. Georgeot, D.L. Shepelyansky, e-print arXiv:quant-ph/0006073

  • P. Terwilliger, J. Algebraic Combin. 1, 363 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz, 51664, Iran

    M. A. Jafarizadeh, R. Sufiani & S. Salimi

  2. Institute for Studies in Theoretical Physics and Mathematics, Tehran, 19395-1795, Iran

    M. A. Jafarizadeh, R. Sufiani & S. Salimi

  3. Research Institute for Fundamental Sciences, Tabriz, 51664, Iran

    M. A. Jafarizadeh

  4. Department of Electrical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran

    S. Jafarizadeh

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  1. M. A. Jafarizadeh
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  2. R. Sufiani
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  3. S. Salimi
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  4. S. Jafarizadeh
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Correspondence to M. A. Jafarizadeh.

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Jafarizadeh, M., Sufiani, R., Salimi, S. et al. Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm . Eur. Phys. J. B 59, 199–216 (2007). https://doi.org/10.1140/epjb/e2007-00281-5

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