Directional quantum random walk induced by coherence

Abstract

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.

Keywords

quantum walk / random walk / ensemble interpretation / directional walking / coherence

References

[1] N. van Kampen, in: Stochastic Processes in Physics and Chemistry, 3rd Ed., North-Holland Personal Library, edited by N. V. Kampen, Elsevier, Amsterdam, 2007, p.ix

[2] Y. Aharonov, L. Davidovich, and N. Zagury, Quantum random walks, Phys. Rev. A 48(2), 1687 (1993)

[3] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing, ACM Press, 2001

[4] V. Kendo, Decoherence in quantum walks – a review, Math. Struct. Comput. Sci. 17(6), 1169 (2007)

[5] S. E. Venegas-Andraca, Quantum walks: A comprehensive review, Quantum Inform. Process. 11(5), 1015 (2012)

[6] G. Grimmett, S. Janson, and P. F. Scudo, Weak limits for quantum random walks, Phys. Rev. E 69(2), 026119 (2004)

[7] G. Abal, R. Siri, A. Romanelli, and R. Donangelo, Quantum walk on the line: Entanglement and nonlocal initial conditions, Phys. Rev. A 73(4), 042302 (2006)

[8] L. Ermann, J. P. Paz, and M. Saraceno, Decoherence induced by a chaotic enviroment: A quantum walker with a complex coin, Phys. Rev. A 73(1), 012302 (2006)

[9] N. Shenvi, J. Kempe, and K. B. Whaley, Quantum random-walk search algorithm, Phys. Rev. A 67(5), 052307 (2003)

[10] A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett. 102(18), 180501 (2009)

[11] N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev. A 81(4), 042330 (2010)

[12] P. Witthaut, Quantum walks and quantum simulations with Bloch-oscillating spinor atoms, Rev. A 82(3), 033602 (2010)

[13] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, Environment-assisted quantum walks in photosynthetic energy transfer, J. Chem. Phys. 129(17), 174106 (2008)

[14] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Observation of topologically protected bound states in photonic quantum walks, Nat. Commun. 3(1), 882 (2012)

[15] K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, and P. Xue, Simulating dynamic quantum phase transitions in photonic quantum walks, Phys. Rev. Lett. 122(2), 020501 (2019)

[16] J. Z. Wu, W. W. Zhang, and B. C. Sanders, Topological quantum walks: Theory and experiments, Front. Phys. 14(6), 61301 (2019)

[17] T. A. Brun, H. A. Carteret, and A. Ambainis, Quantum walks driven by many coins, Phys. Rev. A 67(5), 052317 (2003)

[18] T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, Quantum walks in higher dimensions, J. Phys. Math. Gen. 35(12), 2745 (2002)

[19] A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, Photons walking the line: A quantum walk with adjustable coin operations, Phys. Rev. Lett. 104(5), 050502 (2010)

[20] S. Panahiyan and S. Fritzsche, Controlling quantum random walk with a step-dependent coin, New J. Phys. 20(8), 083028 (2018)

[21] M. Karski, L. Forster, J. M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Quantum walk in position space with single optically trapped atoms, Science 325(5937), 174 (2009)

[22] F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Realization of a quantum walk with one and two trapped ions, Phys. Rev. Lett. 104(10), 100503 (2010)

[23] H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Quantum walk of a trapped ion in phase space, Phys. Rev. Lett. 103(9), 090504 (2009)

[24] P. Xue, B. C. Sanders, and D. Leibfried, Quantum walk on a line for a trapped ion, Phys. Rev. Lett. 103(18), 183602 (2009)

[25] M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, Discrete single-photon quantum walks with tunable decoherence, Phys. Rev. Lett. 104(15), 153602 (2010)

[26] A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, Quantum walks of correlated photons, Science 329(5998), 1500 (2010)

[27] H. Tang, X. F. Lin, Z. Feng, J. Y. Chen, J. Gao, K. Sun, C. Y. Wang, P. C. Lai, X.-Y. Xu, Y. Wang, L. F. Qiao, A. L. Yang, and X. M. Jin, Experimental twodimensional quantum walk on a photonic chip, Sci. Adv. 4(5), eaat3174 (2018)

[28] Z. Yan, Y. R. Zhang, M. Gong, Y. Wu, Y. Zheng, S. Li, C. Wang, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, C. Z. Peng, K. Xia, H. Deng, H. Rong, J. Q. You, F. Nori, H. Fan, X. Zhu, and J. W. Pan, Strongly correlated quantum walks with a 12-qubit superconducting processor, Science 364(6442), 753 (2019)

[29] T. A. Brun, H. A. Carteret, and A. Ambainis, Quantum random walks with decoherent coins, Phys. Rev. A 67(3), 032304 (2003)

[30] T. A. Brun, H. A. Carteret, and A. Ambainis, Quantum to classical transition for random walks, Phys. Rev. Lett. 91(13), 130602 (2003)

[31] K. Zhang, Limiting distribution of decoherent quantum random walks, Phys. Rev. A 77(6), 062302 (2008)

[32] J. D. Whitfield, C. A. Rodríguez-Rosario, and A. Aspuru-Guzik, Quantum stochastic walks: A generalization of classical random walks and quantum walks, Phys. Rev. A 81(2), 022323 (2010)

[33] J. Košík, V. Bužek, and M. Hillery, Quantum walks with random phase shifts, Phys. Rev. A 74(2), 022310 (2006)

[34] P. Ribeiro, P. Milman, and R. Mosseri, Aperiodic quantum random walks, Phys. Rev. Lett. 93(19), 190503 (2004)

[35] L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79(2), 325 (1997)

RIGHTS & PERMISSIONS

2020 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature