Abstract
In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its 2l nearest neighbors (l on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.
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Feynman R.P., Leighton R.B., Sands M.: Feynman Lectures on Physics. Addison Wesley, Boston (1964)
Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC’01), pp. 37–49. ACM Press, New York (2001)
Konno N.: One-dimensional discrete-time quantum walks on random environments. Quant. Inf. Process. 8(5), 387–399 (2009)
Krovi H., Brun T.A.: Quantum walks on quotient graphs. Phys. Rev. A 75, 062332 (2007)
Mülken O., Blumen A.: Continuous-time quantum walks in phase space. Phys. Rev. A 73, 012105 (2006)
Chandrashekar, C.M.: Discrete time quantum walk model for single and entangled particles to retain entanglement in coin space. arXiv: quant-ph/0609113V4 (2006)
Gottlieb A.D.: Convergence of continuous-time quantum walks on the line. Phys. Rev. E 72, 047102 (2005)
Avraham D., Bollt E., Tamon C.: One-dimensional continuous-time quantum walks. Quant. Inf. Process. 3, 295 (2004)
Salimi S.: Continuous-time quantum walks on star graphs. Ann. Phys. 324, 1185–1193 (2009)
Xu X.: Exact analytical results for quantum walks on star graph. J. Phys. A. Math. Theor. 42, 115205 (2009)
Salimi S., Jafarizadeh M.: Continuous-time classical and quantum random walk on direct product of Cayley graphs. Commun. Theor. Phys. 51, 1003–1009 (2009)
Salimi S.: Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quant. Inf. Process. 9, 75–91 (2008)
Salimi S.: Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory. Int. J. Theor. Phys. 47, 3298–3309 (2008)
Jafarizadeh M.A., Salimi S.: Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322, 1005–1033 (2007)
Konno N.: Continuous-time quantum walks on trees in quantum probability theory. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 9(2), 287–297 (2006)
Xu X.: Coherent exciton transport and trapping on long-range interacting cycles. Phys. Rev. E 79, 011117 (2009)
Konno N.: Continuous-time quantum walks on ultrametric spaces. Int. J. Quant. Inf. 4(6), 1023–1035 (2006)
Drezgić M., Hines A.P., Sarovar M., Sastry Sh.: Complete Characterization of mixing time for the continuous quantum walk on the hypercube with Markovian decoherence model. Quant. Inf. Comput 9, 854 (2009)
Kendon V.: Decoherence in quantum walks—a review. Math. Struct. Comput. Sci. 17(6), 1169–1220 (2006)
Strauch F.W.: Reexamination of decoherence in quantum walks on the hypercube. Phys. Rev. A 79, 032319 (2009)
Alagic G., Russell A.: Decoherence in quantum walks on the hypercube. Phys. Rev. A 72, 062304 (2005)
Romanelli A., Siri R., Abal G., Auyuanet A., Donangelo R.: Decoherence in the quantum walk on the line. J. Phys. A 347, 137–152 (2005)
Kendon V., Tregenna B.: Decoherence can be useful in quantum walks. Phy. Rev. A 67, 042315 (2003)
Salimi S., Radgohar R.: Mixing and decoherence in continuous-time quantum walks on long-range interacting cycles. J. Phys. A Math. Theor. 42, 475302 (2009)
Salimi S., Radgohar R.: The effect of large decoherence on mixing time in continuous-time quantum walks on long-range interacting cycles. J. Phys. B. At. Mol. Opt. Phys. 43, 025503 (2010)
Fedichkin L., Solenov D., Tamon C.: Mixing and decoherence in continuous-time quantum walks on cycles. Quant. Inf. Comput. 6(3), 263–276 (2006)
Dür W.: Quantum walks in optical lattices. Phys. Rev. A 66, 052319 (2002)
Côté R.: Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys. 8, 156 (2006)
Salimi S., Radgohar R.: The effect of decoherence on mixing time in continuous-time quantum walks on one-dimensional regular networks. Int. J. Quant. Inf. 8(5), 795–806 (2010)
Xu X.: Continuous-time quantum walks on one-dimensional regular networks. Phys. Rev. E 77, 061127 (2008)
Strogatz S.H., Stewart I.: Coupled oscillators and biological synchronization. Sci. Am. 269, 102 (1993)
Wiesenfeld K.: New results on frequency-locking dynamics of disordered Josephson arrays. Phys. B 222, 315 (1996)
Belykh I.V., Belykh V.N., Hasler M.: Connection graph stability method for synchronized coupled chaotic systems. Phys. D 195, 159–187 (2004)
Watts D.J., Strogatz S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998)
Childs A.M., Goldstone J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314–022324 (2004)
Mülken O., Blumen A.: Slow transport by continuous time quantum walks. Phys. Rev. E 71, 016101–016106 (2005)
Volta A., Mülken O., Blumen A.: Quantum transport on two-dimensional regular graphs. J. Phys. A 39, 14997–15012 (2006)
Montroll E.W., Weiss G.H.: Random walks on lattices. II. J. Math. Phys 6, 167–181 (1965)
Childs A.M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quant. Inf. Process. 1, 35 (2002)
Mülken O., Blumen A.: Spacetime structures of continuous-time quantum walks. Phys. Rev. E 71, 036128 (2005)
Ziman J.M.: Principles of the Theory of Solids. Cambridge University Press, Cambridge (1972)
Solenov D., Fedichkin L.: Continuous-time quantum walks on a cycle graph. Phys. Rev. A 73, 012313 (2006)
de la Torre A.C., Mártin H.O., Goyeneche D.: Quantum diffusion on a cyclic one-dimensional lattice. Phys. Rev. E 68, 031103 (2003)
Pioro-Ladriere M., Abolfath R., Zawadzki P., Lapointe J., Studenikin S.A., Sachrajda A.S., Hawrylak P.: Charge sensing of an artificial H +2 molecule in lateral quantum dots. Phys. Rev. B 72, 125307 (2005)
Hines A.P., Stamp P.C.E.: Quantum walks, quantum gates, and quantum computers. Phys. Rev. A 75, 062321 (2007)
Gurvitz S.A.: Rate equations for quantum transport in multidot systems. Phys. Rev. B 57, 6602 (1998)
Gurvitz S.A.: Measurements with a noninvasive detector and dephasing mechanism. Phys. Rev. B 56, 15215 (1997)
Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of ACM Symposium on Theory of Computation (STOC 01), pp. 50–59 (2001)
Arfken G.B., Weber H.J.: Mathematical Methods for Physicists, Chapter 5. Harcourt Academic Press, San Diego (1972)
Gurvitz S.A.: Quantum description of classical apparatus: zeno effect and decoherence. Quant. Inf. Process. 2, 15 (2003)
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Radgohar, R., Salimi, S. Mixing-time and large-decoherence in continuous-time quantum walks on one-dimension regular networks. Quantum Inf Process 12, 303–320 (2013). https://doi.org/10.1007/s11128-012-0377-8
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DOI: https://doi.org/10.1007/s11128-012-0377-8