Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda; Hironobu Sasaki; Etsuo Segawa; Akito Suzuki; Kanako Suzuki

doi:10.3934/dcds.2018159

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July 2018, 38(7): 3687-3703. doi: 10.3934/dcds.2018159

Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda ^1, , Hironobu Sasaki ^1, , Etsuo Segawa ^2, , Akito Suzuki ^3, and Kanako Suzuki ^4,

1.	Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan
2.	Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
3.	Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Nagano 380-8553, Japan
4.	College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan

Received December 2017 Published April 2018

Fund Project: M.M. was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851 and JP17H02853. H.S. was supported by JSPS KAKENHI Grant Number JP17K05311. E.S. acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637, No. 16K03939). A. S. was supported by JSPS KAKENHI Grant Number JP26800054. K.S acknowledges JSPS the Grant-in-Aid for Scientific Research (C) 26400156.

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QWs.

Keywords: Quantum walks, scattering theory, dispersive estimates, Strichartz estimates, nonlinear scattering.

Mathematics Subject Classification: Primary: 81Q05; Secondary: 35Q55.

Citation: Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159

References:

[1]	A. Ambainis, J. Kempe and A. Rivosh, Coins make quantum walks faster, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, 1099-1108.
[2]	A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, One-dimensional quantum walks, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, 37-49. doi: 10.1145/380752.380757.
[3]	P. Arnault and F. Debbasch, Quantum walks and discrete gauge theories, Phys. Rev. A, 93 (2016), 052301. doi: 10.1103/PhysRevA.93.052301.
[4]	P. Arnault, G. DiMolfetta, M. Brachet and F. Debbasch, Quantum walks and non-abelian discrete gauge theory, Phys. Rev. A, 94 (2016), 012335, 6pp. doi: 10.1103/PhysRevA.94.012335.
[5]	J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B, 88 (2013), 121406.
[6]	J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
[7]	R. Carles and I. Gallagher, Analyticity of the scattering operator for semilinear dispersive equations, Comm. Math. Phys., 286 (2009), 1181-1209. doi: 10.1007/s00220-008-0599-x.
[8]	C. Cedzich, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner and R. F. Werner, Bulk-edge correspondence of one-dimensional quantum walks, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 21LT01, 12pp. doi: 10.1088/1751-8113/49/21/21LT01.
[9]	A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett., 102 (2009), 180501, 4pp. doi: 10.1103/PhysRevLett.102.180501.
[10]	T. Endo, N. Konno, H. Obuse and E. Segawa, Sensitivity of quantum walks to a boundary of two-dimensional lattices: Approaches based on the cgmv method and topological phases, Journal of Physics A: Mathematical and Theoretical, 50 (2017), 455302, 40pp.
[11]	R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed., Dover Publications, Inc., Mineola, NY, 2010, Emended and with a preface by Daniel F. Styer.
[12]	Y. Gerasimenko, B. Tarasinski and C. W. J. Beenakker, Attractor-repeller pair of topological zero modes in a nonlinear quantum walk, Phys. Rev. A, 93 (2016), 022329. doi: 10.1103/PhysRevA.93.022329.
[13]	L. Grafakos, Classical Fourier Analysis, second ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008.
[14]	G. Grimmett, S. Janson and P. F. Scudo, Weak limits for quantum random walks, Phys. Rev. E, 69 (2004), 026119. doi: 10.1103/PhysRevE.69.026119.
[15]	D. Gross, V. Nesme, H. Vogts and R.F. Werner, Index theory of one dimensional quantum walks and cellular automata, Communications in Mathematical Physics, 310 (2012), 419-454. doi: 10.1007/s00220-012-1423-1.
[16]	S. P. Gudder, Quantum Probability, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988.
[17]	M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede and A. Widera, Quantum walk in position space with single optically trapped atoms, Science, 325 (2009), 174-177. doi: 10.1126/science.1174436.
[18]	A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics, 321 (2006), 2-111, January Special Issue. doi: 10.1016/j.aop.2005.10.005.
[19]	T. Kitagawa, Topological phenomena in quantum walks: Elementary introduction to the physics of topological phases, Quantum Information Processing, 11 (2012), 1107-1148. doi: 10.1007/s11128-012-0425-4.
[20]	T. Kitagawa, M. S. Rudner, E. Berg and E. Demler, Exploring topological phases with quantum walks, Phys. Rev. A, 82 (2010), 033429. doi: 10.1103/PhysRevA.82.033429.
[21]	C. -W. Lee, P. Kurzyński and H. Nha, Quantum walk as a simulator of nonlinear dynamics: Nonlinear dirac equation and solitons, Phys. Rev. A, 92 (2015), 052336. doi: 10.1103/PhysRevA.92.052336.
[22]	M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Weak Limit Theorem for a Nonlinear Quantum Walk, preprint, arXiv: 1801.06625.
[23]	K. Manouchehri and J. Wang, Physical Implementation of Quantum Walks, Quantum Science and Technology, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-36014-5.
[24]	D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), 551-574. doi: 10.1007/BF02199356.
[25]	A. Mielke and C. Patz, Dispersive stability of infinite-dimensional {H}amiltonian systems on lattices, Appl. Anal., 89 (2010), 1493-1512. doi: 10.1080/00036810903517605.
[26]	T. Mizumachi and D. Pelinovsky, On the asymptotic stability of localized modes in the discrete nonlinear {S}chrödinger equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 971-987. doi: 10.3934/dcdss.2012.5.971.
[27]	G. Di Molfetta, M. Brachet and F. Debbasch, Quantum walks in artificial electric and gravitational fields, Physica A: Statistical Mechanics and its Applications, 397 (2014), 157-168. doi: 10.1016/j.physa.2013.11.036.
[28]	G. DiMolfetta and F. Debbasch, Discrete-time quantum walks: Continuous limit and symmetries, Journal of Mathematical Physics, 53 (2012), 123302, 10pp. doi: 10.1063/1.4764876.
[29]	G. DiMolfetta, F. Debbasch and M. Brachet, Nonlinear optical galton board: Thermalization and continuous limit, Phys. Rev. E, 92 (2015), 042923. doi: 10.1103/PhysRevE.92.042923.
[30]	C. Morawetz and W. Strauss, On a nonlinear scattering operator, Comm. Pure Appl. Math., 26 (1973), 47-54. doi: 10.1002/cpa.3160260104.
[31]	C. Navarrete-Benlloch, A. Pérez and E. Roldán, Nonlinear optical galton board, Phys. Rev. A, 75 (2007), 062333. doi: 10.1103/PhysRevA.75.062333.
[32]	R. Portugal, Quantum Walks and Search Algorithms, Quantum Science and Technology, Springer, New York, 2013. doi: 10.1007/978-1-4614-6336-8.
[33]	H. Sasaki, Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly, J. Differential Equations, 252 (2012), 2004-2023. doi: 10.1016/j.jde.2011.07.022.
[34]	H. Sasaki, Small data scattering for the one-dimensional nonlinear Dirac equation with power nonlinearity, Comm. Partial Differential Equations, 40 (2015), 1959-2004. doi: 10.1080/03605302.2015.1081608.
[35]	H. Sasaki and A. Suzuki, An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory, Hokkaido Math. J., 40 (2011), 149-186. doi: 10.14492/hokmj/1310042826.
[36]	A. Schreiber, K.N. Cassemiro, V. Potoček, A. Gábris, I. Jex and Ch. Silberhorn, Photonic quantum walks in a fiber based recursion loop, AIP Conference Proceedings, 1363 (2011), 155-158. doi: 10.1063/1.3630170.
[37]	A. Schreiber, A. Gábris, P.P. Rohde, K. Laiho, M. Štefňák, V. Potoček, C. Hamilton, I. Jex and C. Silberhorn, A 2d quantum walk simulation of two-particle dynamics, Science, 336 (2012), 55-58. doi: 10.1126/science.1218448.
[38]	Y. Shikano, T. Wada and J. Horikawa, Discrete-time quantum walk with feed-forward quantum coin, Sci Rep., 4 (2014), 4427. doi: 10.1038/srep04427.
[39]	A. Stefanov and P.G. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18 (2005), 1841-1857. doi: 10.1088/0951-7715/18/4/022.
[40]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ.
[41]	W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Math. Physics, Reidel, Dordrecht, 9 (1974), 53-78. doi: 10.1007/978-94-010-2147-0_3.
[42]	S. Succi, F. Fillion-Gourdeau and S. Palpacelli, Quantum lattice boltzmann is a quantum walk, EPJ Quantum Technology, 2 (2015), p12. doi: 10.1140/epjqt/s40507-015-0025-1.
[43]	T. Sunada and T. Tate, Asymptotic behavior of quantum walks on the line, J. Funct. Anal., 262 (2012), 2608-2645. doi: 10.1016/j.jfa.2011.12.016.
[44]	A. Suzuki, Asymptotic velocity of a position-dependent quantum walk, Quantum Inf. Process., 15 (2016), 103-119. doi: 10.1007/s11128-015-1183-x.
[45]	R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103. doi: 10.1080/03605309708821332.
[46]	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 (2010), 100503.

show all references

References:

[1]	A. Ambainis, J. Kempe and A. Rivosh, Coins make quantum walks faster, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, 1099-1108.
[2]	A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, One-dimensional quantum walks, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, 37-49. doi: 10.1145/380752.380757.
[3]	P. Arnault and F. Debbasch, Quantum walks and discrete gauge theories, Phys. Rev. A, 93 (2016), 052301. doi: 10.1103/PhysRevA.93.052301.
[4]	P. Arnault, G. DiMolfetta, M. Brachet and F. Debbasch, Quantum walks and non-abelian discrete gauge theory, Phys. Rev. A, 94 (2016), 012335, 6pp. doi: 10.1103/PhysRevA.94.012335.
[5]	J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B, 88 (2013), 121406.
[6]	J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
[7]	R. Carles and I. Gallagher, Analyticity of the scattering operator for semilinear dispersive equations, Comm. Math. Phys., 286 (2009), 1181-1209. doi: 10.1007/s00220-008-0599-x.
[8]	C. Cedzich, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner and R. F. Werner, Bulk-edge correspondence of one-dimensional quantum walks, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 21LT01, 12pp. doi: 10.1088/1751-8113/49/21/21LT01.
[9]	A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett., 102 (2009), 180501, 4pp. doi: 10.1103/PhysRevLett.102.180501.
[10]	T. Endo, N. Konno, H. Obuse and E. Segawa, Sensitivity of quantum walks to a boundary of two-dimensional lattices: Approaches based on the cgmv method and topological phases, Journal of Physics A: Mathematical and Theoretical, 50 (2017), 455302, 40pp.
[11]	R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed., Dover Publications, Inc., Mineola, NY, 2010, Emended and with a preface by Daniel F. Styer.
[12]	Y. Gerasimenko, B. Tarasinski and C. W. J. Beenakker, Attractor-repeller pair of topological zero modes in a nonlinear quantum walk, Phys. Rev. A, 93 (2016), 022329. doi: 10.1103/PhysRevA.93.022329.
[13]	L. Grafakos, Classical Fourier Analysis, second ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008.
[14]	G. Grimmett, S. Janson and P. F. Scudo, Weak limits for quantum random walks, Phys. Rev. E, 69 (2004), 026119. doi: 10.1103/PhysRevE.69.026119.
[15]	D. Gross, V. Nesme, H. Vogts and R.F. Werner, Index theory of one dimensional quantum walks and cellular automata, Communications in Mathematical Physics, 310 (2012), 419-454. doi: 10.1007/s00220-012-1423-1.
[16]	S. P. Gudder, Quantum Probability, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988.
[17]	M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede and A. Widera, Quantum walk in position space with single optically trapped atoms, Science, 325 (2009), 174-177. doi: 10.1126/science.1174436.
[18]	A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics, 321 (2006), 2-111, January Special Issue. doi: 10.1016/j.aop.2005.10.005.
[19]	T. Kitagawa, Topological phenomena in quantum walks: Elementary introduction to the physics of topological phases, Quantum Information Processing, 11 (2012), 1107-1148. doi: 10.1007/s11128-012-0425-4.
[20]	T. Kitagawa, M. S. Rudner, E. Berg and E. Demler, Exploring topological phases with quantum walks, Phys. Rev. A, 82 (2010), 033429. doi: 10.1103/PhysRevA.82.033429.
[21]	C. -W. Lee, P. Kurzyński and H. Nha, Quantum walk as a simulator of nonlinear dynamics: Nonlinear dirac equation and solitons, Phys. Rev. A, 92 (2015), 052336. doi: 10.1103/PhysRevA.92.052336.
[22]	M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Weak Limit Theorem for a Nonlinear Quantum Walk, preprint, arXiv: 1801.06625.
[23]	K. Manouchehri and J. Wang, Physical Implementation of Quantum Walks, Quantum Science and Technology, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-36014-5.
[24]	D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), 551-574. doi: 10.1007/BF02199356.
[25]	A. Mielke and C. Patz, Dispersive stability of infinite-dimensional {H}amiltonian systems on lattices, Appl. Anal., 89 (2010), 1493-1512. doi: 10.1080/00036810903517605.
[26]	T. Mizumachi and D. Pelinovsky, On the asymptotic stability of localized modes in the discrete nonlinear {S}chrödinger equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 971-987. doi: 10.3934/dcdss.2012.5.971.
[27]	G. Di Molfetta, M. Brachet and F. Debbasch, Quantum walks in artificial electric and gravitational fields, Physica A: Statistical Mechanics and its Applications, 397 (2014), 157-168. doi: 10.1016/j.physa.2013.11.036.
[28]	G. DiMolfetta and F. Debbasch, Discrete-time quantum walks: Continuous limit and symmetries, Journal of Mathematical Physics, 53 (2012), 123302, 10pp. doi: 10.1063/1.4764876.
[29]	G. DiMolfetta, F. Debbasch and M. Brachet, Nonlinear optical galton board: Thermalization and continuous limit, Phys. Rev. E, 92 (2015), 042923. doi: 10.1103/PhysRevE.92.042923.
[30]	C. Morawetz and W. Strauss, On a nonlinear scattering operator, Comm. Pure Appl. Math., 26 (1973), 47-54. doi: 10.1002/cpa.3160260104.
[31]	C. Navarrete-Benlloch, A. Pérez and E. Roldán, Nonlinear optical galton board, Phys. Rev. A, 75 (2007), 062333. doi: 10.1103/PhysRevA.75.062333.
[32]	R. Portugal, Quantum Walks and Search Algorithms, Quantum Science and Technology, Springer, New York, 2013. doi: 10.1007/978-1-4614-6336-8.
[33]	H. Sasaki, Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly, J. Differential Equations, 252 (2012), 2004-2023. doi: 10.1016/j.jde.2011.07.022.
[34]	H. Sasaki, Small data scattering for the one-dimensional nonlinear Dirac equation with power nonlinearity, Comm. Partial Differential Equations, 40 (2015), 1959-2004. doi: 10.1080/03605302.2015.1081608.
[35]	H. Sasaki and A. Suzuki, An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory, Hokkaido Math. J., 40 (2011), 149-186. doi: 10.14492/hokmj/1310042826.
[36]	A. Schreiber, K.N. Cassemiro, V. Potoček, A. Gábris, I. Jex and Ch. Silberhorn, Photonic quantum walks in a fiber based recursion loop, AIP Conference Proceedings, 1363 (2011), 155-158. doi: 10.1063/1.3630170.
[37]	A. Schreiber, A. Gábris, P.P. Rohde, K. Laiho, M. Štefňák, V. Potoček, C. Hamilton, I. Jex and C. Silberhorn, A 2d quantum walk simulation of two-particle dynamics, Science, 336 (2012), 55-58. doi: 10.1126/science.1218448.
[38]	Y. Shikano, T. Wada and J. Horikawa, Discrete-time quantum walk with feed-forward quantum coin, Sci Rep., 4 (2014), 4427. doi: 10.1038/srep04427.
[39]	A. Stefanov and P.G. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18 (2005), 1841-1857. doi: 10.1088/0951-7715/18/4/022.
[40]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ.
[41]	W. A. Strauss, Nonlinear scattering theory, Scattering Theory in Math. Physics, Reidel, Dordrecht, 9 (1974), 53-78. doi: 10.1007/978-94-010-2147-0_3.
[42]	S. Succi, F. Fillion-Gourdeau and S. Palpacelli, Quantum lattice boltzmann is a quantum walk, EPJ Quantum Technology, 2 (2015), p12. doi: 10.1140/epjqt/s40507-015-0025-1.
[43]	T. Sunada and T. Tate, Asymptotic behavior of quantum walks on the line, J. Funct. Anal., 262 (2012), 2608-2645. doi: 10.1016/j.jfa.2011.12.016.
[44]	A. Suzuki, Asymptotic velocity of a position-dependent quantum walk, Quantum Inf. Process., 15 (2016), 103-119. doi: 10.1007/s11128-015-1183-x.
[45]	R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103. doi: 10.1080/03605309708821332.
[46]	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 (2010), 100503.

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