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

Scattering and inverse scattering for nonlinear quantum walks

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.

Citation: Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

P. Arnault and F. Debbasch, Quantum walks and discrete gauge theories, Phys. Rev. A, 93 (2016), 052301. doi: 10.1103/PhysRevA.93.052301.  Google Scholar

[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.  Google Scholar

[5]

J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B, 88 (2013), 121406. Google Scholar

[6]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett., 102 (2009), 180501, 4pp. doi: 10.1103/PhysRevLett.102.180501.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, second ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[15]

D. GrossV. NesmeH. 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.  Google Scholar

[16]

S. P. Gudder, Quantum Probability, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[17]

M. KarskiL. FörsterJ.-M. ChoiA. SteffenW. AltD. Meschede and A. Widera, Quantum walk in position space with single optically trapped atoms, Science, 325 (2009), 174-177.  doi: 10.1126/science.1174436.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Weak Limit Theorem for a Nonlinear Quantum Walk, preprint, arXiv: 1801.06625. Google Scholar

[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.  Google Scholar

[24]

D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), 551-574.  doi: 10.1007/BF02199356.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Di MolfettaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. Morawetz and W. Strauss, On a nonlinear scattering operator, Comm. Pure Appl. Math., 26 (1973), 47-54.  doi: 10.1002/cpa.3160260104.  Google Scholar

[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.  Google Scholar

[32]

R. Portugal, Quantum Walks and Search Algorithms, Quantum Science and Technology, Springer, New York, 2013. doi: 10.1007/978-1-4614-6336-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. SchreiberK.N. CassemiroV. PotočekA. GábrisI. 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.  Google Scholar

[37]

A. SchreiberA. GábrisP.P. RohdeK. LaihoM. ŠtefňákV. PotočekC. HamiltonI. Jex and C. Silberhorn, A 2d quantum walk simulation of two-particle dynamics, Science, 336 (2012), 55-58.  doi: 10.1126/science.1218448.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103.  doi: 10.1080/03605309708821332.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

P. Arnault and F. Debbasch, Quantum walks and discrete gauge theories, Phys. Rev. A, 93 (2016), 052301. doi: 10.1103/PhysRevA.93.052301.  Google Scholar

[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.  Google Scholar

[5]

J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B, 88 (2013), 121406. Google Scholar

[6]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett., 102 (2009), 180501, 4pp. doi: 10.1103/PhysRevLett.102.180501.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Grafakos, Classical Fourier Analysis, second ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[15]

D. GrossV. NesmeH. 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.  Google Scholar

[16]

S. P. Gudder, Quantum Probability, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[17]

M. KarskiL. FörsterJ.-M. ChoiA. SteffenW. AltD. Meschede and A. Widera, Quantum walk in position space with single optically trapped atoms, Science, 325 (2009), 174-177.  doi: 10.1126/science.1174436.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Weak Limit Theorem for a Nonlinear Quantum Walk, preprint, arXiv: 1801.06625. Google Scholar

[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.  Google Scholar

[24]

D.A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), 551-574.  doi: 10.1007/BF02199356.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Di MolfettaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. Morawetz and W. Strauss, On a nonlinear scattering operator, Comm. Pure Appl. Math., 26 (1973), 47-54.  doi: 10.1002/cpa.3160260104.  Google Scholar

[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.  Google Scholar

[32]

R. Portugal, Quantum Walks and Search Algorithms, Quantum Science and Technology, Springer, New York, 2013. doi: 10.1007/978-1-4614-6336-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. SchreiberK.N. CassemiroV. PotočekA. GábrisI. 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.  Google Scholar

[37]

A. SchreiberA. GábrisP.P. RohdeK. LaihoM. ŠtefňákV. PotočekC. HamiltonI. Jex and C. Silberhorn, A 2d quantum walk simulation of two-particle dynamics, Science, 336 (2012), 55-58.  doi: 10.1126/science.1218448.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103.  doi: 10.1080/03605309708821332.  Google Scholar

[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. Google Scholar

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