Top

Quantum Information Processing

Published in:

01-09-2021

An index theorem for one-dimensional gapless non-unitary quantum walks

Authors: Keisuke Asahara, Daiju Funakawa, Motoki Seki, Yohei Tanaka

Published in: Quantum Information Processing | Issue 9/2021

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a unitary time evolution U and a \(\mathbb {Z}_2\)-grading operator \(\varGamma \) satisfying the chiral symmetry condition, \(U^* = \varGamma U \varGamma .\) In the present article, we extend this index theory to encompass non-unitary time evolutions U. The existing literature for unitary U assumes that U is essentially gapped to define the associated index, that is, the essential spectrum of U contains neither \(-1\) nor \(+1\). We show this assumption is not necessary if U fails to be unitary. As a concrete example, we consider the well-known non-unitary quantum walk model on the integer lattice \(\mathbb {Z}\) introduced by Mochizuki–Kim–Obuse.

previous article Learning bounds for quantum circuits in the agnostic setting

next article A novel quantum (t, n) threshold group signature based on d-dimensional quantum system

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing—STOC’01, New York, New York, USA, pp. 50–59. ACM Press (2001)

Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1(4), 507–518 (2003)CrossRef

Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33rd ACM Symposium of the Theory of Computing, pp. 37–49. ACM Press (2001)

Arai, A.: Analysis on Fock Spaces and Mathematical Theory of Quantum Fields: An Introduction to Mathematical Analysis of Quantum Fields. World Scientific Publishing Company (2017)

Asbóth, J.K., Edge, J.M.: Edge-state-enhanced transport in a two-dimensional quantum walk. Phys. Rev. A 91, 022324 (2015)CrossRefADS

Asbóth, J.K., Obuse, H.: Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88(12), 121406 (2013)CrossRefADS

Bernd, T.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)

Boussaïd, N., Comech, A.: Nonlinear Dirac Equation: Spectral Stability of Solitary Waves. Mathematical Surveys and Monographs, American Mathematical Society (2019)

Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: One-dimensional quantum walks with one defect. Rev. Math. Phys. 24(02), 1250002 (2012)MathSciNetCrossRef

10.

Cedzich, C., Grünbaum, F.A., Stahl, C., Velázquez, L., Werner, A.H., Werner, R.F.: Bulk-edge correspondence of one-dimensional quantum walks. J. Phys. A 49(21), 21LT01 (2016)MathSciNetCrossRef

11.

Cedzich, C., Geib, T., Grünbaum, F.A., Stahl, C., Velázquez, L., Werner, A.H., Werner, R.F.: The topological classification of one-dimensional symmetric quantum walks. Ann. Henri Poincaré 19(2), 325–383 (2018)MathSciNetCrossRefADS

12.

Cedzich, C., Geib, T., Stahl, C., Velázquez, L., Werner, A.H., Werner, R.F.: Complete homotopy invariants for translation invariant symmetric quantum walks on a chain. Quantum 2, 95 (2018)CrossRef

13.

Cedzich, C, Geib, T., Werner, A.H., Werner, R.F.: Chiral floquet systems and quantum walks at half period. arXiv:2006.04634 (2020)

14.

Fuda, T., Funakawa, D., Suzuki, A.: Localization of a multi-dimensional quantum walk with one defect. Quantum Inf. Process. 16(8) (2017)

15.

Fuda, T., Funakawa, D., Suzuki, A.: Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations. J. Math. Phys. 59(8), 082201 (2018)MathSciNetCrossRefADS

16.

Fuda, T., Funakawa, D., Suzuki, A.: Weak limit theorem for a one-dimensional split-step quantum walk. Rev. Roumaine Math. Pures Appl. 64, 157–165 (2019)MathSciNetMATH

17.

Funakawa, D., Matsuzawa, Y., Sasaki, I., Suzuki, A., Teranishi, N.: Time operators for quantum walks. Lett. Math. Phys. 110(9), 2471–2490 (2020)MathSciNetCrossRefADS

18.

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

19.

Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC’96, New York, NY, USA, pp. 212–219. Association for Computing Machinery (1996)

20.

Inui, N., Konishi, Y., Konno, N.: Localization of two-dimensional quantum walks. Phys. Rev. A 69(5), 052323 (2004)CrossRefADS

21.

Kitagawa, T.: Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf. Process. 11(5), 1107–1148 (2012)MathSciNetCrossRef

22.

Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)CrossRefADS

23.

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

24.

Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002)MathSciNetCrossRef

25.

Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9(3), 405–418 (2010)MathSciNetCrossRef

26.

Maeda, M., Suzuki, A.: Continuous limits of linear and nonlinear quantum walks. Rev. Math. Phys. 32(04), 2050008 (2019)MathSciNetCrossRef

27.

Maeda, A., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Weak limit theorem for a nonlinear quantum walk. Quantum Inf. Process. 17(9) (2018)

28.

Maeda, M., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Scattering and inverse scattering for nonlinear quantum walks. Discrete Contin. Dyn. Syst. 38(7), 3687–3703 (2018)MathSciNetCrossRef

29.

Maeda, M., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Dynamics of solitons for nonlinear quantum walks. J. Phys. Commun. 3(7), 075002 (2019)CrossRef

30.

Matsuzawa, Y.: An index theorem for split-step quantum walks. Quantum Inf. Process. 19(8) (2020)

31.

Mochizuki, K., Kim, D., Obuse, H.: Explicit definition of PT symmetry for nonunitary quantum walks with gain and loss. Phys. Rev. A 93(6), 062116 (2016)CrossRefADS

32.

Mochizuki, K., Kim, D., Kawakami, N., Obuse, H.: Bulk-edge correspondence in nonunitary floquet systems with chiral symmetry. Phys. Rev. A 102(6), 062202 (2020)MathSciNetCrossRefADS

33.

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

34.

Morioka, H.: Generalized eigenfunctions and scattering matrices for position-dependent quantum walks. Rev. Math. Phys. 31(07), 1950019 (2019)MathSciNetCrossRef

35.

Narimatsu, A., Ohno, H., Wada, K.: Unitary equivalence classes of split-step quantum walks. arXiv:2104.13529 (2021)

36.

Obuse, H., Kawakami, N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84(19), 195139 (2011)CrossRefADS

37.

Peruzzo, A., Lobino, M., Matthews, J.C.F., Matsud, N., Politi, A., Poulios, K., Zhou, X., Lahini, Y., Ismail, N., Wörhoff, K., Bromberg, Y., Silberberg, Y., Thompson, M.G., OBrien, J.L.: Quantum walks of correlated photons. Science 329(5998), 1500–1503 (2010)

38.

Portugal, R.: Staggered quantum walks on graphs. Phys. Rev. A 93(6), 062335 (2016)MathSciNetCrossRefADS

39.

Regensburger, A., Bersch, C., Miri, M., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488(7410), 167–171 (2012)CrossRefADS

40.

Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin i: spectral theory. Lett. Math. Phys. 108(2), 331–357 (2017)MathSciNetCrossRefADS

41.

Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin ii: scattering theory. Lett. Math. Phys. 109(1), 61–88 (2018)MathSciNetCrossRefADS

42.

Sambou, D., Tiedra de Aldecoa, R.: Quantum time delay for unitary operators: general theory. Rev. Math. Phys. 31(06), 1950018 (2019)MathSciNetCrossRef

43.

Sasaki, I., Suzuki, A.: Essential spectrum of the discrete Laplacian on a perturbed periodic graph. J. Math. Anal. Appl. 446(2), 1863–1881 (2017)MathSciNetCrossRef

44.

Segawa, E.: Localization of quantum walks induced by recurrence properties of random walks. J. Comput. Theor. Nanos. 10 (2011)

45.

Segawa, E., Suzuki, A.: Generator of an abstract quantum walk. Quantum Stud. Math. Found. 3(1), 11–30 (2016)MathSciNetCrossRef

46.

Segawa, E., Suzuki, A.: Spectral mapping theorem of an abstract quantum walk. Quantum Inf. Process. 18(11) (2019)

47.

Suzuki, A.: Asymptotic velocity of a position-dependent quantum walk. Quantum Inf. Process. 15(2), 103–119 (2016)MathSciNetCrossRefADS

48.

Suzuki, A.: Supersymmetry for chiral symmetric quantum walks. Quantum Inf. Process. 18(12) (2019)

49.

Suzuki, A., Tanaka, Y.: The Witten index for 1d supersymmetric quantum walks with anisotropic coins. Quantum Inf. Process. 18(12) (2019)

50.

Tanaka, Y.: A constructive approach to topological invariants for one-dimensional strictly local operators. J. Math. Anal. Appl. 500(2), 125072 (2021)MathSciNetCrossRef

51.

Wada, K.: A weak limit theorem for a class of long-range-type quantum walks in 1d. Quantum Inf. Process. 19(1) (2020)

Title: An index theorem for one-dimensional gapless non-unitary quantum walks
Authors: Keisuke Asahara
Daiju Funakawa
Motoki Seki
Yohei Tanaka
Publication date: 01-09-2021
Publisher: Springer US
Published in: Quantum Information Processing / Issue 9/2021
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-021-03212-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 9/2021

Optimal approximation to unitary quantum operators with linear optics

Amplitude estimation via maximum likelihood on noisy quantum computer

Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models

ADA-QKDN: a new quantum key distribution network routing scheme based on application demand adaptation

Classification witness operator for the classification of different subclasses of three-qubit GHZ class

Genuinely entangling uncorrelated atoms via Jaynes–Cummings interactions