Top

Quantum Information Processing

Published in:

01-05-2019

Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian

Authors: Jalil Khatibi Moqadam, M. C. de Oliveira

Published in: Quantum Information Processing | Issue 5/2019

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

search-config

AI-assisted search

Off

Abstract

The discrete-time quantum walk dynamics can be generated by a time-dependent Hamiltonian, repeatedly switching between the coin and the shift generators. We change the model and consider the case where the Hamiltonian is time-independent, including both the coin and the shift terms in all times. The eigenvalues and the related Bloch vectors for the time-independent Hamiltonian are then compared with the corresponding quantities for the effective Hamiltonian generating the quantum walk dynamics. Restricted to the non-localized initial quantum walk states, we optimize the parameters in the time-independent Hamiltonian such that it generates a dynamics similar to the Hadamard quantum walk. We find that the dynamics of the walker probability distribution and the corresponding standard deviation, the coin-walker entanglement, and the quantum-to-classical transition of the discrete-time quantum walk model can be approximately generated by the optimized time-independent Hamiltonian. We, further, show both dynamics are equivalent in the classical regime, as expected.

previous article Zero transfer in continuous-time quantum walks

next article Color image storage and retrieval using quantum mechanics

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

Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)MATHCrossRef

Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012)MathSciNetMATHCrossRef

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

Genske, M., Alt, W., Steffen, A., Werner, A.H., Werner, R.F., Meschede, D., Alberti, A.: Electric quantum walks with individual atoms. Phys. Rev. Lett. 110, 190601 (2013)ADSCrossRef

Cedzich, C., Rybár, T., Werner, A.H., Alberti, A., Genske, M., Werner, R.F.: Propagation of quantum walks in electric fields. Phys. Rev. Lett. 111, 160601 (2013)ADSCrossRef

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

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

Asbóth, J.K.: Symmetries, topological phases, and bound states in the one-dimensional quantum walk. Phys. Rev. B 86, 195414 (2012)ADSCrossRef

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

10.

Obuse, H., Asbóth, J.K., Nishimura, Y., Kawakami, N.: Unveiling hidden topological phases of a one-dimensional hadamard quantum walk. Phys. Rev. B 92, 045424 (2015)ADSCrossRef

11.

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

12.

Schumacher, B., Werner, R.F.: Reversible quantum cellular automata. arXiv preprint arXiv:quant-ph/0405174 (2004)

13.

Arrighi, P., Grattage, J.: Partitioned quantum cellular automata are intrinsically universal. Nat. Comput. 11(1), 13–22 (2012)MathSciNetMATHCrossRef

14.

D’Ariano, G.M., Perinotti, P.: Derivation of the dirac equation from principles of information processing. Phys. Rev. A 90, 062106 (2014)ADSCrossRef

15.

Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Weyl, dirac and maxwell quantum cellular automata. Found. Phys. 45(10), 1203–1221 (2015)ADSMathSciNetMATHCrossRef

16.

Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, Emended edn. Dover Publications Inc, Mineola (2005)MATH

17.

D’Ariano, G.M., Mosco, N., Perinotti, P., Tosini, A.: Path-integral solution of the one-dimensional dirac quantum cellular automaton. Phys. Lett. A 378(43), 3165–3168 (2014)ADSMATHCrossRef

18.

D’Ariano, G.M., Mosco, N., Perinotti, P., Tosini, A.: Discrete feynman propagator for the weyl quantum walk in 2 + 1 dimensions. EPL 109(4), 40012 (2015)ADSCrossRef

19.

D’Ariano, G.M., Mosco, N., Perinotti, P., Tosini, A.: Path-sum solution of the weyl quantum walk in 3 + 1 dimensions. Philos. Trans. R. Soc. A 375(2106), 20160394 (2017)ADSMathSciNetMATHCrossRef

20.

Bisio, A., D’Ariano, G., Mosco, N., Perinotti, P., Tosini, A.: Solutions of a two-particle interacting quantum walk. Entropy 20(6), 435 (2018)ADSMathSciNetCrossRef

21.

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2010)MATHCrossRef

22.

Travaglione, B., Milburn, G.: Implementing the quantum random walk. Phys. Rev. A 65(3), 032310 (2002)ADSCrossRef

23.

D’Ariano, G.M., Mosco, N., Perinotti, P., Tosini, A.: Discrete time dirac quantum walk in 3+1 dimensions. Entropy 18(6), 228 (2016)ADSMathSciNetMATHCrossRef

24.

Strauch, F.W.: Relativistic quantum walks. Phys. Rev. A 73, 054302 (2006)ADSMathSciNetCrossRef

25.

Strauch, F.W.: Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks. J. Math. Phys. 48(8), 082102 (2007)ADSMathSciNetMATHCrossRef

26.

Suzuki, M.: Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics. J. Math. Phys. 26, 601 (1985)ADSMathSciNetMATHCrossRef

27.

Suzuki, M.: Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun. Math. Phys. 51(2), 183–190 (1976)ADSMathSciNetMATHCrossRef

28.

Pollard, D.: A User’s Guide to Measure Theoretic Probability, vol. 8. Cambridge University Press, Cambridge (2002)MATH

29.

Alagic, G., Russell, A.: Decoherence in quantum walks on the hypercube. Phys. Rev. A 72, 062304 (2005)ADSCrossRef

30.

Drezgich, M., Hines, A.P., Sarovar, M., Sastry, S.: Complete characterization of mixing time for the continuous quantum walk on the hypercube with markovian decoherence model. Quantum Inf. Comput. 9(9), 856–878 (2009)MathSciNetMATH

31.

Artiles, L.M., Gill, R.D., Guta, M.I.: An invitation to quantum tomography. J. R. Stat. Soc. B 67(1), 109–134 (2005)MathSciNetMATHCrossRef

32.

Dajka, J., Łuczka, J., Hänggi, P.: Distance between quantum states in the presence of initial qubit-environment correlations: a comparative study. Phys. Rev. A 84, 032120 (2011)ADSCrossRef

33.

Belavkin, V.P., D’Ariano, G.M., Raginsky, M.: Operational distance and fidelity for quantum channels. J. Math. Phys. 46(6), 062106 (2005)ADSMathSciNetMATHCrossRef

34.

Marian, P., Marian, T.A.: Hellinger distance as a measure of Gaussian discord. J. Phys. A Math. Theor. 48(11), 115301 (2015)ADSMathSciNetMATHCrossRef

35.

Roga, W., Spehner, D., Illuminati, F.: Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations. J. Phys. A Math. Theor. 49(23), 235301 (2016)ADSMathSciNetMATHCrossRef

36.

Suciu, S., Isar, A.: Gaussian geometric discord in terms of hellinger distance. AIP Conf. Proc. 1694, 020013 (2015)CrossRef

37.

Girolami, D., Tufarelli, T., Adesso, G.: Characterizing nonclassical correlations via local quantum uncertainty. Phys. Rev. Lett. 110, 240402 (2013)ADSCrossRef

38.

Chang, L., Luo, S.: Remedying the local ancilla problem with geometric discord. Phys. Rev. A 87, 062303 (2013)ADSCrossRef

39.

Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks. Springer, Berlin (2014)MATHCrossRef

40.

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

41.

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

42.

Sanders, B.C., Bartlett, S.D., Tregenna, B., Knight, P.L.: Quantum quincunx in cavity quantum electrodynamics. Phys. Rev. A 67(4), 042305 (2003)ADSCrossRef

43.

Hardal, A.Ü., Xue, P., Shikano, Y., Müstecaplıoğlu, Ö.E., Sanders, B.C.: Discrete-time quantum walk with nitrogen-vacancy centers in diamond coupled to a superconducting flux qubit. Phys. Rev. A 88(2), 022303 (2013)ADSCrossRef

44.

Moqadam, J.K., Portugal, R., de Oliveira, M.C.: Quantum walks on a circle with optomechanical systems. Quantum Inf. Process. 14(10), 3595–3611 (2015)ADSMathSciNetMATHCrossRef

45.

Ramasesh, V.V., Flurin, E., Rudner, M., Siddiqi, I., Yao, N.Y.: Direct probe of topological invariants using Bloch oscillating quantum walks. Phys. Rev. Lett. 118, 130501 (2017)ADSMathSciNetCrossRef

46.

Flurin, E., Ramasesh, V.V., Hacohen-Gourgy, S., Martin, L.S., Yao, N.Y., Siddiqi, I.: Observing topological invariants using quantum walks in superconducting circuits. Phys. Rev. X 7, 031023 (2017)

47.

Suzuki, M.: On the convergence of exponential operators—the Zassenhaus formula, BCH formula and systematic approximants. Commun. Math. Phys. 57(3), 193–200 (1977)ADSMathSciNetMATHCrossRef

48.

Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)ADSCrossRef

49.

Kendon, V.: Decoherence in quantum walks—a review. Math. Struct. Comput. Sci. 17(06), 1169–1220 (2007)MathSciNetMATHCrossRef

50.

Blais, A., Huang, R.S., Wallraff, A., Girvin, S.M., Schoelkopf, R.J.: Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004)ADSCrossRef

Title: Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian
Authors: Jalil Khatibi Moqadam
M. C. de Oliveira
Publication date: 01-05-2019
Publisher: Springer US
Published in: Quantum Information Processing / Issue 5/2019
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-019-2262-1

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 5/2019

Quantum teleportation of an arbitrary two-qubit state by using two three-qubit GHZ states and the six-qubit entangled state

Noise resistance of activation of the violation of the Svetlichny inequality

Preparing tunable Bell-diagonal states on a quantum computer

Arbitrated quantum signature scheme with quantum walk-based teleportation

Tuning the thermal entanglement in a Ising-XXZ diamond chain with two impurities

Parametrization of quantum states based on the quantum state discrimination problem