Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Quantum Information Processing
Published in: Quantum Information Processing 8/2016

01-08-2016

Quantum walking in curved spacetime

Authors: Pablo Arrighi, Stefano Facchini, Marcelo Forets

Published in: Quantum Information Processing | Issue 8/2016

Log in

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

search-config
loading …

Abstract

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g., the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (\(1+1\)) curved spacetime. Therefore, a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically, we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
previous article Uncertainty-like relations of the relative entropy of coherence
next article Erratum to: Algebraic synthesis of time-optimal unitaries in SU(2) with alternating controls

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
Appendix
Available only for authorised users
Footnotes
1
Recall that if \(A \in \mathbb {C}^{n\times n}\), its real and imaginary parts are \(\mathfrak {R}A := \frac{1}{2}( A+A^\dagger )\) and \(\mathfrak {I}A := \frac{1}{2\mathrm {i}} (A-A^\dagger )\), respectively.
 
2
In this case the dyads are \(e^0_0=(1-2M/x)^{-1/2}\), \(e^1_1=(1-2M/x)^{1/2}\), and \(e^0_1=e^1_0=0\).
 
Literature
1.
2.
Bialynicki-Birula, I.: Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D. 49(12), 6920–6927 (1994)ADSMathSciNetCrossRef
3.
4.
5.
6.
7.
Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A Math. Theor. 47(46), 465302–465316 (2014)ADSMathSciNetCrossRefMATH
8.
D’Ariano, G.M., Perinotti, P.: Derivation of the Dirac equation from principles of information processing. Phys. Rev. A 90(6), 062106-1–062106-18 (2014)ADS
9.
Arrighi, P., Facchini, S., Forets, M.: Discrete Lorentz covariance for quantum walks and quantum cellular automata. New J. Phys. 16(9), 093007–093040 (2014)MathSciNetCrossRef
10.
Arrighi, P., Facchini, S.: Decoupled quantum walks, models of the Klein–Gordon and wave equations. EPL (Europhysics Letters) 104(6), 60004–60008 (2013)ADSCrossRef
11.
Farrelly, T.C., Short, A.J.: Causal fermions in discrete space-time. Phys. Rev. A 89(1), 012302-1–012302-15 (2014)ADSCrossRef
12.
Farrelly, T.C., Short, A.J.: Discrete spacetime and relativistic quantum particles. Phys. Rev. A 89(6), 062109-1–062109-6 (2014)ADSCrossRef
13.
14.
15.
Bracken, A.J., Ellinas, D., Smyrnakis, I.: Free-Dirac-particle evolution as a quantum random walk. Phys. Rev. A 75(2), 022322-1–022322-7 (2007)ADSCrossRef
16.
D’Ariano, G.M.: The Dirac quantum automaton: a preview. AIP Conf. Proc. 1508, 146–155 (2012)ADSCrossRef
17.
Bisio, A., D’Ariano, G.M., Tosini, A.: Dirac quantum cellular automaton in one dimension: Zitterbewegung and scattering from potential. Phys. Rev. A 88(3), 032301-1–032301-7 (2013)ADSCrossRef
18.
Shikano, Y.: From discrete time quantum walk to continuous time quantum walk in limit distribution. J. Comput. Theor. Nanosci. 10(7), 1558–1570 (2013)CrossRef
19.
20.
Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks as massless Dirac fermions in curved spacetime. Phys. Rev. A 88(4), 042301-1–042301-5 (2013)ADSCrossRef
21.
Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks in artificial electric and gravitational fields. Phys. A Stat. Mech. Appl. 397, 157–168 (2014)MathSciNetCrossRef
22.
Succi, S., Fillion-Gourdeau, F., Palpacelli, S.: Quantum lattice Boltzmann is a quantum walk. EPJ Quantum Technol. 2(1), 1–17 (2015)CrossRef
23.
24.
Arrighi, P., Facchini, S., Forets, M.: Three discrete models for the \((1+1)\) curved Dirac equation. Unpublished manuscript (2015)
25.
Succi, S.: The Lattice Boltzmann equation for fluid dynamics and beyond. Clarendon, Oxford (2001)MATH
26.
27.
London, D.: A note on matrices with positive definite real part. In: Proceedings of the American Mathematical Society. pp. 322–324 (1981)
28.
Metadata
Title
Quantum walking in curved spacetime
Authors
Pablo Arrighi
Stefano Facchini
Marcelo Forets
Publication date
01-08-2016
Publisher
Springer US
Published in
Quantum Information Processing / Issue 8/2016
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-016-1335-7

Other articles of this Issue 8/2016

Quantum Information Processing 8/2016 Go to the issue

OriginalPaper

Quantum state transfer between an optomechanical cavity and a diamond nuclear spin ensemble

OriginalPaper

A quantum walk on the half line with a particular initial state

OriginalPaper

Influence of environmental noise on the weak value amplification

OriginalPaper

Protecting tripartite entanglement in non-Markovian environments via quantum partially collapsing measurements

OriginalPaper

Quantum private query based on single-photon interference

OriginalPaper

Remote two-qubit state creation and its robustness