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Cited by 111Quantum Computing Algorithms and ArchitectureQuantum Mechanics and ApplicationsNoncommutative and Quantum Gravity Theories

Relationship between quantum walks and relativistic quantum mechanics

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Chandrashekar, C. M., Banerjee, Subhashish, Srikanth, R.·2010·Physical Review A
DOI 10.1103/physreva.81.062340
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Abstract

Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This article revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled forms of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schr\"odinger form. By showing the coin to be a means to make the walk reversible and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modeled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. The Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of the quantum walk, the maximum speed of walk propagation, and earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two-state system to which the study can be extended.

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Crossref · 10.1103/physreva.81.062340

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  • Our analysis agrees with the works [42-46] that studied the continuum limit of QWs in comparison with the Dirac or the Klein-Gordon equations.

    “We notice that our analysis agrees with the works [42-46] that studied the continuum limit, i.e. when the lattice spacings and the time steps are sent to 0, of QWs in comparison with the Dirac or the Klein-Gordon equations 3 .”

    From Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension (2015) · cites this paper as [45]

    Section IV.A (p. 5) derives a Klein–Gordon-like form for a discrete-time quantum walk, and Section IV.C (pp. 7–8) compares its coupled form with the Dirac equation. But these passages do not show the paper taking lattice spacing and time step to zero: Section IV.A and Appendix B (p. 11) instead set the steps to 1. The specific continuum-limit description is therefore unsupported for this source. (LLM, conf=0.86)

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  • Consistency of the (1+1)-dimensional Dirac QW has been argued in [6], and for the (1+1)-dimensional massless case in [8].

    “Consistency of the + (1 1)-dimensional Dirac QW has been argued in [6], and for the + (1 1)-dimensional massless case in [8].”

    From The Dirac equation as a quantum walk: higher dimensions, observational convergence (2014) · cites this paper as [8]

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Research works printed but not linked: 8

  1. [6]A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proceeding of the 33rd ACM Symposium on Theory of Computing (ACM Press, New York, 2001), p.60.
  2. [7]A. Nayak and A. Vishwanath, DIMACS Technical Report, No. 2000-43 (2001) ; arXiv:quant-ph/0010117. · arXiv
  3. [8]A. Ambainis, J. Kempe, and A. Rivosh, Proceedings of ACM-SIAM Symp. on Discrete Algorithms (SODA) , (AMC Press, New York, 2005), pp.1099-1108.
  4. [9]A. Ambainis, Int. Journal of Quantum Information, 1 , No.4, 507-518 (2003).
  5. [10]A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann and D. A. Spielman, in Proceedings of the 35th ACM Symposium on Theory of Computing (ACM Press, New York, 2003), p.59.
  6. [35]Yutaka Shikano, Kota Chisaki, Etsuo Segawa, and Norio Konno, arXiv:1001.3989 (2010). · arXiv
  7. [40]R. Clifton, J. Bub, H. Halvorson, Found. Phys. 33 , 1561-1591 (2003); see also: arXiv:quant-ph/0311065. · arXiv
  8. [50]B. Nachtergaele, Y. Ogata and R. Sims, Journal of Statistical Phys. 124 , 1 (2006).

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Related

Authors
Chandrashekar, C. M., Banerjee, S., & Srikanth, R.
Affiliations
  • University of Waterloo (CA)
  • Chennai Mathematical Institute (IN)
  • Raman Research Institute (IN)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2010-06-25
Volume
81
Issue
6
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.81.062340
OpenAlex
W2089766148
Cited by
109
References
48
Citations by year
2026: 1 · 2025: 5 · 2024: 7 · 2023: 1 · 2022: 9 · 2021: 10 · 2020: 11 · 2019: 12 · 2018: 9 · 2017: 4 · 2016: 7 · 2015: 5 · 2014: 6 · 2013: 8 · 2012: 8
FWCI
7.45
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
Keywords
quantum walks, relativistic quantum mechanics, Klein-Gordon equation, Dirac equation, discrete-time quantum walks, continuous-time quantum walks, coin operator, reversibility, information conservation, arrow of time, decoherence, entropy (information theory), Lieb-Robinson bound, causal structure, two-dimensional quantum walk, two-level system

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