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Cited by 120Quantum Computing Algorithms and ArchitectureQuantum Mechanics and ApplicationsQuantum chaos and dynamical systems

Quantum walks on a random environment

Open access
Yin, Yue, Katsanos, D. E., Evangelou, S. N.·2008·Physical Review A
DOI 10.1103/physreva.77.022302
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Abstract

Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength $W$, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.

AI snapshot

Written from the paper’s full text.

TL;DR

Quantum walks in a one-dimensional random medium are stopped by static disorder via Anderson localization, while dynamic disorder causes decoherence and a quantum-to-classical crossover at \(t_c \propto W^{-2}\) into diffusive random-walk spreading. The paper demonstrates these effects in quantum carpets and in a two-level qubit, finding a critical disorder \(W_c=\sqrt{24}\) for coherence loss.

Key findings

  • Without disorder, the quantum walk is ballistic: \(P(x,t)=(J_x(2t))^2\), sharp fronts at \(x=\pm 2t\), and \(\sigma^2(t)=2t^2\).
  • Static disorder causes Anderson localization and stops the quantum walk; for \(W=5\) and \(N=101\), the localization length is \(\xi \sim 100 W^{-2} \sim 4\), much less than the system size.
  • Dynamic disorder produces decoherence: after initial ballistic behavior with \(\sigma^2(t)\propto t^2\) and \(C(t)\propto t^{-1}\), the asymptotic classical laws \(\sigma^2(t)\propto t\) and \(C(t)\propto t^{-1/2}\) set in, with crossover \(t_c \propto W^{-2}\).
  • For a qubit with dynamic disorder, the averaged density matrix approaches half the unit matrix; coherence remains for \(W<W_c\) and the phase randomizes for \(W>W_c\), with \(W_c=\sqrt{24}\).

Methods

  • Continuous-time quantum walk modeled as a tight-binding particle in one dimension via the linear wave equation with static or dynamic random site energies \(\epsilon(x,t)\), using initial \(\delta\)-function or uniform states.
  • Static disorder solved via stationary eigenstates; time-dependent disorder solved with a fourth-order Runge-Kutta algorithm. Simulations used finite chains (e.g., \(N=101\) for Figs. 1–2) and a self-expanding chain for later figures to avoid boundaries.
  • Analysis used probability density \(P(x,t)=|\Psi(x,t)|^2\), second moment \(\sigma^2(t)=\sum_x |x|^2 P(x,t)\), autocorrelation \(C(t)=\frac{1}{t}\int_0^t P(0,t')dt'\), and for the qubit the averaged density-matrix element \(\rho_{12}\) and its phase.

Limitations

  • The paper has no dedicated limitations section. It states that finite-chain simulations in Figs. 1–2 show boundary scattering from hard ends, becoming more prominent for broad wave-packets; later figures use a self-expanding chain to avoid boundaries.
  • It states that higher-dimensional behavior is expected but not computed; the study is one-dimensional.
  • Evident from the text: results are numerical simulations for specific disorder models (uniform static disorder, rapidly updated or sinusoidal dynamic disorder), and the qubit treatment uses an averaged density-matrix decoupling approximation; no experimental validation is reported.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.77.022302

Cites

28 of 28 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 13 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Quantum Walks on a Random Environment · p. 1

    Static disorder in one-dimensional quantum walks causes Anderson localization that completely stops the quantum walk.

    “Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk”
    • Also stated in: § static disorder · p. 3 · plain
      “for static disorder they can stop completely due to Anderson localization”
    • Also stated in: § IV. DISCUSSION · p. 7 · plain
      “Anderson localization from negative quantum interference completely stops the quantum walk”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [19] P.W. Anderson, Phys. Rev. 109 , 1492 (1958).

      Not checked in the free run.

    • cites [20] source

      Not checked in the free run.

  • Findingown data
    § static disorder · p. 3

    For a uniform initial state with static disorder, larger amplitudes remain on many sites indefinitely.

    “For the uniform initial state of Fig. 2(b) the larger amplitudes remain on many sites indefinitely.”
    Quote in the paperSubstantial claim
  • Findingown data
    § decoherence in an N -site chain · p. 4

    For sinusoidal dynamic disorder, the approach to the classical limit is even faster than for random updating.

    “The results are similar to Fig. 4 with the approach to the classical limit even faster in this case.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § decoherence in an N -site chain · p. 5

    The complete phase diagram t_c versus W displays a wide crossover grey region where t_c proportional to W^-2 is approximately obeyed.

    “The complete phase diagram is summarized in Fig. 7 with t c vs the strength of dynamic disorder W which displays a wide crossover grey color region where the law t c ∝ W -2 is approximately obeyed.”
    Quote in the paperSubstantial claim
  • Findingown data
    § dynamic disorder · p. 4

    For dynamic disorder at long times, the quantum walk becomes classical diffusive spreading, indistinguishable from an ordinary random walk.

    “A quantum walk can operate in the presence of dynamic disorder but only for short times since for longer times its motion becomes entirely classical, indistinguishable from an ordinary random walk.”
    • Also stated in: § decoherence in an N -site chain · p. 4 · plain
      “Eventually, the classical asymptotic laws σ 2 ( t ) ∝ t and C ( t ) ∝ t -1 / 2 set in after an initial period of quantum ballistic motion where σ 2 ( t ) ∝ t 2 and C ( t ) ∝ t -1 .”
    • Also stated in: § IV. DISCUSSION · p. 7 · plain
      “the quantum walks become classical via a quantum to classical crossover.”
    Quote in the paperSubstantial claim
  • May combine two claims. 12
  • Classified as probably not a substantial claim. 5

  • May combine two claims.

    Findingnone given
    § static disorder · p. 4

    For a uniform initial state with static disorder W=5, the localization length is ξ ∼ 100 W^-2 ∼ 4, much less than the system size N=101.

    “For static disorder of strength W = 5 Anderson localization occurs with localization length ξ ∼ 100 W -2 ∼ 4 much less than the system size N = 101.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § static disorder · p. 3

    For static disorder, the probability density stays around the middle site where the initial wave-packet has maximum amplitude and remains there for longer times.

    “The probability density of Fig. 1(b) is shown to stay around the middle site where the initial wave-packet has maximum amplitude and it remains there for longer times.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § dynamic disorder · p. 4

    In Fig. 1(c), quantum motion quickly disappears; in Fig. 1(d), classical diffusive motion arises.

    “In Fig. 1(c) the quantum motion seen on the left hand side of the figure quickly disappears and this also happens in Fig. 1(d) where classical diffusive motion is seen to arise.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § decoherence in an N -site chain · p. 4

    The crossover region between the ballistic law for small times and the diffusive law for long times is smooth and has mixed quantum and classical character.

    “we find that the crossover region between the ballistic law for small times and the diffusive law for long times is smooth having a mixed quantum and classical character.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § static disorder · p. 4

    For dynamic disorder W=5, quantum interference effects vanish and high probability regions keep changing, leading to randomization so the particle can still move classically.

    “In the presence of dynamic disorder W = 5 the quantum interference effects vanish. The main difference between dynamic disorder (c) and static disorder (b) is that in (c) the regions with high values of P ( x, t ) keep changing leading to randomization so that the particle can still move but in a 'classical' fashion.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data
    § dynamic disorder · p. 4

    In Fig. 2(c), randomization effects of dynamic disorder become more obvious and can be contrasted with quantum localization for static disorder.

    “In Fig. 2(c) the randomization effects of dynamic disorder become more obvious and could be contrasted with quantum localization for static disorder (Fig. 2(b)).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. DISCUSSION · p. 7

    In random media, quantum walks can perform worse than classical counterparts because Anderson localization completely stops the quantum walk while ordinary random walks can still move despite infinitely slowly.

    “In random media quantum walks can perform even worse than their classical counterparts since Anderson localization from negative quantum interference completely stops the quantum walk although the corresponding ordinary random walks in the presence of disorder can still move despite infinitely slowly.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    In the absence of disorder, the probability density is P(x,t)=(J_x(2t))2 and displays two sharp ballistic fronts at x=±2t.

    “In the absence of disorder ( ǫ ( x, t ) = 0) the amplitude is given by the Bessel function so that P ( x, t ) = ( J x (2 t )) 2”
    • Also stated in: § I. INTRODUCTION · p. 1 · plain
      “The evolution of the wave-packet at time t was shown[8] to display two sharp ballistic fronts at x = ± 2 t .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [8] D.E. Katsanos, S.N. Evangelou and Xiong, Phys. Rev. B 51 , 895 (1995).

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data
    § IV. DISCUSSION · p. 7

    Anderson localization and decoherence are the main enemies of quantum walks in static and dynamic disorder, respectively, destroying their quadratic or exponential speed-up.

    “In conclusion, our computations show Anderson localization or decoherence as the main enemies of quantum walks in the presence of static and dynamic disorder, respectively, which destroy their wellknown quadratic or exponential speed-up.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § IV. DISCUSSION · p. 7

    In higher dimensions, quantum walks are expected to operate for weak disorder to avoid quantum localization.

    “In higher dimensions quantum walks are also expected to operate for weak disorder to avoid quantum localization.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Interpretationnone given
    § static disorder · p. 3

    In higher dimensions, the Anderson transition from extended to localized states is expected via an intermediate chaotic regime which is better for quantum propagation.

    “In higher dimensions the Anderson transition from extended to localized states is expected, via an intermediate chaotic regime which is rather better for quantum propagation.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § IV. DISCUSSION · p. 7

    For dynamic disorder, there is no benefit from quantum walks either, since for longer times ballistic evolution crosses over to classical diffusion and quantum walks become classical.

    “For dynamic disorder we have no benefit from quantum walks either, since for longer times the ballistic evolution for smallt crosses over to classical diffusion for longt and the quantum walks become classical via a quantum to classical crossover.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § I. INTRODUCTION · p. 2

    Random walks in random media, the Sinai walk, lead to ultra slow classical evolution σ2(t) proportional to ln t^4.

    “This is the so-called random random or Sinai walk[17, 18] which leads to ultra slow classical evolution σ 2 ( t ) ∝ ln t 4 , very close to a complete cease.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [17] Ya.G. Sinai, Theory Prob. Appl. 27 , 247 (1982). source

      Not checked in the free run.

    • cites [18] J.P. Bouchaud and A. Georges, Phys. Rep. 195 , 127 (1990). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The quadratic speed-up is a general feature of quantum search algorithms and is familiar from tight-binding electron waves on a periodic lattice.

    “This quadratic speed-up is a general feature of quantum search algorithms[1] and is also familiar from standard quantum evolution of tight-binding electron waves on a periodic lattice[8].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [1] L.K. Grover, Phys. Rev. Lett. 79 , 325 (1997). source

      Not checked in the free run.

    • cites [8] D.E. Katsanos, S.N. Evangelou and Xiong, Phys. Rev. B 51 , 895 (1995).

      Not checked in the free run.

14 of 14

  • The paper relies on this without support.

    Disorder is unavoidable in most quantum systems.

    “The main reason for examining the robustness of quantum walks in the presence of noise is because disorder is unavoidable in most quantum systems.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • By considering a generic phase φ different from 0 and π, it is possible to simulate the behaviour of the quantum walk of two anyons.

    “Furthermore, by considering a generic phase φ (dif- ferent from 0 and π), it is possible to simulate the be- haviour of the quantum walk of two anyons, particles with a non semi-integer spin that represent a generaliza- tion of fermions and bosons [30].”

    From Two-Particle Bosonic-Fermionic Quantum Walk via Integrated Photonics (2012) · cites this paper as [30]

    The Introduction (p. 1) describes a continuous-time quantum walk of a particle on a one-dimensional lattice, and the Discussion (p. 7) addresses disorder and decoherence. Searches for anyons and a generic phase φ yielded no passage about simulating a two-anyon walk, so our copy does not verify the attributed claim. (LLM, conf=0.74)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Sinusoidal disorder varies the phase despite the phase being fixed

    Minor

    The method says the frequency is varied randomly while the phase is fixed at zero, but the description of Fig. 5 says the phase varies randomly.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. P. W. Shor, 1997, SIAM J. Comput., 26, 1184
  2. A. M. Childs, 2003, Proceedings of the 35th ACM Symposium on Theory of Computing
  3. Ya. G. Sinai, 1982, Theor. Probab. Appl., 27, 247
  4. A. A. Ovchinikov, 1975, Sov. Phys. JETP, 40, 733

Cited by

Sorted by citations

Related

Authors
Yin, Y., Katsanos, D. E., & Evangelou, S. N.
Affiliations
University of Ioannina (GR)
ORCID
Yin, Yue
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2008-02-05
Volume
77
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.77.022302
OpenAlex
W2086011823
Cited by
120
References
28
Citations by year
2026: 1 · 2025: 3 · 2024: 5 · 2023: 4 · 2022: 7 · 2021: 8 · 2020: 7 · 2019: 10 · 2018: 5 · 2017: 5 · 2016: 9 · 2015: 5 · 2014: 9 · 2013: 8 · 2012: 12
FWCI
11.61
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • 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 › Quantum chaos and dynamical systems
Keywords
quantum walks, random environment, Anderson localization, dynamic disorder, decoherence, quantum-to-classical transition, diffusive spreading, quantum carpets, static disorder

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