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Quantum walks on a random environment
Open accessAbstract
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.
Overview
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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.
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Claims the paper makes
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In the paper
“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”
What the sources say
- Not checked yetcites [19] P.W. Anderson, Phys. Rev. 109 , 1492 (1958).
- Not checked yetcites [20] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § static disorder · p. 3 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For the uniform initial state of Fig. 2(b) the larger amplitudes remain on many sites indefinitely.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The results are similar to Fig. 4 with the approach to the classical limit even faster in this case.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § decoherence in an N -site chain · p. 4 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We shall show that static disorder is responsible for exponentially suppressed quantum evolution with variance σ 2 ( t ) reaching a time-independent limit for long t , depending on the strength of static disorder and space dimensionality.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For static disorder of strength W = 1 . 5 quantum interference causes Anderson localization which stops the quantum”
- Also stated in: § static disorder · p. 3 · plain
“From Figs. 1(b) and 2(b) we can see how strong static disorder with ǫ ( x ) chosen from a uniform probability distribution within [ -W/ 2 , + W/ 2] causes destructive interference with Anderson localization.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § static disorder · p. 3 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The quantum-toclassical crossover takes place after a characteristic time t c ∝ W -2 .”
- Also stated in: § decoherence in a qubit · p. 7 · plain
“The quantum to classical crossover for dynamic disorder of strength W occurs at t c ∝ W -2 .”
- Also stated in: § decoherence in an N -site chain · p. 5 · hedged
“where the law t c ∝ W -2 is approximately obeyed.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § decoherence in a qubit · p. 7 · plain
In the paper
“The decoherence effect of dynamical disorder which turns the quantum wave propagation into classical diffusion is shown in Fig. 3”
- Also stated in: § dynamic disorder · p. 4 · plain
“The effect of dynamic disorder is equivalent to introducing coin chaos which makes the quantum coherence disappear[22].”
What the sources say
- Not checked yetcites [22] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § dynamic disorder · p. 4 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“From Figs. 4 and 5, except for no difference between the two types of dynamic disorder, 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.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes
- It may state two separate claims as one. 12
- Probably not a substantial claim. 5
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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)).”
Strength:hedged· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“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 .”
What the sources say
- Not checked yetcites [8] D.E. Katsanos, S.N. Evangelou and Xiong, Phys. Rev. B 51 , 895 (1995).
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § I. INTRODUCTION · p. 1 · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In higher dimensions quantum walks are also expected to operate for weak disorder to avoid quantum localization.”
Strength:hedged· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“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].”
What the sources say
- Not checked yetcites [1] L.K. Grover, Phys. Rev. Lett. 79 , 325 (1997). source
- Not checked yetcites [8] D.E. Katsanos, S.N. Evangelou and Xiong, Phys. Rev. B 51 , 895 (1995).
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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.”
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?
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“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]
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
MinorThe 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.
“We chose to vary the frequency ω at random uniformly within the interval [0 , 2 π ] and fixed the phase φ x to zero.”
“In Fig. 5 the effect of sinusoidal dynamic disorder is considered with constant amplitude and randomly varying the phase.”
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Metadata
- 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
- 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
- green
- arxiv.org
- 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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