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Cited by 72Quantum Computing Algorithms and ArchitectureQuantum Mechanics and ApplicationsNumerical methods for differential equations

The Dirac equation as a quantum walk: higher dimensions, observational convergence

Open access
Arrighi, Pablo, Nesme, Vincent, Forets, Marcelo·2014·Journal of Physics A: Mathematical and Theoretical
DOI 10.1088/1751-8113/47/46/465302
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Abstract

The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann–Wigner equations and symmetric hyperbolic systems in general. We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation. We do so by adapting a powerful method from standard numerical analysis, which is of general interest to the field of quantum simulation. At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme. At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proves that a discrete-time quantum walk models the Dirac equation in 2+1 and 3+1 dimensions, and that its solutions converge to the Dirac Cauchy problem as the discretization step ε→0. It adapts numerical-analysis consistency, stability, and convergence arguments to give error bounds and an observational interpretation.

Key findings

  • The Dirac quantum walk \(W_\varepsilon = C(H\otimes T_1)(HF\otimes T_2)(F^\dagger\otimes T_3)\) (and lower-dimensional versions) models the Dirac equation; consistency is proved to order \(r=1\), stability follows from unitarity, and hence the model is convergent to first order.
  • For any time \(x_0\) and sufficiently regular initial condition \(\phi\), the probability of observing a discrepancy between the iterated walk \(W_\varepsilon^l\,\mathrm{reconstruct}(\mathrm{discretize}(\phi))\) and the Dirac solution \(T(x_0)\phi\) goes to zero quadratically as \(\varepsilon\to 0\); in norm, the error is bounded as \(O(\varepsilon)\) with \(l\varepsilon=x_0\).
  • The method generalizes to Bargmann–Wigner equations and symmetric hyperbolic systems in general, and more generally to equations whose Hamiltonians can be expressed as sums of logarithms of quantum walks.
  • In the observational-equivalence case \(s=0\), the Sobolev norm coincides with the quantum-theory norm, and the probability of discrepancy is bounded above by a term proportional to \(\varepsilon^2\|\psi(0)\|^2\), diminishing quadratically as \(\varepsilon\to 0\).

Methods

  • Informal derivations are given in 2+1 and 3+1 dimensions; formal analysis is carried out in Fourier space using Sobolev spaces \(H^s_m\), with well-posedness of the Dirac Cauchy problem as a symmetric hyperbolic system.
  • Consistency of order \(r=1\) is proved by Taylor expansion of the Fourier-space walk operator \(W_\varepsilon(k)\) in \(\varepsilon\); stability is proved via unitarity; convergence is obtained via the Lax theorem and an inlined quantified proof.
  • Space discretization is handled by low-pass filtering, sampling, renormalization, and reconstruction (Whittaker–Kotelnikov–Shannon), yielding an overall error bound combining discretization and walk convergence.

Limitations

  • Stated: the paper leaves for future work the study of Poincaré-invariance of such discrete models and a decohered version of the quantum-walk model.
  • Stated/evident: convergence requires sufficiently regular initial conditions, specifically \(\phi\in H^{s+2}_m\), and the discretization uses ideal low-pass filtering and renormalization; the result is for the free Dirac equation and symmetric hyperbolic systems, not interacting quantum field theory.
  • Evident from the text: no experimental data or physical implementation is provided; the observational equivalence is a mathematical probability bound, and the paper explicitly asks whether experimentalists could tell the difference.

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

Integrity signals

Registry

Crossref · 10.1088/1751-8113/47/46/465302

Cites

35 of 35 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

8 of 24 cited sources checked · 3 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    The (2+1)-dimensional Dirac QW models the (2+1)-dimensional Dirac equation.

    “This Dirac QW [4-6] models the + (2 1)-dimensional Dirac equation.”
    Quote in the paperSubstantial claim2 of 3 sources checked
    • cites [4] source
    • cites [5] source

      In “Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata,” p. 4, the paper says its lattice evolution has the Dirac equation as its continuum limit. Page 3 separately mentions a discretized Weyl equation in two spatial dimensions, but the copy does not establish that the cited model is a (2+1)-dimensional Dirac quantum walk. (LLM, conf=0.85)

      Experimental — accuracy not yet measured
    • cites [6] source

      Section 4 (pp. 9, 11) says the quantum particle automaton approximates a Dirac particle and yields the Dirac propagator in the continuum limit. But the Introduction (p. 2) explicitly limits the paper to one dimension; it does not support the claim about a (2+1)-dimensional Dirac quantum walk. (LLM, conf=0.89)

      Experimental — accuracy not yet measured
  • Premisecitation

    Convergence of the Dirac QW has been proved in (1+1)-dimensions in [13, 14, 16].

    “It has been proved in + (1 1)-dimensions in [13, 14, 16].”
    Quote in the paperNot a substantial claim1 of 3 sources checked
    • cites [13] source
    • cites [14] source
    • cites [16] source

      Section 4 (p. 10) derives dynamics of a discrete-time quantum walk that correspond to the Dirac equation in 1+1 dimensions, and Section 6 (p. 18) describes this as asymptotic behavior. The paper does not explicitly establish the claimed convergence of a Dirac quantum walk; its stated weak-convergence results concern quantum-walk limit distributions. (LLM, conf=0.80)

      Experimental — accuracy not yet measured
  • Findingreasoning

    Consistency is ensured to first order.

    “consistency is ensured to first order”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The reconstruction procedure satisfies reconstruct(discretize(φ)) = φ_LP.

    “Notice that ϕ ϕ = ( ) ( ) reconstruct discretize LP LP”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The overall error of the scheme (discretize, walk, reconstruct) is bounded by (C+C') ε ||φ||_{H^{s+2}} for all φ.

    “And thus the bound on the overall error is:”
    • Also stated in: § 4. Space discretization · plain
      “ϕ ε ϕ ε ϕ ∥ - ∥ ⩽ + ′ ε + ( ) W T l lC C reconstruct (discretize( )) ( ) ( ) . l H H 2 m s m s 2”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 13
  • May combine two claims. 8
  • The quote only partly states this claim. 3

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

    Premisecitation

    The difficulty to analyse the (3+1)-dimensional Dirac QW is mentioned in [14, 17-19].

    “The difficulty to analyse the + (3 1)-dimensional Dirac QW is mentioned in [14, 17-19].”
    Quote in the paperNot a substantial claim2 of 4 sources checked
    • cites [14] source
    • cites [17] source

      On p. 4, Strauch says a calculation for a four-component quantum walk on a three-dimensional lattice breaks down because its operators do not commute. This supports a difficulty in analyzing a related higher-dimensional walk, but the passage concerns a particular continuous-time limit and does not identify the example as a (3+1)-dimensional Dirac quantum walk or claim that analyzing it generally is difficult. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
    • cites [18] source
    • cites [19] Cha M, 2011

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation

    Consistency of the (1+1)-dimensional Dirac QW has been observed numerically in [9] and in (3+1)-dimensions in [10-12].

    “It has been observed numerically in + (1 1) -dimensions in [9] and in + (3 1)-dimensions in [10-12].”
    Quote in the paperNot a substantial claim1 of 4 sources checked
    • cites [9] source

      The Introduction and Quantum Lattice-Gas Model sections say that the one-dimensional quantum lattice gas yields the Dirac propagator in the continuum limit. The Simulations section reports numerical evolution, but does not describe it as a numerical test of consistency with the Dirac equation. The paper treats models in multiple dimensions as future work, so it does not support the (3+1)-dimensional part. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
    • cites [10] Palpacelli S, 2009

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [11] source
    • cites [12] source

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

    Premisecitation

    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].”
    Quote in the paperNot a substantial claim2 of 2 sources checked

  • The quote only partly states this claim.

    Findingreasoning

    For every s≥0, the loss induced by low-pass filtering is bounded: ||φ - φ_LP||_{H^s} ≤ C' ε ||φ||_{H^{s+2}}.

    “Low-pass filtering.For every ⩾ s 0, we have”
    • Also stated in: § 4. Space discretization · plain
      “Second, the loss induced by low-pass filtering is small, as needed below in order to bound the overall error.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingnone given

    The paper does not limit itself to the massless case, nor to the (1+1)-dimensional case.

    “Contrary to previous works, we do not limit ourselves to the massless case, nor to the + (1 1)-dimensional case.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning

    For any time x0 and a sufficiently regular initial condition φ, the probability of observing a discrepancy between the iterated walk W reconstruct(discretize(φ)) and the solution of the Dirac equation ψ=T(x0)φ goes to zero quadratically as the discretization step ε goes to zero.

    “We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given

    The result can be generalized to other first-order PDEs.

    “It can also be generalized to other first-order PDEs”
    Quote in the paperNot a substantial claim

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

    Interpretationnone given

    This method will apply equally well to more complicated QWs, e.g. Bargmann-Wigner equations and symmetric hyperbolic systems in general.

    “This is a key point: this method will apply equally well to more complicated QWs, e.g.Bargmann-Wigner equations and symmetric hyperbolic systems in general.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    The convergence result reinforces the status of this QW as a simple, discrete toy model of relativistic particles.

    “At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The Dirac Hamiltonian is a sum of logarithms of QWs, which enables modeling it as the product of these QWs.

    “Ultimately, it is the fact the Dirac Hamiltonian is a sum of logarithms of QWs, which enables us to model it as the product of these QWs.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given

    The result can be generalized to PDEs whose Hamiltonians can be expressed as a sum of logarithms of QWs.

    “as well as to PDEs whose Hamiltonians can be expressed as a sum of logarithms of QWs”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given

    The matrices σ2⊗σ0 and -iσ3 are hermitian, square to the identity, and anticommute.

    “Indeed, one can check that the matrices σ σ ⊗ 2 0 and σ σ -⊗ ( ) i 3 are hermitian, that they square to the identity, and that they anticommute.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    For quantum simulation schemes without known solutions, the procedure will still apply.

    “for quantum simulation schemes without known solutions, the procedure will still apply.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Interpretationreasoning

    The probability of observing a discrepancy between the iterated walk and the Dirac solution is bounded above by ε^2 C^2 ||ψ(0)||^2 / 2, diminishing quadratically as ε→0.

    “Simple trigonometric reasoning shows that this is bounded above by ε ψ x C || (0)|| 2 0 2 2 2 , i.e. it diminishes quadratically as ε goes to zero.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The probability of observing a discrepancy between two quantum states is sin^2 of the angle between them.

  • May combine two claims.

    Premisenone given

    From (2+1) to (3+1) dimensions the Dirac equation changes form, the spin degree of freedom goes to degree four.

    “From + (2 1) to + (3 1)-dimensions the Dirac equation changes form, the spin degree of freedom goes to degree four.”
    Quote in the paperSubstantial claim

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

    Interpretationreasoning

    The method works equally well for any symmetric hyperbolic systems with rational eigenvalues.

    “The method would work equally well for any symmetric hyperbolic systems with rational eigenvalues, i.e. equations of the form”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    The result can be specialized elegantly to lower dimensions.

    “The result can be specialized elegantly to lower dimensions.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The method avoids the trouble of previous works which rely on solving the QW in order to compare its solution against that of the Dirac equation.

    “This method avoids the trouble of previous works (e.g.[14]) which rely on solving the QW in order to compare its solution against that of the Dirac equation.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [14] source

  • May combine two claims.

    Interpretationreasoning

    The convergence result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme.

    “At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme.”
    Quote in the paperSubstantial claim

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

    Premisereasoning

    Convergence implies consistency, but the converse does not always hold.

    “Of course convergence implies consistency, but the converse does not always hold.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    The Dirac operator is just a pointwise multiplication in Fourier space.

    “because the Dirac operator is just a pointwise multiplication in Fourier space, most of our derivations will use it.”
    Quote in the paperNot a substantial claim
21 of 21

  • The paper relies on this without support.

    The Dirac equation is a symmetric hyperbolic system.

    “Since the Dirac equation is a symmetric hyperbolic system”

  • The paper relies on this without support.

    The probability of observing a discrepancy between two quantum states is sin^2 of the angle between them.

    “According to quantum theory the probability of observing through a measurement a discrepancy between the iterated walk ϕ ε W l l and the solution of the Dirac equation ψ x ( ) 0 is given by θ sin ( ) 2 , with θ the angle between both vectors.”

  • The paper relies on this without support.

    Quantum simulation of physical processes will need to be validated.

    “With the future development of quantum computers, the quantum simulation of physical processes will also need to be validated.”
3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. 10.1063/1.2759837 · DOI · Paper page
  2. Sobolev S, 1938, Rec. Math. [Mat. Sbornik] N.S., 4, 471

Cited by

Sorted by citations

Authors
Arrighi, P., Nesme, V., & Forets, M.
Affiliations
Université Joseph Fourier (FR)
ORCID
Venue
Journal of Physics A: Mathematical and Theoretical
Publisher
IOP Publishing
Published
2014-11-04
Volume
47
Issue
46
Pages
465302
ISSN
  • 1751-8113 (ISSN-L)
  • 1751-8121
DOI
10.1088/1751-8113/47/46/465302
OpenAlex
W1978704488
Semantic Scholar
5a92eed31ed21facac183b2ff50d473edc2b8ada
Cited by
72
References
31
Citations by year
2026: 2 · 2025: 6 · 2024: 5 · 2023: 4 · 2022: 8 · 2021: 5 · 2020: 7 · 2019: 10 · 2018: 5 · 2017: 6 · 2016: 7 · 2015: 5 · 2014: 1 · 2013: 1
FWCI
6.27
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Physics A Mathematical and Theoretical (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
Agence Nationale de la Recherche (ANR-10-JCJC-0208, ANR-10, ANR-10-JCJC)
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 › Mathematics › Numerical Analysis › Numerical methods for differential equations
Keywords
Dirac equation, quantum walks, Bargmann-Wigner equations, symmetric hyperbolic systems, quantum simulation, convergence analysis, error bounds, convergence rate, relativistic particles, lattice discretization, Cauchy problem

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