CoChat
Sign Up
Cited by 128Quantum Information and CryptographyQuantum Computing Algorithms and ArchitectureQuantum many-body systems

Quantum speed limit for perfect state transfer in one dimension

Open access
Yung, Man-Hong·2006·Physical Review A
DOI 10.1103/physreva.74.030303
Open this paper in CoChat — free accountOpen PDF

Abstract

The basic idea of spin-chain engineering for perfect quantum state transfer (QST) is to find a set of coupling constants in the Hamiltonian such that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical controls. The minimal possible evolution time represents a speed limit for QST. We prove that the optimal solution is the one simulating the precession of a spin in a static magnetic field. We also argue that, at least for solid-state systems where interactions are local, it is more realistic to characterize the computation power by the couplings than the initial energy.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proves that the fastest possible perfect quantum state transfer (QST) through a one-dimensional engineered spin chain is achieved by the linear eigenvalue spectrum, equivalent to spin precession in a static magnetic field. It establishes a quantum speed limit for QST and argues that, for local solid-state systems, computational power is better characterized by Hamiltonian couplings than by initial energy.

Key findings

  • The optimal perfect-QST solution is the linear spectrum \(E_k = -(N-1)/2 + k - 1\), \(k=1,\dots,N\), which gives perfect transfer at \(\tau=\pi\) and is the same coupling set as spin precession in a static magnetic field.
  • The paper proves \(\eta \ge 1\) for all engineered spin chains, where \(\eta\) compares evolution time under normalized maximum coupling to the linear reference chain, so the linear solution is optimal and sets the one-dimensional QST speed limit.
  • Table I compares efficiencies: linear \(\eta=1\); quadratic \(\eta=q(2\alpha+N)\); gapped linear I \(\eta=(1+4q/N)\); gapped linear II \(\eta=q\) for \(N=4\) only; cosine \(\eta=14.0\) for \(N=31\) with no analytic solution given.
  • Engineered chains can be roughly twice as fast as a series-of-swaps protocol: \(1 \le \eta_{\rm sos} < 2\) for \(N \ge 2\).
  • The paper argues that the Margolus-Levitin theorem, based on initial energy, does not give an appropriate bound for transfer to a particular state; for local solid-state systems, couplings in the Hamiltonian are a more realistic measure of computational power.

Methods

  • Theoretical study of a one-dimensional spin-1/2 chain Hamiltonian with site-dependent couplings \(\omega_j\) and \(\lambda_j\), restricted to the single-excitation subspace and mirror-symmetric chains.
  • Perfect QST is formulated as an inverse eigenvalue problem: given an eigenvalue spectrum satisfying the transfer condition, the couplings are uniquely determined.
  • The proof uses the Hellmann-Feynman theorem, eigenvalue interlacing, trace differences between symmetric subspaces, and a recursive relation for \(\omega_m\) to show \(\omega_{\max} \ge \tilde{\omega}_{\max}\) for even and odd \(N\).

Limitations

  • The paper states that the speed limit could be overcome if not confined to nearest-neighbor interactions in one dimension, by increasing either network complexity or Hamiltonian complexity; details are deferred elsewhere.
  • The comparison table includes restricted cases: gapped linear II is for \(N=4\) only, and the cosine spectrum is for \(N=31\) with no analytic solution given.
  • The proof relies on mirror-symmetric chains and non-degenerate eigenvalues; the coupling-based computational-power argument is explicitly framed for solid-state systems where interactions are local.

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

Integrity signals

Registry

Crossref · 10.1103/physreva.74.030303

Cites

30 of 30 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 19 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 2

    For the linear spectrum, perfect state transfer can be achieved for a time period of τ = π.

    “From Eq. (4) and (5), perfect state transfer can be achieved for a time period of τ = π .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    η ≥ 1 for all possible engineered spin chains.

    “This completes our proof for η ≥ 1 for all possible engineered spin chains.”
    Quote in the paperSubstantial claim
  • Interpretationnone given
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 3

    Under scaling such that ̃∆_min = ∆_min, η ≥ 1 if ω_max ≥ ̃ω_max.

    “Suppose we scale the eigenvalue spectra such that ˜ ∆ min = ∆ min , then η ≥ 1 if ω max ≥ ˜ ω max . We shall show that it is indeed true.”
    Quote in the paperNot a substantial claim
  • Interpretationreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 3

    From Eq. (8) and (10), ω_max ≥ ̃ω_max.

    “One can then show from Eq. (8) and (10) that ω max ≥ ˜ ω max .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The terms λ_j play no role in the eigenvalue spectrum except a constant shift.
  • Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 2

    Given an eigenvalue spectrum satisfying Eq. (5), the corresponding coupling constants can be uniquely determined.

    “Given an eigenvalue spectrum { E k } satisfying Eq. (5), the corresponding coupling constants { ω j , λ j } can be uniquely determined.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [24] D. Boley and G. H. Golub, Inverse Problems 3 , 595 (1987). source

      Not checked in the free run.

    • cites [25] G. M. L. Gladwell, Inverse problems in vibration (Kluwer Academic, Boston, 1986). source

      Not checked in the free run.

  • May combine two claims. 6
  • Classified as probably not a substantial claim. 9

  • May combine two claims.

    Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1

    The optimal solution for perfect state transfer is the one proposed in Ref.[12,22] (the linear spectrum), which is the same as the set of couplings describing spin precession in a static magnetic field.

    “We prove that the optimal solution is the one simulating the precession of a spin in a static magnetic field.”
    • Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1 · boosted
      “In the following, we shall show that the optimal solution for perfect state transfer is exactly the one proposed in Ref.[12, 22].”
    • Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1 · plain
      “Amusingly, this optimal set of couplings is the same as that describing spin precession in a static magnetic field.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [12] M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. 92 , 187902 (2004); M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71 , 032312 (2005). source

      Not checked in the free run.

    • cites [22] As pointed out in Ref.[17], the same set of coupling was first proposed by Cook and Shore in 1979 [R. J. Cook and B. W. Shore, Phys. Rev. A 20 , 539 (1979)] in the context of coherent population transfer between different atomic levels. source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    For odd N ≥ 5, ω_max ≥ ω_m ≥ ̃ω_max.

    “Since ω 2 m is a positive sum of the products ∆ j ∆ k , it is minimized when all ∆ k = ˜ ∆ min , and hence ω max ≥ ω m ≥ ˜ ω max .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 3

    For even N, ω_max ≥ ω_{N/2} ≥ ̃ω_max.

    “Hence we can conclude that ω max ≥ ω N/ 2 ≥ ˜ ω max for even N .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    For the SOS protocol, η_sos = (N-1)/2 ̃ω_max, which is an increasing function of N, and 1 ≤ η_sos < 2 for N ≥ 2.

    “Since it will totally take N -1 steps to transfer a state through a chain of N spins, we can generalize our definition of the efficiency for this case as: η sos = ( N -1) / 2˜ ω max , which is an increasing function of N , and one can easily show that 1 ≤ η sos &lt; 2 for N ≥ 2.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: In the SOS protocol, interactions between pairs of qubits can be arbitrarily switched on and off.

  • May combine two claims.

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 2

    For a mirror-symmetric Hamiltonian, < ̄x|e_k> = (-1)^k <x|e_k>, and hence |< ̄x|U(τ)|x>| = 1 if Eq. (5) holds.

    “Provided that the Hamiltonian in Eq. (1) or Eq. (3) is mirror symmetrical, i.e. λ j = λ ¯ j and ω j = ω N -j , one can show [10] that 〈 ¯ x | e k 〉 = ( -1) k 〈 x | e k 〉 , and hence |〈 ¯ x | U ( τ ) | x 〉| = 1 if”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [10] M.-H. Yung and S. Bose, Phys. Rev. A 71 , 032310 (2005). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1

    Imperfect QST over a uniform spin chain is impossible for long chains.

    “Imperfect QST [8, 9] over a uniform spin chain was studied numerically, showing that perfect state transfer is impossible for long chains.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [8] S. Bose, Phys. Rev. Lett. 91 , 207901 (2003). source

      Not checked in the free run.

    • cites [9] M.-H. Yung, D. W. Leung, and S. Bose, Quantum Inf. Commun. 4 , 174 (2003). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1

    Perfect QST without dynamical controls is possible if couplings between spins are non-uniform.

    “However, it has been suggested [12] that perfect QST, without dynamical controls, is possible if we allow the couplings between the spins to be non-uniform.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [12] M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. 92 , 187902 (2004); M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71 , 032312 (2005). source

      Not checked in the free run.

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

    Premisenone given
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 2

    There are an unlimited number of eigenvalue spectra satisfying Eq. (5).

    “Apparently, there are unlimited number of eigenvalue spectra satisfying the condition in Eq. (5).”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 2

    The linear spectrum is the simplest solution to Eq. (5).

    “Perhaps the simplest solution to the Eq. (5) is the linear spectrum [12].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [12] M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. 92 , 187902 (2004); M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71 , 032312 (2005). source

      Not checked in the free run.

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

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1

    There are many possible solutions for constructing spin chains that allow perfect state transfer, and searching for them is in general an inverse eigenvalue problem.

    “Moreover, it has also been pointed out [10] that there are many possible solutions for constructing spin chains which allow perfect state transfer, and the searching for the solutions is in general an inverse eigenvalue problem.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [10] M.-H. Yung and S. Bose, Phys. Rev. A 71 , 032310 (2005). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1

    The Margolus-Levitin theorem states that the shortest time for a quantum state to evolve into an orthogonal state is limited by the initial energy.

    “The Margolus-Levitin theorem [6] suggests that the shortest time for a quantum state to evolve into an orthogonal state is limited by the initial energy.”
    • Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4 · plain
      “According to the Margolus-Levitin theorem [6], the minimum time for a quantum state | ψ i 〉 to evolve into an orthogonal state | ψ f 〉 , where 〈 ψ f | ψ i 〉 =0, is limited by the initial energy: max ( π 2( E -E 0 ) , π 2∆ E ) , where E = 〈 ψ i | H | ψ i 〉 is the initial energy, E 0 is the ground state energy and ∆ E = √ 〈 ψ i | ( H -E ) 2 | ψ i 〉 is the energy uncertainty of the initial state.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [6] N. Margolus and L. B. Levitin, Physica D 120 , 188 (1998); P. Kosi´ nski and M. Zych, Phys. Rev. A 73 , 024303 (2006). source

      Not checked in the free run.

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

    Interpretationcitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    The speed limit for 1D QST could be overcome by increasing either network complexity or Hamiltonian complexity if not confined to nearest-neighbor interactions in one dimension.

    “We lastly comment that such speed limit, if not confined with nearest-neighbor interactions in one dimension, could be overcome by increasing either the network complexity [9] or the Hamiltonian complexity [26].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [9] M.-H. Yung, D. W. Leung, and S. Bose, Quantum Inf. Commun. 4 , 174 (2003). source

      Not checked in the free run.

    • cites [26] V. Giovannetti, S. Lloyd, L. Maccone, Europhys. Lett. 62 , 615 (2003). source

      Not checked in the free run.

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

    Interpretationnone given
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    Systematic errors are expected to be significantly minimized with the reduction of dynamical controls.

    “More practically, systematic errors are expected to be significantly minimized with the reduction of dynamical controls.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The SOS protocol necessarily introduces errors due to dynamical controls, causing the fidelity of transferred states to decay exponentially as the number of swaps increases.

  • Classified as probably not a substantial claim.

    Interpretationcitation
    § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4

    Quantum algorithms implemented in solid state devices can be optimized by employing the concept of spin chain engineering.

    “This work also implies that quantum algorithms, when implemented in solid state devices, can be optimized, if we employ the concept of spin chain engineering [11].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [11] source

      Not checked in the free run.

14 of 14

  • The paper relies on this without support.

    The SOS protocol necessarily introduces errors due to dynamical controls, causing the fidelity of transferred states to decay exponentially as the number of swaps increases.

    “However, even in the absence of decoherence, the intrinsic problem of this protocol is that each swap would necessarily introduce some errors due to dynamical controls, the fidelity of the transferred states would therefore decay exponentially as the number of swaps increases.”

  • The authors state this as an assumption.

    The terms λ_j play no role in the eigenvalue spectrum except a constant shift.

    “Of course, this argument is based on the assumption that the terms λ j play no role in the eigenvalue spectrum except a constant shift.”

  • The authors state this as an assumption.

    In the SOS protocol, interactions between pairs of qubits can be arbitrarily switched on and off.

    “In these scenarios, a typical assumption usually made is that the interactions between pairs of qubits can be arbitrarily switched on and off [4, 5].”

  • The paper relies on this without support.

    The relative phase factor in the transferred state can be corrected by local operations and is usually ignored.

    “Although the transferred state in general differs from the input state by a relative phase factor, i.e. αe iφ | 1 〉 + β | 0 〉 , the phase factor could be corrected by some local operations and is thus usually ignored.”
4 of 4

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?

  • Analytic expressions for geodesic distance are known only for the quantum Fisher information metric and the Wigner-Yanase information metric.

    “In fact, analytic expressions are known only for the geodesic distance related to the quantum Fisher information metric [56],”

    From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [50]

  • The geodesic distance for the Wigner-Yanase information metric is L_WY(ρ,σ)=arccos[A(ρ,σ)], where A is quantum affinity.

    “L WY ðρ; σÞ ¼ arccos½Aðρ; σÞ;ð6Þ”

    From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [50]

Cites

Flagged first, then by citations

Cited by

Sorted by citations

Related

Authors
Yung, M.-H.
Affiliations
University of Illinois Urbana-Champaign (US)
Corresponding author
Yung, Man-Hong
ORCID
Yung, Man-Hong
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2006-09-14
Volume
74
Issue
3
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.74.030303
OpenAlex
W2073278667
Cited by
128
References
30
Citations by year
2026: 4 · 2025: 5 · 2024: 5 · 2023: 4 · 2022: 11 · 2021: 7 · 2020: 5 · 2019: 4 · 2018: 8 · 2017: 10 · 2016: 9 · 2015: 6 · 2014: 3 · 2013: 6 · 2012: 6
FWCI
8.70
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)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
quantum speed limit, perfect state transfer, spin chains, one-dimensional systems, local interactions, solid-state quantum computing

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review