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Quantum speed limit for perfect state transfer in one dimension
Open accessAbstract
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.
Overview
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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.
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Claims the paper makes
13 Not checked yet · 8 Own interpretation · 7 No basis given · 1 Supported · 1 Partly supported · accuracy not yet measured
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
“From Eq. (4) and (5), perfect state transfer can be achieved for a time period of τ = π .”
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In the paper
“This completes our proof for η ≥ 1 for all possible engineered spin chains.”
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In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“One can then show from Eq. (8) and (10) that ω max ≥ ˜ ω max .”
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- The authors state this as an assumption: The terms λ_j play no role in the eigenvalue spectrum except a constant shift.
None of the sources cited for this claim have been checked yet.
In the paper
“Given an eigenvalue spectrum { E k } satisfying Eq. (5), the corresponding coupling constants { ω j , λ j } can be uniquely determined.”
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“We also argue that, at least for solidstate systems where interactions are local, it is more realistic to characterize the computation power by the couplings than the initial energy.”
- Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4 · hedged
“Here we shall argue that, at least for solid state systems where the interactions are local, it is more realistic to characterize the computational power by the couplings in the Hamiltonian than the initial energy.”
- Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4 · plain
“We have argued that for solid state systems it is more realistic to characterize the computation power explicitly by the coupling terms in the Hamiltonian than the initial energy.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
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- Based: passes — The paper gives a reason why it holds.
- Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4 · hedged
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
“However, this theorem does not give an appropriate bound on the speed limit for the initial state to evolve into a particular state, for example the one dimensional quantum state transfer (QST) problem [cf. Eq. (2)].”
plain- Asserted: passes — The authors put it forward as true.
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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
“By mapping the states | x 〉 to be the eigenstates of the J z angular momentum operator, this solution also describes the precession of a spin J =( N -1) / 2 under a constant magnetic field pointing along the x -direction.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“On the other hand, it is known [25] that the eigenvector | e k 〉 has exactly k -1 sign changes in the position basis {| j 〉} .”
What the sources say
- Not checked yetcites [25] G. M. L. Gladwell, Inverse problems in vibration (Kluwer Academic, Boston, 1986). source
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“From Eq. (5), the evolution time is limited by the minimum energy interval ∆ min ≡ min { ∆ k } , τ ≥ π/ ∆ min , while ˜ τ = π/ ˜ ∆ min .”
What the sources say
- Not checked yetcites [11] source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Probably not a substantial claim. 9
- It may state two separate claims as one. 6
Every source the paper cites for this claim was checked and supports it.
In the paper
“Perhaps the simplest solution to the Eq. (5) is the linear spectrum [12].”
What the sources say
- Supportedcites [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
Section III (p. 4) gives a spectral condition for perfect state transfer: ratios of eigenvalue differences must be rational. Sections VII–VIII (pp. 8–9) present an engineered spin chain with an evenly spaced spin spectrum and show that it achieves perfect transfer. The paper does not rank solutions by simplicity, but the citing sentence hedges that characterization with “perhaps.” (LLM, conf=0.83)
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Supported
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In the paper
“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].”
What the sources say
- Partly supportedcites [11] sourceExperimental — accuracy not yet measured
The paper describes engineered spin chains that enable perfect quantum-state transfer over arbitrarily long distances (p. 1) and identifies a quantum-computer data bus as a possible application (p. 1). It does not establish that this engineering optimizes quantum algorithms implemented in solid-state devices; its result concerns state transfer, not algorithm performance. (LLM, conf=0.82)
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Probably not a substantial claim.
- Partly supported
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
- Not checked yetcites [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 yetcites [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
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 1 · boosted
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
“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 .”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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
“Hence we can conclude that ω max ≥ ω N/ 2 ≥ ˜ ω max for even N .”
boosted- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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
“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 < 2 for N ≥ 2.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: In the SOS protocol, interactions between pairs of qubits can be arbitrarily switched on and off.
In the paper
“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”
What the sources say
- Not checked yetcites [10] M.-H. Yung and S. Bose, Phys. Rev. A 71 , 032310 (2005). source
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“Imperfect QST [8, 9] over a uniform spin chain was studied numerically, showing that perfect state transfer is impossible for long chains.”
What the sources say
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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Probably not a substantial claim.
In the paper
“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.”
What the sources say
- Not checked yetcites [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
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
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
“Apparently, there are unlimited number of eigenvalue spectra satisfying the condition in Eq. (5).”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“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.”
What the sources say
- Not checked yetcites [10] M.-H. Yung and S. Bose, Phys. Rev. A 71 , 032310 (2005). source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“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.”
What the sources say
- Not checked yetcites [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
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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Probably not a substantial claim.
- Also stated in: § Quantum Speed Limit for Perfect State Transfer in One Dimension · p. 4 · plain
None of the sources cited for this claim have been checked yet.
In the paper
“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].”
What the sources say
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
“More practically, systematic errors are expected to be significantly minimized with the reduction of dynamical controls.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- 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.
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.”
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?
In the citing paper
“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]
In the citing paper
“L WY ðρ; σÞ ¼ arccos½Aðρ; σÞ;ð6Þ”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [50]
Citations
Cites
A silicon-based nuclear spin quantum computer
Kane, B. E. (1998) · Nature · doi:10.1038/30156 · Cited by 3,779
Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network
Cirac, J. I., Zoller, P., Kimble, H. J., & Mabuchi, H. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.3221 · Cited by 2,215
Quantum Communication through an Unmodulated Spin Chain
Bose, S. (2003) · Physical Review Letters · doi:10.1103/physrevlett.91.207901 · Cited by 1,407
Perfect State Transfer in Quantum Spin Networks
Christandl, M., Datta, N., Ekert, A., & Landahl, A. J. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.187902 · Cited by 1,179
Ultimate physical limits to computation
Lloyd, S. (2000) · Nature · doi:10.1038/35023282 · Cited by 1,120
The Physics of Quantum Information
(2000) · doi:10.1007/978-3-662-04209-0 · Cited by 1,086
The maximum speed of dynamical evolution
Margolus, N., & Levitin, L. B. (1998) · Physica D: Nonlinear Phenomena · doi:10.1016/s0167-2789(98)00054-2 · Cited by 887
Perfect transfer of arbitrary states in quantum spin networks
Christandl, M., Datta, N., Dorlas, T. C., Ekert, A., Kay, A., & Landahl, A. J. (2005) · Physical Review A · doi:10.1103/physreva.71.032312 · Cited by 512
The maximum speed of dynamical evolution
Margolus, N., & Levitin, L. B. (1998) · Cited by 475
Gladwell, G. M. L. (1986) · Applied Mechanics Reviews · doi:10.1115/1.3149517 · Cited by 439
Cited by
Sorted by citations
Optimal Control at the Quantum Speed Limit
Caneva, T., Murphy, M., Calarco, T., Fazio, R., Montangero, S., Giovannetti, V., & Santoro, G. E. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.240501 · Cited by 509
Quantum communication through spin chain dynamics: an introductory overview
Bose, S. (2007) · Contemporary Physics · doi:10.1080/00107510701342313 · Cited by 484
Quantum communication through spin chain dynamics: an introductory overview
Bose, S. (2007) · UCL Discovery (University College London) · Cited by 443
PERFECT, EFFICIENT, STATE TRANSFER AND ITS APPLICATION AS A CONSTRUCTIVE TOOL
Kay, A. (2010) · International Journal of Quantum Information · doi:10.1142/s0219749910006514 · Cited by 350
Generalized Geometric Quantum Speed Limits
Pires, D. P., Cianciaruso, M., Céleri, L. C., Adesso, G., & Soares-Pinto, D. O. (2016) · Physical Review X · doi:10.1103/physrevx.6.021031 · Cited by 265
Metadata
- 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
- 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
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › 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
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