Perfect State Transfer in Quantum Spin Networks
Open accessAbstract
We propose a class of qubit networks that admit the perfect state transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to N-qubit spin networks of identical qubit couplings, we show that 2log3N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
Verified 2026-10-06 · 14 references not checked yet
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Written from the paper’s full text.
TL;DR
The paper proposes qubit networks that achieve perfect transfer of an arbitrary quantum state in a fixed time without switching qubit couplings. It shows perfect transfer in hypercubes and, with engineered fixed couplings, over arbitrarily long linear chains.
Key findings
- Perfect state transfer is possible between antipodes of one-link and two-link hypercubes, with constant transfer times \(t=\pi/2\) and \(t=\pi/\sqrt{2}\), respectively, independent of dimension.
- For \(N\)-qubit spin networks with identical couplings, the maximal perfect communication distance for hypercube geometries is \(2\log_3 N\); perfect transfer between antipodes of \(N\)-link hypercubes for \(N\ge 3\) is impossible.
- In an XY linear chain, perfect transfer occurs only for \(N=2\) and \(N=3\), with amplitudes \(F(t)=-i\sin(t)\) and \(F(t)=-[\sin(t\sqrt{2})]^2\); no perfect transfer occurs for \(N\ge 4\).
- With fixed but engineered couplings \(J_n=\frac{\lambda}{2}\sqrt{n(N-n)}\), perfect transfer over a chain of any length occurs in constant time \(t=\pi/\lambda\); a Heisenberg model with local magnetic fields can give the same dynamics.
Methods
- Theoretical analysis of continuous-time quantum walks on graphs: the network is a graph \(G\) with adjacency matrix \(A(G)\), and state-transfer dynamics is confined to the \(N\)-dimensional single-excitation subspace \(S_G\), where the Hamiltonian equals \(A(G)\).
- Perfect transfer is assessed via the amplitude \(F(t)=\langle N|e^{-iH_G t}|1\rangle\), with perfect transfer when \(|F(t)|=1\).
- The paper diagonalizes the XY linear chain, uses Cartesian products of two- and three-qubit chains to build hypercubes, and maps an engineered chain to a fictitious spin-\((N-1)/2\) particle governed by \(H=\lambda S_x\). No experimental data are used.
Limitations
- Stated: “we do not consider the effects of any other source of errors in this paper”; the paper also calls it an open problem whether an identical-coupling network can exceed the \(2\log_3 N\) distance.
- Stated: perfect transfer in an unmodulated XY linear chain is limited to \(N=2,3\), and perfect transfer between antipodes of \(N\)-link hypercubes for \(N\ge 3\) is impossible.
- Evident from the text: the scheme assumes idealized time-independent Hamiltonians and no dynamical control, requires pre-engineering couplings for arbitrary-length chains, and provides no experimental implementation or error analysis.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 4 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ Perfect state transfer in quantum spin networks · p. 1
For N-qubit spin networks with identical qubit couplings, the maximal perfect communication distance for hypercube geometries is 2 log 3 N.
“When restricted to N -qubit spin networks of identical qubit couplings, we show that 2 log 3 N is the maximal perfect communication distance for hypercube geometries.”
- Also stated in: § Perfect state transfer in quantum spin networks · p. 3 · boosted
“Thus we have shown that for a two-link hypercube of N sites, the maximum distance of perfect quantum communication is 2 log 3 N .”
Quote in the paperSubstantial claim - Also stated in: § Perfect state transfer in quantum spin networks · p. 3 · boosted
- Findingnone given§ Perfect state transfer in quantum spin networks · p. 1
If fixed but different couplings between qubits are allowed, perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
“if one allows fixed but different couplings between the qubits then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.”
- Also stated in: § Perfect state transfer in quantum spin networks · p. 4 · boosted
“In addition, we have shown that a quantum state can be transferred perfectly over a chain of any length as long as one can pre-engineer inter-qubit interactions.”
Quote in the paperSubstantial claim - Also stated in: § Perfect state transfer in quantum spin networks · p. 4 · boosted
- Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
Perfect state transfer from one end of the XY linear chain to another is possible only for N = 2 and N = 3.
“Perfect state transfer from one end of the chain to another is possible only for N = 2 and N = 3, with F ( t ) = -i sin( t ) and F ( t ) = -[ sin ( t √ 2 )] 2 respectively.”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 1
A class of qubit networks admits perfect transfer of any quantum state in a fixed period of time.
“We propose a class of qubit networks that admit perfect transfer of any quantum state in a fixed period of time.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
The state |1> must evolve into a superposition of states with exactly one spin up and all other spins down.
“This means that the state | 1 〉 ≡ | 1 A 00 · · · 00 B 〉 must evolve into a superposition of states with exactly one spin 'up' and all other spins 'down'.”
Quote in the paperSubstantial claim
- Findingreasoning§ Perfect state transfer in quantum spin networks · p. 4
In H_col, the matrix elements of H_G are the same as in the engineered chain.
“the same as in the engineered chain.”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 3
The matrix Eq. (13) is identical to the representation of the Hamiltonian H of a fictitious spin S = 1/2(N-1) particle: H = λS_x.
“The above matrix, Eq. (13), is identical to the representation of the Hamiltonian H of a fictitious spin S = 1 2 ( N -1) particle: H = λS x , where S x is its angular momentum operator and λ is some constant.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
The coefficient α does not change in time.
“The coefficient α does not change in time, as | 0 〉 is the zero-energy eigenstate of H G .”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 2
The eigenstates of the XY linear chain have corresponding eigenvalues E_k = -2 cos(kπ/(N+1)) for all k = 1, ..., N.
“with corresponding eigenvalues E k = -2 cos kπ N +1 for all k = 1 , . . . , N .”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 3
The d-fold Cartesian product of a one-link chain and a two-link chain lead to one-link and two-link hypercubes with |F(t)| given by the stated expressions.
“The d -fold Cartesian product of a one-link chain (two qubits) and a two-link chain (three qubits) lead to onelink and two-link hypercubes with | F ( t ) | given, respectively, by”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
The initial state |0> is the zero-energy eigenstate of H_G.
“This is an eigenstate of the Hamiltonian (2) corresponding to zero energy.”
Quote in the paperSubstantial claim - Findingown data§ Perfect state transfer in quantum spin networks · p. 4
Perfect quantum state transfer between antipodal points of one-link and two-link hypercubes is possible.
“Perfect quantum state transfer between antipodal points of one-link and two-link hypercubes is possible”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 4
The method presented works equally well for mixed states.
“Our analysis is not restricted to pure states; the method presented here works equally well for mixed states.”
Quote in the paperNot a substantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 2
The Hamiltonian H_G, when restricted to S_G, is represented by an N×N matrix identical to the adjacency matrix A(G) of the underlying graph G.
“The Hamiltonian H G , when restricted to this subspace, is represented by an N × N matrix that is identical to the adjacency matrix A ( G ), Eq. (1), of the underlying graph G .”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 3
Perfect transfer of a quantum state between the two antipodes A and B is obtained in a constant time t = π/λ.
“Thus perfect transfer of a quantum state between the two antipodes A and B is obtained in a constant time t = π/λ .”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 4
The transfer time on these hypercubes is independent of their dimension.
“The transfer time on these hypercubes is independent of their dimension.”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 3
For a two-link hypercube of N sites, the maximum distance of perfect quantum communication is 2 log 3 N.
“Thus we have shown that for a two-link hypercube of N sites, the maximum distance of perfect quantum communication is 2 log 3 N .”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 2
If the strength of couplings between qubits can be engineered, perfect state transfer can be performed between the two ends of a linear chain.
“Moreover, if we can engineer the strength of the couplings between the qubits then perfect state transfer can be also performed between the two ends of a linear chain.”
Quote in the paperSubstantial claim - Findingown data§ Perfect state transfer in quantum spin networks · p. 4
Perfect quantum state transfer between antipodal points of N-link hypercubes for N ≥ 3 is impossible.
“perfect quantum state transfer between antipodal points of N -link hypercubes for N ≥ 3 is impossible.”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 4
The evolution of a state at A under H_G remains in the column space H_col.
“The evolution of a state at A (the first column) under H G (eq. (2)) remains in the column space H col ⊆ H G , spanned by”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 2
A simple XY coupling allows perfect state transfer between antipodes of a hypercube.
“allows perfect state transfer between antipodes of a hypercube.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 4
There is an entire family of graphs G that project to the engineered chain.
“In fact, there is an entire family of such graphs G that project to this chain.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
The state transfer dynamics is completely determined by the evolution in the N-dimensional subspace S_G.
“the state transfer dynamics is completely determined by the evolution in the N -dimensional subspace S G”
Quote in the paperSubstantial claim - Findingown data§ Perfect state transfer in quantum spin networks · p. 3
The corresponding quantum hitting time between antipodes of one- and two-link hypercubes is constant.
“In contrast, as we have shown, the corresponding quantum hitting time is constant.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
The absence of perfect state transfer for N ≥ 4 can be proved by showing explicitly that the rational-ratio condition is not satisfied.
“The absence of perfect state transfer for N ≥ 4 can be proved by showing explicitly that the above condition is not satisfied.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2
For perfect state transfer in a chain, it is necessary that the ratios of the differences of eigenvalues of the related adjacency matrix A(G) are rational numbers.
“For perfect state transfer in a chain, it is necessary that the ratios of the differences of eigenvalues of the related adjacency matrix A ( G ) are rational numbers.”
Quote in the paperSubstantial claim - Findingreasoning§ Perfect state transfer in quantum spin networks · p. 3
The matrix elements J_n are equal to λ/2 sqrt(n(N-n)).
“In this case, the matrix elements J n are equal to λ 2 √ n ( N -n ).”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 4
A quantum state can be transferred perfectly over a chain of any length as long as one can pre-engineer inter-qubit interactions.
“In addition, we have shown that a quantum state can be transferred perfectly over a chain of any length as long as one can pre-engineer inter-qubit interactions.”
Quote in the paperSubstantial claim - Findingnone given§ Perfect state transfer in quantum spin networks · p. 4
The Heisenberg model with local magnetic fields gives exactly the same state transfer dynamics as the XY model.
“gives exactly the same state transfer dynamics as the XY model.”
Quote in the paperNot a substantial claim - Interpretationnone given§ Perfect state transfer in quantum spin networks · p. 1
These networks do not require qubit couplings to be switched on and off.
“these networks do not require qubit couplings to be switched on and off.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 2
The time evolution of the network in the S_G subspace can be expressed as a continuous-time quantum walk on G.
“Due to this, one may express the time evolution of the network in the S G subspace as a continuous-time quantum walk on G (first considered by Farhi and Gutmann in 1998 [10]).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] E. Farhi and S. Gutmann, Phys. Rev. A 57 , 2403 (1998). source
Not checked in the free run.
- Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 3
There is an exponential separation between the classical and quantum mean hitting times between the antipodes of one- and two-link hypercubes.
“This gives an exponential separation between the classical and quantum mean hitting times between the antipodes of one- and two-link hypercubes.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 4
These networks require no dynamical control, unlike many other quantum communication proposals.
“These networks are especially appealing as they require no dynamical control, unlike many other quantum communication proposals.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 3
The evolution of the network represents a rotation of this fictitious particle.
“of the network represents a rotation of this fictitious particle.”
Quote in the paperSubstantial claim - Interpretationnone given§ Perfect state transfer in quantum spin networks · p. 3
Each engineered qubit chain can be viewed as a projection from a graph having identical qubit couplings.
“Each such engineered qubit chain can be viewed as a projection from a graph having identical qubit couplings.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 2
The mechanism of time-evolution under a time-independent Hamiltonian avoids possible errors arising from dynamical control of inter-qubit interactions.
“This mechanism avoids possible errors arising from dynamical control of inter-qubit interactions.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 2
A chain of two or three qubits can serve as basic building blocks for networks that can perfectly transfer a quantum state over longer distances.
“A chain of two or three qubits can serve as basic building blocks for networks that can perfectly transfer a quantum state over longer distances.”
Quote in the paperSubstantial claim
- May combine two claims. 6
- Classified as probably not a substantial claim. 6
May combine two claims.
Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2For N = 2 and N = 3, the transfer amplitudes are F(t) = -i sin(t) and F(t) = -[sin(t√2)]^2 respectively.
“Perfect state transfer from one end of the chain to another is possible only for N = 2 and N = 3, with F ( t ) = -i sin( t ) and F ( t ) = -[ sin ( t √ 2 )] 2 respectively.”
Quote in the paperSubstantial claimMay combine two claims.
Findingcitation§ Perfect state transfer in quantum spin networks · p. 3Any quantum state can be perfectly transferred between the two antipodes of the one-link and two-link hypercubes of any dimensions in constant time t = π/2 and t = π/√2 respectively.
“Any quantum state can be perfectly transferred between the two antipodes of the one-link and two-link hypercubes of any dimensions in constant time t = π/ 2 and t = π/ √ 2 respectively [14].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] Note that continuous time quantum walk on the 1-link hypercube has recently been studied by C. Moore and A. Russell in Randomization and Approximation Techniques: 6th International Workshop, (RANDOM 2002), Lecture Notes in Computer Science , (Springer-Verlag, Heidelberg), vol. 2483, p. 164. source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ Perfect state transfer in quantum spin networks · p. 2The Hilbert space H_G associated with a network of N qubits is of dimension 2^N.
“The Hilbert space H G associated with a network of N qubits is of dimension 2 N .”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ Perfect state transfer in quantum spin networks · p. 2The operator of the total z-component of spin commutes with H_G, leading to conservation of the total z-component of spin.
“commutes with H G , which leads to the conservation of the total z -component of spin.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ Perfect state transfer in quantum spin networks · p. 2These tasks can also be accomplished with the Heisenberg interaction by a suitable modulation of the network, e.g., by placing it in a static but non-uniform external magnetic field.
“These tasks can be also accomplished with the Heisenberg, or exchange, interaction by a suitable modulation of the network, e.g., by placing it in a static but, in general, non-uniform external magnetic field.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ Perfect state transfer in quantum spin networks · p. 1For fixed N, a linear chain of N qubits with A and B at opposite ends maximizes the distance between A and B.
“For fixed N it maximizes the distance between A and B .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ Perfect state transfer in quantum spin networks · p. 3An improvement of the perfect quantum communication distance to N is possible if one allows for different, but fixed, couplings between qubits on a chain.
“An improvement of the perfect quantum communication distance to N is, however, possible if one allows for different, but fixed, couplings between qubits on a chain.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ Perfect state transfer in quantum spin networks · p. 3The classical mean hitting time between the antipodes in a one-link and two-link d-dimensional hypercube is given, for large d, by 2^d and 3^d, respectively.
“The classical mean hitting time between the antipodes in a one-link and two-link d -dimensional hypercube is given, for large d , by 2 d and 3 d , respectively.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] A. Bovier, Markov processes and meta-stability: Lecture notes for the Technical University of Berlin (2003). source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ Perfect state transfer in quantum spin networks · p. 4The method can be used to transfer or to distribute quantum entanglement.
“It can also be used to transfer or to distribute quantum entanglement.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ Perfect state transfer in quantum spin networks · p. 1Quantum communication over short distances through an unmodulated spin chain has been studied in detail, and an expression for the fidelity of quantum state has been obtained.
“Quantum communication over short distances through an unmodulated spin chain has been studied in detail, and an expression for the fidelity of quantum state has been obtained [9].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9] S. Bose, Phys. Rev. Lett. 91 , 207901 (2003); V. Subrahmanyam, Phys. Rev. A 69 , 034304 (2004). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductown data§ Perfect state transfer in quantum spin networks · p. 2The paper does not consider the effects of any other source of errors.
“However, note that we do not consider the effects of any other source of errors in this paper.”
Quote in the paperNot a substantial claim
The authors state this as an assumption.
Initially all spins point down along a prescribed z-axis, i.e., the network is in the state |0> = |0_A 00...00_B>.
“we shall assume that initially all spins point 'down' along a prescribed z -axis, i.e. the network is in the state | 0 〉 = | 0 A 00 · · · 00 B 〉 .”
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 tailoring hopping rates between waveguides of an array to match the matrix elements of the J_x angular momentum operator, perfect quantum state transfer becomes possible.
“Inspired by spin chains [100-102] (see figure 12(a)), by tailoring the hopping rates between the waveguides of an array to match the matrix elements of the J x angular momentum operator, such a task becomes possible.”
From Integrated photonic quantum walks (2020) · cites this paper as [102]
The paper shows that engineering inter-qubit couplings in a linear spin chain to match an angular-momentum operator yields perfect state transfer (p. 3). It does not study hopping rates between waveguides, so the waveguide-array application is not established by this source. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe evolution time is limited by the minimum energy interval: τ ≥ π/∆_min.
“From Eq. (5), the evolution time is limited by the minimum energy interval ∆ min ≡ min { ∆ k } , τ ≥ π/ ∆ min , while ˜ τ = π/ ˜ ∆ min .”
From Quantum speed limit for perfect state transfer in one dimension (2006) · cites this paper as [11]
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].”
From Quantum speed limit for perfect state transfer in one dimension (2006) · cites this paper as [11]
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.
Claimed chain communication distance counts vertices instead of edges
MinorThe proposed chain has N qubits with its endpoints at opposite ends, so their shortest path has N−1 edges. Under the paper's definition of distance, its perfect communication distance is N−1, not N.
“An improvement of the perfect quantum communication distance to N is, however, possible if one allows for different, but fixed, couplings between qubits on a chain.”
“We define the distance between A and B to be the number of edges constituting the shortest path between them.”
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- 1978, Selected Topics in Graph Theory
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- Authors
- Christandl, M., Datta, N., Ekert, A., & Landahl, A. J.
- Affiliations
- University of Cambridge (GB)
- National University of Singapore (SG)
- Hewlett-Packard (United States) (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2004-05-04
- Volume
- 92
- Issue
- 18
- Pages
- 187902-187902
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2028212109
- Semantic Scholar
- eb739158478cb8205c1ed010d906ee69335abf55
- Cited by
- 1,179
- References
- 22
- Citations by year
- 2026: 48 · 2025: 67 · 2024: 45 · 2023: 57 · 2022: 60 · 2021: 54 · 2020: 57 · 2019: 62 · 2018: 48 · 2017: 47 · 2016: 56 · 2015: 50 · 2014: 54 · 2013: 53 · 2012: 54
- FWCI
- 50.92
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- National University of Singapore (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 Mechanics and Applications
- Keywords
- perfect state transfer, quantum spin networks, qubit networks, linear chains, quantum communication, quantum computing, qubit coupling, communication distance
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