Perspective on exchange-coupled quantum-dot spin chains

Kandel, Yadav P.; Qiao, Haifeng; Nichol, John

doi:10.1063/5.0055908

Cited by 15 publications

(6 citation statements)

References 190 publications

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“…The XX chains, on the contrary, allow perfect QST using different strategies [8][9][10][11] or near perfect QST at transfer times close to the minimal time allowed by the quantum speed limit [12][13][14]. Regrettably, more and more experimental implementations can be modeled only by using effective Heisenberg chain Hamiltonians [15][16][17][18][19][20][21][22][23][24][25] (or more complicated interactions [26][27][28]) while the XX chains remain as an option only for simplistic highly anisotropic situations. Of course, there is always room for renewed proposals that involve the XX Hamiltonian, as is the case in some XX chains that show topological states [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

Serra¹,

Ferrón²,

Osenda³

2022

J. Phys. A: Math. Theor.

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We construct the exact solution for a family of one-half spin chains explicitly. The spin chains Hamiltonian corresponds to an isotropic Heisenberg Hamiltonian, with staggered exchange couplings that take only two diﬀerent values. We work out the exact solutions in the one- excitation sub-space. Regarding the problem of quantum state transfer, we use the solution and some theorems concerning the approximation of irrational numbers, to show the appearance of conclusive pretty good transmission for chains with particular lengths. We present numerical evidence that pretty good transmission is achieved by chains whose length is not a power of two. The set of spin chains that shows pretty good transmission is a subset of the family with an exact solution. Using perturbation theory, we thoroughly analyze the case when one of the exchange coupling strengths is orders of magnitude larger than the other. This strong coupling limit allows us to study, in a simple way, the appearance of pretty good transmission. The use of analytical closed expressions for the eigenvalues, eigenvectors, and transmission probabilities allows us to obtain the precise asymptotic behavior of the time where the pretty good transmission is observed. Moreover, we show that this time scales as a power law whose exponent is an increasing function of the chain length. We also discuss the crossover behavior obtained for the pretty good transmission time between the regimes of strong coupling limit and the one observed when the exchange couplings are of the same order of magnitude.

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Section: Introductionmentioning

confidence: 99%

Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

Serra¹,

Ferrón²,

Osenda³

2022

J. Phys. A: Math. Theor.

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“…The time period and the strength of the interaction can both be controlled by tuning the modulation. In semiconductor qubits, exchange interactions can be induced in a similar fashion by tuning tunnelling barriers [18][19][20]. Alternatively, they can be mediated by a cavity-mode [21] where the strength and duration can be controlled by varying the detuning.…”

Section: Introductionmentioning

confidence: 99%

Spreading entanglement through pairwise exchange interactions

Theerthagiri¹,

Ganesh²

2023

Preprint

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The spread of entanglement is a problem of great interest. It is particularly relevant to quantum state synthesis, where an initial direct-product state is sought to be converted into a highly entangled target state. In devices based on pairwise exchange interactions, such a process can be carried out and optimized in various ways. As a benchmark problem, we consider the task of spreading one excitation among N two-level atoms or qubits. Starting from an initial state where one qubit is excited, we seek a target state where all qubits have the same excitation-amplitude -a generalized-W state. This target is to be reached by suitably chosen pairwise exchange interactions. For example, we may have a a setup where any pair of qubits can be brought into proximity for a controllable period of time. We describe three protocols that accomplish this task, each with N − 1 tightlyconstrained steps. In the first, one atom acts as a flying qubit that sequentially interacts with all others. In the second, qubits interact pairwise in sequential order. In these two cases, the required interaction times follow a pattern with an elegant geometric interpretation. They correspond to angles within the spiral of Theodorus -a construction known for more than two millennia. The third protocol follows a divide-and-conquer approach -dividing equally between two qubits at each step. For large N , the flying-qubit protocol yields a total interaction time that scales as √ N , while the sequential approach scales linearly with N . For the divide-and-conquer approach, the time has a lower bound that scales as log N . With any such protocol, we show that the phase differences in the final state cannot be independently controlled. For instance, a W-state (where all phases are equal) cannot be generated by pairwise exchange.

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“…They are also easier to control using external time-dependent pulses (see [12] and references therein). On the other hand, the Heisenberg Hamiltonian, or XXZ Hamiltonians, are effective Hamiltonians that model most experimental situations [13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

The scaling law of the arrival time of spin systems that present pretty good transmission

Serra,

Ferrón,

Osenda

2023

J. Phys. A: Math. Theor.

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The pretty good transmission scenario implies that the probability of sending one excitation from one extreme of a spin chain to the other can reach values arbitrarily close to the unity just by waiting a time long enough. The conditions that ensure the appearance of this scenario are known for chains with different interactions and lengths. Sufficient conditions for the presence of pretty good transmission depend on the spectrum of the Hamiltonian of the spin chain. Some works suggest that the time $t_{\varepsilon}$ at which the pretty good transmission takes place scales as $1/(|\varepsilon|)^{f(N)}$, where $\varepsilon$ is the difference between the probability that a single excitation propagates from one extreme of the chain to the other and the unity, while $f(N)$ is an unknown function of the chain length. In this paper, we show that the exponent is not a simple function of the chain length but a power law of the number of linearly independent irrational eigenvalues of the one-excitation block of the Hamiltonian that enter into the expression of the probability of transmission of one excitation. We explicitly provide examples of a chain showing that the exponent changes when the couplings between the spins change while the length remains fixed. For centrosymmetric spin chains the exponent is at most $N/2$.

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Perspective on exchange-coupled quantum-dot spin chains

Cited by 15 publications

References 190 publications

Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

Spreading entanglement through pairwise exchange interactions

The scaling law of the arrival time of spin systems that present pretty good transmission

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