Fast and efficient long-distance quantum state transfer in long-range spin-$\frac{1}{2}$ models

TL;DR

This work shows that fast, high-fidelity quantum state transfer is achievable in long-range spin-$\frac{1}{2}$ chains by exploiting the dispersion properties of next-nearest-neighbour interactions. By centering the transferred excitation at the inflection point of the linear region of the dispersion via an end-site magnetic field and optimizing only a few end couplings with a genetic algorithm, the authors realize ballistic transfer with $\langle F(t)\rangle$ exceeding $0.99$ and transfer times scaling quasi-linearly with chain length. The approach outperforms existing NNN PST schemes in speed and robustness, and is applicable across several experimental platforms (nanomechanical lattices, superconducting qubits, cold atoms, and waveguide arrays). This offers a practical route to scalable quantum information transfer in systems with longer-range interactions, while highlighting a fidelity–speed trade-off as $J_2$ is varied.

Abstract

Quantum state transfer is investigated beyond the nearest-neighbour coupling scheme in long spin-$\frac{1}{2}$ linear chains. Exploiting the properties of the next-nearest neighbour Hamiltonian's dispersion relation, it is shown that with minimal engineering, i.e., an on-site magnetic field on the two end sites and only a few symmetrically-modified end inter-site couplings, an average transfer fidelity above $99\%$ can be achieved. To leading order, the required time scales linearly with the length of the chain. Such a fast, high-quality quantum state transfer is based on the ballistic propagation of the wave packet centred in the linear region of the dispersion relation by means of the on-site magnetic field. At the same time, the wave packet width, modulated by the inter-site couplings at the chain ends, whose values are found via a carefully designed genetic algorithm, is constrained mostly in the linear region of the dispersion relation. Our coupling scheme is shown to hold for arbitrary values of the next-nearest inter-site coupling and can be straightforwardly applied to longer range coupling schemes.

Figures (9)

• Figure 1: Next-nearest neighbour spin chain with uniform bulk couplings $J_1$ and $J_2$. End site couplings are symmetrical rescaled by $\{\alpha_i\}$ for NN couplings and $\{\beta_i\}$ for NNN couplings. In the figure $i=1,2,3,4$ as our results show that modifying the couplings of just four end sites is sufficient for fast and high-quality QST for long chains. The black arrow on the first and last site represent equal transverse applied magnetic fields, which generate an on-site energy $h$. The sender and the receiver qubit, respectively the red and the blue sphere, are located at the ends.
• Figure 2: Dispersion relation, Eq. \ref{['eq:oddcirculanteigs2']}, for different values of $J_2$, as labelled. The vertical dashed lines indicate the inflection point $\theta_1$ in Eq. \ref{['eq_inflection_points']} for each corresponding NNN coupling. By increasing $J_2$, the linear region of the dispersion relation shifts towards higher energies $\omega$ and, at the same time, the group velocity $\frac{d\omega}{d\theta}$ increases.
• Figure 3: Average Fidelity (orange) and transfer time (blue), in units of ($t \cdot J_{max}$), against the number of pairs of optimised sites for a 51 site spin chain. Inset shows the nearest neighbour (black), $\alpha_i \cdot \frac{J_1}{4}$, and next-nearest neighbour (blue), $\beta_i \cdot \frac{J_2}{4}$, coupling configuration for four pairs of optimised sites at one end (mirrored at the other end) with i = 5 corresponding to the unmodified bulk coupling value.
• Figure 3: Optimised system parameters for various chain lengths $N$, including on-site field $h$. Maximal fidelity scores are attained within time window $t \cdot J_{\text{max}} \in [0, N]$.
• Figure 4: Energy spectra (black), in units of $J_{1}$, for a 51-site next-nearest neighbour spin chain with different numbers of optimised couplings: (a) and (b) shows the cases where no couplings are optimised for both zero (a) and one (b) set of optimised on-site energies $h$, while (c) and (d) correspond to the cases with one and four pairs of optimised couplings respectively, in addition to $h$. The initial wave packet (blue, dash-dot), $v_{1k}^2$, and group velocity (green, dashed), $\frac{d \omega_k}{dk}$, are appropriately rescaled and overlaid on top of the spectra. All spectra are identical to those reported for $J_2 = \frac{1}{2}$ in Fig. \ref{['fig:dispersion_relation']}, but ordered in increasing energy values.