Robustness of spin-coupling distributions for perfect quantum state transfer

The transmission of quantum information between different parts of a quantum computer is of fundamental importance. Spin chains have been proposed as quantum channels for transferring information. Different configurations for the spin couplings were proposed in order to optimize the transfer. As imperfections in the creation of these specific spin-coupling distributions can never be completely avoided, it is important to find out which systems are optimally suited for information transfer by assessing their robustness against imperfections or disturbances. We analyze different spin coupling distributions of spin chain channels designed for perfect quantum state transfer. In particular, we study the transfer of an initial state from one end of the chain to the other end. We quantify the robustness of different coupling distributions against perturbations and we relate it to the properties of the energy eigenstates and eigenvalues. We find that the localization properties of the systems play an important role for robust quantum state transfer.

Figure 1

(a) Energy eigenvalue distributions ${\omega }_k(k_{\beta },\alpha )$. The symbols represent the energy values and the lines give the exact functional dependence of Eq. (5). (b) Exchange couplings determined by solving the inverse eigenvalue problem for each of the spectra given in (a).

Figure 2

Averaged fidelity of the state transfer in a $N=31$ spin chain for the different energy distributions shown in Fig. 1 as a function of time. The linear ${\omega }_k(k_c,1)$ and quadratic ${\omega }_k(k_c,2)$ distributions achieve the perfect transmission with the first echo, while the other cases achieve it after several echoes. In panels (d) and (e), the black regions are due to fast oscillations because of the coupling strength oscillations shown in Fig. 1 b. The transfer times are given by $t_{\mathrm{PST}}^{\mathrm{linear}}=\frac{\pi N}{4J_{\max }}$ and $t_{\mathrm{PST}}\sim \gamma t_{\mathrm{PST}}^{\mathrm{linear}}$ with $\gamma =15.4,17,15,14.5$ for a $quadratic,{\omega }_k(k_b,\frac{1}{2}),{\omega }_k(k_c,\frac{1}{2})$ and ${\omega }_k(k_b,2)$ distribution respectively.

Figure 3

Averaged fidelity of the state transfer in a $N=31$ spin chain with random perturbations of strength ${\varepsilon }_J={10}^{-2}$ averaged over $N_{av}={10}^2$ realizations for the different energy distributions from Fig. 1 as a function of time. The colored lines in (c), (d) and (e) show $F\left(t\right)$ for the systems without perturbation (${\varepsilon }_J=0$).

Figure 4

Averaged fidelity of the state transfer in a $N=31$ spin chain with random perturbations of strength ${\varepsilon }_J={10}^{-2}$ averaged over $N_{av}={10}^2$ realizations for eigenvalue distributions ${\omega }_k(k_c,n)$. Shown is the range of times around the 9th echo of the PST in the unperturbed chain, for $n=1,2,10$ (black lines), and $n=3$ (faint colored line very close to the $n=10$ results).

Figure 5

Averaged fidelity at time $t_{\mathrm{PST}}$ as a function of the perturbation strength ${\varepsilon }_J$ for the different energy distributions from Fig. 1 with $N=31$ and $N_{av}={10}^2$.

Figure 6

Averaged fidelity at odd multiples of $t_{\mathrm{PST}}$ (symbols) for the linear and quadratic energy eigenvalue distributions, ${\omega }_k(k_c,n)$ $(n=1,2)$. Chain length is $N=31$, averages were performed over $N_{av}={10}^2$ realizations. Perturbation strengths are $\varepsilon =0.01,$ 0.02,..., 0.1.

Figure 7

Eigenvector probability $P_{k,i}$ of the site (computational) states $|i\rangle$. $P_{k,i}=a_{k,i}^2$, where $|i\rangle ={\sum }_k{a}_{k,i}|{\Psi }_k\rangle$. The top part of each panel shows the probabilities $P_{k,1}$ of the initial state $|{\Psi }_0\rangle =|1\rangle$, and thus shows which energy eigenstates contribute to the state transfer. The panel labels refer to the different energy distributions given in Fig. 1, where (a) ${\omega }_k(k_c,2)$ quadratic, (b) ${\omega }_k(k_c,1)$ linear (c) ${\omega }_k(k_b,\frac{1}{2})$, (d) ${\omega }_k(k_c,\frac{1}{2})$, and (e) ${\omega }_k(k_b,2)$.

Figure 8

Standard deviation of the energy levels $|\Delta {\omega }_{{\varepsilon }_J}|=|{\omega }_{{\varepsilon }_J}-{\omega }_0|$ due to the perturbation with strength ${\varepsilon }_J$ for the different energy distributions of Fig. 1. For weak perturbations (small ${\varepsilon }_J$) the standard deviation turns out to be proportional to ${\varepsilon }_J{\omega }_{\max }$, which we use as a unit here. The data shown are for ${\varepsilon }_J<0.1$; $N_{av}={10}^3$ realizations were used for the calculations.