Heisenberg spin bus as a robust transmission line for quantum-state transfer

We study quantum-state transfer (QST) through a strongly coupled antiferromagnetic spin chain (acting as a spin bus), between weakly coupled external qubits. By treating the weak coupling as a perturbation, we find that QST is enabled specifically by the second-order terms in the perturbative expansion. We show that QST is robust against disorder in the couplings, either within the bus or to the external qubits. We find that the protocol works when the qubits are attached to any node on an even-size bus or to the antiferromagnetic nodes on an odd-size bus. The optimal time for QST is found to depend nonmonotonically on qubit separation.

Figure 1

QST fidelity $F_B\left(t\right)$ vs time $t$ for an odd-size spin bus ($N=9$). (a) Short times, $t\sim 1/\lambda$. (b) Long times, $t\sim 1/{\lambda }^2$. Here, the two qubits $A$ and $B$ are attached to the opposite ends of the bus, with the initial state of qubit $A$ given by $|{\psi }_A\rangle =\frac{1}{\sqrt{2}}\left(\right|0_A\rangle +|1_A\rangle )$. The solid and dot-dashed lines represent numerical simulations using the full Hamiltonian (1), while the dashed line represents the analytical expression, Eq. (B4).

Figure 2

(a) Optimal times for state transfer, $t_0$ and $t_0^{\prime }$ (scaled by $J_{\alpha ,i}^{\left(1\right)}/\pi \sim \lambda$), as a function of the polar angle $\theta$ of the initial state, with an odd-size ($N=9$) bus. (b) Maximum fidelity $F_B$ as a function of $\theta$. (c) Infidelity $[1-F_B\left(t_0^{\prime }\right)]$ vs $\lambda$ for $N=5$ (open squares) and $N=9$ (solid squares). The line represents ${\lambda }^2$.

Figure 3

(a) Fidelity $F_B\left(t\right)$ vs $t$ when qubits $A$ and $B$ are attached to the opposite ends of an even-size chain ($N=10$). The initial state of qubit $A$ is $|{\psi }_A\rangle =\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$. The solid and dashed lines represent numerical simulations using the full Hamiltonian (1), while the dot-dashed line is the analytical expression, Eq. (9). (b) Infidelity $[1-F_B\left(t_0\right)]$ vs $\lambda$ for $N=4$ (open squares) and $N=10$ (solid squares). The line represents ${\lambda }^3$.

Figure 4

Couplings $J_{1,j}^{\left(2\right)}$, scaled by $J_{A,1}{J}_{B,j}$, and the corresponding optimal times $t_0$, scaled by $\pi /J_{1,1}^{\left(2\right)}$, as a function of position $j$ for even-size buses. Here $J_{A,1}=J_{B,j}=0.1$.

Figure 5

Average QST infidelity $1-{\overline{F}}_B\left(t_0\right)$ [or $1-{\overline{F}}_B\left(t_0^{\prime }\right)$] as a function of exchange variation, represented by the variance ${\sigma }_J$, for a Gaussian distribution around the target value $J_0$, taken from 100 random configurations. Here $t_0$ ($t_0^{\prime }$) is the QST time calibrated for nonuniform coupling strengths, $\lambda =0.05$, and $N$ is the bus size.

Figure 6

Numerical results for an odd-size bus of size $N=13$. (a) The local magnetic moments (in units of Bohr magneton) $m_j=\langle 0_C|{\sigma }_{zj}|0_C\rangle$ for one of the bus ground states. (b) Solid (blue) circles represent the effective couplings $J_{1,j}^{\left(2\right)}/\left(J_{A,i}{J}_{B,j}\right)$ between qubit $A$, connected to the bus at site 1, and qubit $B$, connected at site $j$. Open (red) squares represent the spin-spin correlation function, $-\langle 0_C\left|{\sigma }_{iz}{\sigma }_{jz}\right|0_C\rangle$.