Superadiabatic quantum state transfer in spin chains

In this paper we propose a superadiabatic protocol where quantum state transfer can be achieved with arbitrarily high accuracy and minimal control across long spin chains with an odd number of spins. The quantum state transfer protocol only requires the control of the couplings between the qubits on the edge and the spin chain. We predict fidelities above 0.99 for an evolution of nanoseconds using typical spin-exchange coupling values of $\mu \mathrm{eV}$. Furthermore, by building a superadiabatic formalism on top of this protocol, we propose an effective superadiabatic protocol that retains the minimal control over the spin chain and further improves the fidelity.

Figure 1

(a) Schematic of the Heisenberg spin bus with $N=5$ sites [Eq. (1)]. Our goal is to transfer the state of the sender qubit (L) to the receiver qubit (R). (b) Diagram of the system modeled by the effective Hamiltonian for an odd size spin bus [Eq. (2)].

Figure 2

Normal adiabatic QST across a spin bus with $N$ sites using the linear and trigonometric time evolution of $J_L\left(t\right)$. (a) Calculation of the fidelity $F$ [Eq. (6)] vs the total evolution time $T$ (in units of ${\mathrm{\Delta }}^{-1}$) for the linear evolution of $J_L\left(t\right)$ [Eq. (4)]. (b) The linear evolution of $J_L\left(t\right)$ induces a linear evolution of $J_R$. (c) Fidelity $F$ vs the total evolution time $T$, for the trigonometric evolution of $J_L\left(t\right)$ [Eq. (5)]. (d) Time evolution of the couplings used in (c). In these calculations $J_M=1$ and $J_{SB}=10$. The points from where onwards fidelity reaches high values ($F>0.99$) are denoted by circles.

Figure 3

(a) Contour plot of fidelity $F$ [Eq. (6)] vs $J_M$ and $\alpha =\frac{J_{SB}}{J_M}$. The region $\alpha >1$ can be described by the effective Hamiltonian [Eq. (2)]. A high-fidelity region appears for $\alpha <1$ (see text for discussion). (b) Contour plot of the energy gap between the ground state and the first excited state for the same parameters as in (a). A spin bus with $N=3$ and linear time evolution with $T=1{\mathrm{\Delta }}^{-1}$ was used in both figures (a) and (b).

Figure 4

Fidelity $F$ for the linear and trigonometric evolution of $J_L\left(t\right)$ as a function of $T$ (in units of ${\mathrm{\Delta }}^{-1}$) for spin bus sizes $N=1,3,5$. Values of $J_M=1$ and $J_{SB}=10$ were used. The black solid lines correspond to a normal adiabatic QST using the Hamiltonian (1); the magenta dashed lines are for the effective superadiabatic Hamiltonian (11). In every case higher fidelities can be seen using the effective superadiabatic quantum state transfer.

Figure 5

Surf plot of the improvement of fidelity due to the effective superadiabatic Hamiltonian for the trigonometric evolution of $J_L\left(t\right)$ and the region $\alpha >1$ described by the effective Hamiltonian. In this calculation $N=5$ and $T=1{\mathrm{\Delta }}^{-1}$.