Counterdiabatic transfer of a quantum state in a tunable Heisenberg spin chain via the variational principle

We present a couple of counterdiabatic (CD) schemes for a rapid and high-fidelity transfer of quantum state across a one-dimensional spin chain, between two weakly coupled external qubits at each end. Employing the effective low-energy Hamiltonian of the system, we first construct the optimal CD terms based on the variational principle and then put forward two experimentally feasible shortcuts, which only need to manipulate couplings between the external qubits and chain. Compared to traditional adiabatic protocol, the resulting schemes allow a drastic increase in state-transfer fidelity in a short time. Furthermore, numerical simulation demonstrates that our speed-up protocols hold robustness against the imperfections of control fields and evolution time. The proposed schemes may be applicable to fast quantum-information transport in various spin-based physical systems.

Figure 1

Schematic of the spin structure. (a) Two external qubits $S$ and $R$ (blue balls) weakly coupled to an odd-size spin chains (yellow balls), namely $J_{1,2}\ll J_0$. (b) The corresponding effective model in the low-energy subspace. The odd-size spin chain in its ground state works as an effective $1/2$ spin, labeled as $C$ (brown ball) and ${\tilde{J}}_{1,2}\left(t\right)$ denote the effective qubit-chain couplings.

Figure 2

Instantaneous fidelity $F\left(t\right)$ of the evolved state using the variational-based CD drivings: ${\mathcal{H}}^{\left(1\right)}\left(t\right)$ (dash-dotted green); ${\mathcal{H}}^{\left(2\right)}\left(t\right)$ (dashed orange); and ${\mathcal{H}}^{\left(3\right)}\left(t\right)$ (solid blue), respectively. As a contrast, the native protocol using ${\mathcal{H}}_0\left(t\right)$ is also described (dotted black). Here we adopt the linear ramp $J_1\left(t\right)=\frac{t}{T}, J_2\left(t\right)=1-\frac{t}{T}$ with $T=1$.

Figure 3

Instantaneous infidelity $1-F\left(t\right)$ for the Floquet-engineering protocol obtained by Eq. (34), with $\omega =20\pi$ (top), $40\pi$ (middle), and $80\pi$ (bottom). The initial sweep functions are set as $J_1\left(t\right)=\frac{t}{T}, J_2\left(t\right)=1-\frac{t}{T}$ with $T=1$.

Figure 4

The parameters ${\gamma }_{1,2}\left(t\right)$ and ${\mathcal{J}}_{1,2}^{xy,z}\left(t\right)$ in Eq. (46) with the different ramps (from left to right): 1. $J_1=\frac{t^3}{T^3},J_2\left(t\right)={(1-\frac{t}{T})}^3$; 2. $J_1={sin}^2\left[\frac{\pi }{2}{sin}^2\left(\frac{\pi t}{2T}\right)\right],J_2=1-J_1$; 3. $J_1=sin\frac{\pi t}{2T},J_2=cos\frac{\pi t}{2T}$; 4. $J_1=\frac{t}{T},J_2=1-\frac{t}{T}$. In all panels $T=1$. Notice that ${\gamma }_{1,2}\left(t\right)$ are taken in the interval $[-\frac{\pi }{4},\frac{\pi }{4}]$ regardless of their periodicities.

Figure 5

The instantaneous fidelity $F\left(t\right)$ of the unitarily equivalent scheme [Eq. (45)] using four sets of sweep functions shown in Fig. 4 for $T=1$. A detailed description is given in the main text.

Figure 6

QST fidelity $\mathcal{F}\left(T\right)$ using (a) Floquet-engineering shortcut [Eq. (34)] and (b) unitarily equivalent shortcut [Eq. (46)] for $N=3$, 5, 7 spin chain. The lines marked with symbols denote the speed-up results, and the lines without symbols are the adiabatic results. Here we set $J_0=70, \omega =20\pi$ and choose the original driving protocol having the form of Eq. (48).

Figure 7

(a), (b) The logarithm of infidelity ${\log }_{10}(1-\mathcal{F})$ versus the imperfections of the control fields. (c), (d) The logarithm of infidelity ${\log }_{10}(1-\mathcal{F})$ versus the errors of evolution time. The subfigures in the left and right columns denote the results of Floquet-engineering and unitarily equivalent shortcuts [Eqs. (34) and (46)], respectively. The values of contour lines are labeled in each subfigure. Here we set $J_0=70, \omega =20\pi$, and use trigonometric ramp [Eq. (47)].