Optimizing edge-state transfer in a Su-Schrieffer-Heeger chain via hybrid analog-digital strategies

The Su-Schrieffer-Heeger (SSH) chain, which serves as a paradigmatic model for comprehending topological phases and their associated edge states, plays an essential role in advancing our understanding of quantum materials and quantum information processing and technology. In this paper, we introduce a hybrid analog-digital protocol designed for the nonadiabatic yet high-fidelity transfer of edge states in an SSH chain, featuring two sublattices, A and B. The core of our approach lies in harnessing the approximate time-dependent counterdiabatic (CD) interaction, derived from adiabatic gauge potentials. However, to enhance transfer fidelity, particularly in long-distance chains, higher-order nested commutators become crucial. To simplify the experimental implementation and navigate computational complexities, we identify the next-to-nearest-neighbor hopping terms between sublattice A sites as dominant CD driving and further optimize them by using variational quantum circuits. Through digital quantum simulation, our protocol showcases the capability to achieve rapid and robust solutions, even in the presence of disorder. This analog-digital transfer protocol, an extension of quantum control methodology, establishes a robust framework for edge-state transfer. Importantly, the optimal CD driving identified can be seamlessly implemented across various quantum registers, highlighting the versatility of our approach.

Figure 1

A schematic configuration of an SSH chain. The edge state can be nonadiabatically transferred from the left side to the right one by adding the hopping between the neighboring sites of sublattice A.

Figure 2

The use of different schemes with the Hamiltonians $H_0$, $H_0+H_{\mathrm{cd}}^{(1)}$, and $H_0+H_{\mathrm{cd}}^{(2)}$ to transfer the edge state of an SSH chain with five sites. (a) The fidelity robustness $F(T)$ in the dependence of the operation time $T$ ranging from the adiabatic ($\mathrm{\Omega }=0.01t_0$) to the nonadiabatic regime (up to $\mathrm{\Omega }=10t_0$). (b) The transfer fidelity $F(t)$ during the transfer, where $T=\pi$ and $\mathrm{\Omega }=t_0$. (c) The variational parameters ${\alpha }_1^{(1)}$, ${\alpha }_1^{(2)}$, and ${\alpha }_2^{(2)}$ derived by minimizing the action $S_l=\mathrm{tr}[G_l^2]$ ($l=1,2$). (d) The sublattice-A hoppings for the edge-state transfer: different strengths ${\kappa }_{31}^{(2)}$, ${\kappa }_{53}^{(2)}$, and ${\kappa }_{51}^{(2)}$ [Eq. (13)] and equal strengths ${\kappa }^{(2)}=-\mathrm{\Lambda }\dot{\lambda }{\alpha }_1^{(2)}$ [Eq. (14)] from the second-order NC in comparison with ${\kappa }^{(1)}=-\mathrm{\Lambda }\dot{\lambda }{\alpha }_1^{(1)}$ derived from the first-order NC and $\mathrm{\Lambda }=2t_0$ with $t_0=1$.

Figure 3

The circuit implementation of the state evolution by using the unitary operator $U(T,0)$ [Eq. (18)] Trotterizing the Hamiltonian, which is the sum of $H_0^{(c)}$ (yellow) and $H_{\mathrm{cd}}^{(c)}$ (purple).

Figure 4

From top to bottom: the results for chains of five, seven, nine, $11$, $13$, and $15$ sites [(a)–(f), respectively] using both equal (orange squares) and different NNN hoppings (rest) on sublattice A after ten runs. Left: the NNN couplings for each chain are represented in the case of $\mathrm{\Omega }=t_0=1$ with the labeled average values and their corresponding standard deviation. Right: the evolution of the cost function $F(t)$ for the chains with five, seven, nine, $11$, $13$, and $15$ sites against the rescaled number of iterations with $k\in [1,2,3,5,6,10]$ in each case, respectively. The shaded areas show the region of the returned minimum and maximum values.

Figure 5

Using the Hellinger distance as the cost function [Eq. (22)], we find optimal CD driving for a five-site chain from running quantum circuits $10$ times, where $\mathrm{\Omega }=t_0=1$, when considering equal NNN hopping terms on sublattice A. (a) ${\kappa }_{\mathrm{H}}^{(c)}$. (b) $p_4$ in terms of the iterations. (c) $p_4$ as a function of the number of measurements $N_{\mathrm{shot}}$. (d) We simultaneously check $F(T)$ in the presence of the corresponding $N_{\mathrm{shot}}$ by introducing the optimal driving obtained from the Hellinger distance. Each point derived at a certain $N_{\mathrm{shot}}$ in (c) and (d) is from running a $1000$-times iteration in the circuit. In (a), (c), and (d), the labeled average values and standard deviation are shown, while the areas of the minimum and maximum returned values are plotted in (b).

Figure 6

The robustness of the fidelity $F(T)$ for an SSH chain of five sites using both different and equal NNN hoppings (see Fig. 4) as well as in the adiabatic regime $\mathrm{\Omega }=0.01t_0$ ($t_0=1$) against (a) the relative strength of the staggered Rice-Mele contribution in Eq. (26) in terms of $\delta /t_0$, setting ${\delta }_j={\tau }_j=0$ for $\mathrm{\Delta }(t)=\delta$ (solid) and $\mathrm{\Delta }(t)=\delta \sin \mathrm{\Omega }t$ (dashed), where the fidelities of both parametrizations almost overlap; (b) the diagonal-disorder contribution in Eq. (26) in terms of ${\sigma }_{\delta }/t_0$, setting $\mathrm{\Delta }(t)={\tau }_j=0$; and (c) the hopping-disorder contribution of Eq. (27) in terms of ${\sigma }_{\tau }/t_0$, setting $\mathrm{\Delta }(t)={\delta }_j=0$. For the latter two [(b) and (c)], the results are averaged over $1000$ runs.

Figure 7

(a) The derived different NNN hopping terms on sublattice A for a chain of $15$ sites, setting $H_0=0$. NNN couplings are represented in the case of $t_0=1$ and $\mathrm{\Omega }=1$ with the labeled average values and their corresponding standard deviation after ten runs. (b) The evolution of the cost function $F(t)$ against the number of iterations, where the shaded areas show the region of the returned minimum and maximum values.

Figure 8

The robustness of the fidelity $F(T)$ for an SSH chain of $15$ sites using different NNN hoppings on sublattice A, setting $H_0=0$ (blue lines), as well as the expected behavior (black lines) in the adiabatic regime $\mathrm{\Omega }=0.01t_0$ ($t_0=1$) against (a) the relative strength of the staggered Rice-Mele contribution in Eq. (26) in terms of $\delta$, setting ${\delta }_j={\tau }_j=0$ for $\mathrm{\Delta }(t)=\delta$ (solid) and $\mathrm{\Delta }(t)=\delta \sin \mathrm{\Omega }t$ (dashed), where the fidelities of the two parametrizations almost overlap; (b) the diagonal-disorder contribution in Eq. (26) in terms of ${\sigma }_{\delta }$, setting $\mathrm{\Delta }(t)={\tau }_j=0$; and (c) the hopping-disorder contribution of Eq. (27) in terms of ${\sigma }_{\tau }$, setting $\mathrm{\Delta }(t)={\delta }_j=0$. For the latter two [(b) and (c)], the results are averaged over $1000$ runs.

Figure 9

The on-site probability time evolution starting from $|{\mathrm{\Phi }}_0(0)⟩$ by solving the time-dependent Schrödinger equation without and with the addition of the second-order NC contribution [(a)–(d) and (e)–(h), respectively]. From top to bottom on both subfigures, $\mathrm{\Omega }$ is equal to $0.1$, $0.2$, $0.5$, and $1$.

Figure 10

The hopping parametrizations addressed throughout the main text with $t_1$ ($t_2$), shown as solid (dashed) lines.