Fast and robust quantum state transfer via a topological chain

The SSH model is the simplest topologically non-trivial system in 1-D that can be obtained by suitably modifying the Hamiltonian of Eq. 1. To do so, the chain has to be dimerized. Meaning, we have to make $J_{i}=J_{odd}$ for $i\in odd$ and $J_{i}=J_{even}$ for $i\in even$. In addition, the magnetic field needs to be constant. We will assume that $B_{i}=0$ $\forall i$. For even-sized chains the topological phase arises when $J_{odd}>J_{even}$ and two edge modes appear at the two ends of the chain. The energies corresponding to the two eigenmodes lie close (above and below) to $E=0$ and are separated from the rest of the modes by a finite energy gap. The size of the gap is related to the ratio between $J_{even}$ and $J_{odd}$ asboth2016short . On the contrary, for odd-sized chains, there is always one edge mode that is localized on the end corresponding to the weaker coupling. Namely, when $J_{odd}<J_{even}$ ($J_{odd}>J_{even}$) the mode is localized near the first (last) site of the chain. In this case the energy of the mode is exactly zero. For odd-sized chains the eigenmode energies (besides the zero energy solution) are given by the following expression:

where $q_{j}=2j\pi/(N+1)$ and $j=1,2,...,[N/2]$ ($j$ counts the number of $\pm$ pairs and $[x]$ gives the greatest integer that is less than equal to $x$) coutant2020robustness . Thus, for an odd-sized chain of length $N$ the energy gap can be analytically determined and is given by:

For the protocols we will consider in this case, we restrict ourselves to odd-sized chains and we assume that we can separately control even and odd-indexed couplings. Initially the system is prepared so that $J_{odd}=0$ and $J_{even}=1$ and the excitation is localized on the first site of the chain which is disconnected from the rest (see Fig. 1 (a)). Therefore, the initial state is an eigenstate of the system ($\ket{100...0}$ wth zero eigenenergy). At the transfer time $t^{*}$ (see Fig. 1 (c)) we end up with the reverse situation (i.e. $J_{even}=0$ and $J_{odd}=1$). The system undergoes a transition and transforms from a topological chain supporting an edge mode on the first site, to a topological chain with an edge mode on the last site, resulting to an excitation transfer from one side to the other. Note here, that in this protocol, there always exists a time where all the couplings acquire the same value $J_{even}=J_{odd}$. In our case we will consider this time to be $t=t^{*}/2$ (see Fig. 1 (b)). For the infinite system, $J_{even}=J_{odd}$ corresponds to the closing of the energy gap separating the zero-energy mode with the rest excited states. However, in finite systems a finite energy difference between any two modes is always present. Thus, the point in the parameter space where $J_{even}=J_{odd}$, corresponds to the minimization of the energy gap.

To explore how the topological nature of the underlying static chain affects the transfer we also proceed to a comparison with a protocol employing a topologically-trivial chain. We consider a protocol where the only couplings that are controlled are the ones connecting the edge sites with the rest of the chain (i.e. $J_{1}$ and $J_{N-1}$). This protocol has been chosen based on its performance in terms of speed and also by the fact that local manipulation of the system’s parameters makes it experimentally more feasible. As was the case for the topological chain, the initial state is localized at the first site and corresponds to the system’s zero-energy eigenstate (see Fig. 2 (a)). Here, $J_{1}=0$ while $J_{i}=J$ $\forall i\neq 1$. During the dynamical evolution, due to the odd-size of the chain, the zero-energy eigenstate is always present. Therefore, gradually switching on $J_{1}$ while $J_{N-1}$ is decreased, results at time $t^{*}$, to the transfer of the excitation at the other end of the chain (see Fig. 2 (c)). An important difference between this protocol and the one described in the previous section, is that in this case, there is no point in the parameter space where all the couplings acquire the same value.

Now let us examine in more detail the QST protocols that we briefly described in the previous section and provide the numerical evidence supporting our claims. We will examine chains of moderate length $N=31$ sites. In the protocol we propose the couplings are driven by an exponential function (see Fig. 3 (b) top panel), where $J_{odd}=(1-e^{-\alpha t/t^{*}})/(1-e^{-\alpha})$ and $J_{even}=(1-e^{-\alpha(t^{*}-t))/t^{*}})/(1-e^{-\alpha})$, while $\alpha=6.0$ is a free parameter that has been fine-tuned to increase the efficiency in terms of speed. We will get back to the role of this free parameter at the end of the current section. The exponential protocol will be compared with a protocol proposed in mei2018robust , where $J_{odd}=b(1-\cos{(\pi t/t^{*})})$ and $J_{even}=b(1+\cos{(\pi t/t^{*})})$ and $b=0.5$ (see Fig. 3 (a) top panel). On the other hand, for the trivial protocol the driving function has the following linear form: $J_{1}=\frac{t}{t^{*}}$, $J_{N-1}=1-\frac{t}{t^{*}}$ and $J_{i}=J_{max}=1$,$\forall i\neq 1,N-1$ (see Fig. 3 (c) top panel).

In Fig. 3, we plot for each protocol on the top panel the driving function for the couplings and on the bottom how the instantaneous eigenspectrum evolves over time. Comparing the two protocols that employ the topological chain (see Fig. 3 (a) and (b)), we can immediately notice their qualitative differences. The cosine function initially for large values of the energy gap, drives the system slowly, meaning the numerator of Eq. 5 is smaller as compared to the exponential. However, it approaches the minimum value of the energy gap with greater slope, while the exponential slows down and drives the system more smoothly in this region. Last but not least, the minimum value of the energy gap is analytically obtained by plugging into Eq. 4, the instantaneous values of the couplings at $t=t^{*}/2$. For the cosine we get $g_{min}^{cos}=0.09$, while for the exponential $g_{min}^{exp}=0.18$. As it was already mentioned, this is because the exponential equates $J_{even}$ and $J_{odd}$ at higher values.

In the trivial protocol on the other hand (Fig. 3 (c)), the evolution is completely different. Namely, the instantaneous energy gap separating the zero-energy mode with the rest of the modes starts from its minimum value ($g_{min}^{triv}=0.1$), slowly increases reaching its maximum ($g_{max}^{triv}=0.14$) at the middle of the time evolution and then returns to its initial value.

Now that the qualitative differences between the protocols have become apparent let us proceed and examine some quantitative results. In Fig. 4, for each protocol, we plot the fidelity $\mathcal{F}$ (Eq. 2) as a function of the transfer time $t^{*}$. To make a comparison in terms of the speed of the transfer we have to set a lower bound in fidelity. In particular, we will consider the time after which the fidelity is stabilized above $0.9$. In this case, the exponential protocol is clearly faster than the cosine protocol, since this occurs for $t^{*}\geq 42$ as compared to the cosine where this happens for $t^{*}\geq 761$. The trivial protocol on the other hand, even though it reaches $\mathcal{F}=0.9$ for $t^{*}=35$, appears to have a strongly oscillatory behavior that prevents its stabilization above $\mathcal{F}=0.9$ till $t^{*}\geq 231$.

For all profiles, in the limit of $t^{*}\to\infty$ the fidelity approaches unity and the excitation is perfectly transferred along the chain. This makes up the adiabatic limit where, during the dynamical evolution, we ”follow” the zero-energy state, without exciting other bulk eigenmodes. The oscillations that appear in the fidelity plot of the trivial protocol are in general unwelcome in QST protocols since they demand great precision when tuning the transfer time kay2010perfect . Moreover, they signify that resonant processes are the underlying mechanism responsible for achieving high values of fidelity in such small transfer times. Taking a closer look at the fidelity plot of the exponential protocol (Fig. 5 for $\alpha=6$), we can notice that small oscillations are also present here, i.e. the fidelity curve does not increase smoothly. This indicates that resonant processes are at work also in this case. Obtaining a suitable basis where these processes that occur during the dynamical evolution can be rigorously tracked down, remains a highly non-trivial task lim1991superadiabatic ; dykhne1960quantum . Nevertheless, as we will now show, the resonant processes can be properly handled to increase the efficiency of the transfer process.

When we introduced the exponential driving function, we mentioned that the $\alpha$ parameter is fine-tuned ($\alpha=6.0$). Smaller values of the $a$ parameter lead to a less steep slope of the driving function and a smaller value of $g_{min}^{exp}$ (i.e. $J_{even}$, $J_{odd}$ equate at a smaller value). In this case, the resonant processes are suppressed and the fidelity smoothens out (see Fig. 5 $\alpha=4$). Consequently, the protocol’s speed is reduced since high fidelity values are obtained for larger transfer times. On the contrary, increasing $\alpha$ above the fine-tuned value results to a stronger slope of the driving function and a greater value of $g_{min}^{exp}$. Therefore, the resonant processes take over for small transfer times and strong oscillations appear at the fidelity plot (see Fig. 5 $\alpha=8$). This once again reduces the speed of the protocol. Thus, the value of this fine-tuned parameter $\alpha=6$ is a trade-off, since it signifies the point up to which we strongly drive the system such that the speed is increased, but gently enough to avoid resonant effects.

Let us also note here that we have performed optimizations in terms of the CRAB (chopped random basis) algorithm doria2011optimal where the guess function was assumed to be the exponential, or the cosine one. The CRAB algorithm gives corrections in terms of a chopped polynomial or Fourier expansion of the correction function. The correction of CRAB to the exponential function was very small, with subtle corrections not changing its basic logic and behavior. On the other hand the cosine function as a guess function in CRAB had a strong correction in the direction of getting closer to the exponential profile. Those calculations indicate that our protocol is close to the optimal one, within the constrains taken in this work.

In this section, we will consider static disorder both on the couplings and on the magnetic field and study its effect on fidelity. Based on the matrix representation of the Hamiltonian the disorder on the couplings is commonly addressed to as off-diagonal disorder, while the disorder on the magnetic field as diagonal disorder. Static disorder can be attributed to manufacturing errors that arise during the experimental implementation. The way each disorder realization is imposed on the system’s parameters is the following:

$\delta J_{i}$ and $\delta B_{i}$ acquire random real values uniformly distributed in the interval $(-d_{s},d_{s})$, while $d_{s}$ corresponds to the disorder strength. When we consider static disorder, for each realization a random profile of perturbations is imposed on the parameters and remains fixed during the time evolution.

In Fig. 6 for each protocol, we plot the mean fidelity as a function of $\log{t^{*}}$ for diagonal and off-diagonal disorder of moderate strength $d_{s}=0.2$. What we can immediately notice, is that in almost all cases (there is one exception that will be discussed later on) the effect of disorder does not ruin completely the transfer process. Instead, the main effect is that in the presence of disorder (diagonal or off-diagonal), the transfer time $t^{*}$ needed to reach high values of fidelity is increased.

Let us turn our attention to the protocols employing the SSH chain. The zero-energy mode of the underlying static chain is known to be robust against perturbations that respect chiral symmetry asboth2016short . Off-diagonal (chiral) disorder may change the mode’s wavefunction however its energy remains pinned down to zero. On the contrary, diagonal disorder breaks chiral symmetry and the energy of the mode is shifted. For the time-driven chain a difference between chiral and non-chiral disorder becomes apparent in the case of the adiabatic cosine protocol (see Fig. 6 (a)). As it was expected from the static case, the fidelity reduces more in the presence of non-chiral disorder mei2018robust ; lang2017topological . The exponential protocol however, seems indifferent to whether the disorder is chiral or not (see Fig. 6 (b)). The reason behind this lies in the higher speed of the exponential protocol. Since the effect of disorder strongly manifests in large time scales, the adiabatic cosine protocol is far more sensitive compared to the exponential. This argument has been recently used to justify the resilience of the counter adiabatic protocol in d2020fast .

When we examine the effect of the disorder on the couplings for the topologically trivial protocol (see Fig. 6 (c)) we observe that the oscillatory behavior of fidelity for small transfer times is suppressed (i.e. less oscillations and the mean fidelity is significantly degraded). Thus, we can deduce that the resonant processes are not so robust to the static off-diagonal disorder.

On the other hand, when considering the effect of the disorder on the magnetic field we distinguish two cases. One where the disorder is imposed on all sites of the chain and another where the first and last sites are exempted (i.e. $\delta B_{1}=\delta B_{N}=0$). In the latter case, the protocol proves to be even more robust than the off-diagonal case (see Fig. 6 (c) diagonal 2). However, in the former, the effect of disorder is severe and the transfer process is completely destroyed (see Fig. 6 (c) diagonal 1). The diagonal disorder on the edges greatly affects the transfer process since it induces an energy difference between the initial and the final state. The system’s initial state is localized on the first site with energy equal to $\delta B_{1}$, while the final state is localized on site $N$ with energy $\delta B_{N}$. Combined with the fact that the energy gap separating them from the rest of the excited states takes its minimum value (which is smaller than the strength of the disorder) during the beginning and the end of the transfer process, explains the high impact of the diagonal disorder on the edges. In conclusion, the topological protection coming from the energy gap, clearly favors the topological channels, which are indifferent to whether the diagonal disorder is imposed on the edge sites.

To sum up, the exponential protocol is quite robust to both on and off-diagonal disorder and clearly outperforms the two other protocols. As opposed to the adiabatic cosine protocol, it is indifferent to whether the disorder is chiral or not and compared with the trivial chain we can deduce that no significant (in terms of affecting the fidelity) resonant processes susceptible to static noise are at work. Finally, the presence of a wide energy gap in the underlying static SSH chain clearly favors the topological quantum channel when the diagonal disorder is imposed on the edge sites of the chain.