Poor man's topological quantum gate based on the Su-Schrieffer-Heeger model

Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. Here, we propose and study a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed $Y$ gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zero-energy modes. With numerical simulations, we demonstrate that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry, and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer; nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is noninteracting and nonsuperconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides.

Figure 1

Adiabatically moving a domain wall and the localized zero-energy state it supports, in a fully dimerized SSH chain. Red circles depict the particle density of the localized state. The sequence (a)-(b)-(c) shows how to move the domain wall and the localized state by two sites. The domain-wall movement time $T_{\text{DW}}$ is the time window between (a) and (c). (d) Time dependence of the hopping amplitudes; see Eq. (3).

Figure 2

Braiding in a Y junction constructed from three fully dimerized SSH chains. (a) shows the initial configuration and the labels associated to the sites. In (b)–(e), the colored sites (red, green) depict the zero-energy edge modes. These edge modes are exchanged by adiabatically moving the domain walls supporting them, following the scheme in Fig. 1. (b) depicts the initial configuration, $t=0$. (c) and (d) depict the two intermediate configurations when the zero-energy modes are localized at the outer ends of the chains, $t=3T_{\text{DW}}$ and $t=6T_{\text{DW}}$, respectively. (e) denotes the final configuration at $t=9T_{\text{DW}}$: the Hamiltonian is the same as the initial one (b), but the edge modes have been exchanged.

Figure 3

Error of the $Y$ gate in the fully dimerized SSH Y junction due to the finite braiding time. (a) Gate error [infidelity; see Eq. (10)] is shown as the function of braiding time. (b) Gate error of panel (a) is shown, rescaled, as the function of the time $T_{\text{DW}}$ required for a single step of domain-wall motion. Errors are induced due to the nonadiabatic character of the braiding; hence they get suppressed as the braiding time is increased.

Figure 4

SSH Y junction away from the fully dimerized limit with hopping disorder. Structure of the initial and final Hamiltonian of the braiding protocol is depicted. Solid (dashed) lines are strong (weak) hopping amplitudes, all having random components. In the initial and final configuration shown here, the defects residing on sites $(\text{L},3)$ and $(\text{R},3)$ are isolated from the rest of the system.

Figure 5

Error of the $Y$ gate in a nondimerized SSH Y junction with real-valued hopping disorder. Disorder-averaged gate error [infidelity; see Eq. (10)] is shown as a function of braiding time $T$ for three different chain lengths $N_c$. The error saturates for long braiding times in each case, and the error minimum decreases exponentially as the chain length is increased, despite the presence of disorder; this indicates topological protection. Each curve is an average for 1000 random realizations.

Figure 6

Error of the $Y$ gate in the presence of on-site disorder. Disorder-averaged gate error [infidelity; see Eq. (10)] is shown as a function of braiding time $T$ for three different chain lengths $N_c$. The $Y$ gate is not robust against on-site disorder: the gate error saturates at a high value for long braiding times, at a value independent of the chain length. Each curve is an average for 1000 random realizations.

Figure 7

Error of the $Y$ gate in the presence of complex-valued hopping disorder. Disorder-averaged gate error [infidelity; see Eq. (10)] is shown as a function of braiding time $T$ for three different chain lengths $N_c$. The error saturates at a value for long braiding times, but that value increases as the chain length is increased. This indicates that the $Y$ gate is not robust against complex-valued hopping disorder. Each curve is an average of 1000 random realizations.

Figure 8

An SSH double Y junction allowing for braiding-based non-Abelian operations. Three defects (1, 2, 3) define a three-dimensional zero-energy subspace. The operation obtained by exchanging defects 1 and 2 first and 2 and 3 second is different from the operation obtained by doing these two steps in the reversed order.

Figure 9

Optical waveguide structure for the demonstration of a braiding-based $Y$ gate. (a) Three-dimensional view of the setup. The three segments (bottom: preparation, middle: braiding, top: analyzer) are separated for better visualization, but they might have direct contact in a real device. (b) Details of the phase-sensitive measurement. Preparation segment creates a 50-50 superposition on the outer waveguides with no phase difference. Braiding segment acts as a $Y$ gate, shifting the relative phase between the two beams by $\pi$. Due to this phase shift and the corresponding complete destructive interference, the beam does not enter into the middle waveguide as passing through the analyzer segment, but comes out with 50-50 intensity in the left and right outputs. (c) Braiding segment. Beam injected in left waveguide comes out from right output as a result of braiding. This measurement alone cannot distinguish between the $X$ gate and a $Y$ gate. Vector $\psi$ describes the input and output beam amplitudes at each segment.