Edge-state-enhanced transport in a two-dimensional quantum walk

Quantum walks on translation-invariant regular graphs spread quadratically faster than their classical counterparts. The same coherence that gives them this quantum speedup inhibits or even stops their spread in the presence of disorder. We ask how to create an efficient transport channel from a fixed source site $\left(A\right)$ to fixed target site $\left(B\right)$ in a disordered two-dimensional discrete-time quantum walk by cutting some of the links. We show that the somewhat counterintuitive strategy of cutting links along a single line connecting $A$ to $B$ creates such a channel. The efficient transport along the cut is due to topologically protected chiral edge states, which exist even though the bulk Chern number in this system vanishes. We give a realization of the walk as a periodically driven lattice Hamiltonian and identify the bulk topological invariant responsible for the edge states as the quasienergy winding of this Hamiltonian.

Figure 1

Layout of the 2D quantum walk, with a source at $A$, a detector at the target site at $B$, and detectors at the edges. For the conditional wave function, the detectors play the role of absorbers.

Figure 2

Layout of the 2D quantum walk, with a source at $A$, a detector at the target site at $B$, and detectors at the edges. To increase the efficiency of transport from $A$ to $B$, the first idea is to cut an island that will form a transport channel, as indicated by the dotted line. All links crossing the dotted line are cut; a particle attempting to hop across a cut link will have its spin flipped instead of hopping.

Figure 3

Mean displacement of the quantum walker (continuous lines) and transmission probability (dotted lines) in the geometry of the “island” of Fig. 2, for a horizontal island of fixed width of 1 and varying island length of 40 (thick light gray), 80 (medium gray), and 160 (thin black). Mean rotation angles are set to ${\theta }_1=0.35\pi ,{\theta }_2=0.15\pi$. Without disorder, $\delta =0$, the wave function spreads ballistically (a) and the transmission probability reaches a value close to 1 as the wave packet arrives (b). To illustrate the effects of disorder, we set $\delta =0.2\pi$, and use a single disorder realization, varying only the distance $n$ between $A$ and $B$ (and, correspondingly, the length of the island). For a large-enough system $\left(n=160\right)$, the mean distance from $A$ saturates at around 30 (c), and in this case there is virtually no transmission [(d) $P_t<{10}^{-4}$ for $n=160$ for all times $t$.

Figure 4

Layout of the 2D quantum walk, with a source at $A$, a detector at the target site at $B$, and detectors at the edges. Links crossing the dotted line are cut: This single line cut creates a transport channel between $A$ and $B$. This is an alternative to the “island” approach of Fig. 2.

Figure 5

Mean displacement of the quantum walker [(a) continuous lines] and transmission probability [(b) dotted lines] in the geometry of the “single cut” of Fig. 4 for a horizontal cut. A single realization of a disordered quantum walk is taken, with mean rotation angles ${\theta }_1=0.35\pi ,{\theta }_2=0.15\pi$, and disorder $\delta =0.3\pi$. The $A\text{−}B$ distance $n$ is varied: $n=40$ (thick light gray), $n=80$ (medium gray), and $n=160$ (thin black). The walker propagates ballistically along the cut (b) and arrives at $B$ with a high probability (c).

Figure 6

For the analytical calculation, we consider a simple geometry with a reflecting edge on top and absorbers on the bottom. An infinite strip (left) can be treated as a 1D chain with twisted boundary conditions, i.e., with periodic boundaries along $x$ with an extra phase of $e^{\pm ikx}$ for right-left hopping. The top three rows with dark (blue) background are defined as the edge region.

Figure 7

Dispersion relation of a 2DQW on a strip with cut links on top, and absorbers at the bottom. Quasienergies of long-lived states (magnitude of Floquet eigenvalue higher than $0.9)$ are shown, with edge states (more than 80% of the weight on the top three rows) highlighted in thick (blue) lines. The bulk gap is closed and reopened by setting the rotation angles to (a) ${\theta }_1=0.35\pi ,{\theta }_2=0.15\pi$; (b) ${\theta }_1={\theta }_2=0.25\pi$; (c) ${\theta }_1=0.15\pi ,{\theta }_2=0.35\pi$; (d) ${\theta }_1=0.65\pi ,{\theta }_2=0.15\pi$; (e) ${\theta }_1=0.75\pi ,{\theta }_2=0.25\pi$; (f) ${\theta }_1=0.85\pi ,{\theta }_2=0.35\pi$.

Figure 8

Parameter space of the split-step 2D discrete-time quantum walk. Along continuous (dotted) lines, the bulk quasienergy gap around $0\left(\pi \right)$ quasienergy closes: Here these lines overlap. Each gapped domain supports edge states near a cut, at both quasienergies 0 and $\pi$. The number of these edge states [equal to the Rudner winding number as per Eq. (20)] is written in bold. The Chern number, which is always 0 due to the sublattice symmetry, is shown in normal typeface. The panels (a)–(f) refer to the parameter sets of Fig. 7.

Figure 9

(Left) The lattice on which the 2DQW is realized as a continuously driven Hamiltonian. Gray shaded unit cells include two sites each. The three types of hoppings allowed are intracell (black), horizontal intercell (red), and vertical intercell (blue). (Right) The drive sequence for the lattice Hamiltonian.

Figure 10

The four-step quantum walk is set on a Lieb lattice (left). The driving sequence of the corresponding continuously driven Hamiltonian consists of nonoverlapping pulses of arbitrary area (right).

Figure 11

Parameter space of the four-step 2DQW as defined in Eq. (30). Gapped domains, with Rudner winding numbers $W$ (boldface) and Chern numbers (normal typeface), are separated by lines, along which the bulk quasienergy gap around $\varepsilon =0$ (continuous lines) or around $\varepsilon =\pi$ (dotted lines) closes. Since sublattice symmetry of the walk is broken by the extra rotations through angles ${\varphi }_1,{\varphi }_2$, the gaps can close independently, and the Chern number can take on nonzero values. The angles shown on the left are ${\varphi }_{+}=\left|{\varphi }_2+{\varphi }_1\right|,{\varphi }_{-}=\left|{\varphi }_2-{\varphi }_1\right|$, assuming both of these are less than $\pi$. In the example shown, ${\varphi }_1=-\pi /10$ and ${\varphi }_2=\pi /5$.

Figure 12

Wave function for a particular disorder realization and the cut for a system size given by $M=10$. The values of rotation angles are ${\theta }_1=0.35\pi ,{\theta }_2=0.15\pi ,\delta =0.1\pi$. The cut is represented by the black line. The starting point for the wave function is just above the black line on the left. The final point at which the wave function gets absorbed is represented by the orange dot on the right-hand side. On the top plot the wave function is plotted for $t=0$. The middle plot is for $t=2M=20$ when a good fraction of the walker is on the conveyor. In the bottom plot the wave function is plotted for $t=10M=100$, long after the bulk of the walker has been absorbed by the orange point B.

Figure 13

(Top) Arrival and survival probabilities for ${\theta }_1=0.35\pi ,{\theta }_2=0.15\pi ,M=30$, and different amounts of rotation-angle disorder $\delta$. Solid lines are cumulative arrival probabilities at point $B$ and dashed lines are the remaining wave-function amplitudes, thus the probability of survival up to time $t$. The solid and dashed lines of the same color and thickness correspond to the same system. We have averaged over 100 different disorder realizations. (Bottom) Plot showing the arrival probability as a function of time for a different system sizes and different disorder strengths. The time axis is scaled with the system size. The curves for different system sizess collapse on one another, showing that the propagation along the cut is ballistic. The plot also shows that we may choose a system size of $M=30$ in order to further investigate the system.

Figure 14

The arrival probability at point B after 322 time steps, averaged over 100 disorder configurations for ${\theta }_1=\frac{\pi }{4}$ as a function of ${\theta }_2$ and $\delta$. The system has the same geometry as in Fig. 12, but is three times larger, having $M=30$. In the plot the azimuthal angle represents ${\theta }_2$ and the radius is related to $\delta$ by $r=1-2\delta /\pi$, such that the largest possible value of $\delta =\pi /2$ is taken at the center at $r=0$, at which point ${\theta }_1$ and ${\theta }_2$ are irrelevant. The black dashed line marks the regime at which $\delta$ becomes large enough for both types of topological invariants to be locally present in the system. Beyond that line transport begins to be suppressed. The magenta diamonds mark the points at which the group velocity becomes too small for the walker to arrive within the simulation time.

Figure 15

Illustration of the conveyor mechanism around a star-shaped figure for a single disorder realization. We have plotted the wave function at different times $t$, starting at $t=0$ and ending at $t=150$, at which point the majority of the wave function amplitude has been absorbed by the final point, marked in orange. Note that the color scale changes between the different panels. ${\theta }_1=0.45\pi ,{\theta }_2=-0.05\pi$, and $\delta =0.1\pi$ were chosen, as, according to Eq. (15), these maximize the propagation velocity along the cut. In the bottom panel we have plotted the arrival probability, which for large times approaches $P_t=0.7$.