Strongly trapped two-dimensional quantum walks

Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a broad scope of applications. In particular, they can be used as computational primitives, and they are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due to their quantumness. In some cases, however, they can remain localized at their initial state (trapping). The trapping and other fundamental properties of DTQWs are determined by the choice of the coin operator. We introduce and analyze a type of walks driven by a coin class leading to strong trapping, complementing the known list of walks. This class of walks exhibits a number of exciting properties with possible applications ranging from light pulse trapping in a medium to topological effects and quantum search.

Figure 1

Quantum circuit representing the coin class $\mathcal{C}$ defined in Eq. (8). The abstract two-qubit space of $|{\psi }_{\text{in(out)}}^{\left(1\right)}\rangle$ and $|{\psi }_{\text{in(out)}}^{\left(2\right)}\rangle$ is spanned by basis vectors $\left\{\right|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$ corresponding to the four-dimensional coin space of the two-dimensional quantum walk with $|L\rangle =|00\rangle ,|D\rangle =|01\rangle ,|U\rangle =|10\rangle ,|R\rangle =|11\rangle$.

Figure 2

Minimum probability of finding the particle at its initial position after 40 steps of a $\mathcal{C}$-class coin [see Eq. (8)] driven quantum walk. The probabilities are determined by numerically searching the initial states yielding the minimum probability. The coin is investigated in the ${\delta }_1,{\delta }_2$ parameter space with ${\alpha }_1={\alpha }_2={\beta }_1={\beta }_2=\frac{\pi }{2}$ and $\phi =\pi$. The red (gray) lines correspond to the condition of Eq. (11) where strong trapping disappears. On these lines the minimum probability is naturally zero due to the appearance of the escaping state of Eq. (12). However, where strong trapping is present, the minimum probability is larger than zero.

Figure 3

Topological map (left plot) and quasienergy spectrum (right plot) of $\mathcal{C}$-class coins [see Eq. (8)]. The light brown (gray) and white areas correspond to different topological phases; i.e., in the boundary between two topologically different bulk regions we found a topologically protected edge state. The thick lines represent the gap closing condition (11), which is also the condition describing the disappearance of strong trapping. We used the definitions (8) and (9) with parameter values ${\alpha }_1={\alpha }_2={\beta }_1={\beta }_2=\frac{\pi }{2}$ and $\phi =\pi$, invoking a real valued subclass of $\mathcal{C}$, which contains the Grover coin for ${\delta }_1={\delta }_2=\frac{\pi }{4}$ (marked by the black dot). The quasienergy $\omega$ spectrum as the function of momentum $k$ on the right plot corresponds to the two-dimensional QW on a torus using two parallel bulk regions with ${\delta }_1=\frac{\pi }{10},{\delta }_2=\frac{4\pi }{10}$ and ${\delta }_1=\frac{4\pi }{10},{\delta }_2=\frac{\pi }{10}$ (denoted by circles on the topological map on the left). The topologically protected edge states are emphasized by shading.

Figure 4

Probability of finding a marked vertex on a $10\times 80$ torus using the original spatial quantum search algorithm (dashed curve), and a modified one (red [gray] curve) employing the coin class $\mathcal{C}$ defined in Eq. (8) with parameters ${\delta }_1={\delta }_2=\frac{31\pi }{90},{\alpha }_1={\beta }_2=\frac{\pi }{2},{\alpha }_2={\beta }_1=-\frac{\pi }{2}$, and $\phi =\pi$. The vertical black line denotes the measuring time defined in the algorithm $\sqrt{2LM}=40$. At that time the probabilities of the original and modified algorithms at the marked vertex are $0.0616$ and $0.0827$ respectively. Note that these probabilities are obtained before the final amplitude amplification step concluding the algorithm.