Quantum walk search for exceptional configurations on one- and two-dimensional lattice with extra long-range edges of Hanoi network of degree four

TL;DR

The paper investigates quantum spatial search for multiple marked vertices on one- and two-dimensional lattices augmented with Hanoi network degree-four long-range edges (HN4) using discrete-time quantum walks. It compares coin operators—$\mathcal{C}_{Grov}$, $\mathcal{C}_{SKW}$, $\mathcal{C}_{l}$, and the modified coin $\mathcal{C}_G$—and demonstrates that $\mathcal{C}_G$ can efficiently search all identified exceptional configurations, while Grover and lackadaisical coins fail in several cases. In 1D, novel exceptional configurations arise from directed self-loops, and in 2D, diagonal configurations lose their exceptional status; new exceptional types include pairs adjacent via long-range edges and vertices with one or two directed self-loops. The results show that in 2D, lackadaisical search with $\mathcal{C}_G$ attains the optimal $O(\sqrt{N/M})$ time, with stationary-state analysis explaining why alternative coins fail, highlighting the practical impact for designing quantum search on complex networks.

Abstract

There exist several types of configurations of marked vertices, referred to as the exceptional configurations, on one- and two-dimensional periodic lattices with additional long-range edges of the Hanoi network of degree four (HN4), which are challenging to find using discrete-time quantum walk algorithms. In this article, we conduct a comparative analysis of the discrete-time quantum walk algorithm utilizing various coin operators to search for these exceptional configurations. First, we study the emergence of several new exceptional configurations/vertices due to the additional long-range edges of the HN4 on both one- and two-dimensional lattices. Second, our study shows that the diagonal configuration on a two-dimensional lattice, which is exceptional in the case without long-range edges, no longer remains an exceptional configuration. Third, it is also shown that a recently proposed modified coin can search all these configurations, including any other configurations in one- and two-dimensional lattices with very high success probability. Additionally, we construct stationary states for the exceptional configurations caused by the additional long-range edges, which explains why the standard and lackadaisical quantum walks with the Grover coin cannot search these configurations.

Figures (9)

• Figure 1: (a) One-dimensional periodic lattice of size $N=16$ with three blue colored marked vertices and (b) Hanoi network of degree four (HN4) with $N=16$ vertices and yellow colored long-range edges.
• Figure 2: Success probability to measure the marked vertex $(n-1,0)=(5,0)= 32$ in $N=64$ one-dimensional periodic lattice with extra long-range edges as a function of the number of iteration steps obtained using (a) $\mathcal{C}_{Grov}$, (b) $\mathcal{C}_{SKW}$, (c) $\mathcal{C}_{l}$, and (d) $\mathcal{C}_{G}$ coins.
• Figure 3: (a) Two-dimensional lattice of size $\sqrt{N} \times \sqrt{N}$ with periodic boundary conditions with a blue colored cluster of marked vertices and (b) Each row and column of the lattice in left panel is additionally added with long-range edges of Hanoi network of degree four as depicted in right panel for a $16\times 16$ square lattice. In lackadaisical quantum walk search, self-loop is also added at each vertex of the two-dimensional lattice.
• Figure 4: Success probability to measure two non-adjacent vertices $(2;1), (8;7)$ on $2^{5} \times 2^{5}$ two-dimensional periodic lattice with extra long-range edges as a function of the number of iteration steps for (a) $\mathcal{C}_{Grov}$, (b) $\mathcal{C}_{SKW}$, (c) $\mathcal{C}_{l}$, and (d) $\mathcal{C}_{G}$ coins.
• Figure 5: Success probability to measure the diagonal configuration on $2^{5} \times 2^{5}$ two-dimensional periodic lattice with extra long-range edges as a function of the number of iteration steps for (a) $\mathcal{C}_{Grov}$, (b) $\mathcal{C}_{SKW}$, (c) $\mathcal{C}_{l}$, and (d) $\mathcal{C}_{G}$ coins.