Quantum Search with a Generalized Laplacian

TL;DR

This work shows that a single excitation in a Heisenberg spin network can implement a generalized Laplacian quantum walk on signed weighted graphs, encapsulated by $H_\alpha=-\gamma L_\alpha$ with $L_\alpha=L+\alpha D$. The authors prove that the Heisenberg model realizes the generalized walk for any real $\alpha$, unifying the Laplacian, adjacency, and signless Laplacian cases, and they apply this to a spatial search on a weighted barbell graph where an oracle is implemented by an external magnetic field. They identify two critical bridge weights $w_+$ and $w_-$ (depending on $\alpha$) that enable amplitude transfer between the cliques, boosting the single-stage success probabilities to about $0.820$ and $0.843$, and they demonstrate a two-stage protocol (notably for $w_+$) that can reach $0.996$. All results scale as $O(\sqrt{N})$ in runtime with different constants, illustrating that bridge weighting and the choice of $\alpha$ can substantially improve quantum search performance on networked spin systems.

Abstract

A single excitation in a quantum spin network described by the Heisenberg model can effect a variety of continuous-time quantum walks on unweighted graphs, including those governed by the discrete Laplacian, adjacency matrix, and signless Laplacian. In this paper, we show that the Heisenberg model can effect these three quantum walks on signed weighted graphs, as well as a generalized Laplacian equal to the discrete Laplacian plus a real-valued multiple of the degree matrix, for which the standard Laplacian, adjacency matrix, and signless Laplacian are special cases. We explore the algorithmic consequence of this generalized Laplacian quantum walk when searching a weighted barbell graph consisting of two equal-sized, unweighted cliques connected by a single signed weighted edge or bridge, with the search oracle constituting an external magnetic field in the spin network. We prove that there are two weights for the bridge (which could both be positive, both negative, or one of each, depending on the multiple of the degree matrix) that allow amplitude to cross from one clique to the other -- except for the standard and signless Laplacians that respectively only have one negative or positive weight bridge -- boosting the success probability from 0.5 to 0.820 or 0.843 for each weight. Moreover, one of the weights leads to a two-stage algorithm that further boosts the success probability to 0.996.

Figures (6)

• Figure 1: A weighted barbell graph of $N = 12$ vertices. Solid edges are unweighted, and the dotted edge has real-valued weight $w$. A vertex is marked, indicated by a double circle.
• Figure 2: A signed weighted graph of $N = 4$ vertices, where $e_{ij}$ is the weight of the edge joining vertices $i$ and $j$ and can be positive or negative. The graph is undirected, so $e_{ji} = e_{ij}$.
• Figure 3: Success probability vs time for search on the weighted barbell graph with $N = 1200$ vertices using the generalized Laplacian quantum walk with (a) $\alpha = -3$, (b) $\alpha = 0$, (c) $\alpha = 1$, (d) $\alpha = 2$, and (e) $\alpha = 4$. In all the left plots, the solid black curve is when the weight of the bridge is $w = 120$, dashed red is $w = 150$, dotted green is $w = 200$, dot-dashed blue is $w = 300$, dot-dot-dashed orange is $w = 600$, and dot-dashed-dashed brown is $w = 1200$. In all the right plots, the respective curves are when the weights are negated, i.e., $w = -120$, $-150$, $-200$, $-300$, $-600$, and $-1200$.
• Figure 4: The weighted barbell graph of $N = 12$ vertices from Fig. \ref{['fig:barbell']}, with identically evolving vertices are identically colored and labeled.
• Figure 5: Search on the weighted barbell graph with $N = 1200$ vertices using the generalized Laplacian quantum walk with $\alpha = 4$. The solid black curve is the probability at the marked $a$ vertex, i.e., the success probability. The dashed red curve is the probability at the $b$ vertices, dotted green is $c$, dot-dashed blue is $d$ (and overlaps with the dotted green $c$ curve), and dot-dot-dashed orange is $e$. The dot-dashed-dashed brown curve is the probability in the marked clique, i.e., the probability in vertices $a$, $b$, or $c$, and the short dashed violet curve is the probability in vertices $a$ and $b$. In (a), $w = w_+ = N/(2\alpha) = 150$ is used throughout. In (b), $w = w_+ = N/(2\alpha) = 150$ is used left of the vertical dashed line, and $w = 1$ is used right of the vertical dashed line.