Graph state extraction from two-dimensional cluster states

TL;DR

The paper tackles the problem of extracting arbitrary graph states from a pre-existing two-dimensional cluster state using only single-qubit measurements and local unitaries. It introduces two graph-state manipulation tools—merging of subgraphs and vertex-degree expansion—and the zipper scheme to connect distant vertices, enabling three extraction strategies (two decentralized variants and centralized MBQC-based generation) and overhead comparisons. It develops decentralized LVDE and OVDE methods and a centralized MBQC-based approach, and compares their resource costs across target graphs, highlighting tradeoffs between flexibility and overhead. The work discusses scalability limits, the NP-complete nature of related graph-transformation problems, resilience to noise, and practical applications in entanglement-based networks and distributed quantum computing.

Abstract

We propose schemes to extract arbitrary graph states from two-dimensional cluster states by locally manipulating the qubits solely via single-qubit measurements. We introduce graph state manipulation tools that allow one to increase the local vertex degree and to merge subgraphs. We utilize these tools together with the previously introduced zipper scheme that generates multiple edges between distant vertices to extract the desired graph state from a two-dimensional cluster state. We show how to minimize overheads by avoiding multiple edges, and compare with a local manipulation strategy based on measurement-based quantum computation together with transport. These schemes have direct applications in entanglement-based quantum networks, sensor networks, and distributed quantum computing in general.

Figures (13)

• Figure 1: The family of graph states are multipartite entangled states which can be represented by graphs such as the green graph in the figure. A prominent graph state is the two-dimensional cluster state, whose underlying graph is a grid, as shown in gray. In this paper, we initially have two-dimensional cluster states and aim to extract general graph states from it.
• Figure 2: The initial graph state, associated with the graph in the center, consists of four vertices (green dots) and four edges (gray lines). The states depicted around the central graph show the effects of local complementation and Pauli measurements. The up-left graphs show the effect of local complementation on vertex $d$. Vertex $d$ is adjacent to all other vertices. Because vertices $a$ and $c$ are adjacent in the initial graph, they get disconnected. Vertex $b$ is initially not adjacent to vertices $a$ and $c$ and therefore gets connected. If we measure qubit $c$ in the $Z$ basis, the resulting state is represented by the bottom-right graph: All edges containing $c$ are deleted. To see the effect of measuring qubit $c$ in the $Y$ basis, we first apply a local complementation on qubit $c$ and then disconnect it from the graph. This is shown in the bottom-left graph. Finally, measuring qubit $c$ in the $X$ basis is represented by a local complementation on the adjacent vertex $d$, then applying the rules of the $Y$ measurement, and finally applying a local complementation on vertex $d$. The resulting graph is shown up-right.
• Figure 3: The two-dimensional cluster states correspond to a two-dimensional grid graph, shown in gray. a) The zipper scheme connects pairs of distant green qubits in the cluster state by measuring the pink qubits at a staircase-shaped path corresponding to the black dashed line. We illustrate the use of the zipper scheme and its iterative application by the boxes highlighted in pink (although the actual path may not be directly along that path). b) The post-measurement state is a tensor product of the two Bell pairs in green, and the graph state corresponding to the remaining grid graph. The zipper scheme mostly restores the underlying two-dimensional cluster state. Qubits represented by vertices adjacent to the end or turning points get measured in Pauli $Z$ basis, those are shown in bright yellow.
• Figure 4: Merging of two subgraphs: a) The yellow path of edges is necessary to merge the two subgraphs $G_1$ (brown) and $G_2$ (green). b) The first step is to measure the qubit associated to the vertex labeled $Y_1$ in the Y basis, which inverts the neighborhood of $Y_1$ and connects it to $Y_3$. c) A $Y$ measurement of $Y_3$ merges $G_1$ and $G_2$ at vertex $Y_2$, while restoring the original neighborhood of vertex $Y_1$. The roles of $Y_1$ and $Y_2$ can be interchanged, leading to a merging at $Y_1$.
• Figure 5: Vertex degree expansion: a-c) The vertex degree expansion increases the degree of the bright-green vertex $Y_2$ from $d=4$ to $d=6$, where the dark green qubits and edges denote the already existing neighbors. a) Two Pauli $Z$ measurements, labeled as bright-yellow $Z$, isolate the edge to be expanded from the grid. Measurement of the yellow qubit $Y_1$, seen in b), and $Y_3$, seen in c), expand the degree of the vertex $Y_2$. d) A $n_{\text{exp}}$ times iterative application of the vertex degree expansion along the horizontal direction enhances the degree of vertex $a$ by $2\times n_{\text{exp}}$.