Efficient Preparation of Graph States using the Quotient-Augmented Strong Split Tree

Abstract

Graph states are a key resource for measurement-based quantum computation and quantum networking, but state-preparation costs limit their practical use. Graph states related by local complement (LC) operations are equivalent up to single-qubit Clifford gates; one may reduce entangling resources by preparing a favorable LC-equivalent representative. However, exhaustive optimization over the LC orbit is not scalable. We address this problem using the split decomposition and its quotient-augmented strong split tree (QASST). For several families of distance-hereditary (DH) graphs, we use the QASST to characterize LC orbits and identify representatives with reduced controlled-Z count or preparation circuit depth. We also introduce a split-fuse construction for arbitrary DH graph states, achieving linear scaling with respect to entangling gates, time steps, and auxiliary qubits. Beyond the DH setting, we discuss a generalized divide-and-conquer split-fuse strategy and a simple greedy heuristic for generic graphs based on triangle enumeration. Together, these methods outperform direct implementations on sufficiently large graphs, providing a scalable alternative to brute-force optimization.

Figures (10)

• Figure 1: Examples of the special families of graphs we characterize, all of which are distance-hereditary.
• Figure 2: (a) The effect of local complement on a graph $G$ with respect to vertex $1\in V(G)$ (green vertex). The edges between the neighbors of 1 (yellow vertices) are complemented: existing edges are deleted, and missing edges are added. (b) The LC orbit of the complete graph on three vertices, locally equivalent to a star with 2 spokes.
• Figure 3: The split decompositions for the three special families of DH graph that we consider. $K_{n,m}$ splits into two quotient graphs, while $K_{n_1,\cdots,n_k}$ and $CS^r_{n_1,\cdots,n_k}$ always split into $k+1$ quotient graphs. We adopt the convention of labeling the central quotient graph $Q_0$ and the others $Q_1,\cdots,Q_k$ matching the index of the vertex group. The LC orbits of these graphs have been fully classified based on the symmetries of quotient graphs in the QASST.
• Figure 4: An example of two LC-equivalent graph states and their preparation circuits. Vertices denote qubits prepared in the $|+\rangle=H|0\rangle$ state, and edges denote entanglement via a controlled-Z (CZ) gate. (a) The graph state corresponding to the complete bipartite graph $K_{3,3}$. (b) The preparation circuit for $|K_{3,3}\rangle$ with CZ operations ordered sequentially. (c) The graph state for a binary-star, locally equivalent to $|K_{3,3}\rangle$ via an edge-pivot. This graph has chromatic index $\chi'(G)=3$, and edges are colored according to a minimal edge-coloring. (d) A minimal-depth preparation circuit corresponding to this edge-coloring, wherein edges of the same color correspond to CZ operations that are performed simultaneously. This is an optimal graph state locally equivalent to $|K_{3,3}\rangle$ (non-unique).
• Figure 5: An example of a Type-II fusion between qubits $q_1$ and $q_2$ from two different graph states $|G_1\rangle$ and $|G_2\rangle$.

Theorems & Definitions (2)

• Proposition 1
• proof