Efficient Photonic Graph State Generation

TL;DR

This work introduces an efficient protocol for generating arbitrary caterpillar graph states (CGSs) in photonic systems using the Linear Quantum Graph (LQG) framework. By leveraging directed unipartite and bipartite graph representations, along with path circuit digraphs $P^{(l)}$ and primate circuit digraphs, CGSs are constructed with fewer photons and significantly higher success probabilities than fusion-based methods. The linear-optical realization relies on heralded single-photon subtractions and multiport splitters to implement the required CZ entanglements, achieving a probabilistic yet scalable route to complex graph states. The approach unifies fusion-based and heralded subtraction strategies under LQG, paving the way for scalable MBQC resources and potential extensions to generalized graph families and integrated photonic platforms.

Abstract

Graph states are central resources for quantum information processing, supporting applications in computation, communication, and error correction. In photonic systems, they are typically assembled from smaller entangled states using probabilistic fusion gates, which demand many photons and suffer from low success rates. We present an optimized scheme for directly generating caterpillar graph states (CGSs) -- essential resource states for constructing high-dimensional lattice graph states -- using only single-photon sources, linear optics, and heralded measurements. Based on the linear quantum graph (LQG) picture, our method produces CGSs efficiently and scalably. For CGSs of length $l\ge 3$, it requires $l-2$ fewer photons and achieves a success rate $2^{l-2}$ times higher than fusion-based approaches. These results demonstrate that the LQG picture provides a powerful and flexible route for realizing complex photonic graph states with minimal resources.

Figures (7)

• Figure 1: Path circuit digraphs $P^{(1)}$, $P^{(2)}$, and $P^{(3)}$. $P^{(l)}$ can be drawn from $P^{(l-1)}$ by adding $A_{l+1}$ and edges according to the iteration relation Eq. \ref{['central_path_adj']}.
• Figure 2: Sculpting bigraphs that generates CGSs. (a) A sculpting digraph for an arbitrary $(k_1+k_2+2)$-partite CGS of path length 1, directly generalizing the digraph \ref{['Bell_di']}. (b) A sculpting digraph for an arbitrary $\sum_{l}(k_l+1)$-partite caperpillar graph state of length $l$ based on $P^{(l)}$.
• Figure 3: Elements that construct a sculpting bigraph for an arbitrary CGSs. (a) path circuitbigraphs $P^{(1)}$, $P^{(2)}$, and $P^{(3)}$ in $G_{ub}$. This is easily drawn from the directed graphs in Fig. \ref{['fig:central_path']} using Table I. Note that all the horizontal lines correspond to the loops of path circuitdigraphs, which will be replaced with the graph element (b), which is now called a primate circuit bigraph.
• Figure 4: The sculpting bigraph that generates an arbitrary CGS of length $l$. The horizontal lines from $A_j$$(j \in \{1,2,\cdots, l+1\})$ in $P^{(l)}$ are all replaced with primate circuit bigraphs. For each $j$th primate circuit bigraph, the open black edge is connected to $A_{j}$, and the open red edge to the dot next to $A_{j}$.
• Figure 5: Linear optical circuit to generate an arbitrary CGS \ref{['caterpillar_arb']}. The circuit elements in the dotted box is from $P^{(l)}$, and those in the dashed boxes are from primate circuit bigraphs (we now call them primate circuits). Each solid box both in $P^{(l)}$ and primate circuits performs a single-photon subtraction. The black boxes with $\bar{p}$ are asymmetryc $p$-partite splitter, in which the relative probability amplitudes are asymmetrically given as in Eq. \ref{['app:asymmetric_amp']}. The black boxes with $p$ are symmetric $p$-partite splitters that perform $p$-level discrete Fourier transforms. This circuit can also generate weighted graph states by relaxing the phase rotation angles.

Theorems & Definitions (1)

• Definition 1