Cost-aware Photonic Graph State Generation: A Graphical Framework

TL;DR

This work addresses the bottleneck of scalable, resource‑efficient generation of photonic graph states by introducing a cost‑aware graphical framework and the Graph Builder algorithm for deterministic emitter‑based state synthesis. It formalizes a complete set of eligibility conditions (I–III) and a universal graph‑transformation toolkit (edge toggles, emission modes, LC) to systematically navigate the generation landscape. The method achieves substantial two‑qubit gate reductions across random and structured graphs, demonstrates explicit circuit‑blocks for important graph families (e.g., encoded 6‑rings, RHG lattices, RGS), and provides a flexible optimization framework with multiple degrees of freedom (emitters, order, PLC equivalence, extra emitters). These advances offer a versatile path toward scalable, resource‑efficient photonic quantum information processing, with potential extensions to fusion‑based approaches and practical quantum networks.

Abstract

Photonic graph states are essential resources for quantum computation and communication. Deterministic emitter-based generation of graph states overcomes the scalability issues of probabilistic approaches; nonetheless, their experimental realization is constrained by technological demands, often expressed by the number of two-qubit gates and the depth and/or width of the quantum circuits used to model the generation process. Here, we introduce a cost-aware framework for the generation of photonic graph states of arbitrary size and shape, built on a complete set of necessary and sufficient conditions and a universal set of elementary graph operations that govern the evolution of the state toward the target. Within this framework, we develop Graph Builder, a deterministic generation algorithm that achieves substantial reductions (up to an order of magnitude) in two-qubit gate usage for both random and structured graphs, compared with alternative approaches. Furthermore, we show that this framework enables the identification of elementary building blocks in specific cases, such as encoded 6-ring states. The algorithm uses the minimum number of emitters possible for a fixed emission sequence, while also supporting the use of extra emitters for controlled trade-offs between emitter count and other cost metrics. Moreover, by systematically identifying the degrees of freedom at each stage of the generation process, this framework fully characterizes the optimization landscape, enabling analytic, heuristic, or exhaustive strategies for further cost reductions. Our approach provides a general and versatile tool for designing and optimizing emitter-based photonic graph state generation protocols, essential for scalable and resource-efficient photonic quantum information processing.

Figures (29)

• Figure 1: Generation of photonic graph states with a quantum emitter. (a) A proposed energy level structure lindner_proposal_2009, made up of doubly degenerate ground and excited states of spin 1/2 and 3/2, respectively. The selection rules then allow the emitter's spin to become entangled with the polarization of the emitted photon (b) A quantum circuit representation of the process. Initially, the emitter and photonic qubits are considered to be in $\ket{0}$ state, a Hadamard gate on the emitter brings it to a superposition state and a cnot is used to model the emission process as described in the dashed box. (c) The outcome of the emission circuit in graph representation: the emitter and the photonic node connected with an edge. The emitter can continue emitting and grow the graph by adding photonic nodes to it. (d) Caterpillar graphs. A class of entangled states obtainable with a single quantum emitter and local operations. A general caterpillar state consists of a main path of $\mathcal{L}$ entangled qubits, each having an arbitrary number ($n_\ell$) of leaf qubits attached to them.
• Figure 2: An example of local complementation. The transformation is applied at node $k$ with the neighborhood $N(k)=\{1,2,3,4\}$ whose subgraph is replaced by its complement.
• Figure 3: An example target graph is shown in its original form (top), in its bi-partitioned form with the first two photons considered as emitted (middle), and as the intermediate physical graph with two emitted photons and two emitters (bottom). The adjacency matrices corresponding to the target and physical graph states, together with the associated biadjacency matrices, $B$ and $B'$, are also presented. The rows of the biadjacency matrix $B$ represent the connectivity vectors ($R$) of the emitted photons to the future nodes according to the target state. The physical biadjacency matrix $B'$ includes the connectivity vectors ($R'$) for each photon to the current set of emitters at this step of the generation.
• Figure 4: The evolution of the biadjacency matrix with emission of each photon at each step. The figure shows the case of $B(n)\to B(n+1)$ for $n=2$. The column to remove shows the connections of the new photons to the inside set. The new row ($R_{new}$) represent the edges that need to be established between the new photon and the future ones to come.
• Figure 5: A representation of condition \ref{['III-2']} satisfied through all steps on the adjacency matrix of the target graph. At step $n$, the inside neighbors of the newly emitted photon are determined by the non-zero entries in the column above the $n$-th diagonal element. In the final step the complete adjacency matrix of the target graph is recovered. Due to symmetry of the adjacency matrix, fixing only one side of the diagonal is enough.

Theorems & Definitions (18)

• Definition
• Proposition 1
• Theorem 1
• Definition
• Definition
• Theorem 2
• Lemma 1
• Proposition 2
• Lemma 2
• Lemma 3