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arXiv:2609.30400

Generation of Photonic Graph States with minimal number of quantum emitters

Konstantinos-Rafail Revis, Nils Tomke Ottink, Pierre-Emmanuel Emeriau, Paul Hilaire

quant-phSubmitted Sep 24, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

Graph states are a fundamental resource for measurement and fusion-based quantum computing, quantum networks, and sensing. Preparing them in a photonic system deterministically is, in principle, possible, but finding efficient schemes to prepare them was a long-standing problem addressed recently. Additionally, heuristic optimization schemes for reducing the required number of two-qubit gates were developed. However, the problem of reducing the number of emitters by optimizing the emission ordering was not addressed, due to its computational complexity, as it is connected to a well-known NP-hard problem from graph theory, the linear rank width computation. In this work, we focus on developing heuristic polynomial algorithms to reduce the number of emitters required. In total, we propose four distinct algorithms, which demonstrate up to $30\%$ emitter reduction on random graphs. Furthermore, we provide numerical and statistical evidence that the combination of our optimization schemes with the preexisting algorithms for optimizing the two-qubit gates of the preparation protocol can further reduce them by around $20\%$. Finally, we examine the developed algorithms for various useful graph state families, such as graphs useful for measurement-based quantum algorithms, and cluster states and graph codes used for quantum error correction, to determine the performance of each algorithm.

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45 pages, 33 figures

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Graph states are a fundamental resource for measurement and fusion-based quantum computing, quantum networks, and sensing. Preparing them in a photonic system deterministically is, in principle, possible, but finding efficient schemes to prepare them was a long-standing problem addressed recently. Additionally, heuristic optimization schemes for reducing the required number of two-qubit gates were developed. However, the problem of reducing the number of emitters by optimizing the emission ordering was not addressed, due to its computational complexity, as it is connected to a well-known NP-hard problem from graph theory, the linear rank width computation. In this work, we focus on developing heuristic polynomial algorithms to reduce the number of emitters required. In total, we propose four distinct algorithms, which demonstrate up to 30%30\%30% emitter reduction on random graphs. Furthermore, we provide numerical and statistical evidence that the combination of our optimization schemes with the preexisting algorithms for optimizing the two-qubit gates of the preparation protocol can further reduce them by around 20%20\%20%. Finally, we examine the developed algorithms for various useful graph state families, such as graphs useful for measurement-based quantum algorithms, and cluster states and graph codes used for quantum error correction, to determine the performance of each algorithm.
Abstract Photonic graph states are important for measurement- and fusion-based quantum computing, quantum networks, and sensing. They can in principle be generated deterministically by using emitters to create the requisite entanglement. Finding ways to minimize the number of entangling gates between emitters and understanding the overall optimization complexity of such protocols is crucial for practical implementations. Here, we address these issues using graph theory concepts. We develop optimizers that minimize the number of entangling gates, reducing them by up to 75% compared to naive schemes for moderately sized random graphs. While the complexity of optimizing emitter-emitter CNOT counts is likely NP-hard, we are able to develop heuristics based on strong connections between graph transformations and the optimization of stabilizer circuits. These patterns allow us to process large graphs and still achieve a reduction of up to 66% in emitter CNOTs, without relying on subtle metrics such as edge density. We find the optimal emission orderings and circuits to prepare unencoded and encoded repeater graph states of any size, achieving global minimization of emitter and CNOT resources despite the average NP-hardness of both optimization problems. We further study the locally equivalent orbit of graphs. Although enumerating orbits is #P complete for arbitrary graphs, we analytically calculate the size of the orbit of repeater graphs and find a procedure to generate the orbit for any repeater size. Finally, we inspect the entangling gate cost of preparing any graph from a given orbit and show that we can achieve the same optimal CNOT count across the orbit.
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.
Abstract Multi-photon entangled graph states are a fundamental resource in quantum communication networks, distributed quantum computing, and sensing. These states can in principle be created deterministically from quantum emitters such as optically active quantum dots or defects, atomic systems, or superconducting qubits. However, finding efficient schemes to produce such states has been a long-standing challenge. Here, we present an algorithm that, given a desired multi-photon graph state, determines the minimum number of quantum emitters and precise operation sequences that can produce it. The algorithm itself and the resulting operation sequence both scale polynomially in the size of the photonic graph state, allowing one to obtain efficient schemes to generate graph states containing hundreds or thousands of photons.
Photonic graph states constitute important resources in quantum information processing. They are used in measurement-based and fusion-based quantum computing and in all-photonic repeaters for quantum networks. However, generating such highly entangled states is challenging due to the fact that photons do not interact with each other. We propose a protocol for generating arbitrary graph states using matter qubits as quantum emitters. Our approach guarantees the minimal number of emitters and provides in polynomial time the quantum circuit for the graph state generation (which includes the non-unitary process of photon emission). We find that the quantum circuit depth for general graphs also scale polynomially with system size.
Fusion-based quantum computing (FBQC) relies on a set of small, typically photonic, resource states that are fused together through Bell state measurements. The main bottleneck of FBQC is the low rate of generating the resource states, which stems from the probabilistic nature of photonic fusion gates. Previous work introduced a general algorithm for constructing circuits that deterministically generate photonic resource states from a minimal number of quantum emitters for a specified photon emission ordering. However, finding the minimal number of emitters and CNOT gates across all possible orderings is an NP-hard problem. Here, we exploit the symmetries present in FBQC resource states to dramatically simplify this optimization problem. We find that logically encoded 24-photon FBQC resource states can be produced using as few as 3 emitters and 11 CNOTs.
Multiqubit entangled photonic graph states are an important ingredient for all-photonic quantum computing, repeaters, and networking. Preparing them using probabilistic stitching of single photons using linear optics presents a formidable resource challenge due to multiplexing needs. Quantum emitters provide a viable solution to prepare photonic graph states, as they enable deterministic production of photons entangled with emitter qubits and deterministic two-qubit interactions among emitters. A handful of emitters often suffice to generate useful-size graph states that would otherwise require millions of emitters used as single photon sources, using the linear-optics method. Photon loss, however, impedes the emitter method due to a large circuit depth and hence loss accrual on the photons of the graph state produced, given the typically large number of slow two-qubit controlled-not (cnot) gates between emitters. We propose an algorithm that can trade the number of emitters with the graph-state depth, while minimizing the number of emitter cnot gates. We apply our algorithm to generate a repeater graph state (RGS) for an alternative all-photonic repeater protocol, which achieves a far superior rate-distance trade-off compared to using the least number of emitters needed to generate the RGS. Yet it needs five orders of magnitude fewer emitters than the multiplexed linear-optics method---with each emitter used as a photon source---to achieve a desired rate-distance performance.