Graph-theoretical optimization of fusion-based graph state generation

Seok-Hyung Lee1,2 and Hyunseok Jeong1

1Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea
2Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia

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Graph states are versatile resources for various quantum information processing tasks, including measurement-based quantum computing and quantum repeaters. Although the type-II fusion gate enables all-optical generation of graph states by combining small graph states, its non-deterministic nature hinders the efficient generation of large graph states. In this work, we present a graph-theoretical strategy to effectively optimize fusion-based generation of any given graph state, along with a Python package OptGraphState. Our strategy comprises three stages: simplifying the target graph state, building a fusion network, and determining the order of fusions. Utilizing this proposed method, we evaluate the resource overheads of random graphs and various well-known graphs. Additionally, we investigate the success probability of graph state generation given a restricted number of available resource states. We expect that our strategy and software will assist researchers in developing and assessing experimentally viable schemes that use photonic graph states.

Graph states, which are quantum states where qubits are entangled in a way instructed by a graph structure, are versatile resource states for quantum computing and communication. In particular, graph states in photonic systems can be used for measurement-based quantum computing and fusion-based quantum computing, which are promising candidates for near-term fault-tolerant quantum computing. In this work, we propose a method to build arbitrary photonic graph states from initial three-photon basic resource states. This is achieved through a series of "fusion" operations, where smaller graph states are probabilistically merged into larger ones via specific photon measurements. The core of our strategy is a graph-theoretical framework designed to minimize the resource requirements of this process, enhancing efficiency and feasibility.

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[2] Sobhan Ghanbari, Jie Lin, Benjamin MacLellan, Luc Robichaud, Piotr Roztocki, and Hoi-Kwong Lo, "Optimization of deterministic photonic graph state generation via local operations", arXiv:2401.00635, (2023).

[3] Seungbeom Chin, "Boson subtraction as an alternative to fusion gates for generating graph states", arXiv:2306.15148, (2023).

[4] Maxime Cautrès, Nathan Claudet, Mehdi Mhalla, Simon Perdrix, Valentin Savin, and Stéphan Thomassé, "Vertex-minor universal graphs for generating entangled quantum subsystems", arXiv:2402.06260, (2024).

[5] Jie Lin, Benjamin MacLellan, Sobhan Ghanbari, Julie Belleville, Khuong Tran, Luc Robichaud, Roger G. Melko, Hoi-Kwong Lo, and Piotr Roztocki, "GraphiQ: Quantum circuit design for photonic graph states", arXiv:2402.09285, (2024).

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