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Efficient Graph State Generation in Linear Optics

Seungbeom Chin and William John Munro

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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. 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. Our method uses only single-photon sources, linear-optical elements, and heralding measurements, so successful preparation is achieved without destroying the generated state. Our approach demonstrates the advantage of employing a broader set of measurement gates beyond conventional fusion gates.

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AbstractGraph 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. 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 to generating complex photonic graph states for efficient quantum information processing.Featured image: Linear quantum graph representation of the heralded scheme for generating caterpillar graph statesPopular summaryPhotonic quantum technologies rely on the ability to create large entangled states of light. Among the most useful are graph states, which support applications ranging from quantum computing and communication to error correction and tests of quantum nonlocality. However, producing large photonic graph states remains challenging, because the conventional approach repeatedly combines smaller entangled states using probabilistic fusion operations, requiring many photons and rapidly reducing the overall success probability. In this work, we develop a more efficient way to generate an important family of graph states known as caterpillar graph states. These states are useful building blocks for constructing larger lattice-like resource states for measurement-based quantum computing and for studying strong forms of multipartite nonlocality. Our method uses only single-photon sources, linear-optical elements, and heralding measurements, so successful preparation is achieved without destroying the generated state. The design is based on the linear quantum graph (LQG) picture, which translates bosonic state-generation problems into graph-theoretic structures. By identifying two key structures, called path circuit graphs and primate circuit graphs, we derive optical circuits that generate arbitrary caterpillar graph states more efficiently than conventional fusion-based constructions. For caterpillar states of length $l\geq 3$, the scheme uses $l-2$ fewer photons and improves the success probability by a factor of $2^{l-2}$. Our approach demonstrates the advantage of employing a broader set of measurement gates beyond conventional fusion gates. More broadly, it demonstrates how the LQG framework can guide the construction of efficient optical circuits for increasingly complex quantum resource states.► BibTeX data@article{Chin2026efficientgraphstate, doi = {10.22331/q-2026-08-13-2189}, url = {https://doi.org/10.22331/q-2026-08-13-2189}, title = {Efficient {G}raph {S}tate {G}eneration in {L}inear {O}ptics}, author = {Chin, Seungbeom and Munro, William John}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2189}, month = aug, year = {2026} }► References [1] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409(6816):46–52, 2001. https://doi.org/10.1038/35051009. https://doi.org/10.1038/35051009 [2] Pieter Kok, William J Munro, Kae Nemoto, Timothy C Ralph, Jonathan P Dowling, and Gerard J Milburn. Linear optical quantum computing with photonic qubits. Reviews of Modern Physics, 79(1):135–174, 2007. https://doi.org/10.1103/revmodphys.79.135. https://doi.org/10.1103/revmodphys.79.135 [3] Jeremy L. O'Brien. Optical quantum computing. Science, 318(5856):1567–1570, 2007. https://doi.org/10.1126/science.1142892. https://doi.org/10.1126/science.1142892 [4] Koji Azuma, Kiyoshi Tamaki, and Hoi-Kwong Lo. All-photonic quantum repeaters. Nature Communications, 6(1):6787, 2015. https://doi.org/10.1038/ncomms7787. https://doi.org/10.1038/ncomms7787 [5] Koji Azuma, Kiyoshi Tamaki, and William J Munro. All-photonic intercity quantum key distribution. Nature Communications, 6(1):10171, 2015. https://doi.org/10.1038/ncomms10171. https://doi.org/10.1038/ncomms10171 [6] Robert Raussendorf and Hans J Briegel. A one-way quantum computer.

Physical Review Letters, 86(22):5188, 2001. https://doi.org/10.1103/physrevlett.86.5188. https://doi.org/10.1103/physrevlett.86.5188 [7] Robert Raussendorf, Daniel E Browne, and Hans J Briegel. Measurement-based quantum computation on cluster states. Physical Review A, 68(2):022312, 2003. https://doi.org/10.1103/physreva.68.022312. https://doi.org/10.1103/physreva.68.022312 [8] Marc Hein, Jens Eisert, and Hans J Briegel. Multiparty entanglement in graph states. Physical Review A, 69(6):062311, 2004. https://doi.org/10.1103/physreva.69.062311. https://doi.org/10.1103/physreva.69.062311 [9] Hans J Briegel, David E Browne, Wolfgang Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5(1):19–26, 2009. https://doi.org/10.1038/nphys1157. https://doi.org/10.1038/nphys1157 [10] Maarten Van den Nest. Universal quantum computation with little entanglement.

Physical Review Letters, 110(6):060504, 2013. https://doi.org/10.1103/physrevlett.110.060504. https://doi.org/10.1103/physrevlett.110.060504 [11] Marc Hein, Wolfgang Dür, Jens Eisert, Robert Raussendorf, M Nest, and H-J Briegel. Entanglement in graph states and its applications. arXiv preprint quant-ph/0602096, 2006. https://doi.org/10.48550/arXiv.quant-ph/0602096. https://doi.org/10.48550/arXiv.quant-ph/0602096 arXiv:quant-ph/0602096 [12] D. Schlingemann and R. F. Werner. Quantum error-correcting codes associated with graphs. Phys. Rev. A, 65:012308, Dec 2001. https://doi.org/10.1103/PhysRevA.65.012308. https://doi.org/10.1103/PhysRevA.65.012308 [13] Damian Markham and Barry C. Sanders. Graph states for quantum secret sharing. Phys. Rev. A, 78:042309, Oct 2008. https://doi.org/10.1103/PhysRevA.78.042309. https://doi.org/10.1103/PhysRevA.78.042309 [14] BA Bell, Damian Markham, DA Herrera-Martí, Anne Marin, WJ Wadsworth, JG Rarity, and MS Tame. Experimental demonstration of graph-state quantum secret sharing. Nature Communications, 5(1):1–12, 2014. https://doi.org/10.1038/ncomms6480. https://doi.org/10.1038/ncomms6480 [15] Xavier Coiteux-Roy, Owidiusz Makuta, Fionnuala Curran, Remigiusz Augusiak, and Marc-Olivier Renou. The genuinely multipartite nonlocality of graph states is model-dependent. npj Quantum Information, 11(1):1–6, 2025. https://doi.org/10.1038/s41534-025-01024-x. https://doi.org/10.1038/s41534-025-01024-x [16] Daniel E Browne and Terry Rudolph. Resource-efficient linear optical quantum computation.

Physical Review Letters, 95(1):010501, 2005. https://doi.org/10.1103/physrevlett.95.010501. https://doi.org/10.1103/physrevlett.95.010501 [17] Michael Varnava, Daniel E Browne, and Terry Rudolph. Loss tolerance in one-way quantum computation via counterfactual error correction.

Physical Review Letters, 97(12):120501, 2006. https://doi.org/10.1103/physrevlett.97.120501. https://doi.org/10.1103/physrevlett.97.120501 [18] Michael Varnava, Daniel E Browne, and Terry Rudolph. How good must single photon sources and detectors be for efficient linear optical quantum computation?

Physical Review Letters, 100(6):060502, 2008. https://doi.org/10.1103/physrevlett.100.060502. https://doi.org/10.1103/physrevlett.100.060502 [19] Ying Li, Peter C Humphreys, Gabriel J Mendoza, and Simon C Benjamin. Resource costs for fault-tolerant linear optical quantum computing. Physical Review X, 5(4):041007, 2015. https://doi.org/10.1103/physrevx.5.041007. https://doi.org/10.1103/physrevx.5.041007 [20] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. Fusion-based quantum computation. Nature Communications, 14(1):912, 2023. https://doi.org/10.1038/s41467-023-36493-1. https://doi.org/10.1038/s41467-023-36493-1 [21] Seok-Hyung Lee and Hyunseok Jeong. Graph-theoretical optimization of fusion-based graph state generation. Quantum, 7:1212, 2023. https://doi.org/10.22331/q-2023-12-20-1212. https://doi.org/10.22331/q-2023-12-20-1212 [22] Seungbeom Chin, Yong-Su Kim, and Sangmin Lee. Graph picture of linear quantum networks and entanglement. Quantum, 5:611, 2021. https://doi.org/10.22331/q-2021-12-23-611. https://doi.org/10.22331/q-2021-12-23-611 [23] Seungbeom Chin, Yong-Su Kim, and Marcin Karczewski. Shortcut to multipartite entanglement generation: A graph approach to boson subtractions. npj Quantum Information, 10(1):67, 2024. https://doi.org/10.1038/s41534-024-00845-6. https://doi.org/10.1038/s41534-024-00845-6 [24] Seungbeom Chin, Marcin Karczewski, and Yong-Su Kim. Heralded optical entanglement generation via the graph picture of linear quantum networks. Quantum, 8:1572, 2024. https://doi.org/10.22331/q-2024-12-18-1572. https://doi.org/10.22331/q-2024-12-18-1572 [25] Seungbeom Chin, Junghee Ryu, and Yong-Su Kim. Exponentially enhanced scheme for the heralded qudit greenberger-horne-zeilinger state in linear optics.

Physical Review Letters, 133(25):253601, 2024. https://doi.org/10.1103/physrevlett.133.253601. https://doi.org/10.1103/physrevlett.133.253601 [26] Paul Hilaire, Leonid Vidro, Hagai S Eisenberg, and Sophia E Economou. Near-deterministic hybrid generation of arbitrary photonic graph states using a single quantum emitter and linear optics. Quantum, 7:992, 2023. https://doi.org/10.22331/q-2023-04-27-992. https://doi.org/10.22331/q-2023-04-27-992 [27] Love A Pettersson, Anders S Sørensen, and Stefano Paesani. Deterministic generation of concatenated graph codes from quantum emitters. PRX Quantum, 6(1):010305, 2025. https://doi.org/10.1103/prxquantum.6.010305. https://doi.org/10.1103/prxquantum.6.010305 [28] Minhyeok Kang, Jaehee Kim, William J Munro, Seungbeom Chin, and Joonsuk Huh. Heralded linear optical generation of dicke states. New Journal of Physics, 28(5):054501, 2026. https://doi.org/10.1088/1367-2630/ae6135. https://doi.org/10.1088/1367-2630/ae6135 [29] Aditi Sen, Ujjwal Sen, Veronica Ahufinger, Hans J Briegel, Anna Sanpera, and Maciej Lewenstein. Quantum-information processing in disordered and complex quantum systems. Physical Review A—Atomic, Molecular, and Optical Physics, 74(6):062309, 2006. https://doi.org/10.1103/physreva.74.062309. https://doi.org/10.1103/physreva.74.062309 [30] Simon Anders, Hans J Briegel, and Wolfgang Dür. A variational method based on weighted graph states. New Journal of Physics, 9(10):361, 2007. https://doi.org/10.1088/1367-2630/9/10/361. https://doi.org/10.1088/1367-2630/9/10/361 [31] Matthias C Löbl, Love A Pettersson, Andrew Jena, Luca Dellantonio, Stefano Paesani, and Anders S Sørensen. Generating graph states with a single quantum emitter and the minimum number of fusions. Physical Review A, 111(5):052604, 2025. https://doi.org/10.1103/physreva.111.052604. https://doi.org/10.1103/physreva.111.052604 [32] Ido Schwartz, Dan Cogan, Emma R Schmidgall, Yaroslav Don, Liron Gantz, Oded Kenneth, Netanel H Lindner, and David Gershoni. Deterministic generation of a cluster state of entangled photons. Science, 354(6311):434–437, 2016. https://doi.org/10.1126/science.aah4758. https://doi.org/10.1126/science.aah4758 [33] Antonio Russo, Edwin Barnes, and Sophia E Economou. Generation of arbitrary all-photonic graph states from quantum emitters. New Journal of Physics, 21(5):055002, 2019. https://doi.org/10.1088/1367-2630/ab193d. https://doi.org/10.1088/1367-2630/ab193d [34] Philip Thomas, Leonardo Ruscio, Olivier Morin, and Gerhard Rempe. Efficient generation of entangled multiphoton graph states from a single atom. Nature, 608(7924):677–681, 2022. https://doi.org/10.1038/s41586-022-04987-5. https://doi.org/10.1038/s41586-022-04987-5 [35] Hassan Shapourian and Alireza Shabani. Modular architectures to deterministically generate graph states. Quantum, 7:935, 2023. https://doi.org/10.22331/q-2023-03-02-935. https://doi.org/10.22331/q-2023-03-02-935 [36] H Huet, PR Ramesh, SC Wein, N Coste, P Hilaire, N Somaschi, M Morassi, A Lemaı̂tre, Isabelle Sagnes, MF Doty, et al. Deterministic and reconfigurable graph state generation with a single solid-state quantum emitter. Nature communications, 16(1):4337, 2025. https://doi.org/10.1038/s41467-025-59693-3. https://doi.org/10.1038/s41467-025-59693-3 [37] David L Moehring, Peter Maunz, Steve Olmschenk, Kelly C Younge, Dzmitry N Matsukevich, L-M Duan, and Christopher Monroe. Entanglement of single-atom quantum bits at a distance. Nature, 449(7158):68–71, 2007. https://doi.org/10.1038/nature06118. https://doi.org/10.1038/nature06118 [38] Stephan Ritter, Christian Nölleke, Carolin Hahn, Andreas Reiserer, Andreas Neuzner, Manuel Uphoff, Martin Mücke, Eden Figueroa, Joerg Bochmann, and Gerhard Rempe. An elementary quantum network of single atoms in optical cavities. Nature, 484(7393):195–200, 2012. https://doi.org/10.1038/nature11023. https://doi.org/10.1038/nature11023 [39] Kae Nemoto, Michael Trupke, Simon J Devitt, Ashley M Stephens, Burkhard Scharfenberger, Kathrin Buczak, Tobias Nöbauer, Mark S Everitt, Jörg Schmiedmayer, and William J Munro. Photonic architecture for scalable quantum information processing in diamond. Physical Review X, 4(3):031022, 2014. https://doi.org/10.1103/physrevx.4.031022. https://doi.org/10.1103/physrevx.4.031022 [40] Stefan Scheel, William J Munro, Jens Eisert, Kae Nemoto, and Pieter Kok. Feed-forward and its role in conditional linear optical quantum dynamics. Physical Review A—Atomic, Molecular, and Optical Physics, 73(3):034301, 2006. https://doi.org/10.1103/physreva.73.034301. https://doi.org/10.1103/physreva.73.034301 [41] Mohammed Dakna, Tiemo Anhut, T Opatrnỳ, Ludwig Knöll, and D-G Welsch. Generating schrödinger-cat-like states by means of conditional measurements on a beam splitter. Physical Review A, 55(4):3184, 1997. https://doi.org/10.1103/physreva.55.3184. https://doi.org/10.1103/physreva.55.3184 [42] Alexei Ourjoumtsev, Rosa Tualle-Brouri, Julien Laurat, and Philippe Grangier. Generating optical schrodinger kittens for quantum information processing. Science, 312(5770):83–86, 2006. https://doi.org/10.1126/science.1122858. https://doi.org/10.1126/science.1122858 [43] A Zavatta, V Parigi, MS Kim, and M Bellini. Subtracting photons from arbitrary light fields: experimental test of coherent state invariance by single-photon annihilation. New Journal of Physics, 10(12):123006, 2008. https://doi.org/10.1088/1367-2630/10/12/123006. https://doi.org/10.1088/1367-2630/10/12/123006 [44] V Averchenko, C Jacquard, V Thiel, C Fabre, and N Treps. Multimode theory of single-photon subtraction. New Journal of Physics, 18(8):083042, 2016. https://doi.org/10.1088/1367-2630/18/8/083042. https://doi.org/10.1088/1367-2630/18/8/083042 [45] Sara Bartolucci, Patrick M Birchall, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Mihir Pant, Terry Rudolph, Jake Smith, Chris Sparrow, and Mihai D Vidrighin. Creation of entangled photonic states using linear optics. arXiv preprint arXiv:2106.13825, 2021. https://doi.org/10.48550/arXiv.2106.13825. https://doi.org/10.48550/arXiv.2106.13825 arXiv:2106.13825 [46] Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator.

Physical Review Letters, 73(1):58, 1994. https://doi.org/10.1103/physrevlett.73.58. https://doi.org/10.1103/physrevlett.73.58 [47] Yuan Liang Lim and Almut Beige. Generalized Hong–Ou–Mandel experiments with bosons and fermions. New Journal of Physics, 7(1):155, 2005. https://doi.org/10.1088/1367-2630/7/1/155. https://doi.org/10.1088/1367-2630/7/1/155Cited byCould not fetch Crossref cited-by data during last attempt 2026-08-13 11:33:12: Could not fetch cited-by data for 10.22331/q-2026-08-13-2189 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-08-13 11:33:12: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractGraph 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. 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 to generating complex photonic graph states for efficient quantum information processing.Featured image: Linear quantum graph representation of the heralded scheme for generating caterpillar graph statesPopular summaryPhotonic quantum technologies rely on the ability to create large entangled states of light. Among the most useful are graph states, which support applications ranging from quantum computing and communication to error correction and tests of quantum nonlocality. However, producing large photonic graph states remains challenging, because the conventional approach repeatedly combines smaller entangled states using probabilistic fusion operations, requiring many photons and rapidly reducing the overall success probability. In this work, we develop a more efficient way to generate an important family of graph states known as caterpillar graph states. These states are useful building blocks for constructing larger lattice-like resource states for measurement-based quantum computing and for studying strong forms of multipartite nonlocality. Our method uses only single-photon sources, linear-optical elements, and heralding measurements, so successful preparation is achieved without destroying the generated state. The design is based on the linear quantum graph (LQG) picture, which translates bosonic state-generation problems into graph-theoretic structures. By identifying two key structures, called path circuit graphs and primate circuit graphs, we derive optical circuits that generate arbitrary caterpillar graph states more efficiently than conventional fusion-based constructions. For caterpillar states of length $l\geq 3$, the scheme uses $l-2$ fewer photons and improves the success probability by a factor of $2^{l-2}$. Our approach demonstrates the advantage of employing a broader set of measurement gates beyond conventional fusion gates. More broadly, it demonstrates how the LQG framework can guide the construction of efficient optical circuits for increasingly complex quantum resource states.► BibTeX data@article{Chin2026efficientgraphstate, doi = {10.22331/q-2026-08-13-2189}, url = {https://doi.org/10.22331/q-2026-08-13-2189}, title = {Efficient {G}raph {S}tate {G}eneration in {L}inear {O}ptics}, author = {Chin, Seungbeom and Munro, William John}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2189}, month = aug, year = {2026} }► References [1] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409(6816):46–52, 2001. https://doi.org/10.1038/35051009. https://doi.org/10.1038/35051009 [2] Pieter Kok, William J Munro, Kae Nemoto, Timothy C Ralph, Jonathan P Dowling, and Gerard J Milburn. Linear optical quantum computing with photonic qubits. Reviews of Modern Physics, 79(1):135–174, 2007. https://doi.org/10.1103/revmodphys.79.135. https://doi.org/10.1103/revmodphys.79.135 [3] Jeremy L. O'Brien. Optical quantum computing. Science, 318(5856):1567–1570, 2007. https://doi.org/10.1126/science.1142892. https://doi.org/10.1126/science.1142892 [4] Koji Azuma, Kiyoshi Tamaki, and Hoi-Kwong Lo. All-photonic quantum repeaters. Nature Communications, 6(1):6787, 2015. https://doi.org/10.1038/ncomms7787. https://doi.org/10.1038/ncomms7787 [5] Koji Azuma, Kiyoshi Tamaki, and William J Munro. All-photonic intercity quantum key distribution. Nature Communications, 6(1):10171, 2015. https://doi.org/10.1038/ncomms10171. https://doi.org/10.1038/ncomms10171 [6] Robert Raussendorf and Hans J Briegel. A one-way quantum computer.

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Efficient Graph State Generation in Linear Optics

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