Entanglement-efficient bipartite-distributed quantum computing

Jun-Yi Wu; Kosuke Matsui; Tim Forrer; Akihito Soeda; Pablo Andrés-Martínez; Daniel Mills; Luciana Henaut; Mio Murao

doi:10.22331/q-2023-12-05-1196

Entanglement-efficient bipartite-distributed quantum computing

Jun-Yi Wu^1,2, Kosuke Matsui³, Tim Forrer³, Akihito Soeda^3,4,5, Pablo Andrés-Martínez⁶, Daniel Mills⁶, Luciana Henaut⁶, and Mio Murao³

¹Department of Physics and Center for Advanced Quantum Computing, Tamkang University, 151 Yingzhuan Rd., New Taipei City 25137, Taiwan, ROC
²Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan, ROC
³The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
⁴Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁵Department of Informatics, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁶Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK

Published:	2023-12-05, volume 7, page 1196
Eprint:	arXiv:2212.12688v3
Doi:	https://doi.org/10.22331/q-2023-12-05-1196
Citation:	Quantum 7, 1196 (2023).

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Abstract

In noisy intermediate-scale quantum computing, the limited scalability of a single quantum processing unit (QPU) can be extended through distributed quantum computing (DQC), in which one can implement global operations over two QPUs by entanglement-assisted local operations and classical communication. To facilitate this type of DQC in experiments, we need an entanglement-efficient protocol. To this end, we extend the protocol in [Eisert et. al., PRA, 62:052317(2000)] implementing each nonlocal controlled-unitary gate locally with one maximally entangled pair to a packing protocol, which can pack multiple nonlocal controlled-unitary gates locally using one maximally entangled pair. In particular, two types of packing processes are introduced as the building blocks, namely the distributing processes and embedding processes. Each distributing process distributes corresponding gates locally with one entangled pair. The efficiency of entanglement is then enhanced by embedding processes, which merge two non-sequential distributing processes and hence save the entanglement cost. We show that the structure of distributability and embeddability of a quantum circuit can be fully represented by the corresponding packing graphs and conflict graphs. Based on these graphs, we derive heuristic algorithms for finding an entanglement-efficient packing of distributing processes for a given quantum circuit to be implemented by two parties. These algorithms can determine the required number of local auxiliary qubits in the DQC. We apply these algorithms for bipartite DQC of unitary coupled-cluster circuits and find a significant reduction of entanglement cost through embeddings. This method can determine a constructive upper bound on the entanglement cost for the DQC of quantum circuits.

Featured image: Packing and conflict edges of a circuit. The red lines induce the packing edges, while the blue and green lines induce intrinsic and extrinsic conflict edges, respectively. The packing graph and conflict graph induced by these edges fully describe the distributability, embeddability, and incompatibility of a quantum circuit, upon which our embedding-enhanced-distributing protocol is built.

Popular summary

The scalability of quantum computing on a single quantum processing unit is physically limited. To break through this bottleneck, one needs to distribute quantum computing over multiple quantum processing units. It is therefore essential to establish an entanglement-economical architecture to release the requirement on the entanglement generation rate of a quantum network.

In our paper, an entanglement-efficient architecture for bipartite distributed quantum computing is established based on a decomposition of a quantum circuit into a set of distributable blocks. In each block, one maximally-entangled state is consumed to distribute nonlocal gates with local operations and classical communication. In previous protocols, a distributing process ends with single qubit gates. To improve entanglement efficiency, we introduce embedding processes to merge two non-sequential distributing processes into one, saving on the amount of entanglement needed. Such an embedding-enhanced-distributing structure of a quantum circuit can be fully described by a packing graph made up of vertices representing gate nodes that could be merged together.

The vertices of a packing graph are the candidates for distributing processes. However, there may be conflicts between two embeddings that prevent us from implementing them simultaneously. We have to remove some embeddings to solve the conflicts, which leads to a split of vertices in a packing graph. Such a conflict can be fully described by a conflict graph.

The packing graph and conflict graph of a circuit fully describe its distributability, embeddability, and incompatibility. To determine the best way to distribute a circuit, we develop a packing algorithm that takes into account both the minimum vertex covering of the packing graph and its corresponding conflict graph to find a distributing strategy that is both entanglement-efficient and conflict-free.

Overall, our protocol can be summarized as an “embedding-enhanced-distributing'' protocol, which is based on two types of entanglement-assisted quantum operation, namely distributing processes and embedding processes. Compared to previous protocols, it significantly reduces the amount of entanglement required for distributed quantum computing, making it more practical for real-world applications. It can be employed to determine a tighter constructive upper bound on the entanglement cost for a unitary being decomposed into local operations and classical communication. The protocol can be extended to multipartite systems and adapted to network topology for distributed quantum computing over the quantum internet. The “embedding-enhanced-distributing'' protocol can therefore facilitate large-scale quantum computing in a quantum network of QPUs.

► BibTeX data

@article{Wu2023entanglement,
  doi = {10.22331/q-2023-12-05-1196},
  url = {https://doi.org/10.22331/q-2023-12-05-1196},
  title = {Entanglement-efficient bipartite-distributed quantum computing},
  author = {Wu, Jun-Yi and Matsui, Kosuke and Forrer, Tim and Soeda, Akihito and Andr{\'{e}}s-Mart{\'{i}}nez, Pablo and Mills, Daniel and Henaut, Luciana and Murao, Mio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1196},
  month = dec,
  year = {2023}
}

► References

[1] J. Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2018 2: 79. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[2] N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 2018 3 (3): 030503. 10.1088/2058-9565/aab822.
https://doi.org/10.1088/2058-9565/aab822

[3] A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta. Validating quantum computers using randomized model circuits. Physical Review A, 2019 100: 032328. 10.1103/PhysRevA.100.032328.
https://doi.org/10.1103/PhysRevA.100.032328

[4] S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral. Proposal for teleportation of an atomic state via cavity decay. Physical Review Letters, 1999 83: 5158–5161. 10.1103/PhysRevLett.83.5158.
https://doi.org/10.1103/PhysRevLett.83.5158

[5] C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller. Creation of entangled states of distant atoms by interference. Physical Review A, 1999 59: 1025–1033. 10.1103/PhysRevA.59.1025.
https://doi.org/10.1103/PhysRevA.59.1025

[6] D. E. Browne, M. B. Plenio, and S. F. Huelga. Robust creation of entanglement between ions in spatially separate cavities. Physical Review Letters, 2003 91: 067901. 10.1103/PhysRevLett.91.067901.
https://doi.org/10.1103/PhysRevLett.91.067901

[7] L.-M. Duan, B. Blinov, D. Moehring, and C. Monroe. Scalable trapped ion quantum computation with a probabilistic ion-photon mapping. Quantum Information and Computation, 2004 4: 165–173. 10.48550/arXiv.quant-ph/0401020.
https://doi.org/10.48550/arXiv.quant-ph/0401020
arXiv:quant-ph/0401020

[8] Y. L. Lim, A. Beige, and L. C. Kwek. Repeat-until-success linear optics distributed quantum computing. Physical Review Letters, 2005 95: 030505. 10.1103/PhysRevLett.95.030505.
https://doi.org/10.1103/PhysRevLett.95.030505

[9] L.-M. Duan, M. J. Madsen, D. L. Moehring, P. Maunz, R. N. Kohn, and C. Monroe. Probabilistic quantum gates between remote atoms through interference of optical frequency qubits. Physical Review A, 2006 73: 062324. 10.1103/PhysRevA.73.062324.
https://doi.org/10.1103/PhysRevA.73.062324

[10] Z.-q. Yin, W. L. Yang, L. Sun, and L. M. Duan. Quantum network of superconducting qubits through an optomechanical interface. Physical Review A, 2015 91: 012333. 10.1103/PhysRevA.91.012333.
https://doi.org/10.1103/PhysRevA.91.012333

[11] K. Koshino, K. Inomata, Z. R. Lin, Y. Tokunaga, T. Yamamoto, and Y. Nakamura. Theory of deterministic entanglement generation between remote superconducting atoms. Physical Review Applied, 2017 7: 064006. 10.1103/PhysRevApplied.7.064006.
https://doi.org/10.1103/PhysRevApplied.7.064006

[12] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe. Entanglement of single-atom quantum bits at a distance. Nature, 2007 449 (7158): 68–71. 10.1038/nature06118.
https://doi.org/10.1038/nature06118

[13] L. Slodička, G. Hétet, N. Röck, P. Schindler, M. Hennrich, and R. Blatt. Atom-atom entanglement by single-photon detection. Physical Review Letters, 2013 110 (8): 083603. 10.1103/physrevlett.110.083603.
https://doi.org/10.1103/physrevlett.110.083603

[14] S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, M. Neuzner, Andreas andUphoff, M. Mucke, E. Figueroa, J. Bochmann, , and G. Rempe. An elementary quantum network of single atoms in optical cavities. Nature, 2012 484: 195–200. 10.1038/nature11023.
https://doi.org/10.1038/nature11023

[15] J. Hofmann, M. Krug, N. Ortegel, L. Gérard, M. Weber, W. Rosenfeld, and H. Weinfurter. Heralded entanglement between widely separated atoms. Science, 2012 337 (6090): 72–75. 10.1126/science.1221856.
https://doi.org/10.1126/science.1221856

[16] H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson. Heralded entanglement between solid-state qubits separated by three metres. Nature, 2013 497 (7447): 86–90. 10.1038/nature12016.
https://doi.org/10.1038/nature12016

[17] A. Delteil, Z. Sun, W. bo Gao, E. Togan, S. Faelt, and A. Imamoğlu. Generation of heralded entanglement between distant hole spins. Nature Physics, 2015 12 (3): 218–223. 10.1038/nphys3605.
https://doi.org/10.1038/nphys3605

[18] R. Stockill, M. J. Stanley, L. Huthmacher, E. Clarke, M. Hugues, A. J. Miller, C. Matthiesen, C. Le Gall, and M. Atatüre. Phase-tuned entangled state generation between distant spin qubits. Physical Review Letters, 2017 119: 010503. 10.1103/PhysRevLett.119.010503.
https://doi.org/10.1103/PhysRevLett.119.010503

[19] A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller, S. O. Mundhada, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret. Robust concurrent remote entanglement between two superconducting qubits. Physical Review X, 2016 6: 031036. 10.1103/PhysRevX.6.031036.
https://doi.org/10.1103/PhysRevX.6.031036

[20] L. J. Stephenson, D. P. Nadlinger, B. C. Nichol, S. An, P. Drmota, T. G. Ballance, K. Thirumalai, J. F. Goodwin, D. M. Lucas, and C. J. Ballance. High-rate, high-fidelity entanglement of qubits across an elementary quantum network. Physical Review Letters, 2020 124: 110501. 10.1103/PhysRevLett.124.110501.
https://doi.org/10.1103/PhysRevLett.124.110501

[21] D. Hucul, I. V. Inlek, G. Vittorini, C. Crocker, S. Debnath, S. M. Clark, and C. Monroe. Modular entanglement of atomic qubits using photons and phonons. Nature Physics, 2014 11 (1): 37–42. 10.1038/nphys3150.
https://doi.org/10.1038/nphys3150

[22] H. J. Kimble. The quantum internet. Nature, 453: 1023–1030, 2008. 10.1038/nature07127.
https://doi.org/10.1038/nature07127

[23] A. Soeda, Y. Kinjo, P. S. Turner, and M. Murao. Quantum computation over the butterfly network. Physical Review A, 2011 84: 012333. 10.1103/PhysRevA.84.012333.
https://doi.org/10.1103/PhysRevA.84.012333

[24] D. Gottesman and I. L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402: 390–393, 1999. 10.1038/46503.
https://doi.org/10.1038/46503

[25] X. Zhou, D. W. Leung, and I. L. Chuang. Methodology for quantum logic gate construction. Physical Review A, 2000 62: 052316. 10.1103/PhysRevA.62.052316.
https://doi.org/10.1103/PhysRevA.62.052316

[26] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio. Optimal local implementation of nonlocal quantum gates. Physical Review A, 2000 62: 052317. 10.1103/PhysRevA.62.052317.
https://doi.org/10.1103/PhysRevA.62.052317

[27] S. F. Huelga, J. A. Vaccaro, A. Chefles, and M. B. Plenio. Quantum remote control: Teleportation of unitary operations. Physical Review A, 2001 63: 042303. 10.1103/PhysRevA.63.042303.
https://doi.org/10.1103/PhysRevA.63.042303

[28] S. F. Huelga, M. B. Plenio, and J. A. Vaccaro. Remote control of restricted sets of operations: Teleportation of angles. Physical Review A, 2002 65: 042316. 10.1103/PhysRevA.65.042316.
https://doi.org/10.1103/PhysRevA.65.042316

[29] L. Jiang, J. M. Taylor, A. S. Sørensen, and M. D. Lukin. Distributed quantum computation based on small quantum registers. Physical Review A, 2007 76: 062323. 10.1103/PhysRevA.76.062323.
https://doi.org/10.1103/PhysRevA.76.062323

[30] R. V. Meter, W. Munro, K. Nemoto, and K. M. Itoh. Arithmetic on a distributed-memory quantum multicomputer. ACM Journal on Emerging Technologies in Computing Systems (JETC), 3 (4): 1–23, 2008. 10.1145/1324177.1324179.
https://doi.org/10.1145/1324177.1324179

[31] M. Caleffi, M. Amoretti, D. Ferrari, D. Cuomo, J. Illiano, A. Manzalini, and A. S. Cacciapuoti. Distributed quantum computing: a survey, 2022. 10.48550/arXiv.2212.10609.
https://doi.org/10.48550/arXiv.2212.10609

[32] K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf. Deterministic teleportation of a quantum gate between two logical qubits. Nature, 2018 561 (7723): 368–373. 10.1038/s41586-018-0470-y.
https://doi.org/10.1038/s41586-018-0470-y

[33] Y. Wan, D. Kienzler, S. D. Erickson, K. H. Mayer, T. R. Tan, J. J. Wu, H. M. Vasconcelos, S. Glancy, E. Knill, D. J. Wineland, A. C. Wilson, and D. Leibfried. Quantum gate teleportation between separated qubits in a trapped-ion processor. Science, 364 (6443): 875–878, 2019. 10.1126/science.aaw9415.
https://doi.org/10.1126/science.aaw9415

[34] P. Andrés-Martínez and C. Heunen. Automated distribution of quantum circuits via hypergraph partitioning. Physical Review A, 2019 100: 032308. 10.1103/PhysRevA.100.032308.
https://doi.org/10.1103/PhysRevA.100.032308

[35] R. G. Sundaram, H. Gupta, and C. R. Ramakrishnan. Efficient Distribution of Quantum Circuits. In S. Gilbert, editor, 35th International Symposium on Distributed Computing (DISC 2021), volume 209 of Leibniz International Proceedings in Informatics (LIPIcs), pages 41:1–41:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. ISBN 978-3-95977-210-5. 10.4230/LIPIcs.DISC.2021.41.
https://doi.org/10.4230/LIPIcs.DISC.2021.41

[36] D. Cuomo, M. Caleffi, K. Krsulich, F. Tramonto, G. Agliardi, E. Prati, and A. S. Cacciapuoti. Optimized compiler for distributed quantum computing. ACM Transactions on Quantum Computing, 2023 4 (2). ISSN 2643-6809. 10.1145/3579367.
https://doi.org/10.1145/3579367

[37] R. G. Sundaram, H. Gupta, and C. R. Ramakrishnan. Distribution of quantum circuits over general quantum networks. In 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 415–425, 2022 Los Alamitos, CA, USA. IEEE Computer Society. 10.1109/QCE53715.2022.00063.
https://doi.org/10.1109/QCE53715.2022.00063

[38] D. Stahlke and R. B. Griffiths. Entanglement requirements for implementing bipartite unitary operations. Physical Review A, 2011 84: 032316. 10.1103/PhysRevA.84.032316.
https://doi.org/10.1103/PhysRevA.84.032316

[39] P. Andres-Martinez, T. Forrer, D. Mills, J.-Y. Wu, L. Henaut, K. Yamamoto, M. Murao, and R. Duncan. Distributing circuits over heterogeneous, modular quantum computing network architectures. 10.48550/arXiv.2305.14148.
https://doi.org/10.48550/arXiv.2305.14148

[40] J. M. Baker, C. Duckering, A. Hoover, and F. T. Chong. Time-sliced quantum circuit partitioning for modular architectures. In Proceedings of the 17th ACM International Conference on Computing Frontiers, CF '20, page 98–107, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450379564. 10.1145/3387902.3392617.
https://doi.org/10.1145/3387902.3392617

[41] S. DiAdamo, M. Ghibaudi, and J. Cruise. Distributed quantum computing and network control for accelerated vqe. IEEE Transactions on Quantum Engineering, 2: 1–21, 2021. 10.1109/TQE.2021.3057908.
https://doi.org/10.1109/TQE.2021.3057908

[42] D. Ferrari, A. S. Cacciapuoti, M. Amoretti, and M. Caleffi. Compiler design for distributed quantum computing. IEEE Transactions on Quantum Engineering, 2: 1–20, 2021. 10.1109/TQE.2021.3053921.
https://doi.org/10.1109/TQE.2021.3053921

[43] A. Ovide, S. Rodrigo, M. Bandic, H. Van Someren, S. Feld, S. Abadal, E. Alarcon, and C. G. Almudever. Mapping quantum algorithms to multi-core quantum computing architectures. In 2023 IEEE International Symposium on Circuits and Systems (ISCAS), pages 1–5, 2023. 10.1109/ISCAS46773.2023.10181589.
https://doi.org/10.1109/ISCAS46773.2023.10181589

[44] D. Ferrari, S. Carretta, and M. Amoretti. A modular quantum compilation framework for distributed quantum computing. IEEE Transactions on Quantum Engineering, 2023 4 (01): 1–13. ISSN 2689-1808. 10.1109/TQE.2023.3303935.
https://doi.org/10.1109/TQE.2023.3303935

[45] A. G. Taube and R. J. Bartlett. New perspectives on unitary coupled-cluster theory. International Journal of Quantum Chemistry, 106 (15): 3393–3401, 2006. 10.1002/qua.21198.
https://doi.org/10.1002/qua.21198

[46] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 2014 5 (1). 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213

[47] D. Jonathan and M. B. Plenio. Entanglement-assisted local manipulation of pure quantum states. Physical Review Letters, 1999 83: 3566–3569. 10.1103/PhysRevLett.83.3566.
https://doi.org/10.1103/PhysRevLett.83.3566

[48] A. Yimsiriwattana and S. J. Lomonaco. Generalized GHZ states and distributed quantum computing, 2004. 10.48550/ARXIV.QUANT-PH/0402148.
https://doi.org/10.48550/ARXIV.QUANT-PH/0402148

[49] T. Uno. Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In Algorithms and Computation, pages 92–101. Springer Berlin Heidelberg. 10.1007/3-540-63890-3_11.
https://doi.org/10.1007/3-540-63890-3_11

[50] M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Mathematical Sciences Series. Freeman, 1979. ISBN 9780716710448.

[51] A. Björklund, T. Husfeldt, and M. Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39 (2): 546–563, 2009. 10.1137/070683933.
https://doi.org/10.1137/070683933

[52] R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Heidelberg, 2005 2005 edition. 10.1007/978-3-662-53622-3.
https://doi.org/10.1007/978-3-662-53622-3

[53] J. W. Moon and L. Moser. On cliques in graphs. Israel journal of Mathematics, 3: 23–28, 1965. 10.1007/BF02760024.
https://doi.org/10.1007/BF02760024

[54] J. E. Hopcroft and R. M. Karp. An $n^{5/2}$ algorithm for maximum matchings in bipartite graphs. 2 (4): 225–231. 10.1137/0202019.
https://doi.org/10.1137/0202019

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[20] Mohamed Shaban, Muhammad Ismail, and Walid Saad, "SPARQ: Efficient Entanglement Distribution and Routing in Space–Air–Ground Quantum Networks", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[21] Iago Fernández Llovo, Guillermo Díaz-Camacho, Natalia Costas Lago, and Andrés Gómez Tato, "Network-Assisted Collective Operations for Efficient Distributed Quantum Computing", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[22] James Ang, Gabriella Carini, Yanzhu Chen, Isaac Chuang, Michael Demarco, Sophia Economou, Alec Eickbusch, Andrei Faraon, Kai-Mei Fu, Steven Girvin, Michael Hatridge, Andrew Houck, Paul Hilaire, Kevin Krsulich, Ang Li, Chenxu Liu, Yuan Liu, Margaret Martonosi, David McKay, Jim Misewich, Mark Ritter, Robert Schoelkopf, Samuel Stein, Sara Sussman, Hong Tang, Wei Tang, Teague Tomesh, Norm Tubman, Chen Wang, Nathan Wiebe, Yongxin Yao, Dillon Yost, and Yiyu Zhou, "ARQUIN: Architectures for Multinode Superconducting Quantum Computers", ACM Transactions on Quantum Computing 5 3, 1 (2024).

[23] Farah A. Abdelrazek, Mohamed Swillam, and Thomas J. G. Mikhail, Frontiers in Optics + Laser Science 2024 (FiO, LS) JW4A.15 (2024) ISBN:978-1-957171-95-1.

[24] Moritz Schmidt, Abhoy Kole, Leon Wichette, Rolf Drechsler, Frank Kirchner, and Elie Mounzer, "Exploration of Design Alternatives for Reducing Idle Time in Shor's Algorithm: A Study on Monolithic and Distributed Quantum Systems", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[25] Guangwu Hou, Changhua Zhu, and Yi Sun, "Multi-objective simulated annealing-based quantum circuit cutting for distributed quantum computation", Physica Scripta 99 11, 115108 (2024).

[26] Eneet Kaur, Shahrooz Pouryousef, Hassan Shapourian, Jiapeng Zhao, Michael Kilzer, Ramana Kompella, and Reza Nejabati, "Optimized Quantum Circuit Partitioning Across Multiple Quantum Processors", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[27] Felix Burt, Kuan-Cheng Chen, and Kin K. Leung, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 173 (2024) ISBN:979-8-3315-4137-8.

[28] Riccardo Mengoni, Walter Nadalin, Mathys Rennela, Jimmy Rotureau, Tom Darras, Julien Laurat, Eleni Diamanti, and Ioannis Lavdas, "Efficient gate reordering for Distributed Quantum Compiling in data centers", EPJ Quantum Technology 13 1, 65 (2026).

[29] Longshan Xu, Edwin Hsing-Mean Sha, Xiulin Cui, and Qingfeng Zhuge, Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis 774 (2025) ISBN:9798400714665.

[30] Thales Gottardello Marrafon and Akihito Soeda, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 476 (2025) ISBN:979-8-3315-5736-2.

[31] Wen-Qiang Liu and Hai-Rui Wei, "Quantum gate teleportation with the superposition of causal order", Physical Review Applied 23 1, 014064 (2025).

[32] George Biswas, Shao-Hua Hu, Jun-Yi Wu, Debasish Biswas, and Anindya Biswas, "Fidelity and entanglement of random bipartite pure states: insights and applications", Physica Scripta 99 7, 075103 (2024).

[33] Felix Burt, Kuan-Cheng Chen, and Kin K. Leung, "A Multilevel Framework for Partitioning Quantum Circuits", arXiv:2503.19082, (2025).

[34] Felix Burt, Kuan-Cheng Chen, and Kin Leung, "Generalised Circuit Partitioning for Distributed Quantum Computing", arXiv:2408.01424, (2024).

[35] Seiseki Akibue, Jisho Miyazaki, and Hiroyuki Osaka, "Optimizing Entanglement Manipulation via Algebraic-Geometric Decompositions and Semidefinite Programming Hierarchies", arXiv:2501.17394, (2025).

[36] Dmitry Filippov, Peter Yang, and Prakash Murali, "Architecting Distributed Quantum Computers: Design Insights from Resource Estimation", arXiv:2508.19160, (2025).

[37] Raúl Santos, Lorenzo Buffoni, and Yasser Omar, "Enhancing Quantum Annealing via entanglement distribution", arXiv:2212.02465, (2022).

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