{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T11:48:23Z","timestamp":1774352903383,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T00:00:00Z","timestamp":1772841600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T00:00:00Z","timestamp":1772841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002551","name":"Seoul National University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100002551","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Modular and networked quantum architectures can scale beyond the qubit count of a single device, but executing a circuit across modules requires implementing non-local two-qubit gates using shared entanglement (ebits) and classical communication, making ebit cost a central resource in distributed execution. The resulting distributed quantum circuit (DQC) problem is combinatorial, motivating prior heuristic approaches such as hypergraph partitioning. In this work, we decouple module allocation from distribution. For a fixed module allocation (i.e., assignment of each qubit to a specific quantum processing unit), we formulate the remaining distribution layer as an exact binary integer programming (BIP). This yields solver optimal distributions for the fixed allocation subproblem and can be used as a post-processing step on top of any existing allocation method. We derive compact BIP formulations for four or more modules and a tighter specialization for three modules. Across a diverse benchmark suite, BIP post-processing reduces ebit cost by up to 20% for random circuits and by more than an order of magnitude for some arithmetic circuits. While the method incurs offline classical overhead, it is amortized when circuits are executed repeatedly.<\/jats:p>","DOI":"10.1007\/s11128-026-05112-5","type":"journal-article","created":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T14:26:55Z","timestamp":1772893615000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Binary integer programming for optimizing ebit cost in distributed quantum circuits with fixed module allocation"],"prefix":"10.1007","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-2933-6989","authenticated-orcid":false,"given":"Hyunho","family":"Cha","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6804-980X","authenticated-orcid":false,"given":"Jungwoo","family":"Lee","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,7]]},"reference":[{"issue":"7779","key":"5112_CR1","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","volume":"574","author":"F Arute","year":"2019","unstructured":"Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., et al.: Quantum supremacy using a programmable superconducting processor. Nature 574(7779), 505\u2013510 (2019)","journal-title":"Nature"},{"issue":"6412","key":"5112_CR2","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1126\/science.aar3106","volume":"362","author":"S Bravyi","year":"2018","unstructured":"Bravyi, S., Gosset, D., K\u00f6nig, R.: Quantum advantage with shallow circuits. Science 362(6412), 308\u2013311 (2018)","journal-title":"Science"},{"issue":"6385","key":"5112_CR3","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1126\/science.aao4309","volume":"360","author":"C Neill","year":"2018","unstructured":"Neill, C., Roushan, P., Kechedzhi, K., Boixo, S., Isakov, S.V., Smelyanskiy, V., et al.: Superconducting qubits: current state of play. Science 360(6385), 195\u2013199 (2018)","journal-title":"Science"},{"issue":"1","key":"5112_CR4","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1146\/annurev-conmatphys-031119-050605","volume":"11","author":"M Kjaergaard","year":"2020","unstructured":"Kjaergaard, M., Schwartz, M.E., Braum\u00fcller, J., Krantz, P., Wang, J.I.J., Gustavsson, S., et al.: Superconducting qubits: current state of play. Annu. Rev. Condens. Matter Phys. 11(1), 369\u2013395 (2020)","journal-title":"Annu. Rev. Condens. Matter Phys."},{"issue":"2","key":"5112_CR5","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1103\/RevModPhys.87.307","volume":"87","author":"BM Terhal","year":"2015","unstructured":"Terhal, B.M.: Quantum error correction for quantum memories. Rev. Mod. Phys. 87(2), 307\u2013346 (2015)","journal-title":"Rev. Mod. Phys."},{"issue":"3","key":"5112_CR6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.86.032324","volume":"86","author":"AG Fowler","year":"2012","unstructured":"Fowler, A.G., Mariantoni, M., Martinis, J.M., Cleland, A.N.: Surface codes: towards practical large-scale quantum computation. Phys. Rev. A-Atomic, Molecular, and Optical Phys. 86(3), 032324 (2012)","journal-title":"Phys. Rev. A-Atomic, Molecular, and Optical Phys."},{"issue":"3","key":"5112_CR7","volume":"6","author":"PJ O\u2019Malley","year":"2016","unstructured":"O\u2019Malley, P.J., Babbush, R., Kivlichan, I.D., Romero, J., McClean, J.R., Barends, R., et al.: Scalable quantum simulation of molecular energies. Phys. Rev. X 6(3), 031007 (2016)","journal-title":"Phys. Rev. X"},{"key":"5112_CR8","doi-asserted-by":"publisher","first-page":"392","DOI":"10.22331\/q-2021-02-04-392","volume":"5","author":"JE Bourassa","year":"2021","unstructured":"Bourassa, J.E., Alexander, R.N., Vasmer, M., Patil, A., Tzitrin, I., Matsuura, T., et al.: Blueprint for a scalable photonic fault-tolerant quantum computer. Quantum 5, 392 (2021)","journal-title":"Quantum"},{"issue":"6","key":"5112_CR9","doi-asserted-by":"publisher","first-page":"4249","DOI":"10.1103\/PhysRevA.59.4249","volume":"59","author":"JI Cirac","year":"1999","unstructured":"Cirac, J.I., Ekert, A., Huelga, S.F., Macchiavello, C.: Distributed quantum computation over noisy channels. Phys. Rev. A 59(6), 4249 (1999)","journal-title":"Phys. Rev. A"},{"issue":"1","key":"5112_CR10","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1109\/MNET.001.1900092","volume":"34","author":"AS Cacciapuoti","year":"2019","unstructured":"Cacciapuoti, A.S., Caleffi, M., Tafuri, F., Cataliotti, F.S., Gherardini, S., Bianchi, G.: Quantum internet: networking challenges in distributed quantum computing. IEEE Network 34(1), 137\u2013143 (2019)","journal-title":"IEEE Network"},{"issue":"2","key":"5112_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3579367","volume":"4","author":"D Cuomo","year":"2023","unstructured":"Cuomo, D., Caleffi, M., Krsulich, K., Tramonto, F., Agliardi, G., Prati, E., et al.: Optimized compiler for distributed quantum computing. ACM Trans. Quantum Comput. 4(2), 1\u201329 (2023)","journal-title":"ACM Trans. Quantum Comput."},{"key":"5112_CR12","doi-asserted-by":"crossref","unstructured":"Yimsiriwattana, A., Lomonaco, Jr. S. J.: Generalized GHZ states and distributed quantum computing. arXiv preprint arXiv:quant-ph\/0402148. (2004)","DOI":"10.1090\/conm\/381\/07096"},{"issue":"5","key":"5112_CR13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.62.052317","volume":"62","author":"J Eisert","year":"2000","unstructured":"Eisert, J., Jacobs, K., Papadopoulos, P., Plenio, M.B.: Optimal local implementation of nonlocal quantum gates. Phys. Rev. A 62(5), 052317 (2000)","journal-title":"Phys. Rev. A"},{"key":"5112_CR14","volume-title":"Quantum Computation and Quantum Information","author":"MA Nielsen","year":"2010","unstructured":"Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2010)"},{"issue":"1","key":"5112_CR15","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/11\/1\/013006","volume":"11","author":"D Sych","year":"2009","unstructured":"Sych, D., Leuchs, G.: A complete basis of generalized Bell states. New J. Phys. 11(1), 013006 (2009)","journal-title":"New J. Phys."},{"issue":"1","key":"5112_CR16","doi-asserted-by":"publisher","first-page":"14641","DOI":"10.1038\/s41598-018-32928-8","volume":"8","author":"F Zaman","year":"2018","unstructured":"Zaman, F., Jeong, Y., Shin, H.: Counterfactual Bell-state analysis. Sci. Rep. 8(1), 14641 (2018)","journal-title":"Sci. Rep."},{"key":"5112_CR17","unstructured":"Sundaram, R., Gupta, H., Ramakrishnan, C.: Efficient distribution of quantum circuits. In: 35th International Symposium on Distributed Computing (DISC 2021). Schloss Dagstuhl-Leibniz-Zentrum f\u00fcr Informatik; p. 41\u20131 (2021)"},{"issue":"3","key":"5112_CR18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.100.032308","volume":"100","author":"P Andres-Martinez","year":"2019","unstructured":"Andres-Martinez, P., Heunen, C.: Automated distribution of quantum circuits via hypergraph partitioning. Phys. Rev. A 100(3), 032308 (2019)","journal-title":"Phys. Rev. A"},{"key":"5112_CR19","doi-asserted-by":"publisher","first-page":"1196","DOI":"10.22331\/q-2023-12-05-1196","volume":"7","author":"JY Wu","year":"2023","unstructured":"Wu, J.Y., Matsui, K., Forrer, T., Soeda, A., Andr\u00e9s-Mart\u00ednez, P., Mills, D., et al.: Entanglement-efficient bipartite-distributed quantum computing. Quantum 7, 1196 (2023)","journal-title":"Quantum"},{"issue":"4","key":"5112_CR20","doi-asserted-by":"publisher","DOI":"10.1088\/2058-9565\/ad6734","volume":"9","author":"P Andres-Martinez","year":"2024","unstructured":"Andres-Martinez, P., Forrer, T., Mills, D., Wu, J.Y., Henaut, L., Yamamoto, K., et al.: Distributing circuits over heterogeneous, modular quantum computing network architectures. Quantum Sci. Technol. 9(4), 045021 (2024)","journal-title":"Quantum Sci. Technol."},{"key":"5112_CR21","volume-title":"Combinatorial Pptimization: Algorithms and Complexity","author":"CH Papadimitriou","year":"1998","unstructured":"Papadimitriou, C.H., Steiglitz, K.: Combinatorial Pptimization: Algorithms and Complexity. Courier Corporation (1998)"},{"key":"5112_CR22","doi-asserted-by":"crossref","unstructured":"Karp, R. M.: Reducibility among combinatorial problems. In: 50 Years of Integer Programming 1958\u20132008: from the Early Years to the State-of-the-Art. Springer; p. 219\u2013241 (2009)","DOI":"10.1007\/978-3-540-68279-0_8"},{"key":"5112_CR23","doi-asserted-by":"publisher","DOI":"10.1002\/9781119606475","volume-title":"Integer Programming","author":"LA Wolsey","year":"2020","unstructured":"Wolsey, L.A.: Integer Programming. John Wiley & Sons (2020)"},{"key":"5112_CR24","unstructured":"Coppersmith, D.: An approximate Fourier transform useful in quantum factoring. arXiv preprint arXiv:quant-ph\/0201067. (2002)"},{"key":"5112_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-017-1603-1","volume":"16","author":"L Ruiz-Perez","year":"2017","unstructured":"Ruiz-Perez, L., Garcia-Escartin, J.C.: Quantum arithmetic with the quantum Fourier transform. Quantum Inf. Process. 16, 1\u201314 (2017)","journal-title":"Quantum Inf. Process."},{"issue":"5","key":"5112_CR26","doi-asserted-by":"publisher","first-page":"3457","DOI":"10.1103\/PhysRevA.52.3457","volume":"52","author":"A Barenco","year":"1995","unstructured":"Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., et al.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457 (1995)","journal-title":"Phys. Rev. A"},{"issue":"8041","key":"5112_CR27","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1038\/s41586-024-08178-2","volume":"636","author":"A Carrera Vazquez","year":"2024","unstructured":"Carrera Vazquez, A., Tornow, C., Riste, D., Woerner, S., Takita, M., Egger, D.J.: Combining quantum processors with real-time classical communication. Nature 636(8041), 75\u201379 (2024)","journal-title":"Nature"},{"issue":"2","key":"5112_CR28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.7.023120","volume":"7","author":"E B\u00e4umer","year":"2025","unstructured":"B\u00e4umer, E., Woerner, S.: Measurement-based long-range entangling gates in constant depth. Phys. Rev. Res. 7(2), 023120 (2025)","journal-title":"Phys. Rev. Res."},{"key":"5112_CR29","unstructured":"Gurobi Optimization, LLC.: Gurobi Optimizer Reference Manual. Available from: https:\/\/www.gurobi.com"},{"issue":"9","key":"5112_CR30","doi-asserted-by":"publisher","first-page":"1889","DOI":"10.1103\/PhysRevLett.86.1889","volume":"86","author":"YS Weinstein","year":"2001","unstructured":"Weinstein, Y.S., Pravia, M., Fortunato, E., Lloyd, S., Cory, D.G.: Implementation of the quantum Fourier transform. Phys. Rev. Lett. 86(9), 1889 (2001)","journal-title":"Phys. Rev. Lett."},{"key":"5112_CR31","doi-asserted-by":"publisher","first-page":"2101417","DOI":"10.1109\/TQE.2025.3623158","volume":"6","author":"E Kaur","year":"2025","unstructured":"Kaur, E., Pouryousef, S., Shapourian, H., Zhao, J., Kilzer, M., Kompella, R., et al.: Optimized quantum circuit partitioning across multiple quantum processors. IEEE Trans. Quantum Eng. 6, 2101417 (2025)","journal-title":"IEEE Trans. Quantum Eng."},{"key":"5112_CR32","doi-asserted-by":"crossref","unstructured":"Neumann, N. M., van Houte, R., Attema, T.: Imperfect distributed quantum phase estimation. In: International Conference on Computational Science. Springer; p. 605\u2013615 (2020)","DOI":"10.1007\/978-3-030-50433-5_46"},{"key":"5112_CR33","unstructured":"Clausen, J.: Branch and bound algorithms-principles and examples. Department of computer science, University of Copenhagen. p. 1\u201330 (1999)"},{"key":"5112_CR34","doi-asserted-by":"crossref","unstructured":"Land, A. H., Doig, A. G.: An automatic method for solving discrete programming problems. In: 50 Years of Integer Programming 1958\u20132008: From the Early Years to the State-of-the-Art. Springer; p. 105\u2013132 (2009)","DOI":"10.1007\/978-3-540-68279-0_5"},{"issue":"3","key":"5112_CR35","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.100.032328","volume":"100","author":"AW Cross","year":"2019","unstructured":"Cross, A.W., Bishop, L.S., Sheldon, S., Nation, P.D., Gambetta, J.M.: Validating quantum computers using randomized model circuits. Phys. Rev. A 100(3), 032328 (2019)","journal-title":"Phys. Rev. A"},{"key":"5112_CR36","unstructured":"Draper, T. G.: Addition on a quantum computer. arXiv preprint arXiv:quant-ph\/0008033. (2000)"},{"key":"5112_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TQE.2023.3303935","volume":"4","author":"D Ferrari","year":"2023","unstructured":"Ferrari, D., Carretta, S., Amoretti, M.: A modular quantum compilation framework for distributed quantum computing. IEEE Trans. Quantum Eng. 4, 1\u201313 (2023)","journal-title":"IEEE Trans. Quantum Eng."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05112-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05112-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05112-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T09:55:13Z","timestamp":1774346113000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05112-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,7]]},"references-count":37,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["5112"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05112-5","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,7]]},"assertion":[{"value":"26 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 March 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"97"}}