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Current quantum hardware is noisy, can only store information for a short time and is limited to a few quantum bits, that is, qubits, typically arranged in a planar connectivity1. However, many applications of quantum computing require more connectivity than the planar lattice offered by the hardware on more qubits than is available on a single quantum processing unit (QPU). The community hopes to tackle these limitations by connecting QPUs using classical communication, which has not yet been proven experimentally. Here we experimentally realize error-mitigated dynamic circuits and circuit cutting to create quantum states requiring periodic connectivity using up to 142 qubits spanning two QPUs with 127 qubits each connected in real time with a classical link. In a dynamic circuit, quantum gates can be classically controlled by the outcomes of mid-circuit measurements within run-time, that is, within a fraction of the coherence time of the qubits. Our real-time classical link enables us to apply a quantum gate on one QPU conditioned on the outcome of a measurement on another QPU. Furthermore, the error-mitigated control flow enhances qubit connectivity and the instruction set of the hardware thus increasing the versatility of our quantum computers. Our work demonstrates that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link. 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Egger","url":"http://orcid.org/0000-0002-5523-9807","affiliation":[{"name":"IBM Research Europe - Zurich","address":{"name":"IBM Quantum, IBM Research Europe - Zurich, Rüschlikon, Switzerland","@type":"PostalAddress"},"@type":"Organization"}],"email":"deg@zurich.ibm.com","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> <link rel="canonical" href="https://www.nature.com/articles/s41586-024-08178-2"> <meta name="journal_id" content="41586"/> <meta name="dc.title" content="Combining quantum processors with real-time classical communication"/> <meta name="dc.source" content="Nature 2024"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Nature Publishing Group"/> <meta name="dc.date" content="2024-11-20"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2024 The Author(s)"/> <meta name="dc.rights" content="2024 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="Quantum computers process information with the laws of quantum mechanics. Current quantum hardware is noisy, can only store information for a short time and is limited to a few quantum bits, that is, qubits, typically arranged in a planar connectivity1. However, many applications of quantum computing require more connectivity than the planar lattice offered by the hardware on more qubits than is available on a single quantum processing unit (QPU). The community hopes to tackle these limitations by connecting QPUs using classical communication, which has not yet been proven experimentally. Here we experimentally realize error-mitigated dynamic circuits and circuit cutting to create quantum states requiring periodic connectivity using up to 142 qubits spanning two QPUs with 127 qubits each connected in real time with a classical link. In a dynamic circuit, quantum gates can be classically controlled by the outcomes of mid-circuit measurements within run-time, that is, within a fraction of the coherence time of the qubits. Our real-time classical link enables us to apply a quantum gate on one QPU conditioned on the outcome of a measurement on another QPU. Furthermore, the error-mitigated control flow enhances qubit connectivity and the instruction set of the hardware thus increasing the versatility of our quantum computers. Our work demonstrates that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link. A 142-qubit processor can be realized by connecting two smaller quantum processors using classical communications and circuit cutting."/> <meta name="prism.issn" content="1476-4687"/> <meta name="prism.publicationName" content="Nature"/> <meta name="prism.publicationDate" content="2024-11-20"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="5"/> <meta name="prism.copyright" content="2024 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://www.nature.com/articles/s41586-024-08178-2"/> <meta name="prism.doi" content="doi:10.1038/s41586-024-08178-2"/> <meta name="citation_pdf_url" content="https://www.nature.com/articles/s41586-024-08178-2.pdf"/> <meta name="citation_fulltext_html_url" content="https://www.nature.com/articles/s41586-024-08178-2"/> <meta name="citation_journal_title" content="Nature"/> <meta name="citation_journal_abbrev" content="Nature"/> <meta name="citation_publisher" content="Nature Publishing Group"/> <meta name="citation_issn" content="1476-4687"/> <meta name="citation_title" content="Combining quantum processors with real-time classical communication"/> <meta name="citation_online_date" content="2024/11/20"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="5"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1038/s41586-024-08178-2"/> <meta name="DOI" content="10.1038/s41586-024-08178-2"/> <meta name="size" content="317301"/> <meta name="citation_doi" content="10.1038/s41586-024-08178-2"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1038/s41586-024-08178-2&api_key="/> <meta name="description" content="Quantum computers process information with the laws of quantum mechanics. Current quantum hardware is noisy, can only store information for a short time and is limited to a few quantum bits, that is, qubits, typically arranged in a planar connectivity1. However, many applications of quantum computing require more connectivity than the planar lattice offered by the hardware on more qubits than is available on a single quantum processing unit (QPU). The community hopes to tackle these limitations by connecting QPUs using classical communication, which has not yet been proven experimentally. Here we experimentally realize error-mitigated dynamic circuits and circuit cutting to create quantum states requiring periodic connectivity using up to 142 qubits spanning two QPUs with 127 qubits each connected in real time with a classical link. In a dynamic circuit, quantum gates can be classically controlled by the outcomes of mid-circuit measurements within run-time, that is, within a fraction of the coherence time of the qubits. Our real-time classical link enables us to apply a quantum gate on one QPU conditioned on the outcome of a measurement on another QPU. Furthermore, the error-mitigated control flow enhances qubit connectivity and the instruction set of the hardware thus increasing the versatility of our quantum computers. Our work demonstrates that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link. 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Phys.; citation_title=The future of quantum computing with superconducting qubits; citation_author=S Bravyi, O Dial, JM Gambetta, D Gil, Z Nazario; citation_volume=132; citation_publication_date=2022; citation_pages=160902; citation_doi=10.1063/5.0082975; citation_id=CR2"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=High-threshold and low-overhead fault-tolerant quantum memory; citation_author=S Bravyi; citation_volume=627; citation_publication_date=2024; citation_pages=778-782; citation_doi=10.1038/s41586-024-07107-7; citation_id=CR3"/> <meta name="citation_reference" content="citation_journal_title=Nat. Commun.; citation_title=A high-fidelity quantum matter-link between ion-trap microchip modules; citation_author=M Akhtar; citation_volume=14; citation_publication_date=2023; citation_doi=10.1038/s41467-022-35285-3; citation_id=CR4"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=A quantum processor based on coherent transport of entangled atom arrays; citation_author=D Bluvstein; citation_volume=604; citation_publication_date=2022; citation_pages=451-456; citation_doi=10.1038/s41586-022-04592-6; citation_id=CR5"/> <meta name="citation_reference" content="citation_journal_title=Appl. Phys. Rev.; citation_title=A quantum engineer’s guide to superconducting qubits; citation_author=P Krantz; citation_volume=6; citation_publication_date=2019; citation_pages=021318; citation_doi=10.1063/1.5089550; citation_id=CR6"/> <meta name="citation_reference" content="citation_journal_title=Appl. Phys. Lett.; citation_title=Superconducting qubits in a flip-chip architecture; citation_author=CR Conner; citation_volume=118; citation_publication_date=2021; citation_pages=232602; citation_doi=10.1063/5.0050173; citation_id=CR7"/> <meta name="citation_reference" content="citation_journal_title=npj Quantum Inf.; citation_title=Entanglement across separate silicon dies in a modular superconducting qubit device; citation_author=A Gold; citation_volume=7; citation_publication_date=2021; citation_doi=10.1038/s41534-021-00484-1; citation_id=CR8"/> <meta name="citation_reference" content="citation_journal_title=Nat. Phys.; citation_title=Violating Bell’s inequality with remotely connected superconducting qubits; citation_author=Y Zhong; citation_volume=15; citation_publication_date=2019; citation_pages=741-744; citation_doi=10.1038/s41567-019-0507-7; citation_id=CR9"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=Deterministic multi-qubit entanglement in a quantum network; citation_author=Y Zhong; citation_volume=590; citation_publication_date=2021; citation_pages=571-575; citation_doi=10.1038/s41586-021-03288-7; citation_id=CR10"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Appl.; citation_title=Enhanced quantum state transfer and Bell-state generation over long-range multimode interconnects via superadiabatic transitionless driving; citation_author=M Malekakhlagh; citation_volume=22; citation_publication_date=2024; citation_pages=024006; citation_doi=10.1103/PhysRevApplied.22.024006; citation_id=CR11"/> <meta name="citation_reference" content="citation_journal_title=ACM Trans. Quantum Comput.; citation_title=ARQUIN: architectures for multinode superconducting quantum computers; citation_author=J Ang; citation_volume=5; citation_publication_date=2024; citation_pages=19; citation_doi=10.1145/3674151; citation_id=CR12"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Exploiting dynamic quantum circuits in a quantum algorithm with superconducting qubits; citation_author=AD Córcoles; citation_volume=127; citation_publication_date=2021; citation_pages=100501; citation_doi=10.1103/PhysRevLett.127.100501; citation_id=CR13"/> <meta name="citation_reference" content="citation_journal_title=PRX Quantum; citation_title=Efficient long-range entanglement using dynamic circuits; citation_author=E Bäumer; citation_volume=5; citation_publication_date=2024; citation_pages=030339; citation_doi=10.1103/PRXQuantum.5.030339; citation_id=CR14"/> <meta name="citation_reference" content="citation_journal_title=J. Phys. A Math. Theor.; citation_title=How to simulate a universal quantum computer using negative probabilities; citation_author=HF Hofmann; citation_volume=42; citation_publication_date=2009; citation_pages=275304; citation_doi=10.1088/1751-8113/42/27/275304; citation_id=CR15"/> <meta name="citation_reference" content="citation_journal_title=New J. Phys.; citation_title=Constructing a virtual two-qubit gate by sampling single-qubit operations; citation_author=K Mitarai, K Fujii; citation_volume=23; citation_publication_date=2021; citation_pages=023021; citation_doi=10.1088/1367-2630/abd7bc; citation_id=CR16"/> <meta name="citation_reference" content="citation_journal_title=IEEE Trans. Inf. Theor.; citation_title=Circuit knitting with classical communication; citation_author=C Piveteau, D Sutter; citation_volume=1; citation_publication_date=2023; citation_pages=2734-2745; citation_id=CR17"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Res.; citation_title=Experimental demonstration of a high-fidelity virtual two-qubit gate; citation_author=AP Singh; citation_volume=6; citation_publication_date=2024; citation_pages=013235; citation_doi=10.1103/PhysRevResearch.6.013235; citation_id=CR18"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations; citation_author=D Gottesman, IL Chuang; citation_volume=402; citation_publication_date=1999; citation_pages=390-393; citation_doi=10.1038/46503; citation_id=CR19"/> <meta name="citation_reference" content="citation_journal_title=Science; citation_title=Quantum gate teleportation between separated qubits in a trapped-ion processor; citation_author=Y Wan; citation_volume=364; citation_publication_date=2019; citation_pages=875; citation_doi=10.1126/science.aaw9415; citation_id=CR20"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Dynamical decoupling of open quantum systems; citation_author=L Viola, E Knill, S Lloyd; citation_volume=82; citation_publication_date=1999; citation_pages=2417-2421; citation_doi=10.1103/PhysRevLett.82.2417; citation_id=CR21"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Error mitigation for short-depth quantum circuits; citation_author=K Temme, S Bravyi, JM Gambetta; citation_volume=119; citation_publication_date=2017; citation_pages=180509; citation_doi=10.1103/PhysRevLett.119.180509; citation_id=CR22"/> <meta name="citation_reference" content="Reilly, D. J. Challenges in scaling-up the control interface of a quantum computer. Preprint at arxiv.org/abs/1912.05114 (2019)."/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Simulating large quantum circuits on a small quantum computer; citation_author=T Peng, AW Harrow, M Ozols, X Wu; citation_volume=125; citation_publication_date=2020; citation_pages=150504; citation_doi=10.1103/PhysRevLett.125.150504; citation_id=CR24"/> <meta name="citation_reference" content="Brenner, L., Piveteau, C. & Sutter, D. Optimal wire cutting with classical communication. Preprint at arxiv.org/abs/2302.03366 (2023)."/> <meta name="citation_reference" content="Pednault, E. An alternative approach to optimal wire cutting without ancilla qubits. Preprint at arxiv.org/abs/2303.08287 (2023)."/> <meta name="citation_reference" content="Zander, R. & Becker, C. K.-U. Benchmarking multipartite entanglement generation with graph states. Preprint at arxiv.org/abs/2402.00766 (2024)."/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Appl.; citation_title=Experimental benchmarking of an automated deterministic error-suppression workflow for quantum algorithms; citation_author=PS Mundada; citation_volume=20; citation_publication_date=2023; citation_pages=024034; citation_doi=10.1103/PhysRevApplied.20.024034; citation_id=CR28"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=Encoding a magic state with beyond break-even fidelity; citation_author=RS Gupta; citation_volume=625; citation_publication_date=2024; citation_pages=259-263; citation_doi=10.1038/s41586-023-06846-3; citation_id=CR29"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Optimal joint cutting of two-qubit rotation gates; citation_author=C Ufrecht; citation_volume=109; citation_publication_date=2024; citation_pages=052440; citation_doi=10.1103/PhysRevA.109.052440; citation_id=CR30"/> <meta name="citation_reference" content="Schmitt, L., Piveteau, C. & Sutter, D. Cutting circuits with multiple two-qubit unitaries. Preprint at arxiv.org/abs/2312.11638 (2023)."/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Quantum state transfer and entanglement distribution among distant nodes in a quantum network; citation_author=JI Cirac, P Zoller, HJ Kimble, H Mabuchi; citation_volume=78; citation_publication_date=1997; citation_pages=3221-3224; citation_doi=10.1103/PhysRevLett.78.3221; citation_id=CR32"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=Long-distance quantum communication with atomic ensembles and linear optics; citation_author=L-M Duan, MD Lukin, JI Cirac, P Zoller; citation_volume=414; citation_publication_date=2001; citation_pages=413-418; citation_doi=10.1038/35106500; citation_id=CR33"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems; citation_author=P Magnard; citation_volume=125; citation_publication_date=2020; citation_pages=260502; citation_doi=10.1103/PhysRevLett.125.260502; citation_id=CR34"/> <meta name="citation_reference" content="citation_journal_title=Nat. Electron.; citation_title=Low-loss interconnects for modular superconducting quantum processors; citation_author=J Niu; citation_volume=6; citation_publication_date=2023; citation_pages=235-241; citation_doi=10.1038/s41928-023-00925-z; citation_id=CR35"/> <meta name="citation_reference" content="citation_journal_title=Quantum Sci. Technol.; citation_title=Engineering electro-optics in SiGe/Si waveguides for quantum transduction; citation_author=J Orcutt; citation_volume=5; citation_publication_date=2020; citation_pages=034006; citation_doi=10.1088/2058-9565/ab84c1; citation_id=CR36"/> <meta name="citation_reference" content="citation_journal_title=Quantum Sci. Technol.; citation_title=Perspectives on quantum transduction; citation_author=N Lauk; citation_volume=5; citation_publication_date=2020; citation_pages=020501; citation_doi=10.1088/2058-9565/ab788a; citation_id=CR37"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Optically heralded entanglement of superconducting systems in quantum networks; citation_author=S Krastanov; citation_volume=127; citation_publication_date=2021; citation_pages=040503; citation_doi=10.1103/PhysRevLett.127.040503; citation_id=CR38"/> <meta name="citation_reference" content="Bechtold, M., Barzen, J., Leymann, F. & Mandl, A. Circuit cutting with non-maximally entangled states. Preprint at arxiv.org/abs/2306.12084 (2023)."/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Purification of noisy entanglement and faithful teleportation via noisy channels; citation_author=CH Bennett; citation_volume=76; citation_publication_date=1996; citation_pages=722-725; citation_doi=10.1103/PhysRevLett.76.722; citation_id=CR40"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Appl.; citation_title=Minimum quantum run-time characterization and calibration via restless measurements with dynamic repetition rates; citation_author=C Tornow, N Kanazawa, WE Shanks, DJ Egger; citation_volume=17; citation_publication_date=2022; citation_pages=064061; citation_doi=10.1103/PhysRevApplied.17.064061; citation_id=CR41"/> <meta name="citation_reference" content="Harrow, A. W. & Lowe, A. Optimal quantum circuit cuts with application to clustered Hamiltonian simulation. Preprint at arxiv.org/abs/2403.01018 (2024)."/> <meta name="citation_reference" content="Caleffi, M. et al. Distributed quantum computing: a survey. Preprint at arxiv.org/abs/2212.10609 (2022)."/> <meta name="citation_reference" content="citation_journal_title=Rev. Mod. Phys.; citation_title=Quantum error mitigation; citation_author=Z Cai; citation_volume=95; citation_publication_date=2023; citation_pages=045005; citation_doi=10.1103/RevModPhys.95.045005; citation_id=CR44"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. X; citation_title=Practical quantum error mitigation for near-future applications; citation_author=S Endo, SC Benjamin, Y Li; citation_volume=8; citation_publication_date=2018; citation_pages=031027; citation_id=CR45"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Efficient Z gates for quantum computing; citation_author=DC McKay, CJ Wood, S Sheldon, JM Chow, JM Gambetta; citation_volume=96; citation_publication_date=2017; citation_pages=022330; citation_doi=10.1103/PhysRevA.96.022330; citation_id=CR46"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. X; citation_title=Hierarchy of linear light cones with long-range interactions; citation_author=MC Tran; citation_volume=10; citation_publication_date=2020; citation_pages=031009; citation_id=CR47"/> <meta name="citation_reference" content="Zettles, G., Willenborg, S., Johnson, B. R., Wack, A. & Allison, B. 26.2 Design considerations for superconducting quantum systems. In 2022 IEEE International Solid-State Circuits Conference (ISSCC), Vol. 65, pp. 1–3 (IEEE, 2022)."/> <meta name="citation_reference" content="citation_journal_title=ACM Trans. Quantum Comput.; citation_title=OpenQASM 3: a broader and deeper quantum assembly language; citation_author=A Cross; citation_volume=3; citation_publication_date=2022; citation_pages=1-50; citation_doi=10.1145/3505636; citation_id=CR49"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Robustness of entanglement; citation_author=G Vidal, R Tarrach; citation_volume=59; citation_publication_date=1999; citation_pages=141-155; citation_doi=10.1103/PhysRevA.59.141; citation_id=CR50"/> <meta name="citation_reference" content="Kraft, D. A Software Package for Sequential Quadratic Programming (DFVLR, 1988)."/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Probabilistic error cancellation for dynamic quantum circuits; citation_author=RS Gupta; citation_volume=109; citation_publication_date=2024; citation_pages=062617; citation_doi=10.1103/PhysRevA.109.062617; citation_id=CR52"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Persistent entanglement in arrays of interacting particles; citation_author=HJ Briegel, R Raussendorf; citation_volume=86; citation_publication_date=2001; citation_pages=910-913; citation_doi=10.1103/PhysRevLett.86.910; citation_id=CR53"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Multiparty entanglement in graph states; citation_author=M Hein, J Eisert, HJ Briegel; citation_volume=69; citation_publication_date=2004; citation_pages=062311; citation_doi=10.1103/PhysRevA.69.062311; citation_id=CR54"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Entanglement witnesses for graph states: general theory and examples; citation_author=B Jungnitsch, T Moroder, O Gühne; citation_volume=84; citation_publication_date=2011; citation_pages=032310; citation_doi=10.1103/PhysRevA.84.032310; citation_id=CR55"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. A; citation_title=Entanglement detection in the stabilizer formalism; citation_author=G Tóth, O Gühne; citation_volume=72; citation_publication_date=2005; citation_pages=022340; citation_doi=10.1103/PhysRevA.72.022340; citation_id=CR56"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. 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</span>orcid.org/0000-0001-8871-4638</a></span><sup class="u-js-hide"><a href="#Aff4">4</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 6 authors for this article" title="Show all 6 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Daniel_J_-Egger-Aff1" data-author-popup="auth-Daniel_J_-Egger-Aff1" data-author-search="Egger, Daniel J." data-corresp-id="c1">Daniel J. Egger<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-5523-9807"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-5523-9807</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Nature</i></a> (<span data-test="article-publication-year">2024</span>)<a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">2027 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">3 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/s41586-024-08178-2/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/quantum-information" data-track="click" data-track-action="view subject" data-track-label="link">Quantum information</a></li><li class="c-article-subject-list__subject"><a href="/subjects/quantum-simulation" data-track="click" data-track-action="view subject" data-track-label="link">Quantum simulation</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Quantum computers process information with the laws of quantum mechanics. Current quantum hardware is noisy, can only store information for a short time and is limited to a few quantum bits, that is, qubits, typically arranged in a planar connectivity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023)." href="/articles/s41586-024-08178-2#ref-CR1" id="ref-link-section-d109866146e348">1</a></sup>. However, many applications of quantum computing require more connectivity than the planar lattice offered by the hardware on more qubits than is available on a single quantum processing unit (QPU). The community hopes to tackle these limitations by connecting QPUs using classical communication, which has not yet been proven experimentally. Here we experimentally realize error-mitigated dynamic circuits and circuit cutting to create quantum states requiring periodic connectivity using up to 142 qubits spanning two QPUs with 127 qubits each connected in real time with a classical link. In a dynamic circuit, quantum gates can be classically controlled by the outcomes of mid-circuit measurements within run-time, that is, within a fraction of the coherence time of the qubits. Our real-time classical link enables us to apply a quantum gate on one QPU conditioned on the outcome of a measurement on another QPU. Furthermore, the error-mitigated control flow enhances qubit connectivity and the instruction set of the hardware thus increasing the versatility of our quantum computers. Our work demonstrates that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41586-023-06505-7/MediaObjects/41586_2023_6505_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41586-023-06505-7?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41586-023-06505-7">Measurement-induced entanglement and teleportation on a noisy quantum processor </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">18 October 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41534-024-00892-z/MediaObjects/41534_2024_892_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41534-024-00892-z?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s41534-024-00892-z">Extending the computational reach of a superconducting qutrit processor </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">14 October 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41534-023-00762-0/MediaObjects/41534_2023_762_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41534-023-00762-0?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41534-023-00762-0">Realization of quantum signal processing on a noisy quantum computer </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">23 September 2023</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732373642, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Main"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Main</h2><div class="c-article-section__content" id="Sec1-content"><p>Quantum computers process information encoded in quantum bits with unitary operations. However, quantum computers are noisy and most large-scale architectures arrange the physical qubits in a planar lattice. Nevertheless, current processors with error mitigation can already simulate hardware-native Ising models with 127 qubits and measure observables at a scale where brute-force approaches with classical computers begin to struggle<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023)." href="/articles/s41586-024-08178-2#ref-CR1" id="ref-link-section-d109866146e369">1</a></sup>. The usefulness of quantum computers hinges on further scaling and overcoming their limited qubit connectivity. A modular approach is important for scaling current noisy quantum processors<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Bravyi, S., Dial, O., Gambetta, J. M., Gil, D. & Nazario, Z. The future of quantum computing with superconducting qubits. J. Appl. Phys. 132, 160902 (2022)." href="/articles/s41586-024-08178-2#ref-CR2" id="ref-link-section-d109866146e373">2</a></sup> and for achieving the large numbers of physical qubits needed for fault tolerance<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. Nature 627, 778–782 (2024)." href="/articles/s41586-024-08178-2#ref-CR3" id="ref-link-section-d109866146e377">3</a></sup>. Trapped ion and neutral atom architectures can achieve modularity by physically transporting the qubits<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Akhtar, M. et al. A high-fidelity quantum matter-link between ion-trap microchip modules. Nat. Commun. 14, 531 (2023)." href="/articles/s41586-024-08178-2#ref-CR4" id="ref-link-section-d109866146e381">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Bluvstein, D. et al. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451–456 (2022)." href="/articles/s41586-024-08178-2#ref-CR5" id="ref-link-section-d109866146e384">5</a></sup>. In the near term, modularity in superconducting qubits<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Krantz, P. et al. A quantum engineer’s guide to superconducting qubits. Appl. Phys. Rev. 6, 021318 (2019)." href="/articles/s41586-024-08178-2#ref-CR6" id="ref-link-section-d109866146e388">6</a></sup> is achieved by short-range interconnects that link adjacent chips<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Conner, C. R. et al. Superconducting qubits in a flip-chip architecture. Appl. Phys. Lett. 118, 232602 (2021)." href="/articles/s41586-024-08178-2#ref-CR7" id="ref-link-section-d109866146e393">7</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gold, A. et al. Entanglement across separate silicon dies in a modular superconducting qubit device. npj Quantum Inf. 7, 142 (2021)." href="/articles/s41586-024-08178-2#ref-CR8" id="ref-link-section-d109866146e396">8</a></sup>.</p><p>In the medium term, long-range gates operating in the microwave regime may be carried out over long conventional cables<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zhong, Y. et al. Violating Bell’s inequality with remotely connected superconducting qubits. Nat. Phys. 15, 741–744 (2019)." href="#ref-CR9" id="ref-link-section-d109866146e403">9</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zhong, Y. et al. Deterministic multi-qubit entanglement in a quantum network. Nature 590, 571–575 (2021)." href="#ref-CR10" id="ref-link-section-d109866146e403_1">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Malekakhlagh, M. et al. Enhanced quantum state transfer and Bell-state generation over long-range multimode interconnects via superadiabatic transitionless driving. Phys. Rev. Appl. 22, 024006 (2024)." href="/articles/s41586-024-08178-2#ref-CR11" id="ref-link-section-d109866146e406">11</a></sup>. This would enable non-planar qubit connectivity suitable for efficient error correction<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. Nature 627, 778–782 (2024)." href="/articles/s41586-024-08178-2#ref-CR3" id="ref-link-section-d109866146e410">3</a></sup>. A long-term alternative is to entangle remote QPUs with an optical link leveraging a microwave to optical transduction<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Ang, J. et al. ARQUIN: architectures for multinode superconducting quantum computers. ACM Trans. Quantum Comput. 5, 19 (2024)." href="/articles/s41586-024-08178-2#ref-CR12" id="ref-link-section-d109866146e414">12</a></sup>, which has not yet been demonstrated, to our knowledge. Moreover, dynamic circuits broaden the set of operations of a quantum computer by performing mid-circuit measurements (MCMs) and classically controlling a gate within the coherence time of the qubits. They enhance algorithmic quality<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Córcoles, A. D. et al. Exploiting dynamic quantum circuits in a quantum algorithm with superconducting qubits. Phys. Rev. Lett. 127, 100501 (2021)." href="/articles/s41586-024-08178-2#ref-CR13" id="ref-link-section-d109866146e418">13</a></sup> and qubit connectivity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Bäumer, E. et al. Efficient long-range entanglement using dynamic circuits. PRX Quantum 5, 030339 (2024)." href="/articles/s41586-024-08178-2#ref-CR14" id="ref-link-section-d109866146e422">14</a></sup>. As we will show, dynamic circuits also enable modularity by connecting QPUs in real time through a classical link.</p><p>We take a complementary approach based on virtual gates to implement long-range interactions in a modular architecture. We connect qubits at arbitrary locations and create the statistics of entanglement through a quasi-probability decomposition (QPD)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hofmann, H. F. How to simulate a universal quantum computer using negative probabilities. J. Phys. A Math. Theor. 42, 275304 (2009)." href="#ref-CR15" id="ref-link-section-d109866146e429">15</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="#ref-CR16" id="ref-link-section-d109866146e429_1">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e432">17</a></sup>. We compare a Local Operations (LO) only scheme<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="/articles/s41586-024-08178-2#ref-CR16" id="ref-link-section-d109866146e436">16</a></sup> to one augmented by Classical Communication (LOCC)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e440">17</a></sup>. The LO scheme, demonstrated in a two-qubit setting<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Singh, A. P. et al. Experimental demonstration of a high-fidelity virtual two-qubit gate. Phys. Rev. Res. 6, 013235 (2024)." href="/articles/s41586-024-08178-2#ref-CR18" id="ref-link-section-d109866146e444">18</a></sup>, requires executing multiple quantum circuits with local operations only. By contrast, to implement LOCC, we consume virtual Bell pairs in a teleportation circuit to create two-qubit gates<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Gottesman, D. & Chuang, I. L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390–393 (1999)." href="/articles/s41586-024-08178-2#ref-CR19" id="ref-link-section-d109866146e448">19</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Wan, Y. et al. Quantum gate teleportation between separated qubits in a trapped-ion processor. Science 364, 875 (2019)." href="/articles/s41586-024-08178-2#ref-CR20" id="ref-link-section-d109866146e451">20</a></sup>. On quantum hardware with sparse and planar connectivity, creating a Bell pair between arbitrary qubits requires a long-range controlled-NOT (CNOT) gate. To avoid these gates, we use a QPD over local operations resulting in cut Bell pairs that the teleportation consumes. LO do not need the classical link and is thus simpler to implement than LOCC. However, as LOCC only requires a single parameterized template circuit, it is more efficient to compile than LO and the cost of its QPD is lower than the cost of the LO scheme.</p><p>Our work makes four key contributions. First, we present the quantum circuits and QPD to create multiple cut Bell pairs to realize the virtual gates in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e458">17</a></sup>. Second, we suppress and mitigate the errors arising from the latency of the classical control hardware in dynamic circuits<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417–2421 (1999)." href="/articles/s41586-024-08178-2#ref-CR21" id="ref-link-section-d109866146e462">21</a></sup> with a combination of dynamical decoupling and zero-noise extrapolation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017)." href="/articles/s41586-024-08178-2#ref-CR22" id="ref-link-section-d109866146e466">22</a></sup>. Third, we leverage these methods to engineer periodic boundary conditions on a 103-node graph state. Fourth, we demonstrate a real-time classical connection between two separate QPUs thereby demonstrating that a system of distributed QPUs can be operated as one through a classical link<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Reilly, D. J. Challenges in scaling-up the control interface of a quantum computer. Preprint at arxiv.org/abs/1912.05114 (2019)." href="/articles/s41586-024-08178-2#ref-CR23" id="ref-link-section-d109866146e470">23</a></sup>. Combined with dynamic circuits, this enables us to operate both chips as a single quantum computer, which we exemplify by engineering a periodic graph state that spans both devices on 142 qubits. We discuss a path forward to create long-range gates and provide our conclusion.</p></div></div></section><section data-title="Circuit cutting"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Circuit cutting</h2><div class="c-article-section__content" id="Sec2-content"><p>We run large quantum circuits that may not be directly executable on our hardware because of limitations in qubit count or connectivity by cutting gates. Circuit cutting decomposes a complex circuit into subcircuits that can be individually executed<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hofmann, H. F. How to simulate a universal quantum computer using negative probabilities. J. Phys. A Math. Theor. 42, 275304 (2009)." href="#ref-CR15" id="ref-link-section-d109866146e482">15</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="#ref-CR16" id="ref-link-section-d109866146e482_1">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e485">17</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Peng, T., Harrow, A. W., Ozols, M. & Wu, X. Simulating large quantum circuits on a small quantum computer. Phys. Rev. Lett. 125, 150504 (2020)." href="#ref-CR24" id="ref-link-section-d109866146e488">24</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Brenner, L., Piveteau, C. & Sutter, D. Optimal wire cutting with classical communication. Preprint at arxiv.org/abs/2302.03366 (2023)." href="#ref-CR25" id="ref-link-section-d109866146e488_1">25</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Pednault, E. An alternative approach to optimal wire cutting without ancilla qubits. Preprint at arxiv.org/abs/2303.08287 (2023)." href="/articles/s41586-024-08178-2#ref-CR26" id="ref-link-section-d109866146e491">26</a></sup>. However, we must run an increased number of circuits, which we call the sampling overhead. The results from these subcircuits are then classically recombined to yield the result of the original circuit (<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec6">Methods</a>).</p><p>As one of the main contributions of our work is implementing virtual gates with LOCC, we show how to create the required cut Bell pairs with local operations. Here, multiple cut Bell pairs are engineered by parameterized quantum circuits, which we call a cut Bell pair factory (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1b,c</a>). Cutting multiple pairs at the same time requires a lower sampling overhead<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e504">17</a></sup>. As the cut Bell pair factory forms two disjoint quantum circuits, we place each subcircuit close to qubits that have long-range gates. The resulting resource is then consumed in a teleportation circuit. For instance, in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1b</a>, the cut Bell pairs are consumed to create CNOT gates on the qubit pairs (0, 1) and (2, 3) (see section ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec11">Cut Bell pair factories</a>’).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Local operations and classical communication."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1: Local operations and classical communication.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="881"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p><b>a</b>, Depiction of an IBM Quantum System Two architecture. Here, two 127 qubit Eagle QPUs are connected with a real-time classical link. Each QPU is controlled by its electronics in its rack. We tightly synchronize both racks to operate both QPUs as one. <b>b</b>, Template quantum circuit to implement virtual CNOT gates on qubit pairs (<i>q</i><sub>0</sub>, <i>q</i><sub>1</sub>) and (<i>q</i><sub>2</sub>, <i>q</i><sub>3</sub>) with LOCC by consuming cut Bell pairs in a teleportation circuit. The purple double lines correspond to the real-time classical link. <b>c</b>, Cut Bell pair factories <i>C</i><sub>2</sub>(<b>θ</b><sup><i>i</i></sup>) for two simultaneously cut Bell pairs. The QPD has a total of 27 different parameter sets <b>θ</b><sup><i>i</i></sup>. Here, <span class="mathjax-tex">$U(\theta ,\phi )=\sqrt{X}{R}_{z}(\theta )\sqrt{X}{R}_{z}(\phi )$</span>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Periodic boundary conditions"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3">Periodic boundary conditions</h2><div class="c-article-section__content" id="Sec3-content"><p>We construct a graph state |<i>G</i><span class="stix">⟩</span> with periodic boundary conditions on ibm_kyiv, an Eagle processor<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023)." href="/articles/s41586-024-08178-2#ref-CR1" id="ref-link-section-d109866146e692">1</a></sup>, going beyond the limits imposed by its physical connectivity (see section ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec13">Graph states</a>’). Here, <i>G</i> has <span class="stix">∣</span><i>V</i><span class="stix">∣</span> = 103 nodes and requires four long-range edges <i>E</i><sub>lr</sub> = {(1, 95), (2, 98), (6, 102), (7, 97)} between the top and bottom qubits of the Eagle processor (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2a</a>). We measure the node stabilizers <i>S</i><sub><i>i</i></sub> at each node <i>i</i> <span class="stix">∈</span> <i>V</i> and the edge stabilizers formed by the product <i>S</i><sub><i>i</i></sub><i>S</i><sub><i>j</i></sub> across each edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i>. From these stabilizers, we build an entanglement witness <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}=(1-\langle {S}_{i}\rangle -\langle {S}_{j}\rangle -\langle {S}_{i}{S}_{j}\rangle )/4$</span>, which is negative if there is bipartite entanglement across the edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> (ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Zander, R. & Becker, C. K.-U. Benchmarking multipartite entanglement generation with graph states. Preprint at arxiv.org/abs/2402.00766 (2024)." href="/articles/s41586-024-08178-2#ref-CR27" id="ref-link-section-d109866146e882">27</a></sup>) (see section ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec14">Entanglement witness</a>’). We focus on bipartite entanglement because this is the resource we wish to recreate with virtual gates. Measuring witnesses of entanglement between more than two parties will measure only the quality of the non-virtual gates and measurements making the impact of the virtual gates less clear.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Periodic boundary conditions."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2: Periodic boundary conditions.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig2_HTML.png?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="426"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p><b>a</b>, The heavy-hexagonal graph is folded on itself into a tubular form by the edges (1, 95), (2, 98), (6, 102) and (7, 97) highlighted in blue. We cut these edges. <b>b</b>, The node stabilizers <i>S</i><sub><i>j</i></sub> (top) and witnesses <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>, (bottom), with 1 standard deviation for the nodes and edges close to the long-range edges. Vertical dashed lines group stabilizers and witnesses by their distance to cut edges. <b>c</b>, Cumulative distribution function of the stabilizer errors. The stars indicate node stabilizers <i>S</i><sub><i>j</i></sub> that have an edge implemented by a long-range gate. In the dropped edge benchmark (dash-dotted red line), the long-range gates are not implemented and the star-indicated stabilizers thus have unit error. The grey region is the probability mass corresponding to node stabilizers affected by the cuts. <b>d</b>–<b>f</b>, In the two-dimensional layouts, the green nodes duplicate nodes 95, 98, 102 and 97 to show the cut edges. The blue nodes in <b>e</b> are qubit resources to create cut Bell pairs. The colour of node <i>i</i> is the absolute error <span class="stix">∣</span><i>S</i><sub><i>i</i></sub> − 1<span class="stix">∣</span> of the measured stabilizer, as indicated by the colour bar. An edge is black if entanglement statistics are detected at a 99% confidence level and violet if not. In <b>d</b>, the long-range gates are implemented with SWAP gates. In <b>e</b>, the same gates are implemented with LOCC. In <b>f</b>, they are not implemented at all.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>We prepare |<i>G</i><span class="stix">⟩</span> using three different methods. The hardware-native edges are always implemented with CNOT gates but the periodic boundary conditions are implemented with (1) SWAP gates, (2) LOCC and (3) LO to connect qubits across the whole lattice. The main difference between LOCC and LO is a feed-forward operation consisting of single-qubit gates conditioned on 2<i>n</i> measurement outcomes, where <i>n</i> is the number of cuts. Each of the 2<sup>2<i>n</i></sup> cases triggers a unique combination of <i>X</i> and/or <i>Z</i> gates on the appropriate qubits. Acquiring the measurement results, determining the corresponding case and acting based on it is performed in real time by the control hardware, at the cost of a fixed added latency. We mitigate and suppress the errors resulting from this latency with zero-noise extrapolation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017)." href="/articles/s41586-024-08178-2#ref-CR22" id="ref-link-section-d109866146e1016">22</a></sup> and staggered dynamical decoupling<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417–2421 (1999)." href="/articles/s41586-024-08178-2#ref-CR21" id="ref-link-section-d109866146e1020">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Mundada, P. S. et al. Experimental benchmarking of an automated deterministic error-suppression workflow for quantum algorithms. Phys. Rev. Appl. 20, 024034 (2023)." href="/articles/s41586-024-08178-2#ref-CR28" id="ref-link-section-d109866146e1023">28</a></sup> (see section ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec10">Error-mitigated quantum circuit switch instructions</a>’).</p><p>We benchmark the SWAP, LOCC and LO implementations of |<i>G</i><span class="stix">⟩</span> with a hardware-native graph state on <i>G</i>′ = (<i>V</i>, <i>E</i>′) obtained by removing the long-range gates, that is, <i>E</i>′ = <i>E</i>\<i>E</i><sub>lr</sub>. The circuit preparing |<i>G</i>′<span class="stix">⟩</span> thus requires only 112 CNOT gates arranged in three layers following the heavy-hexagonal topology of the Eagle processor. This circuit will report large errors when measuring the node and edge stabilizers of |<i>G</i><span class="stix">⟩</span> for nodes on a cut gate because it is designed to implement |<i>G</i>′<span class="stix">⟩</span>. We refer to this hardware-native benchmark as the dropped edge benchmark. The swap-based circuit requires an additional 262 CNOT gates to create the long-range edges <i>E</i><sub>lr</sub>, which drastically reduces the value of the measured stabilizers (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b–d</a>). By contrast, the LOCC and LO implementation of the edges in <i>E</i><sub>lr</sub> does not require SWAP gates. The errors of their node and edge stabilizers for nodes not involved in a cut gate closely follow the dropped edge benchmark (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b,c</a>). Conversely, the stabilizers involving a virtual gate have a lower error than the dropped edge benchmark and the swap implementation (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2c</a>, star markers). As an overall quality metric, we first report the sum of absolute errors on the node stabilizers, that is, ∑<sub><i>i</i><span class="stix">∈</span><i>V</i></sub><span class="stix">∣</span><i>S</i><sub><i>i</i></sub> − 1<span class="stix">∣</span> (Extended Data Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41586-024-08178-2#Tab1">1</a>). The large SWAP overhead is responsible for the 44.3 sum absolute error. The 13.1 error on the dropped edge benchmark is dominated by the eight nodes on the four cuts (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2c</a>, star markers). By contrast, the LO and LOCC errors are affected by MCMs. We attribute the 1.9 additional error of LOCC over LO to the delays and the CNOT gates in the teleportation circuit and cut Bell pairs. In the SWAP-based results, <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> does not detect entanglement across 35 of the 116 edges at the 99% confidence level (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b,d</a>). For the LO and LOCC implementation, <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> witnesses the statistics of bipartite entanglement across all edges in <i>G</i> at the 99% confidence level (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2e</a>). These metrics show that virtual long-range gates produce stabilizers with smaller errors than their decomposition into SWAPs. Furthermore, they keep the variance low enough to verify the statistics of entanglement.</p></div></div></section><section data-title="Operating two QPUs as one"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4">Operating two QPUs as one</h2><div class="c-article-section__content" id="Sec4-content"><p>We now combine two Eagle QPUs with 127 qubits each into a single QPU through a real-time classical connection. Operating the devices as a single, larger processor consists of executing quantum circuits spanning the larger qubit register. Apart from unitary gates and measurements running concurrently on the merged QPU, we use dynamic circuits to perform gates that act on qubits on both devices. This is enabled by a tight synchronization and fast classical communication between physically separate instruments required to collect measurement results and determine the control flow across the whole system<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024)." href="/articles/s41586-024-08178-2#ref-CR29" id="ref-link-section-d109866146e1187">29</a></sup>.</p><p>We test this real-time classical connection by engineering a graph state on 134 qubits built from heavy-hexagonal rings that wind through both QPUs (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig3">3</a>). These rings were chosen by excluding qubits plagued by two-level systems and readout issues to ensure a high-quality graph state. This graph forms a ring in three dimensions and requires four long-range gates that we implement with LO and LOCC. As before, the LOCC protocol thus requires two additional qubits per cut gate for the cut Bell pairs. As in the previous section, we benchmark our results to a graph that does not implement the edges that span both QPUs. As there is no quantum link between the two devices, a benchmark with SWAP gates is impossible. All edges exhibit the statistics of bipartite entanglement when we implement the graph with LO and LOCC at a 99% confidence level. Furthermore, the LO and LOCC stabilizers have the same quality as the dropped edge benchmark for nodes that are not affected by a long-range gate (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig3">3c</a>). Stabilizers affected by long-range gates have a large reduction in error compared with the dropped edge benchmark. The sum of absolute errors on the node stabilizers ∑<sub><i>i</i><span class="stix">∈</span><i>V</i></sub><span class="stix">∣</span><i>S</i><sub><i>i</i></sub> − 1<span class="stix">∣</span>, is 21.0, 19.2 and 12.6 for the dropped edge benchmark, LOCC and LO, respectively. As before, we attribute the 6.6 additional errors of LOCC over LO to the delays and the CNOT gates in the teleportation circuit and cut Bell pairs. The LOCC results demonstrate how a dynamic quantum circuit in which two subcircuits are connected by a real-time classical link can be executed on two otherwise disjoint QPUs. The LO results could be obtained on a single device with 127 qubits at the cost of an additional factor of 2 in run-time as the subcircuits can be run successively.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Two connected QPUs with LOCC."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3: Two connected QPUs with LOCC.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig3_HTML.png?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="369"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p><b>a</b>, Graph state with periodic boundaries shown in three dimensions. The blue edges are the cut edges. <b>b</b>, Coupling map of two Eagle QPUs operated as a single device with 254 qubits. The purple nodes are the qubits forming the graph state in <b>a</b> and the blue nodes are used for cut Bell pairs. <b>c</b>,<b>d</b>, Absolute error on the stabilizers (<b>c</b>) and edge witnesses (<b>d</b>) implemented with LOCC (solid green) and LO (solid orange) and on a dropped edge benchmark graph (dotted-dashed red) for the graph state in <b>a</b>. In <b>c</b> and <b>d</b>, the stars show stabilizers and edge witnesses that are affected by the cuts. In <b>c</b> and <b>d</b>, the grey region is the probability mass corresponding to node stabilizers and edge witnesses, respectively, affected by the cut. In <b>c</b> and <b>d</b>, we observe that the LO implementation outperforms the dropped edge benchmark, which we attribute to better device conditions as these data were taken on a different day from the benchmark and LOCC data.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41586-024-08178-2/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Discussion and conclusion"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5">Discussion and conclusion</h2><div class="c-article-section__content" id="Sec5-content"><p>We implement long-range gates with LO and LOCC. With these gates, we engineer periodic boundary conditions on a 103-node planar lattice and connect two Eagle processors in real time to create a graph state on 134 qubits, going beyond the abilities of a single chip. Here, we chose to implement graph states as an application to highlight the scalable properties of dynamic circuits. Our cut Bell pair factories enable the LOCC scheme presented in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e1284">17</a></sup>. Both the LO and LOCC protocols deliver high-quality results that closely match a hardware-native benchmark. Circuit cutting increases the variance of measured observables. We can keep the variance under control in both the LO and LOCC schemes as indicated by the statistical tests on the witnesses. An in-depth discussion of the measured variance is found in the <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41586-024-08178-2#MOESM1">Supplementary Information</a>.</p><p>The variance increase from the QPD is why research now focuses on reducing the sampling overhead. It was recently shown that cutting multiple two-qubit gates in parallel results in optimal LO QPDs with the same sampling overhead as LOCC but requires an additional ancilla qubit and possibly reset<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Ufrecht, C. et al. Optimal joint cutting of two-qubit rotation gates. Phys. Rev. A 109, 052440 (2024)." href="/articles/s41586-024-08178-2#ref-CR30" id="ref-link-section-d109866146e1294">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Schmitt, L., Piveteau, C. & Sutter, D. Cutting circuits with multiple two-qubit unitaries. Preprint at arxiv.org/abs/2312.11638 (2023)." href="/articles/s41586-024-08178-2#ref-CR31" id="ref-link-section-d109866146e1297">31</a></sup>. In LOCC, the QPD is required only to cut the Bell pairs. This costly QPD could be removed, that is, no shot overhead, by distributing entanglement across multiple chips<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221–3224 (1997)." href="/articles/s41586-024-08178-2#ref-CR32" id="ref-link-section-d109866146e1301">32</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Duan, L.-M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001)." href="/articles/s41586-024-08178-2#ref-CR33" id="ref-link-section-d109866146e1304">33</a></sup>. In the near to medium term, this could be done by operating gates in the microwave regime over conventional cables<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Zhong, Y. et al. Deterministic multi-qubit entanglement in a quantum network. Nature 590, 571–575 (2021)." href="/articles/s41586-024-08178-2#ref-CR10" id="ref-link-section-d109866146e1308">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Magnard, P. et al. Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems. Phys. Rev. Lett. 125, 260502 (2020)." href="/articles/s41586-024-08178-2#ref-CR34" id="ref-link-section-d109866146e1311">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Niu, J. et al. Low-loss interconnects for modular superconducting quantum processors. Nat. Electron. 6, 235–241 (2023)." href="/articles/s41586-024-08178-2#ref-CR35" id="ref-link-section-d109866146e1314">35</a></sup> or, in the long term, with an optical-to-microwave transduction<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Orcutt, J. et al. Engineering electro-optics in SiGe/Si waveguides for quantum transduction. Quantum Sci. Technol. 5, 034006 (2020)." href="#ref-CR36" id="ref-link-section-d109866146e1318">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lauk, N. et al. Perspectives on quantum transduction. Quantum Sci. Technol. 5, 020501 (2020)." href="#ref-CR37" id="ref-link-section-d109866146e1318_1">37</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Krastanov, S. et al. Optically heralded entanglement of superconducting systems in quantum networks. Phys. Rev. Lett. 127, 040503 (2021)." href="/articles/s41586-024-08178-2#ref-CR38" id="ref-link-section-d109866146e1321">38</a></sup>. Entanglement distribution is typically noisy and may result in non-maximally entangled states. However, gate teleportation requires a maximally entangled resource. Nevertheless, non-maximally entangled states could lower the sampling cost of the QPD<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Bechtold, M., Barzen, J., Leymann, F. & Mandl, A. Circuit cutting with non-maximally entangled states. Preprint at arxiv.org/abs/2306.12084 (2023)." href="/articles/s41586-024-08178-2#ref-CR39" id="ref-link-section-d109866146e1325">39</a></sup> and multiple copies of non-maximally entangled states could be distilled into a pure state for teleportation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Bennett, C. H. et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722–725 (1996)." href="/articles/s41586-024-08178-2#ref-CR40" id="ref-link-section-d109866146e1330">40</a></sup> either during the execution of a quantum circuit or possibly during the delays between consecutive shots, which may be as large as 250 μs for resets<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Tornow, C., Kanazawa, N., Shanks, W. E. & Egger, D. J. Minimum quantum run-time characterization and calibration via restless measurements with dynamic repetition rates. Phys. Rev. Appl. 17, 064061 (2022)." href="/articles/s41586-024-08178-2#ref-CR41" id="ref-link-section-d109866146e1334">41</a></sup>. Combined with these settings, our error-mitigated and suppressed dynamic circuits would enable a modular quantum computing architecture without the sampling overhead of circuit cutting.</p><p>In an application setting, circuit cutting could benefit Hamiltonian simulation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Harrow, A. W. & Lowe, A. Optimal quantum circuit cuts with application to clustered Hamiltonian simulation. Preprint at arxiv.org/abs/2403.01018 (2024)." href="/articles/s41586-024-08178-2#ref-CR42" id="ref-link-section-d109866146e1341">42</a></sup>. Here, the cost of circuit cutting is exponential in the strength of the cut bonds times the evolution time. This cost may thus be reasonable for weak bonds and/or short evolution times. Furthermore, the LO scheme presented in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Harrow, A. W. & Lowe, A. Optimal quantum circuit cuts with application to clustered Hamiltonian simulation. Preprint at arxiv.org/abs/2403.01018 (2024)." href="/articles/s41586-024-08178-2#ref-CR42" id="ref-link-section-d109866146e1345">42</a></sup> requires ancilla qubits in a Hadamard test, which would require a reset through a dynamic circuit if the same bond is cut multiple times in a Trotterized time evolution.</p><p>Circuit cutting can be applied to both wires and gates. The resulting quantum circuits have a similar structure making our approach applicable to both cases. Our real-time classical link implements long-range gates and classically couples disjoint quantum processors. The cut Bell pairs that we present have values beyond our work. For example, these pairs are directly usable to cut circuits in measurement-based quantum computing, which relies on dynamic circuits<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Bäumer, E. et al. Efficient long-range entanglement using dynamic circuits. PRX Quantum 5, 030339 (2024)." href="/articles/s41586-024-08178-2#ref-CR14" id="ref-link-section-d109866146e1352">14</a></sup>. This could also be accomplished with LO; the result would be an execution setting identical to ours with dynamic circuits. Furthermore, the combination of staggered dynamical decoupling with zero-noise extrapolation mitigates the lengthy delays of the feed-forward operations, which enables a high-quality implementation of dynamic circuits. Our work sheds light on the noise sources, such as <i>Z</i><i>Z</i> cross-talk occurring during the latency, that a transpiler for distributed superconducting quantum computers must consider<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Caleffi, M. et al. Distributed quantum computing: a survey. Preprint at arxiv.org/abs/2212.10609 (2022)." href="/articles/s41586-024-08178-2#ref-CR43" id="ref-link-section-d109866146e1361">43</a></sup>. In summary, we demonstrate that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6">Methods</h2><div class="c-article-section__content" id="Sec6-content"><h3 class="c-article__sub-heading" id="Sec7">Circuit cutting</h3><p>The gates in a quantum circuit are quantum channels acting on density matrices <i>ρ</i>. A single quantum channel <span class="mathjax-tex">${\mathcal{E}}(\rho )$</span> is cut by expressing it as a sum over <i>I</i> quantum channels <span class="mathjax-tex">${{\mathcal{E}}}_{i}(\rho )$</span> resulting in the QPD</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\mathcal{E}}(\rho )=\mathop{\sum }\limits_{i=0}^{I-1}{a}_{i}{{\mathcal{E}}}_{i}(\rho ).$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>The channels <span class="mathjax-tex">${{\mathcal{E}}}_{i}(\rho )$</span> are easier to implement than <span class="mathjax-tex">${\mathcal{E}}(\rho )$</span> and are built from LO<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="/articles/s41586-024-08178-2#ref-CR16" id="ref-link-section-d109866146e1644">16</a></sup> or LOCC<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e1648">17</a></sup> (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1</a>). As some of the coefficients <i>a</i><sub><i>i</i></sub> are negative, we introduce <i>γ</i> = ∑<sub><i>i</i></sub><span class="stix">∣</span><i>a</i><sub><i>i</i></sub><span class="stix">∣</span> and <i>P</i><sub><i>i</i></sub> = <span class="stix">∣</span><i>a</i><sub><i>i</i></sub><span class="stix">∣</span>/<i>γ</i> to recover a valid probability distribution with probabilities <i>P</i><sub><i>i</i></sub> over the channels <span class="mathjax-tex">${{\mathcal{E}}}_{i}$</span>. Here, <i>γ</i> can be seen as the amount by which the QPD deviates from a true probability distribution and is thus a cost to pay to implement the QPD. Without a QPD an observable is estimated by <span class="mathjax-tex">$\langle O\rangle ={\rm{Tr}}\,\{O{\mathcal{E}}(\rho )\}$</span>. However, when using this QPD, we build an unbiased Monte Carlo estimator of <i>O</i> as</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\langle O\rangle }_{{\rm{QPD}}}=\gamma \mathop{\sum }\limits_{i=0}^{I-1}{P}_{i}{\rm{sign}}({a}_{i})\,\text{Tr}\,\{O{{\mathcal{E}}}_{i}(\rho )\}.$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>The variance of the QPD estimator <span class="stix">⟨</span><i>O</i><span class="stix">⟩</span><sub>QPD</sub> is a factor of <i>γ</i><sup>2</sup> larger than the variance of the non-cut estimator <span class="stix">⟨</span><i>O</i><span class="stix">⟩</span> (ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Cai, Z. et al. Quantum error mitigation. Rev. Mod. Phys. 95, 045005 (2023)." href="/articles/s41586-024-08178-2#ref-CR44" id="ref-link-section-d109866146e1960">44</a></sup>). When cutting <i>n</i> > 1 identical channels, we can build an estimator by taking the product of the QPDs for each individual channel, resulting in a <i>γ</i><sup>2<i>n</i></sup> rescaling factor<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017)." href="/articles/s41586-024-08178-2#ref-CR22" id="ref-link-section-d109866146e1975">22</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Endo, S., Benjamin, S. C. & Li, Y. Practical quantum error mitigation for near-future applications. Phys. Rev. X 8, 031027 (2018)." href="/articles/s41586-024-08178-2#ref-CR45" id="ref-link-section-d109866146e1978">45</a></sup>. This exponential increase in variance is compensated by a corresponding increase in the number of measured shots. Therefore, <i>γ</i><sup>2<i>n</i></sup> is called the sampling overhead and indicates that circuit cutting must be used sparingly. Details of the LO and LOCC quantum channels <span class="mathjax-tex">${{\mathcal{E}}}_{i}$</span> and their coefficients <i>a</i><sub><i>i</i></sub> are provided in sections ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec8">Virtual gates implemented with LO</a>’ and ‘<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41586-024-08178-2#Sec9">Virtual gates implemented with LOCC</a>’, respectively.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec8">Virtual gates implemented with LO</h3><p>Here, we discuss how to implement virtual CZ gates with LO<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="/articles/s41586-024-08178-2#ref-CR16" id="ref-link-section-d109866146e2039">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Singh, A. P. et al. Experimental demonstration of a high-fidelity virtual two-qubit gate. Phys. Rev. Res. 6, 013235 (2024)." href="/articles/s41586-024-08178-2#ref-CR18" id="ref-link-section-d109866146e2042">18</a></sup>. We follow ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="/articles/s41586-024-08178-2#ref-CR16" id="ref-link-section-d109866146e2046">16</a></sup> and, therefore, decompose each cut CZ gate into local operations and a sum over six different circuits defined by</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{array}{l}2{\rm{CZ}}\,=\sum _{\alpha \in \{\pm 1\}}{R}_{z}\left(\alpha \frac{\pi }{2}\right)\otimes {R}_{z}\left(\alpha \frac{\pi }{2}\right)\\ \,\,\,-\sum _{{\alpha }_{1},{\alpha }_{2}\in \{\pm 1\}}{\alpha }_{1}{\alpha }_{2}{R}_{z}\left(-\frac{{\alpha }_{1}+1}{2}\pi \right)\otimes \left(\frac{I+{\alpha }_{2}Z}{2}\right)\\ \,\,\,-\sum _{{\alpha }_{1},{\alpha }_{2}\in \{\pm 1\}}{\alpha }_{1}{\alpha }_{2}\left(\frac{I+{\alpha }_{1}Z}{2}\right)\otimes {R}_{z}\left(-\frac{{\alpha }_{2}+1}{2}\pi \right),\end{array}$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>where <span class="mathjax-tex">${R}_{z}(\theta )=\exp \left(-{\rm{i}}\frac{\theta }{2}Z\right)$</span> are virtual Z rotations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient Z gates for quantum computing. Phys. Rev. A 96, 022330 (2017)." href="/articles/s41586-024-08178-2#ref-CR46" id="ref-link-section-d109866146e2673">46</a></sup>. The factor 2 in front of CZ is for readability. Each of the possible six circuits is thus weighted by a 1/6 probability (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig4">1</a>). The operations (<i>I</i> + <i>Z</i>)/2 and (<i>I</i> − <i>Z</i>)/2 correspond to the projectors |0<span class="stix">⟩</span> <span class="stix">⟨</span>0| and |1<span class="stix">⟩</span> <span class="stix">⟨</span>1|, respectively. They are implemented by MCMs and classical post-processing. More specifically, when computing the expectation value of an observable <span class="stix">⟨</span><i>O</i><span class="stix">⟩</span> = ∑<sub><i>i</i></sub><i>a</i><sub><i>i</i></sub><span class="stix">⟨</span><i>O</i><span class="stix">⟩</span><sub><i>i</i></sub> with the LO QPD, we multiply the expectation values <span class="stix">⟨</span><i>O</i><span class="stix">⟩</span><sub><i>i</i></sub> by 1 and −1 when the outcome of an MCM is 0 and 1, respectively.</p><p>In the experiments that implement graph states with LO in the main text, we implement the CZ gate with six circuits built from <i>R</i><sub><i>z</i></sub> gates and MCMs<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. New J. Phys. 23, 023021 (2021)." href="/articles/s41586-024-08178-2#ref-CR16" id="ref-link-section-d109866146e2729">16</a></sup>. Cutting four CZ gates with LO thus requires <i>I</i> = 6<sup>4</sup> = 1,296 circuits. However, as the node and edge stabilizers of the graph states are at most in the light cone<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Tran, M. C. et al. Hierarchy of linear light cones with long-range interactions. Phys. Rev. X 10, 031009 (2020)." href="/articles/s41586-024-08178-2#ref-CR47" id="ref-link-section-d109866146e2738">47</a></sup> of one virtual gate, we instead implement two QPDs in parallel, which requires <i>I</i> = 6<sup>2</sup> = 36 LO circuits per expectation value. In general, sampling from a QPD results in an overhead of <span class="mathjax-tex">${({\sum }_{i=0}^{I-1}| {a}_{i}| )}^{2}$</span>, where <i>I</i> is the number of circuits in the QPD and the <i>a</i><sub><i>i</i></sub> are the QPD coefficients<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Cai, Z. et al. Quantum error mitigation. Rev. Mod. Phys. 95, 045005 (2023)." href="/articles/s41586-024-08178-2#ref-CR44" id="ref-link-section-d109866146e2837">44</a></sup>. However, as the LO QPDs in our experiments have only 36 circuits, we fully enumerate the QPDs by executing all 36 circuits. The sampling cost of full enumeration is <span class="mathjax-tex">$I({\sum }_{i=0}^{I-1}| {a}_{i}{| }^{2})$</span>. Furthermore, as <span class="stix">∣</span><i>a</i><sub><i>i</i></sub><span class="stix">∣</span> = 1/2 <span class="stix">∀</span> <i>i</i> = 0, …, <i>I</i> − 1, sampling from the QPD and fully enumerating it both have the same shot overhead.</p><p>The decomposition in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ3">3</a>) with <i>γ</i><sup>2</sup> = 9 is optimal with respect to the sampling overhead for a single gate<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Piveteau, C. & Sutter, D. Circuit knitting with classical communication. IEEE Trans. Inf. Theor. 1, 2734–2745 (2023)." href="/articles/s41586-024-08178-2#ref-CR17" id="ref-link-section-d109866146e2949">17</a></sup>. Recently, refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Ufrecht, C. et al. Optimal joint cutting of two-qubit rotation gates. Phys. Rev. A 109, 052440 (2024)." href="/articles/s41586-024-08178-2#ref-CR30" id="ref-link-section-d109866146e2953">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Schmitt, L., Piveteau, C. & Sutter, D. Cutting circuits with multiple two-qubit unitaries. Preprint at arxiv.org/abs/2312.11638 (2023)." href="/articles/s41586-024-08178-2#ref-CR31" id="ref-link-section-d109866146e2956">31</a></sup> found a new protocol that achieves the same <i>γ</i> overhead as LOCC when cutting multiple gates in parallel. The proofs in refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Ufrecht, C. et al. Optimal joint cutting of two-qubit rotation gates. Phys. Rev. A 109, 052440 (2024)." href="/articles/s41586-024-08178-2#ref-CR30" id="ref-link-section-d109866146e2964">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Schmitt, L., Piveteau, C. & Sutter, D. Cutting circuits with multiple two-qubit unitaries. Preprint at arxiv.org/abs/2312.11638 (2023)." href="/articles/s41586-024-08178-2#ref-CR31" id="ref-link-section-d109866146e2967">31</a></sup> are theoretical demonstrating the existence of a decomposition.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec9">Virtual gates implemented with LOCC</h3><p>We now discuss the implementation of the dynamic circuits that enable the virtual gates with LOCC. We first present an error suppression and mitigation of dynamic circuits with dynamical decoupling (DD) and zero-noise extrapolation (ZNE). Second, we discuss the methodology to create the cut Bell pairs and present the circuits to implement one, two and three cut Bell pairs. Finally, we propose a simple benchmark experiment to assess the quality of a virtual gate.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec10">Error-mitigated quantum circuit switch instructions</h4><p>All quantum circuits presented in this work are written in Qiskit. The feed-forward operations of the LOCC circuits are executed with a quantum circuit switch instruction, hereafter referred to as a switch. A switch defines a set of cases in which the quantum circuit can branch depending on the outcome of a corresponding set of measurements. This branching occurs in real time for each experimental shot, with the measurement outcomes being collected by a central processor, which in turn broadcasts the selected case (here corresponding to a combination of <i>X</i> and <i>Z</i> gates) to all control instruments.</p><p>As quantum computing scales, the control electronics become tailored to its QPU and are no longer built from off-the-shelf components. Recent IBM devices have a single QPU with a rack of dedicated and tailored control electronics, as shown in refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024)." href="/articles/s41586-024-08178-2#ref-CR29" id="ref-link-section-d109866146e2995">29</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Zettles, G., Willenborg, S., Johnson, B. R., Wack, A. & Allison, B. 26.2 Design considerations for superconducting quantum systems. In 2022 IEEE International Solid-State Circuits Conference (ISSCC), Vol. 65, pp. 1–3 (IEEE, 2022)." href="/articles/s41586-024-08178-2#ref-CR48" id="ref-link-section-d109866146e2998">48</a></sup>. The realization of the feed-forward we present builds upon the work in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024)." href="/articles/s41586-024-08178-2#ref-CR29" id="ref-link-section-d109866146e3002">29</a></sup> and advances its scalability in two main ways. First, our development enables the synchronization and inter-communication between separate experimental setups. Not only are the control instruments for the two sub-QPUs located in different racks, but they are also configurable in software to operate on them independently for the LO experiments and recombined for LOCC. This architecture is extensible to multiple racks and QPUs. It overcomes several of the challenges in operating a distributed control system as pointed out in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Reilly, D. J. Challenges in scaling-up the control interface of a quantum computer. Preprint at arxiv.org/abs/1912.05114 (2019)." href="/articles/s41586-024-08178-2#ref-CR23" id="ref-link-section-d109866146e3006">23</a></sup>. Second, the duration of the conditional operation is independent of the measurement results, of which qubits are measured, and which qubits are subject to the conditional operations (apart from minor differences due to cable lengths). This enables the scheduling and execution of programs equally across the combined QPU as if it were a single one.</p><p>The feed-forward process results in a latency of the order of 0.5 μs (independent of the selected case) during which no gates can be applied (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2a</a>, red area). Free evolution during this period (<i>τ</i>), often dominated by static <i>Z</i><i>Z</i> cross-talk in the Hamiltonian, typically with a strength ranging from about 10<sup>3</sup> Hz to 10<sup>4</sup> Hz, substantially deteriorates results. To cancel this unwanted interaction and any other constant or slowly fluctuating <i>I</i><i>Z</i> or <i>Z</i><i>I</i> terms, we precede the conditional gates with a staggered DD <i>X</i>–<i>X</i> sequence, adding 3<i>τ</i> to the switch duration (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2a</a>). The value of <i>τ</i> is determined by the longest latency path from one QPU to the other and is fine-tuned by maximizing the signal on such a DD sequence. Furthermore, we mitigate the effect of the overall delay on the observables of interest with ZNE<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017)." href="/articles/s41586-024-08178-2#ref-CR22" id="ref-link-section-d109866146e3055">22</a></sup>. To do this, we first stretch the switch duration by a factor <i>c</i> = (<i>τ</i> + <i>δ</i>)/<i>τ</i>, where <i>δ</i> is a variable delay added before each <i>X</i> gate in the DD sequence (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2a</a>). Second, we extrapolate the stabilizer values to the zero-delay limit <i>c</i> = 0 with a linear fit. In many cases, an exponential fit can be justified<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023)." href="/articles/s41586-024-08178-2#ref-CR1" id="ref-link-section-d109866146e3084">1</a></sup>; however, we observe in our benchmark experiments that a linear fit is appropriate (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2</a>). Without DD, we observe strong oscillations in the measured stabilizers that prevent an accurate ZNE (see the <i>X</i><i>Z</i> stabilizer in Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2c</a>). As seen in the main text, this error suppression and mitigation reduce the error on the stabilizers affected by virtual gates.</p><p>The error suppression and mitigation that we implement for the switch also apply to other control flow statements. The switch is not the only instruction capable of representing control flow. For instance, OpenQASM3<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Cross, A. et al. OpenQASM 3: a broader and deeper quantum assembly language. ACM Trans. Quantum Comput. 3, 1–50 (2022)." href="/articles/s41586-024-08178-2#ref-CR49" id="ref-link-section-d109866146e3103">49</a></sup> supports if/else statements. Our scheme is done by (1) adding DD sequences to the latency (possibly by adding delays if the control electronics cannot emit pulses during the latency); (2) stretching the delay; and (3) extrapolating to the zero-delay limit.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec11">Cut Bell pair factories</h4><p>Here, we discuss the quantum circuits to prepare the cut Bell pairs needed to realize virtual gates with LOCC. To create a factory for <i>k</i> cut Bell pairs, we must find a linear combination of circuits with two disjoint partitions with <i>k</i> qubits each to reproduce the statistics of Bell pairs. We create the state <i>ρ</i><sub><i>k</i></sub> of the Bell pairs following ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Vidal, G. & Tarrach, R. Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999)." href="/articles/s41586-024-08178-2#ref-CR50" id="ref-link-section-d109866146e3127">50</a></sup> such that <span class="mathjax-tex">${\rho }_{k}=(1+{t}_{k}){\rho }_{k}^{+}-{t}_{k}{\rho }_{k}^{-}$</span>, where <i>t</i><sub><i>k</i></sub> = 2<sup><i>k</i></sup> − 1. Here, <span class="mathjax-tex">${\rho }_{k}^{\pm }$</span> are mixed states separable with respect to the partitions <i>A</i> and <i>B</i>. Note that <i>ρ</i><sub><i>k</i></sub> entangles the qubit partitions <i>A</i> and <i>B</i>, shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1c</a>, but <span class="mathjax-tex">${\rho }_{k}^{\pm }$</span> do not. The total cost of this QPD with two states is determined by <i>γ</i><sub><i>k</i></sub> = 2<i>t</i><sub><i>k</i></sub> + 1. Next, we realize <span class="mathjax-tex">${\rho }_{k}^{\pm }$</span> from a probabilistic mixture of pure states <span class="mathjax-tex">${\rho }_{k,i}^{\pm }$</span>, that is, valid probability distributions. The state <span class="mathjax-tex">${\rho }_{k}^{-}$</span> is easily implemented by a uniform mixture of all basis states that correspond to a 0 entry on the diagonal of the density matrix <i>ρ</i><sub><i>k</i></sub>. The basis states themselves do not appear in <i>ρ</i><sub><i>k</i></sub>. We thus implement <span class="mathjax-tex">${\rho }_{k}^{-}$</span> as a diagonal density matrix of <span class="mathjax-tex">${n}_{k}^{-}={4}^{k}-{2}^{k}$</span> basis states. The state <span class="mathjax-tex">${\rho }_{k}^{+}$</span> is harder to engineer. It requires a probabilistic mixture of intricate states with entanglement within each partition <i>A</i> and <i>B</i> but not between them. To engineer <span class="mathjax-tex">${\rho }_{k}^{+}$</span>, we thus build a parametric quantum circuit <i>C</i><sub><i>k</i></sub>(<b>θ</b><sup><i>i</i></sup>) with parameters <b>θ</b><sup><i>i</i></sup> in which no two-qubit gate connects qubits between <i>A</i> and <i>B</i>. Following ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Vidal, G. & Tarrach, R. Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999)." href="/articles/s41586-024-08178-2#ref-CR50" id="ref-link-section-d109866146e3668">50</a></sup>, we need <span class="mathjax-tex">${n}_{k}^{+}={2}^{{2}^{k}}-1$</span> pure states to realize <span class="mathjax-tex">${\rho }_{k}^{+}$</span>. The exact form of <span class="mathjax-tex">${\rho }_{k}^{+}$</span>, omitted here for brevity, is given in Appendix B of ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Vidal, G. & Tarrach, R. Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999)." href="/articles/s41586-024-08178-2#ref-CR50" id="ref-link-section-d109866146e3805">50</a></sup>. Therefore, the total number of parameter sets <span class="mathjax-tex">$I={n}_{k}^{+}+{n}_{k}^{-}$</span> required to implement one, two and three cut Bell pairs is 5, 27 and 311, respectively. Finally, the coefficients <i>a</i><sub><i>i</i>,<i>k</i></sub> of all the circuits in the QPD in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ1">1</a>) that implement <span class="mathjax-tex">${\rho }_{k}^{\pm }$</span> are</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${a}_{i,k}=\frac{1+{t}_{k}}{{n}_{k}^{+}},\,\,{\rm{for}}\,\,i\in \{0,...,{n}_{k}^{+}-1\},\,{\rm{and}}$$</span></div><div class="c-article-equation__number"> (4) </div></div><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${a}_{i,k}=-\frac{{t}_{k}}{{n}_{k}^{-}},\,\,{\rm{for}}\,\,i\in \{{n}_{k}^{+},...,{n}_{k}^{+}+{n}_{k}^{-}-1\}.$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>For <i>k</i> = 2, the resulting weights, <span class="stix">∣</span><i>a</i><sub><i>i</i>,<i>k</i></sub><span class="stix">∣</span>/<i>γ</i><sub><i>k</i></sub> are approximately all equal. There is thus no practical difference between sampling and enumerating the <i>k</i> = 2 QPD when executing it on hardware. More precisely, for the factories with two cut Bell pairs that we run on hardware, the cost of sampling the QPD is <span class="mathjax-tex">${({\sum }_{i=0}^{I-1}| {a}_{i,2}| )}^{2}={\gamma }_{2}^{2}(1+1.6\times 1{0}^{-7})$</span> and the cost of fully enumerating the QPD is <span class="mathjax-tex">$I({\sum }_{i=0}^{I-1}| {a}_{i,2}{| }^{2})={\gamma }_{2}^{2}(1+1.0\times 1{0}^{-3})$</span>, where <i>γ</i><sub>2</sub> = 7.</p><p>We construct all pure states <span class="mathjax-tex">${\rho }_{k,i}^{\pm }$</span> from the same template variational quantum circuit <i>C</i><sub><i>k</i></sub>(<b>θ</b><sup><i>i</i></sup>) with parameters <b>θ</b><sup><i>i</i></sup>, where the index <i>i</i> = 0, …, <i>I</i> − 1 runs over the <i>I</i> elements of the probabilistic mixtures defining <span class="mathjax-tex">${\rho }_{k}^{\pm }$</span>. The parameters <b>θ</b><sup><i>i</i></sup> in the template circuits <i>C</i><sub><i>k</i></sub>(<b>θ</b><sup><i>i</i></sup>) are optimized by the SLSQP classical optimizer<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Kraft, D. A Software Package for Sequential Quadratic Programming (DFVLR, 1988)." href="/articles/s41586-024-08178-2#ref-CR51" id="ref-link-section-d109866146e4685">51</a></sup> by minimizing the <i>L</i><sub>2</sub>-norm with respect to the <i>I</i> pure target states needed to represent <span class="mathjax-tex">${\rho }_{k}^{+}$</span>, where the norm is evaluated with a classical state vector simulation. After testing various approaches, we find that those provided in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1c</a> and Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig6">3</a> enable us to achieve an error, based on the <i>L</i><sub>2</sub> norm, of less than 10<sup>−8</sup> for each state while having minimal hardware requirements. To enable rapid execution of the QPD with parametric updates, all the parameters are the angles of virtual <i>Z</i> rotations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient Z gates for quantum computing. Phys. Rev. A 96, 022330 (2017)." href="/articles/s41586-024-08178-2#ref-CR46" id="ref-link-section-d109866146e4746">46</a></sup> (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1c</a>). As <span class="mathjax-tex">${\rho }_{k}^{-}$</span> is built from basis states, we analytically derive the parameters. Therefore, we could also significantly simplify the ansatz <i>C</i><sub><i>k</i></sub>(<b>θ</b><sup><i>i</i></sup>), for example, by cancelling CNOT gates. However, we keep the same template for compilation and execution efficiency. On first inspection, the parameters entering <span class="mathjax-tex">${\rho }_{k}^{+}$</span> do not have any usable structure. We thus leave it up to future research to further investigate whether these parameters have any structure that could be leveraged to simplify the cut Bell pair factories.</p><p>A single-cut Bell pair is engineered by applying the gates <i>U</i>(<i>θ</i><sub>0</sub>, <i>θ</i><sub>1</sub>) and <i>U</i>(<i>θ</i><sub>2</sub>, <i>θ</i><sub>3</sub>) on qubits 0 and 1. Here, and in the figures, the gate <i>U</i>(<i>θ</i>, <i>ϕ</i>) corresponds to <span class="mathjax-tex">$\sqrt{X}{R}_{z}(\theta )\sqrt{X}{R}_{z}(\phi )$</span>. The QPD of a single-cut Bell pair requires five sets of parameters given by {[π/2, 0, π/2, 0], [π/2, −2π/3, π/2, 2π/3], [π/2, 2π/3, π/2, −2π/3], [π, 0, 0, 0], [0, 0, π, 0]} which could also be derived analytically. The circuits to simultaneously create two and three cut Bell pairs are shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1c</a> and Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig6">3</a>, respectively. The circuits and the values of the parameters as obtained by the optimizer are available on GitHub (<a href="http://www.github.com/eggerdj/cut_graph_state_data">www.github.com/eggerdj/cut_graph_state_data</a>).</p><p>In the experiments that implement graph states with LOCC in the main text, we construct two QPDs in parallel with <i>I</i> = 27 circuits, each QPD implementing two long-range CZ gates. This execution is similar to the LO execution in which we also execute two QPDs in parallel.</p><h4 class="c-article__sub-heading c-article__sub-heading--small" id="Sec12">Benchmarking qubits for LOCC</h4><p>The quality of a CNOT gate implemented with dynamic circuits depends on hardware properties. For example, qubit relaxation, dephasing and static <i>Z</i><i>Z</i> cross-talk all negatively affect the qubits during the idle time of the switch. Furthermore, measurement quality also affects virtual gates implemented with LOCC. Errors on MCMs are harder to correct than errors on final measurements as they propagate to the rest of the circuit through the conditional gates<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Gupta, R. S. et al. Probabilistic error cancellation for dynamic quantum circuits. Phys. Rev. A 109, 062617 (2024)." href="/articles/s41586-024-08178-2#ref-CR52" id="ref-link-section-d109866146e4985">52</a></sup>. For instance, assignment errors during readout result in an incorrect application of a single-qubit <i>X</i> or <i>Z</i> gate. Given the variability in these qubit properties, care must be taken in selecting those to act as cut Bell pairs. To determine which qubits will perform well as cut Bell pairs, we develop a fast characterization experiment on four qubits that does not require a QPD or error mitigation. This experiment creates a graph state between qubits 0 and 3 by consuming an uncut Bell pair created on qubits 1 and 2 with a Hadamard and a CNOT gate. We measure the stabilizers <i>Z</i><i>X</i> and <i>X</i><i>Z</i> which require two different measurement bases. The resulting circuit, shown in Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig7">4a</a>, is structurally equivalent to half of the circuit that consumes two cut Bell pairs, for example, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig1">1c</a>. We execute this experiment on all qubit chains of length four on the devices that we use and report the mean squared error (MSE), that is, [(<span class="stix">⟨</span><i>Z</i><i>X</i><span class="stix">⟩</span> − 1)<sup>2</sup> + (<span class="stix">⟨</span><i>X</i><i>Z</i><span class="stix">⟩</span> − 1)<sup>2</sup>]/2 as a quality metric. The lower the MSE is the better the set of qubits act as cut Bell pairs. With this experiment we benchmark, ibm_kyiv (the device used to create the graph state with 103 nodes), and ibm_pinguino-1a and ibm_pinguino-1b (the two Eagle QPUs combined into a single device, named ibm_pinguino-2a, used to create the graph state with 134 nodes). We observe more than an order of magnitude variation in MSE across each device (Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig7">4b</a>).</p><p>The qubits we chose to act as cut Bell pairs are a tradeoff between the graph we want to engineer and the quality of the MSE benchmark. For example, the graphs with periodic boundary conditions presented in the main text were designed first based on the desired shape of |<i>G</i><span class="stix">⟩</span> and second based on the MSE of the Bell pair quality test.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec13">Graph states</h3><p>A graph state |<i>G</i><span class="stix">⟩</span> is created from a graph <i>G</i> = (<i>V</i>, <i>E</i>) with nodes <i>V</i> and edges <i>E</i> by applying an initial Hadamard gate to each qubit, corresponding to a node in <i>V</i>, and then CZ gates to each pair of qubits (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> (refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title="Briegel, H. J. & Raussendorf, R. Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910–913 (2001)." href="/articles/s41586-024-08178-2#ref-CR53" id="ref-link-section-d109866146e5076">53</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="Hein, M., Eisert, J. & Briegel, H. J. Multiparty entanglement in graph states. Phys. Rev. A 69, 062311 (2004)." href="/articles/s41586-024-08178-2#ref-CR54" id="ref-link-section-d109866146e5079">54</a></sup>). The resulting state |<i>G</i><span class="stix">⟩</span> has <span class="stix">∣</span><i>V</i><span class="stix">∣</span> first-order stabilizers, one for each node <i>i</i> <span class="stix">∈</span> <i>V</i>, defined by <i>S</i><sub><i>i</i></sub> = <i>X</i><sub><i>i</i></sub>∏<sub><i>k</i><span class="stix">∈</span><i>N</i>(<i>i</i>)</sub><i>Z</i><sub><i>k</i></sub>. Here, <i>N</i>(<i>i</i>) is the neighbourhood of node <i>i</i> defined by <i>E</i>. These stabilizers satisfy <i>S</i><sub><i>i</i></sub>|<i>G</i><span class="stix">⟩</span> = |<i>G</i><span class="stix">⟩</span>. By construction, any product of stabilizers is also a stabilizer. If an edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> is not implemented by a CZ gate, the corresponding stabilizers drop to zero, that is, <span class="stix">⟨</span><i>S</i><sub><i>i</i></sub><span class="stix">⟩</span> = <span class="stix">⟨</span><i>S</i><sub><i>j</i></sub><span class="stix">⟩</span> = 0. This effect can be seen in the dropped edge benchmark, see, for example, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b</a>.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec14">Entanglement witness</h3><p>We now define a success criterion for the implementation of a graph state with entanglement witnesses<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Jungnitsch, B., Moroder, T. & Gühne, O. Entanglement witnesses for graph states: general theory and examples. Phys. Rev. A 84, 032310 (2011)." href="/articles/s41586-024-08178-2#ref-CR55" id="ref-link-section-d109866146e5185">55</a></sup>. A witness <span class="mathjax-tex">${\mathcal{W}}$</span> is designed to detect a certain form of entanglement. As we cut edges in the graph state, we focus on witnesses <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> over two nodes <i>i</i> and <i>j</i> connected by an edge in <i>E</i>. An edge (<i>i</i>, <i>j</i>) of our graph state |<i>G</i><span class="stix">⟩</span> presents entanglement if the expectation value <span class="mathjax-tex">$\langle {{\mathcal{W}}}_{i,j}\rangle < 0$</span>. The witness does not detect entanglement if <span class="mathjax-tex">$\langle {{\mathcal{W}}}_{i,j}\rangle \ge 0$</span>. The first-order stabilizers of nodes <i>i</i> and <i>j</i> with (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> are</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${S}_{i}={Z}_{j}{X}_{i}\prod _{k\in N(i)\backslash j}{Z}_{k}\,\text{and}\,{S}_{j}={X}_{j}{Z}_{i}\prod _{k\in N(j)\backslash i}{Z}_{k}.$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>Here, <i>N</i>(<i>i</i>) is the neighbourhood of node <i>i</i>, which includes <i>j</i> because (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i>. Thus, <i>N</i>(<i>i</i>)\<i>j</i> is the neighbourhood of node <i>i</i> excluding <i>j</i>. Following refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Jungnitsch, B., Moroder, T. & Gühne, O. Entanglement witnesses for graph states: general theory and examples. Phys. Rev. A 84, 032310 (2011)." href="/articles/s41586-024-08178-2#ref-CR55" id="ref-link-section-d109866146e5614">55</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="Tóth, G. & Gühne, O. Entanglement detection in the stabilizer formalism. Phys. Rev. A 72, 022340 (2005)." href="/articles/s41586-024-08178-2#ref-CR56" id="ref-link-section-d109866146e5617">56</a></sup>, we build an entanglement witness for edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> as</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{\mathcal{W}}}_{i,j}=\frac{1}{4}{\mathbb{I}}-\frac{1}{4}(\langle {S}_{i}\rangle +\langle {S}_{j}\rangle +\langle {S}_{i}{S}_{j}\rangle ).$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>This witness is zero or positive if the states are separable. Alternatively, as in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Zander, R. & Becker, C. K.-U. Benchmarking multipartite entanglement generation with graph states. Preprint at arxiv.org/abs/2402.00766 (2024)." href="/articles/s41586-024-08178-2#ref-CR27" id="ref-link-section-d109866146e5802">27</a></sup>, a witness for bi-separability is also given by</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{\mathcal{W}}}_{i,j}^{{\prime} }={\mathbb{I}}-\langle {S}_{i}\rangle -\langle {S}_{j}\rangle .$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>Here, we consider both witnesses. The data in the main text are presented for <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>. As discussed in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="Tóth, G. & Gühne, O. Entanglement detection in the stabilizer formalism. Phys. Rev. A 72, 022340 (2005)." href="/articles/s41586-024-08178-2#ref-CR56" id="ref-link-section-d109866146e5943">56</a></sup>, <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> is more robust to noise than <span class="mathjax-tex">${{\mathcal{W}}}_{i,\,j}^{{\prime} }$</span>. However, <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> requires more experimental effort to measure than <span class="mathjax-tex">${{\mathcal{W}}}_{i,\,j}^{{\prime} }$</span> because of the stabilizer <i>S</i><sub><i>i</i></sub><i>S</i><sub><i>j</i></sub>.</p><p>For completeness, we now show how a witness can detect entanglement by focusing on <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>. A separable state satisfies <span class="mathjax-tex">$\langle {P}_{1}...{P}_{n}\rangle ={\prod }_{i}\langle {P}_{i}\rangle $</span>, where <i>P</i><sub><i>i</i></sub> are single-qubit Pauli operators. Therefore, we can show, using the Cauchy–Schwarz inequality, that <span class="mathjax-tex">$\langle {S}_{i}\rangle +\langle {S}_{j}\rangle +\langle {S}_{i}{S}_{j}\rangle \le 1$</span> and that <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}\ge 0$</span> for separable states.</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\langle {S}_{i}\rangle +\langle {S}_{j}\rangle +\langle {S}_{i}{S}_{j}\rangle =\langle {Z}_{j}\rangle \langle {X}_{i}\rangle \prod _{k\in N(i)\backslash j}\langle {Z}_{k}\rangle $$</span></div><div class="c-article-equation__number"> (9) </div></div><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$+\langle {X}_{j}\rangle \langle {Z}_{i}\rangle \prod _{k\in N(j)\backslash i}\langle {Z}_{k}\rangle +\langle {Y}_{i}\rangle \langle {Y}_{j}\rangle \prod _{k\in M(i,j)}\langle {Z}_{k}\rangle $$</span></div><div class="c-article-equation__number"> (10) </div></div><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\le | \langle {Z}_{j}\rangle | | \langle {X}_{i}\rangle | +| \langle {X}_{j}\rangle | | \langle {Z}_{i}\rangle | +| \langle {Y}_{j}\rangle | | \langle {Y}_{i}\rangle | $$</span></div><div class="c-article-equation__number"> (11) </div></div><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\le \sqrt{{\langle {X}_{i}\rangle }^{2}+{\langle {Y}_{i}\rangle }^{2}+{\langle {Z}_{i}\rangle }^{2}}\sqrt{{\langle {X}_{j}\rangle }^{2}+{\langle {Y}_{j}\rangle }^{2}+{\langle {Z}_{j}\rangle }^{2}}$$</span></div><div class="c-article-equation__number"> (12) </div></div><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\le 1.$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>The step from equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ10">10</a>) to equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ11">11</a>) relies on ∏<sub><i>i</i></sub><i>a</i><sub><i>i</i></sub> ≤ ∏<sub><i>i</i></sub> <span class="stix">∣</span><i>a</i><sub><i>i</i></sub><span class="stix">∣</span> and that <span class="mathjax-tex">${\prod }_{k}| \langle {Z}_{k}\rangle | \le 1$</span>, where the product runs over nodes that do not contain <i>i</i> or <i>j</i>. The step from equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ11">11</a>) to equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ12">12</a>) is based on the Cauchy–Schwarz inequality. The final step relies on the fact that <span class="mathjax-tex">${\langle {X}_{i}\rangle }^{2}+{\langle {Y}_{i}\rangle }^{2}+{\langle {Z}_{i}\rangle }^{2}\le 1$</span> with pure states equal to one. Therefore, the witness <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> will be negative if the state is not separable.</p><p>In the graph states presented in the main text, we execute a statistical test at a 99% confidence level to detect entanglement. As discussed in the <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41586-024-08178-2#MOESM1">Supplementary Information</a> and shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b</a>, some witnesses may go below −1/2 because of readout error mitigation, the QPD and Switch ZNE. We, therefore, consider an edge to have the statistics of entanglement if the deviation from −1/2 is not statistically greater than ±1/2. Based on a one-tailed test, we consider that edge (<i>i</i>, <i>j</i>) is bi-partite entangled if</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$-\frac{1}{2}+\left|\langle {{\mathcal{W}}}_{i,j}\rangle +\frac{1}{2}\right|+{z}_{99 \% }{\sigma }_{{\mathcal{W}},i,j} < 0.$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>Similarly, we form a success criterion based on <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}^{{\prime} }$</span> as</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$-1+| \langle {{\mathcal{W}}}_{i,j}^{{\prime} }\rangle +1| +{z}_{99 \% }{\sigma }_{{{\mathcal{W}}}^{{\prime} },i,j} < 0.$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>This criterion penalizes any deviation from −1, that is, the most negative value that <span class="mathjax-tex">${{\mathcal{W}}}_{i,\,j}^{{\prime} }$</span> can have. Here, <i>z</i><sub>99%</sub> = 2.326 is the <i>z</i>-score of a Gaussian distribution at a 99% confidence level and <span class="mathjax-tex">${\sigma }_{{\mathcal{W}},i,j}$</span> is the standard deviation of edge witness <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>. These tests are conservative as they penalize any deviation from the ideal values. Moreover, these tests are most suitable for circuit cutting because the QPD may increase the variance <span class="mathjax-tex">${\sigma }_{{{\mathcal{W}}}_{i,j}}$</span> of the measured witnesses. Therefore, the statistics of entanglement are detected only if the mean of a witness is sufficiently negative and its standard deviation is sufficiently small. An edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i> fails the criteria if equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ14">14</a>) or equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41586-024-08178-2#Equ15">15</a>) is not satisfied. All edges in <i>E</i>, including the cut edges, pass the test based on <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span> when implemented with LO and LOCC (Extended Data Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41586-024-08178-2#Tab2">2</a>). However, some edges fail the test based on <span class="mathjax-tex">${{\mathcal{W}}}_{i,\,j}^{{\prime} }$</span> because of the lower noise robustness of <span class="mathjax-tex">${{\mathcal{W}}}_{i,\,j}^{{\prime} }$</span> compared with <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec15">Circuit count for stabilizer measurements</h3><p>Obtaining the bipartite entanglement witnesses requires measuring the expectation values of <span class="stix">⟨</span><i>S</i><sub><i>i</i></sub><span class="stix">⟩</span>, <span class="stix">⟨</span><i>S</i><sub><i>j</i></sub><span class="stix">⟩</span> and <span class="stix">⟨</span><i>S</i><sub><i>i</i></sub><i>S</i><sub><i>j</i></sub><span class="stix">⟩</span> of each edge (<i>i</i>, <i>j</i>) <span class="stix">∈</span> <i>E</i>. For the 103- and 134-node graphs presented in the main text, all 219- and 278-node and edge stabilizers, respectively, can be measured in <i>N</i><sub>S</sub> = 7 groups of commuting observables. To mitigate final measurement readout errors, we use twirled readout error extinction (TREX) with <i>N</i><sub>TREX</sub> samples<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="van den Berg, E., Minev, Z. K. & Temme, K. Model-free readout-error mitigation for quantum expectation values. Phys. Rev. A 105, 032620 (2022)." href="/articles/s41586-024-08178-2#ref-CR57" id="ref-link-section-d109866146e8307">57</a></sup>. When virtual gates are used with LO and LOCC, we require <i>I</i><sub>LO</sub> and <i>I</i><sub>LOCC</sub> more circuits, respectively. In this work, we fully enumerate the QPD. Furthermore, for LOCC, we mitigate the delay of the switch instruction with ZNE based on <i>N</i><sub>ZNE</sub> stretch factors. Therefore, the four types of experiments are executed with the following number of circuits.</p><ul class="u-list-style-none"> <li> <p>Swaps: <i>N</i><sub>S</sub><i>N</i><sub>TREX</sub></p> </li> <li> <p>Dropped edge: <i>N</i><sub>S</sub><i>N</i><sub>TREX</sub></p> </li> <li> <p>LO: <i>N</i><sub>S</sub><i>N</i><sub>TREX</sub><i>I</i><sub>LO</sub></p> </li> <li> <p>LOCC: <i>N</i><sub>S</sub><i>N</i><sub>TREX</sub><i>I</i><sub>LOCC</sub><i>N</i><sub>ZNE</sub></p> </li> </ul><p>In the experiments for the 103- and 134-node graph states, we use <i>N</i><sub>TREX</sub> = 5 and 3 TREX samples, respectively. Therefore, measuring the stabilizers without a QPD requires <i>N</i><sub>S</sub> × <i>N</i><sub>TREX</sub> = 35 circuits for the 103-node graph. For LO and LOCC, measuring the stabilizers for the graphs in the main text requires 6<sup>4</sup> and 27<sup>2</sup> circuits, respectively. However, owing to the graph structure, each edge witness is only ever in the light cone of two cut gates at most. We may thus execute a total of <i>I</i><sub>LO</sub> = 6<sup>2</sup> and <i>I</i><sub>LOCC</sub> = 27 circuits for LO and LOCC, respectively, based on the light cone of the gates. For higher-weight observables, this corresponds to sampling the diagonal terms of a joint QPD. Therefore, measuring the stabilizers with LO requires <i>N</i><sub>S</sub> × <i>N</i><sub>TREX</sub> × <i>I</i><sub>LO</sub> = 1,260 circuits. For LOCC, we further perform error mitigation of the switch with <i>N</i><sub>ZNE</sub> = 5 stretch factors. We, therefore, execute <i>N</i><sub>S</sub> × <i>N</i><sub>TREX</sub> × <i>I</i><sub>LOCC</sub> × <i>N</i><sub>ZNE</sub> = 4,725 circuits to measure the error-mitigated stabilizers needed to compute <span class="mathjax-tex">${{\mathcal{W}}}_{i,j}$</span>. Each circuit is executed with a total of 1,024 shots.</p><p>To reconstruct the value of the measured observables, we first merge the shots from the TREX samples. To do this, we flip the classical bits in the measured bit strings corresponding to measurements for which TREX prepended an <i>X</i> gate. These processed bit strings are then aggregated in a count dictionary with 1,024 × <i>N</i><sub>TREX</sub> counts. Next, to obtain the value of a stabilizer, we identify which of the <i>N</i><sub>S</sub> measurement bases we need to use. The value of a stabilizer and its corresponding standard deviation are then obtained by resampling the corresponding 1,024 × <i>N</i><sub>TREX</sub> counts. Here, we randomly select 10% of the shots to compute an expectation value. Ten such expectation values are averaged and reported as the measured stabilizer value. The standard deviation of these 10 measurements is reported as the standard deviation of the stabilizer, shown as error bars in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig2">2b</a>. Finally, if the stabilizer is in the light cone of a virtual gate implemented with LOCC, we linearly fit the value of the stabilizer obtained at the <i>N</i><sub>ZNE</sub> = 5 switch stretch factors. This fit, shown in Extended Data Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41586-024-08178-2#Fig5">2d</a>, enables us to report the stabilizer at the extrapolated zero-delay switch.</p></div></div></section> </div> <div> <section data-title="Data availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>The code to analyse the counts, reproduce the plots in this paper and produce the circuits for the cut Bell pairs are available on GitHub (<a href="https://github.com/eggerdj/cut_graph_state_data">https://github.com/eggerdj/cut_graph_state_data</a>). The raw counts are unavailable on GitHub because of size constraints but are available upon reasonable request.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. <i>Nature</i> <b>618</b>, 500–505 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41586-023-06096-3" data-track-item_id="10.1038/s41586-023-06096-3" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41586-023-06096-3" aria-label="Article reference 1" data-doi="10.1038/s41586-023-06096-3">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2023Natur.618..500K" aria-label="ADS reference 1">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3sXht1ejtrfO" aria-label="CAS reference 1">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=37316724" aria-label="PubMed reference 1">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC10266970" aria-label="PubMed Central reference 1">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=Evidence%20for%20the%20utility%20of%20quantum%20computing%20before%20fault%20tolerance&journal=Nature&doi=10.1038%2Fs41586-023-06096-3&volume=618&pages=500-505&publication_year=2023&author=Kim%2CY"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2">Bravyi, S., Dial, O., Gambetta, J. M., Gil, D. & Nazario, Z. The future of quantum computing with superconducting qubits. <i>J. Appl. Phys.</i> <b>132</b>, 160902 (2022).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1063/5.0082975" data-track-item_id="10.1063/5.0082975" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1063%2F5.0082975" aria-label="Article reference 2" data-doi="10.1063/5.0082975">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2022JAP...132p0902B" aria-label="ADS reference 2">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB38XislClsLfM" aria-label="CAS reference 2">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=The%20future%20of%20quantum%20computing%20with%20superconducting%20qubits&journal=J.%20Appl.%20Phys.&doi=10.1063%2F5.0082975&volume=132&publication_year=2022&author=Bravyi%2CS&author=Dial%2CO&author=Gambetta%2CJM&author=Gil%2CD&author=Nazario%2CZ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3">Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. <i>Nature</i> <b>627</b>, 778–782 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41586-024-07107-7" data-track-item_id="10.1038/s41586-024-07107-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41586-024-07107-7" aria-label="Article reference 3" data-doi="10.1038/s41586-024-07107-7">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2024Natur.627..778B" aria-label="ADS reference 3">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXmvFeqtbg%3D" aria-label="CAS reference 3">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=38538939" aria-label="PubMed reference 3">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC10972743" aria-label="PubMed Central reference 3">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=High-threshold%20and%20low-overhead%20fault-tolerant%20quantum%20memory&journal=Nature&doi=10.1038%2Fs41586-024-07107-7&volume=627&pages=778-782&publication_year=2024&author=Bravyi%2CS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4">Akhtar, M. et al. A high-fidelity quantum matter-link between ion-trap microchip modules. <i>Nat. Commun.</i> <b>14</b>, 531 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41467-022-35285-3" data-track-item_id="10.1038/s41467-022-35285-3" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41467-022-35285-3" aria-label="Article reference 4" data-doi="10.1038/s41467-022-35285-3">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2023NatCo..14..531A" aria-label="ADS reference 4">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3sXjtVegtrg%3D" aria-label="CAS reference 4">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=36754957" aria-label="PubMed reference 4">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC9908934" aria-label="PubMed Central reference 4">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 4" href="http://scholar.google.com/scholar_lookup?&title=A%20high-fidelity%20quantum%20matter-link%20between%20ion-trap%20microchip%20modules&journal=Nat.%20Commun.&doi=10.1038%2Fs41467-022-35285-3&volume=14&publication_year=2023&author=Akhtar%2CM"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5">Bluvstein, D. et al. A quantum processor based on coherent transport of entangled atom arrays. <i>Nature</i> <b>604</b>, 451–456 (2022).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41586-022-04592-6" data-track-item_id="10.1038/s41586-022-04592-6" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41586-022-04592-6" aria-label="Article reference 5" data-doi="10.1038/s41586-022-04592-6">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2022Natur.604..451B" aria-label="ADS reference 5">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB38XhtVGrsbnJ" aria-label="CAS reference 5">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=35444318" aria-label="PubMed reference 5">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC9021024" aria-label="PubMed Central reference 5">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=A%20quantum%20processor%20based%20on%20coherent%20transport%20of%20entangled%20atom%20arrays&journal=Nature&doi=10.1038%2Fs41586-022-04592-6&volume=604&pages=451-456&publication_year=2022&author=Bluvstein%2CD"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6">Krantz, P. et al. A quantum engineer’s guide to superconducting qubits. <i>Appl. Phys. Rev.</i> <b>6</b>, 021318 (2019).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1063/1.5089550" data-track-item_id="10.1063/1.5089550" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1063%2F1.5089550" aria-label="Article reference 6" data-doi="10.1063/1.5089550">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2019ApPRv...6b1318K" aria-label="ADS reference 6">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=A%20quantum%20engineer%E2%80%99s%20guide%20to%20superconducting%20qubits&journal=Appl.%20Phys.%20Rev.&doi=10.1063%2F1.5089550&volume=6&publication_year=2019&author=Krantz%2CP"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7">Conner, C. R. et al. Superconducting qubits in a flip-chip architecture. <i>Appl. Phys. Lett.</i> <b>118</b>, 232602 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1063/5.0050173" data-track-item_id="10.1063/5.0050173" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1063%2F5.0050173" aria-label="Article reference 7" data-doi="10.1063/5.0050173">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021ApPhL.118w2602C" aria-label="ADS reference 7">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3MXhtlSrurbE" aria-label="CAS reference 7">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Superconducting%20qubits%20in%20a%20flip-chip%20architecture&journal=Appl.%20Phys.%20Lett.&doi=10.1063%2F5.0050173&volume=118&publication_year=2021&author=Conner%2CCR"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8">Gold, A. et al. Entanglement across separate silicon dies in a modular superconducting qubit device. <i>npj Quantum Inf.</i> <b>7</b>, 142 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41534-021-00484-1" data-track-item_id="10.1038/s41534-021-00484-1" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41534-021-00484-1" aria-label="Article reference 8" data-doi="10.1038/s41534-021-00484-1">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021npjQI...7..142G" aria-label="ADS reference 8">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Entanglement%20across%20separate%20silicon%20dies%20in%20a%20modular%20superconducting%20qubit%20device&journal=npj%20Quantum%20Inf.&doi=10.1038%2Fs41534-021-00484-1&volume=7&publication_year=2021&author=Gold%2CA"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9">Zhong, Y. et al. Violating Bell’s inequality with remotely connected superconducting qubits. <i>Nat. Phys.</i> <b>15</b>, 741–744 (2019).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41567-019-0507-7" data-track-item_id="10.1038/s41567-019-0507-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41567-019-0507-7" aria-label="Article reference 9" data-doi="10.1038/s41567-019-0507-7">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BC1MXosVCiurw%3D" aria-label="CAS reference 9">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Violating%20Bell%E2%80%99s%20inequality%20with%20remotely%20connected%20superconducting%20qubits&journal=Nat.%20Phys.&doi=10.1038%2Fs41567-019-0507-7&volume=15&pages=741-744&publication_year=2019&author=Zhong%2CY"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10">Zhong, Y. et al. Deterministic multi-qubit entanglement in a quantum network. <i>Nature</i> <b>590</b>, 571–575 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41586-021-03288-7" data-track-item_id="10.1038/s41586-021-03288-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41586-021-03288-7" aria-label="Article reference 10" data-doi="10.1038/s41586-021-03288-7">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021Natur.590..571Z" aria-label="ADS reference 10">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3MXltFaltLY%3D" aria-label="CAS reference 10">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=33627810" aria-label="PubMed reference 10">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=Deterministic%20multi-qubit%20entanglement%20in%20a%20quantum%20network&journal=Nature&doi=10.1038%2Fs41586-021-03288-7&volume=590&pages=571-575&publication_year=2021&author=Zhong%2CY"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11">Malekakhlagh, M. et al. Enhanced quantum state transfer and Bell-state generation over long-range multimode interconnects via superadiabatic transitionless driving. <i>Phys. Rev. Appl.</i> <b>22</b>, 024006 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevApplied.22.024006" data-track-item_id="10.1103/PhysRevApplied.22.024006" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevApplied.22.024006" aria-label="Article reference 11" data-doi="10.1103/PhysRevApplied.22.024006">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXisVyis7%2FJ" aria-label="CAS reference 11">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 11" href="http://scholar.google.com/scholar_lookup?&title=Enhanced%20quantum%20state%20transfer%20and%20Bell-state%20generation%20over%20long-range%20multimode%20interconnects%20via%20superadiabatic%20transitionless%20driving&journal=Phys.%20Rev.%20Appl.&doi=10.1103%2FPhysRevApplied.22.024006&volume=22&publication_year=2024&author=Malekakhlagh%2CM"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12">Ang, J. et al. ARQUIN: architectures for multinode superconducting quantum computers. <i>ACM Trans. Quantum Comput.</i> <b>5</b>, 19 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1145/3674151" data-track-item_id="10.1145/3674151" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1145%2F3674151" aria-label="Article reference 12" data-doi="10.1145/3674151">Article</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=ARQUIN%3A%20architectures%20for%20multinode%20superconducting%20quantum%20computers&journal=ACM%20Trans.%20Quantum%20Comput.&doi=10.1145%2F3674151&volume=5&publication_year=2024&author=Ang%2CJ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13">Córcoles, A. D. et al. Exploiting dynamic quantum circuits in a quantum algorithm with superconducting qubits. <i>Phys. Rev. Lett.</i> <b>127</b>, 100501 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.127.100501" data-track-item_id="10.1103/PhysRevLett.127.100501" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.127.100501" aria-label="Article reference 13" data-doi="10.1103/PhysRevLett.127.100501">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021PhRvL.127j0501C" aria-label="ADS reference 13">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=34533358" aria-label="PubMed reference 13">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 13" href="http://scholar.google.com/scholar_lookup?&title=Exploiting%20dynamic%20quantum%20circuits%20in%20a%20quantum%20algorithm%20with%20superconducting%20qubits&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.127.100501&volume=127&publication_year=2021&author=C%C3%B3rcoles%2CAD"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14">Bäumer, E. et al. Efficient long-range entanglement using dynamic circuits. <i>PRX Quantum</i> <b>5</b>, 030339 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PRXQuantum.5.030339" data-track-item_id="10.1103/PRXQuantum.5.030339" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPRXQuantum.5.030339" aria-label="Article reference 14" data-doi="10.1103/PRXQuantum.5.030339">Article</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 14" href="http://scholar.google.com/scholar_lookup?&title=Efficient%20long-range%20entanglement%20using%20dynamic%20circuits&journal=PRX%20Quantum&doi=10.1103%2FPRXQuantum.5.030339&volume=5&publication_year=2024&author=B%C3%A4umer%2CE"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15">Hofmann, H. F. How to simulate a universal quantum computer using negative probabilities. <i>J. Phys. A Math. Theor.</i> <b>42</b>, 275304 (2009).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1088/1751-8113/42/27/275304" data-track-item_id="10.1088/1751-8113/42/27/275304" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1088%2F1751-8113%2F42%2F27%2F275304" aria-label="Article reference 15" data-doi="10.1088/1751-8113/42/27/275304">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2512131" aria-label="MathSciNet reference 15">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 15" href="http://scholar.google.com/scholar_lookup?&title=How%20to%20simulate%20a%20universal%20quantum%20computer%20using%20negative%20probabilities&journal=J.%20Phys.%20A%20Math.%20Theor.&doi=10.1088%2F1751-8113%2F42%2F27%2F275304&volume=42&publication_year=2009&author=Hofmann%2CHF"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16">Mitarai, K. & Fujii, K. Constructing a virtual two-qubit gate by sampling single-qubit operations. <i>New J. Phys.</i> <b>23</b>, 023021 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1088/1367-2630/abd7bc" data-track-item_id="10.1088/1367-2630/abd7bc" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1088%2F1367-2630%2Fabd7bc" aria-label="Article reference 16" data-doi="10.1088/1367-2630/abd7bc">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021NJPh...23b3021M" aria-label="ADS reference 16">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4225362" aria-label="MathSciNet reference 16">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=Constructing%20a%20virtual%20two-qubit%20gate%20by%20sampling%20single-qubit%20operations&journal=New%20J.%20Phys.&doi=10.1088%2F1367-2630%2Fabd7bc&volume=23&publication_year=2021&author=Mitarai%2CK&author=Fujii%2CK"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17">Piveteau, C. & Sutter, D. Circuit knitting with classical communication. <i>IEEE Trans. Inf. Theor.</i> <b>1</b>, 2734–2745 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4730064" aria-label="MathSciNet reference 17">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=Circuit%20knitting%20with%20classical%20communication&journal=IEEE%20Trans.%20Inf.%20Theor.&volume=1&pages=2734-2745&publication_year=2023&author=Piveteau%2CC&author=Sutter%2CD"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18">Singh, A. P. et al. Experimental demonstration of a high-fidelity virtual two-qubit gate. <i>Phys. Rev. Res.</i> <b>6</b>, 013235 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevResearch.6.013235" data-track-item_id="10.1103/PhysRevResearch.6.013235" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevResearch.6.013235" aria-label="Article reference 18" data-doi="10.1103/PhysRevResearch.6.013235">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXotFCkt70%3D" aria-label="CAS reference 18">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=Experimental%20demonstration%20of%20a%20high-fidelity%20virtual%20two-qubit%20gate&journal=Phys.%20Rev.%20Res.&doi=10.1103%2FPhysRevResearch.6.013235&volume=6&publication_year=2024&author=Singh%2CAP"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19">Gottesman, D. & Chuang, I. L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. <i>Nature</i> <b>402</b>, 390–393 (1999).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/46503" data-track-item_id="10.1038/46503" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2F46503" aria-label="Article reference 19" data-doi="10.1038/46503">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1999Natur.402..390G" aria-label="ADS reference 19">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DyaK1MXnvVyktLc%3D" aria-label="CAS reference 19">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 19" href="http://scholar.google.com/scholar_lookup?&title=Demonstrating%20the%20viability%20of%20universal%20quantum%20computation%20using%20teleportation%20and%20single-qubit%20operations&journal=Nature&doi=10.1038%2F46503&volume=402&pages=390-393&publication_year=1999&author=Gottesman%2CD&author=Chuang%2CIL"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="20."><p class="c-article-references__text" id="ref-CR20">Wan, Y. et al. Quantum gate teleportation between separated qubits in a trapped-ion processor. <i>Science</i> <b>364</b>, 875 (2019).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1126/science.aaw9415" data-track-item_id="10.1126/science.aaw9415" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1126%2Fscience.aaw9415" aria-label="Article reference 20" data-doi="10.1126/science.aaw9415">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2019Sci...364..875W" aria-label="ADS reference 20">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BC1MXhtVKgt7jN" aria-label="CAS reference 20">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=31147517" aria-label="PubMed reference 20">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC11533827" aria-label="PubMed Central reference 20">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 20" href="http://scholar.google.com/scholar_lookup?&title=Quantum%20gate%20teleportation%20between%20separated%20qubits%20in%20a%20trapped-ion%20processor&journal=Science&doi=10.1126%2Fscience.aaw9415&volume=364&publication_year=2019&author=Wan%2CY"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="21."><p class="c-article-references__text" id="ref-CR21">Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. <i>Phys. Rev. Lett.</i> <b>82</b>, 2417–2421 (1999).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.82.2417" data-track-item_id="10.1103/PhysRevLett.82.2417" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.82.2417" aria-label="Article reference 21" data-doi="10.1103/PhysRevLett.82.2417">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1999PhRvL..82.2417V" aria-label="ADS reference 21">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1676105" aria-label="MathSciNet reference 21">MathSciNet</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DyaK1MXhvFWjs70%3D" aria-label="CAS reference 21">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 21" href="http://scholar.google.com/scholar_lookup?&title=Dynamical%20decoupling%20of%20open%20quantum%20systems&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.82.2417&volume=82&pages=2417-2421&publication_year=1999&author=Viola%2CL&author=Knill%2CE&author=Lloyd%2CS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="22."><p class="c-article-references__text" id="ref-CR22">Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. <i>Phys. Rev. Lett.</i> <b>119</b>, 180509 (2017).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.119.180509" data-track-item_id="10.1103/PhysRevLett.119.180509" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.119.180509" aria-label="Article reference 22" data-doi="10.1103/PhysRevLett.119.180509">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2017PhRvL.119r0509T" aria-label="ADS reference 22">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3730724" aria-label="MathSciNet reference 22">MathSciNet</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=29219599" aria-label="PubMed reference 22">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 22" href="http://scholar.google.com/scholar_lookup?&title=Error%20mitigation%20for%20short-depth%20quantum%20circuits&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.119.180509&volume=119&publication_year=2017&author=Temme%2CK&author=Bravyi%2CS&author=Gambetta%2CJM"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="23."><p class="c-article-references__text" id="ref-CR23">Reilly, D. J. Challenges in scaling-up the control interface of a quantum computer. Preprint at <a href="https://arxiv.org/abs/1912.05114" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/1912.05114">arxiv.org/abs/1912.05114</a> (2019).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="24."><p class="c-article-references__text" id="ref-CR24">Peng, T., Harrow, A. W., Ozols, M. & Wu, X. Simulating large quantum circuits on a small quantum computer. <i>Phys. Rev. Lett.</i> <b>125</b>, 150504 (2020).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.125.150504" data-track-item_id="10.1103/PhysRevLett.125.150504" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.125.150504" aria-label="Article reference 24" data-doi="10.1103/PhysRevLett.125.150504">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2020PhRvL.125o0504P" aria-label="ADS reference 24">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3cXit1ShtbjP" aria-label="CAS reference 24">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=33095634" aria-label="PubMed reference 24">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 24" href="http://scholar.google.com/scholar_lookup?&title=Simulating%20large%20quantum%20circuits%20on%20a%20small%20quantum%20computer&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.125.150504&volume=125&publication_year=2020&author=Peng%2CT&author=Harrow%2CAW&author=Ozols%2CM&author=Wu%2CX"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="25."><p class="c-article-references__text" id="ref-CR25">Brenner, L., Piveteau, C. & Sutter, D. Optimal wire cutting with classical communication. Preprint at <a href="https://arxiv.org/abs/2302.03366" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2302.03366">arxiv.org/abs/2302.03366</a> (2023).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="26."><p class="c-article-references__text" id="ref-CR26">Pednault, E. An alternative approach to optimal wire cutting without ancilla qubits. Preprint at <a href="https://arxiv.org/abs/2303.08287" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2303.08287">arxiv.org/abs/2303.08287</a> (2023).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="27."><p class="c-article-references__text" id="ref-CR27">Zander, R. & Becker, C. K.-U. Benchmarking multipartite entanglement generation with graph states. Preprint at <a href="https://arxiv.org/abs/2402.00766" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2402.00766">arxiv.org/abs/2402.00766</a> (2024).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="28."><p class="c-article-references__text" id="ref-CR28">Mundada, P. S. et al. Experimental benchmarking of an automated deterministic error-suppression workflow for quantum algorithms. <i>Phys. Rev. Appl.</i> <b>20</b>, 024034 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevApplied.20.024034" data-track-item_id="10.1103/PhysRevApplied.20.024034" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevApplied.20.024034" aria-label="Article reference 28" data-doi="10.1103/PhysRevApplied.20.024034">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2023PhRvP..20b4034M" aria-label="ADS reference 28">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3sXhvFOitLrM" aria-label="CAS reference 28">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 28" href="http://scholar.google.com/scholar_lookup?&title=Experimental%20benchmarking%20of%20an%20automated%20deterministic%20error-suppression%20workflow%20for%20quantum%20algorithms&journal=Phys.%20Rev.%20Appl.&doi=10.1103%2FPhysRevApplied.20.024034&volume=20&publication_year=2023&author=Mundada%2CPS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="29."><p class="c-article-references__text" id="ref-CR29">Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. <i>Nature</i> <b>625</b>, 259–263 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41586-023-06846-3" data-track-item_id="10.1038/s41586-023-06846-3" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41586-023-06846-3" aria-label="Article reference 29" data-doi="10.1038/s41586-023-06846-3">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2024Natur.625..259G" aria-label="ADS reference 29">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXpsFWjuw%3D%3D" aria-label="CAS reference 29">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=38200302" aria-label="PubMed reference 29">PubMed</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed central reference" data-track-action="pubmed central reference" href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC10781628" aria-label="PubMed Central reference 29">PubMed Central</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 29" href="http://scholar.google.com/scholar_lookup?&title=Encoding%20a%20magic%20state%20with%20beyond%20break-even%20fidelity&journal=Nature&doi=10.1038%2Fs41586-023-06846-3&volume=625&pages=259-263&publication_year=2024&author=Gupta%2CRS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="30."><p class="c-article-references__text" id="ref-CR30">Ufrecht, C. et al. Optimal joint cutting of two-qubit rotation gates. <i>Phys. Rev. A</i> <b>109</b>, 052440 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.109.052440" data-track-item_id="10.1103/PhysRevA.109.052440" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.109.052440" aria-label="Article reference 30" data-doi="10.1103/PhysRevA.109.052440">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2024PhRvA.109e2440U" aria-label="ADS reference 30">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4764328" aria-label="MathSciNet reference 30">MathSciNet</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXitl2ltL7P" aria-label="CAS reference 30">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 30" href="http://scholar.google.com/scholar_lookup?&title=Optimal%20joint%20cutting%20of%20two-qubit%20rotation%20gates&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.109.052440&volume=109&publication_year=2024&author=Ufrecht%2CC"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="31."><p class="c-article-references__text" id="ref-CR31">Schmitt, L., Piveteau, C. & Sutter, D. Cutting circuits with multiple two-qubit unitaries. Preprint at <a href="https://arxiv.org/abs/2312.11638" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2312.11638">arxiv.org/abs/2312.11638</a> (2023).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="32."><p class="c-article-references__text" id="ref-CR32">Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. <i>Phys. Rev. Lett.</i> <b>78</b>, 3221–3224 (1997).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.78.3221" data-track-item_id="10.1103/PhysRevLett.78.3221" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.78.3221" aria-label="Article reference 32" data-doi="10.1103/PhysRevLett.78.3221">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1997PhRvL..78.3221C" aria-label="ADS reference 32">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DyaK2sXislWkur0%3D" aria-label="CAS reference 32">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 32" href="http://scholar.google.com/scholar_lookup?&title=Quantum%20state%20transfer%20and%20entanglement%20distribution%20among%20distant%20nodes%20in%20a%20quantum%20network&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.78.3221&volume=78&pages=3221-3224&publication_year=1997&author=Cirac%2CJI&author=Zoller%2CP&author=Kimble%2CHJ&author=Mabuchi%2CH"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="33."><p class="c-article-references__text" id="ref-CR33">Duan, L.-M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. <i>Nature</i> <b>414</b>, 413–418 (2001).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/35106500" data-track-item_id="10.1038/35106500" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2F35106500" aria-label="Article reference 33" data-doi="10.1038/35106500">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2001Natur.414..413D" aria-label="ADS reference 33">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BD3MXovFanurw%3D" aria-label="CAS reference 33">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=11719796" aria-label="PubMed reference 33">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 33" href="http://scholar.google.com/scholar_lookup?&title=Long-distance%20quantum%20communication%20with%20atomic%20ensembles%20and%20linear%20optics&journal=Nature&doi=10.1038%2F35106500&volume=414&pages=413-418&publication_year=2001&author=Duan%2CL-M&author=Lukin%2CMD&author=Cirac%2CJI&author=Zoller%2CP"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="34."><p class="c-article-references__text" id="ref-CR34">Magnard, P. et al. Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems. <i>Phys. Rev. Lett.</i> <b>125</b>, 260502 (2020).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.125.260502" data-track-item_id="10.1103/PhysRevLett.125.260502" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.125.260502" aria-label="Article reference 34" data-doi="10.1103/PhysRevLett.125.260502">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2020PhRvL.125z0502M" aria-label="ADS reference 34">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3MXivVOksrY%3D" aria-label="CAS reference 34">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=33449744" aria-label="PubMed reference 34">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 34" href="http://scholar.google.com/scholar_lookup?&title=Microwave%20quantum%20link%20between%20superconducting%20circuits%20housed%20in%20spatially%20separated%20cryogenic%20systems&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.125.260502&volume=125&publication_year=2020&author=Magnard%2CP"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="35."><p class="c-article-references__text" id="ref-CR35">Niu, J. et al. Low-loss interconnects for modular superconducting quantum processors. <i>Nat. Electron.</i> <b>6</b>, 235–241 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/s41928-023-00925-z" data-track-item_id="10.1038/s41928-023-00925-z" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fs41928-023-00925-z" aria-label="Article reference 35" data-doi="10.1038/s41928-023-00925-z">Article</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 35" href="http://scholar.google.com/scholar_lookup?&title=Low-loss%20interconnects%20for%20modular%20superconducting%20quantum%20processors&journal=Nat.%20Electron.&doi=10.1038%2Fs41928-023-00925-z&volume=6&pages=235-241&publication_year=2023&author=Niu%2CJ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="36."><p class="c-article-references__text" id="ref-CR36">Orcutt, J. et al. Engineering electro-optics in SiGe/Si waveguides for quantum transduction. <i>Quantum Sci. Technol.</i> <b>5</b>, 034006 (2020).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1088/2058-9565/ab84c1" data-track-item_id="10.1088/2058-9565/ab84c1" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1088%2F2058-9565%2Fab84c1" aria-label="Article reference 36" data-doi="10.1088/2058-9565/ab84c1">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2020QS%26T....5c4006O" aria-label="ADS reference 36">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 36" href="http://scholar.google.com/scholar_lookup?&title=Engineering%20electro-optics%20in%20SiGe%2FSi%20waveguides%20for%20quantum%20transduction&journal=Quantum%20Sci.%20Technol.&doi=10.1088%2F2058-9565%2Fab84c1&volume=5&publication_year=2020&author=Orcutt%2CJ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="37."><p class="c-article-references__text" id="ref-CR37">Lauk, N. et al. Perspectives on quantum transduction. <i>Quantum Sci. Technol.</i> <b>5</b>, 020501 (2020).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1088/2058-9565/ab788a" data-track-item_id="10.1088/2058-9565/ab788a" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1088%2F2058-9565%2Fab788a" aria-label="Article reference 37" data-doi="10.1088/2058-9565/ab788a">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2020QS%26T....5b0501L" aria-label="ADS reference 37">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 37" href="http://scholar.google.com/scholar_lookup?&title=Perspectives%20on%20quantum%20transduction&journal=Quantum%20Sci.%20Technol.&doi=10.1088%2F2058-9565%2Fab788a&volume=5&publication_year=2020&author=Lauk%2CN"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="38."><p class="c-article-references__text" id="ref-CR38">Krastanov, S. et al. Optically heralded entanglement of superconducting systems in quantum networks. <i>Phys. Rev. Lett.</i> <b>127</b>, 040503 (2021).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.127.040503" data-track-item_id="10.1103/PhysRevLett.127.040503" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.127.040503" aria-label="Article reference 38" data-doi="10.1103/PhysRevLett.127.040503">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2021PhRvL.127d0503K" aria-label="ADS reference 38">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3MXhslWnsrjM" aria-label="CAS reference 38">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=34355947" aria-label="PubMed reference 38">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 38" href="http://scholar.google.com/scholar_lookup?&title=Optically%20heralded%20entanglement%20of%20superconducting%20systems%20in%20quantum%20networks&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.127.040503&volume=127&publication_year=2021&author=Krastanov%2CS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="39."><p class="c-article-references__text" id="ref-CR39">Bechtold, M., Barzen, J., Leymann, F. & Mandl, A. Circuit cutting with non-maximally entangled states. Preprint at <a href="https://arxiv.org/abs/2306.12084" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2306.12084">arxiv.org/abs/2306.12084</a> (2023).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="40."><p class="c-article-references__text" id="ref-CR40">Bennett, C. H. et al. Purification of noisy entanglement and faithful teleportation via noisy channels. <i>Phys. Rev. Lett.</i> <b>76</b>, 722–725 (1996).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.76.722" data-track-item_id="10.1103/PhysRevLett.76.722" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.76.722" aria-label="Article reference 40" data-doi="10.1103/PhysRevLett.76.722">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1996PhRvL..76..722B" aria-label="ADS reference 40">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DyaK28XntVCktA%3D%3D" aria-label="CAS reference 40">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=10061534" aria-label="PubMed reference 40">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 40" href="http://scholar.google.com/scholar_lookup?&title=Purification%20of%20noisy%20entanglement%20and%20faithful%20teleportation%20via%20noisy%20channels&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.76.722&volume=76&pages=722-725&publication_year=1996&author=Bennett%2CCH"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="41."><p class="c-article-references__text" id="ref-CR41">Tornow, C., Kanazawa, N., Shanks, W. E. & Egger, D. J. Minimum quantum run-time characterization and calibration via restless measurements with dynamic repetition rates. <i>Phys. Rev. Appl.</i> <b>17</b>, 064061 (2022).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevApplied.17.064061" data-track-item_id="10.1103/PhysRevApplied.17.064061" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevApplied.17.064061" aria-label="Article reference 41" data-doi="10.1103/PhysRevApplied.17.064061">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2022PhRvP..17f4061T" aria-label="ADS reference 41">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB38XhvVSrsrnM" aria-label="CAS reference 41">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 41" href="http://scholar.google.com/scholar_lookup?&title=Minimum%20quantum%20run-time%20characterization%20and%20calibration%20via%20restless%20measurements%20with%20dynamic%20repetition%20rates&journal=Phys.%20Rev.%20Appl.&doi=10.1103%2FPhysRevApplied.17.064061&volume=17&publication_year=2022&author=Tornow%2CC&author=Kanazawa%2CN&author=Shanks%2CWE&author=Egger%2CDJ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="42."><p class="c-article-references__text" id="ref-CR42">Harrow, A. W. & Lowe, A. Optimal quantum circuit cuts with application to clustered Hamiltonian simulation. Preprint at <a href="https://arxiv.org/abs/2403.01018" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2403.01018">arxiv.org/abs/2403.01018</a> (2024).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="43."><p class="c-article-references__text" id="ref-CR43">Caleffi, M. et al. Distributed quantum computing: a survey. Preprint at <a href="https://arxiv.org/abs/2212.10609" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="https://arxiv.org/abs/2212.10609">arxiv.org/abs/2212.10609</a> (2022).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="44."><p class="c-article-references__text" id="ref-CR44">Cai, Z. et al. Quantum error mitigation. <i>Rev. Mod. Phys.</i> <b>95</b>, 045005 (2023).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/RevModPhys.95.045005" data-track-item_id="10.1103/RevModPhys.95.045005" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FRevModPhys.95.045005" aria-label="Article reference 44" data-doi="10.1103/RevModPhys.95.045005">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2023RvMP...95d5005C" aria-label="ADS reference 44">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4704107" aria-label="MathSciNet reference 44">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 44" href="http://scholar.google.com/scholar_lookup?&title=Quantum%20error%20mitigation&journal=Rev.%20Mod.%20Phys.&doi=10.1103%2FRevModPhys.95.045005&volume=95&publication_year=2023&author=Cai%2CZ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="45."><p class="c-article-references__text" id="ref-CR45">Endo, S., Benjamin, S. C. & Li, Y. Practical quantum error mitigation for near-future applications. <i>Phys. Rev. X</i> <b>8</b>, 031027 (2018).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BC1MXltFSnsLw%3D" aria-label="CAS reference 45">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 45" href="http://scholar.google.com/scholar_lookup?&title=Practical%20quantum%20error%20mitigation%20for%20near-future%20applications&journal=Phys.%20Rev.%20X&volume=8&publication_year=2018&author=Endo%2CS&author=Benjamin%2CSC&author=Li%2CY"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="46."><p class="c-article-references__text" id="ref-CR46">McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient <i>Z</i> gates for quantum computing. <i>Phys. Rev. A</i> <b>96</b>, 022330 (2017).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.96.022330" data-track-item_id="10.1103/PhysRevA.96.022330" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.96.022330" aria-label="Article reference 46" data-doi="10.1103/PhysRevA.96.022330">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2017PhRvA..96b2330M" aria-label="ADS reference 46">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 46" href="http://scholar.google.com/scholar_lookup?&title=Efficient%20Z%20gates%20for%20quantum%20computing&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.96.022330&volume=96&publication_year=2017&author=McKay%2CDC&author=Wood%2CCJ&author=Sheldon%2CS&author=Chow%2CJM&author=Gambetta%2CJM"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="47."><p class="c-article-references__text" id="ref-CR47">Tran, M. C. et al. Hierarchy of linear light cones with long-range interactions. <i>Phys. Rev. X</i> <b>10</b>, 031009 (2020).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB3cXitV2hsLbM" aria-label="CAS reference 47">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 47" href="http://scholar.google.com/scholar_lookup?&title=Hierarchy%20of%20linear%20light%20cones%20with%20long-range%20interactions&journal=Phys.%20Rev.%20X&volume=10&publication_year=2020&author=Tran%2CMC"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="48."><p class="c-article-references__text" id="ref-CR48">Zettles, G., Willenborg, S., Johnson, B. R., Wack, A. & Allison, B. 26.2 Design considerations for superconducting quantum systems. In <i>2022 IEEE International Solid-State Circuits Conference (ISSCC)</i>, Vol. 65, pp. 1–3 (IEEE, 2022).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="49."><p class="c-article-references__text" id="ref-CR49">Cross, A. et al. OpenQASM 3: a broader and deeper quantum assembly language. <i>ACM Trans. Quantum Comput.</i> <b>3</b>, 1–50 (2022).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1145/3505636" data-track-item_id="10.1145/3505636" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1145%2F3505636" aria-label="Article reference 49" data-doi="10.1145/3505636">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4460004" aria-label="MathSciNet reference 49">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 49" href="http://scholar.google.com/scholar_lookup?&title=OpenQASM%203%3A%20a%20broader%20and%20deeper%20quantum%20assembly%20language&journal=ACM%20Trans.%20Quantum%20Comput.&doi=10.1145%2F3505636&volume=3&pages=1-50&publication_year=2022&author=Cross%2CA"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="50."><p class="c-article-references__text" id="ref-CR50">Vidal, G. & Tarrach, R. Robustness of entanglement. <i>Phys. Rev. A</i> <b>59</b>, 141–155 (1999).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.59.141" data-track-item_id="10.1103/PhysRevA.59.141" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.59.141" aria-label="Article reference 50" data-doi="10.1103/PhysRevA.59.141">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1999PhRvA..59..141V" aria-label="ADS reference 50">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1670562" aria-label="MathSciNet reference 50">MathSciNet</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DyaK1MXit1GitQ%3D%3D" aria-label="CAS reference 50">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 50" href="http://scholar.google.com/scholar_lookup?&title=Robustness%20of%20entanglement&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.59.141&volume=59&pages=141-155&publication_year=1999&author=Vidal%2CG&author=Tarrach%2CR"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="51."><p class="c-article-references__text" id="ref-CR51">Kraft, D. <i>A Software Package for Sequential Quadratic Programming</i> (DFVLR, 1988).</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="52."><p class="c-article-references__text" id="ref-CR52">Gupta, R. S. et al. Probabilistic error cancellation for dynamic quantum circuits. <i>Phys. Rev. A</i> <b>109</b>, 062617 (2024).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.109.062617" data-track-item_id="10.1103/PhysRevA.109.062617" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.109.062617" aria-label="Article reference 52" data-doi="10.1103/PhysRevA.109.062617">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4776849" aria-label="MathSciNet reference 52">MathSciNet</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BB2cXisVCmsr7F" aria-label="CAS reference 52">CAS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 52" href="http://scholar.google.com/scholar_lookup?&title=Probabilistic%20error%20cancellation%20for%20dynamic%20quantum%20circuits&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.109.062617&volume=109&publication_year=2024&author=Gupta%2CRS"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="53."><p class="c-article-references__text" id="ref-CR53">Briegel, H. J. & Raussendorf, R. Persistent entanglement in arrays of interacting particles. <i>Phys. Rev. Lett.</i> <b>86</b>, 910–913 (2001).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.86.910" data-track-item_id="10.1103/PhysRevLett.86.910" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.86.910" aria-label="Article reference 53" data-doi="10.1103/PhysRevLett.86.910">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2001PhRvL..86..910B" aria-label="ADS reference 53">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="cas reference" data-track-action="cas reference" href="/articles/cas-redirect/1:CAS:528:DC%2BD3MXms1OgtA%3D%3D" aria-label="CAS reference 53">CAS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="pubmed reference" data-track-action="pubmed reference" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=11177971" aria-label="PubMed reference 53">PubMed</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 53" href="http://scholar.google.com/scholar_lookup?&title=Persistent%20entanglement%20in%20arrays%20of%20interacting%20particles&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.86.910&volume=86&pages=910-913&publication_year=2001&author=Briegel%2CHJ&author=Raussendorf%2CR"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="54."><p class="c-article-references__text" id="ref-CR54">Hein, M., Eisert, J. & Briegel, H. J. Multiparty entanglement in graph states. <i>Phys. Rev. A</i> <b>69</b>, 062311 (2004).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.69.062311" data-track-item_id="10.1103/PhysRevA.69.062311" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.69.062311" aria-label="Article reference 54" data-doi="10.1103/PhysRevA.69.062311">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2004PhRvA..69f2311H" aria-label="ADS reference 54">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2081877" aria-label="MathSciNet reference 54">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 54" href="http://scholar.google.com/scholar_lookup?&title=Multiparty%20entanglement%20in%20graph%20states&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.69.062311&volume=69&publication_year=2004&author=Hein%2CM&author=Eisert%2CJ&author=Briegel%2CHJ"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="55."><p class="c-article-references__text" id="ref-CR55">Jungnitsch, B., Moroder, T. & Gühne, O. Entanglement witnesses for graph states: general theory and examples. <i>Phys. Rev. A</i> <b>84</b>, 032310 (2011).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.84.032310" data-track-item_id="10.1103/PhysRevA.84.032310" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.84.032310" aria-label="Article reference 55" data-doi="10.1103/PhysRevA.84.032310">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2011PhRvA..84c2310J" aria-label="ADS reference 55">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 55" href="http://scholar.google.com/scholar_lookup?&title=Entanglement%20witnesses%20for%20graph%20states%3A%20general%20theory%20and%20examples&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.84.032310&volume=84&publication_year=2011&author=Jungnitsch%2CB&author=Moroder%2CT&author=G%C3%BChne%2CO"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="56."><p class="c-article-references__text" id="ref-CR56">Tóth, G. & Gühne, O. Entanglement detection in the stabilizer formalism. <i>Phys. Rev. A</i> <b>72</b>, 022340 (2005).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.72.022340" data-track-item_id="10.1103/PhysRevA.72.022340" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.72.022340" aria-label="Article reference 56" data-doi="10.1103/PhysRevA.72.022340">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2005PhRvA..72b2340T" aria-label="ADS reference 56">ADS</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 56" href="http://scholar.google.com/scholar_lookup?&title=Entanglement%20detection%20in%20the%20stabilizer%20formalism&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.72.022340&volume=72&publication_year=2005&author=T%C3%B3th%2CG&author=G%C3%BChne%2CO"> Google Scholar</a>  </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="57."><p class="c-article-references__text" id="ref-CR57">van den Berg, E., Minev, Z. K. & Temme, K. Model-free readout-error mitigation for quantum expectation values. <i>Phys. Rev. A</i> <b>105</b>, 032620 (2022).</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevA.105.032620" data-track-item_id="10.1103/PhysRevA.105.032620" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevA.105.032620" aria-label="Article reference 57" data-doi="10.1103/PhysRevA.105.032620">Article</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2022PhRvA.105c2620V" aria-label="ADS reference 57">ADS</a>  <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4412660" aria-label="MathSciNet reference 57">MathSciNet</a>  <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 57" href="http://scholar.google.com/scholar_lookup?&title=Model-free%20readout-error%20mitigation%20for%20quantum%20expectation%20values&journal=Phys.%20Rev.%20A&doi=10.1103%2FPhysRevA.105.032620&volume=105&publication_year=2022&author=Berg%2CE&author=Minev%2CZK&author=Temme%2CK"> Google Scholar</a>  </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/s41586-024-08178-2?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team. We acknowledge B. Donovan, I. Hincks and K. Barton for circuit compilation and execution. We also thank D. Sutter, J. Orcutt, E. van den Berg, K. Temme, L. Bishop and P. Seidler for their discussions.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">IBM Quantum, IBM Research Europe - Zurich, Rüschlikon, Switzerland</p><p class="c-article-author-affiliation__authors-list">Almudena Carrera Vazquez, Caroline Tornow, Stefan Woerner & Daniel J. 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Egger</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Daniel%20J.%20Egger" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Daniel%20J.%20Egger" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Daniel%20J.%20Egger%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>A.C.V. gathered the LO data and studied the error mitigation of MCMs. C.T. studied the error mitigation of dynamic circuits and graph states. D.J.E. gathered the LOCC data. A.C.V., C.T. and D.J.E. analysed the circuit-cutting data. D.R. implemented the error suppression for dynamic circuits and optimized their execution on hardware. S.W. designed the cut Bell pairs. M.T. coordinated hardware resources and experiment execution. D.J.E. and M.T. designed the graph state experiments. D.J.E. designed the switch error mitigation. All authors discussed the results and improved and approved the paper.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:deg@zurich.ibm.com">Daniel J. Egger</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar2">Competing interests</h3> <p>A patent (application no. 18/523211) was filed on 29 November 2023 with listed inventors D.J.E., C.T., D.R., A.C.V., S.W. and M.T. The authors declare no other competing financial or non-financial interests.</p> </div></div></section><section data-title="Peer review"><div class="c-article-section" id="peer-review-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="peer-review">Peer review</h2><div class="c-article-section__content" id="peer-review-content"> <h3 class="c-article__sub-heading" id="FPar1">Peer review information</h3> <p><i>Nature</i> thanks the anonymous reviewers for their contribution to the peer review of this work. <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41586-024-08178-2#MOESM2">Peer reviewer reports</a> are available.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p><b>Publisher’s note</b> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section data-title="Extended data figures and tables"><div class="c-article-section" id="Sec17-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec17">Extended data figures and tables</h2><div class="c-article-section__content" id="Sec17-content"><div data-test="supplementary-info"><div id="figshareContainer" class="c-article-figshare-container" data-test="figshare-container"></div><div class="c-article-supplementary__item js-c-reading-companion-figures-item" data-test="supp-item" id="Fig4"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="extended data fig. 1 lo decomposition of a cz gate" href="/articles/s41586-024-08178-2/figures/4" data-supp-info-image="//media.springernature.com/lw685/springer-static/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig4_ESM.jpg">Extended Data Fig. 1 LO decomposition of a CZ gate.</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>A single CZ gate can be simulated through local operations by sampling from the shown QPD and applying classical post-processing to the results. Each of the six circuits has a sampling probability of 1/(2<i>γ</i>) = 1/6. For the four circuits featuring mid-circuit measurements, the corresponding QPD coefficient is adjusted by a factor of + 1 for outcome 0 and a factor of − 1 for outcome 1. To optimize the execution, these six circuits are consolidated into three parametrized circuits to enable a parametric circuit execution. The green, red, and yellow circuits correspond to the three template circuits generated from cutting a single CZ gate with the LO protocol. Here, the presence or absence of a mid-circuit measurement changes the pulse-level payload which thus requires compilation.</p></div></div><div class="c-article-supplementary__item js-c-reading-companion-figures-item" data-test="supp-item" id="Fig5"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="extended data fig. 2 zero-noise extrapolation of a" href="/articles/s41586-024-08178-2/figures/5" data-supp-info-image="//media.springernature.com/lw685/springer-static/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig5_ESM.jpg">Extended Data Fig. 2 Zero-noise extrapolation of a switch.</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p><b>a</b>, Implementation of a switch with DD. The conditional gates (not shown) are executed after the last DD <i>X</i> gate. The red delay of <i>τ</i> shows the duration in which no gates can be executed as the control electronics is busy, see Sec. VIC 1. The three additional delays of <i>τ</i> enable staggered DD. The four additional and variable delays of <i>δ</i> allow us to vary the duration of the switch for ZNE. <b>b</b>,<b>c</b>, The <i>Z</i><i>X</i> and <i>X</i><i>Z</i> correlators measured on <i>ibm_peekskill</i> as a function of the switch stretch factor <i>c</i> for a two-qubit graph state on <i>G</i> = ({0, 1}, {(0, 1)}). <b>d</b>, Example correlators of the 103 node graph extrapolated with ZNE.</p></div></div><div class="c-article-supplementary__item js-c-reading-companion-figures-item" data-test="supp-item" id="Fig6"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="extended data fig. 3 quantum circuit of three cut " href="/articles/s41586-024-08178-2/figures/6" data-supp-info-image="//media.springernature.com/lw685/springer-static/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig6_ESM.jpg">Extended Data Fig. 3 Quantum circuit of three cut Bell pairs.</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>A sum over the right set of parameter vectors {<b><i>θ</i></b>} results in three cut Bell pairs between qubit pairs (<i>q</i><sub>0</sub>, <i>q</i><sub>3</sub>), (<i>q</i><sub>1</sub>, <i>q</i><sub>4</sub>), and (<i>q</i><sub>2</sub>, <i>q</i><sub>5</sub>). The gate <i>U</i>(<i>θ</i>, <i>ϕ</i>) corresponds to the gate sequence <span class="mathjax-tex">$\sqrt{X}{R}_{z}(\theta )\sqrt{X}{R}_{z}(\phi )$</span>. The blue and red shaded regions correspond to the two disjoint portions of the quantum circuit.</p></div></div><div class="c-article-supplementary__item js-c-reading-companion-figures-item" data-test="supp-item" id="Fig7"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="extended data fig. 4 locc bell pair benchmark." href="/articles/s41586-024-08178-2/figures/7" data-supp-info-image="//media.springernature.com/lw685/springer-static/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_Fig7_ESM.jpg">Extended Data Fig. 4 LOCC Bell pair benchmark.</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p><b>a</b>, Quantum circuit that creates an uncut Bell pair on qubits (1, 2) and consume it in a teleportation circuit to create a Bell state on qubits (0, 3). <b>b</b>, Cumulative distribution function of the MSE of <span class="stix">⟨</span><i>Z</i><i>X</i><span class="stix">⟩</span> and <span class="stix">⟨</span><i>X</i><i>Z</i><span class="stix">⟩</span> for all groups of four linearly connected qubits on each device. The stars correspond to the qubits used in the 103- and 134-node graph states presented in the main text. The numbers in brackets indicate the qubit numbers corresponding to (<i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>) in panel (a).</p></div></div><div class="c-article-supplementary__item" data-test="supp-item"><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Extended Data Table 1 Circuit structure and node error</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/articles/s41586-024-08178-2/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div><div class="c-article-supplementary__item" data-test="supp-item"><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-2"><figure><figcaption class="c-article-table__figcaption"><b id="Tab2" data-test="table-caption">Extended Data Table 2 Witness tests</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/articles/s41586-024-08178-2/tables/2" aria-label="Full size table 2"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></div></div></section><section data-title="Supplementary information"><div class="c-article-section" id="Sec18-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec18">Supplementary information</h2><div class="c-article-section__content" id="Sec18-content"><div data-test="supplementary-info"><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM1"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary information" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_MOESM1_ESM.pdf" data-supp-info-image="">Supplementary Information</a></h3></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM2"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="peer review file" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41586-024-08178-2/MediaObjects/41586_2024_8178_MOESM2_ESM.pdf" data-supp-info-image="">Peer Review File</a></h3></div></div></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. 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