{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T08:00:57Z","timestamp":1782979257475,"version":"3.54.5"},"reference-count":75,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T00:00:00Z","timestamp":1764201600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"\n We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about\n \n 50<\/mml:mn>\n <\/mml:math>\n qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as a\n \n [<\/mml:mo>\n [<\/mml:mo>\n 48<\/mml:mn>\n ,<\/mml:mo>\n 4<\/mml:mn>\n ,<\/mml:mo>\n 7<\/mml:mn>\n ]<\/mml:mo>\n ]<\/mml:mo>\n <\/mml:math>\n BB5 code. For a physical error rate of\n \n \n 10<\/mml:mn>\n \n &#x2212;<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n , the\n \n [<\/mml:mo>\n [<\/mml:mo>\n 48<\/mml:mn>\n ,<\/mml:mo>\n 4<\/mml:mn>\n ,<\/mml:mo>\n 7<\/mml:mn>\n ]<\/mml:mo>\n ]<\/mml:mo>\n <\/mml:math>\n BB5 code achieves a logical error rate per logical qubit of\n \n 5<\/mml:mn>\n &#x22C5;<\/mml:mo>\n \n 10<\/mml:mn>\n \n &#x2212;<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n , which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.\n <\/jats:p>","DOI":"10.22331\/q-2025-11-27-1920","type":"journal-article","created":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T06:21:42Z","timestamp":1764224502000},"page":"1920","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":["Quantum error correction for long chains of trapped ions"],"prefix":"10.22331","volume":"9","author":[{"given":"Min","family":"Ye","sequence":"first","affiliation":[{"name":"IonQ Inc."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"IonQ Inc."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,11,27]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Google Quantum AI. 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Toward a 2d local implementation of quantum low-density parity-check codes. PRX Quantum, 6 (1): 010306, 2025. 10.1103\/PRXQuantum.6.010306.","DOI":"10.1103\/PRXQuantum.6.010306"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. Logical quantum processor based on reconfigurable atom arrays. Nature, 626 (7997): 58\u201365, 2024. 10.1038\/s41586-023-06927-3.","DOI":"10.1038\/s41586-023-06927-3"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Hector Bombin and Miguel Angel Martin-Delgado. Topological quantum distillation. Physical review letters, 97 (18): 180501, 2006. 10.1103\/PhysRevLett.97.180501.","DOI":"10.1103\/PhysRevLett.97.180501"},{"key":"6","doi-asserted-by":"publisher","unstructured":"J Pablo Bonilla Ataides, David K Tuckett, Stephen D Bartlett, Steven T Flammia, and Benjamin J Brown. 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Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, D. Gresh, A. Hall, A. Hankin, A. Hansen, N. Hewitt, I. Hoffman, C. Holliman, R. B. Hutson, T. Jacobs, J. Johansen, P. J. Lee, E. Lehman, D. Lucchetti, D. Lykov, I. S. Madjarov, B. Mathewson, K. Mayer, M. Mills, P. Niroula, J. M. Pino, C. Roman, M. Schecter, P. E. Siegfried, B. G. Tiemann, C. Volin, J. Walker, R. Shaydulin, M. Pistoia, S. A. Moses, D. Hayes, B. Neyenhuis, R. P. Stutz, and M. Foss-Feig. Computational power of random quantum circuits in arbitrary geometries. Phys. Rev. X, 15: 021052, May 2025. 10.1103\/PhysRevX.15.021052.","DOI":"10.1103\/PhysRevX.15.021052"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. 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Nature, 598 (7880): 281\u2013286, 2021. 10.1038\/s41586-021-03928-y.","DOI":"10.1038\/s41586-021-03928-y"},{"key":"22","unstructured":"Laird Nicholas Egan. Scaling quantum computers with long chains of trapped ions. PhD thesis, University of Maryland, College Park, 2021."},{"key":"23","doi-asserted-by":"publisher","unstructured":"Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A\u2014Atomic, Molecular, and Optical Physics, 86 (3): 032324, 2012. 10.1103\/PhysRevA.86.032324.","DOI":"10.1103\/PhysRevA.86.032324"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Craig Gidney. Stim: a fast stabilizer circuit simulator. Quantum, 5: 497, 2021. 10.22331\/q-2021-07-06-497.","DOI":"10.22331\/q-2021-07-06-497"},{"key":"25","unstructured":"Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997."},{"key":"26","unstructured":"Virgile Guemard and Gilles Z\u00e9mor. Moderate-length lifted quantum tanner codes. arXiv:2502.20297, 2025."},{"key":"27","doi-asserted-by":"publisher","unstructured":"J\u00e9r\u00e9mie Guillaud and Mazyar Mirrahimi. Repetition cat qubits for fault-tolerant quantum computation. Physical Review X, 9 (4): 041053, 2019. 10.1103\/PhysRevX.9.041053.","DOI":"10.1103\/PhysRevX.9.041053"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Hartmut H\u00e4ffner, Christian F Roos, and Rainer Blatt. Quantum computing with trapped ions. Physics reports, 469 (4): 155\u2013203, 2008. 10.1016\/j.physrep.2008.09.003.","DOI":"10.1016\/j.physrep.2008.09.003"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Oscar Higgott and Craig Gidney. Sparse Blossom: correcting a million errors per core second with minimum-weight matching. 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Reviews of Modern Physics, 93 (2): 025001, 2021. 10.1103\/RevModPhys.93.025001.","DOI":"10.1103\/RevModPhys.93.025001"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Thomas Monz, Philipp Schindler, Julio T Barreiro, Michael Chwalla, Daniel Nigg, William A Coish, Maximilian Harlander, Wolfgang H\u00e4nsel, Markus Hennrich, and Rainer Blatt. 14-qubit entanglement: Creation and coherence. Physical Review Letters, 106 (13): 130506, 2011. 10.1103\/PhysRevLett.106.130506.","DOI":"10.1103\/PhysRevLett.106.130506"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Thomas Monz, Daniel Nigg, Esteban A Martinez, Matthias F Brandl, Philipp Schindler, Richard Rines, Shannon X Wang, Isaac L Chuang, and Rainer Blatt. Realization of a scalable Shor algorithm. Science, 351 (6277): 1068\u20131070, 2016. 10.1126\/science.aad9480.","DOI":"10.1126\/science.aad9480"},{"key":"44","doi-asserted-by":"publisher","unstructured":"CJ Myatt, EA Burt, RW Ghrist, Eric A Cornell, and CE Wieman. Production of two overlapping Bose-Einstein condensates by sympathetic cooling. Physical Review Letters, 78 (4): 586, 1997. 10.1103\/PhysRevLett.78.586.","DOI":"10.1103\/PhysRevLett.78.586"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Yunseong Nam, Jwo-Sy Chen, Neal C Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R Brown, Stewart Allen, Jason M Amini, Joel Apisdorf, et al. Ground-state energy estimation of the water molecule on a trapped-ion quantum computer. npj Quantum Information, 6 (1): 33, 2020. 10.1038\/s41534-020-0259-3.","DOI":"10.1038\/s41534-020-0259-3"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. 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