Quantum error correction for long chains of trapped ions

Min Ye; Nicolas Delfosse

doi:10.22331/q-2025-11-27-1920

Quantum error correction for long chains of trapped ions

Min Ye and Nicolas Delfosse

IonQ Inc.

Published:	2025-11-27, volume 9, page 1920
Editor:	Philipp Schindler
Eprint:	arXiv:2503.22071v3
Doi:	https://doi.org/10.22331/q-2025-11-27-1920
Citation:	Quantum 9, 1920 (2025).

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Abstract

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 $50$ 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 $[[48, 4, 7]]$ BB5 code. For a physical error rate of $10^{-3}$, the $[[48, 4, 7]]$ BB5 code achieves a logical error rate per logical qubit of $5 \cdot 10^{-5}$, 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.

Featured image: We propose a model for quantum computing with long chains of trapped ions and construct new quantum codes that outperform the state-of-the-art for chains of approximately 50 qubits. Specifically, we introduce a new variant of BB codes defined by weight-five measurements, which we call BB5 codes. We identify specific BB5 codes that achieve a better minimum distance than any existing BB code with the same number of logical and data qubits, such as $[[30, 4, 5]]$ and $[[48, 4, 7]]$ BB5 codes.

In the figure, we compare two sets of BB and BB5 codes with identical code lengths and logical qubit counts. For length-30 codes with 4 logical qubits, our new BB5 code achieves a 10x improvement over standard BB codes. For length-48 codes with 4 logical qubits, this advantage grows to a 20x improvement. These results are validated through circuit-level noise simulations under the ion chain model.

► BibTeX data

@article{Ye2025quantumerror,
  doi = {10.22331/q-2025-11-27-1920},
  url = {https://doi.org/10.22331/q-2025-11-27-1920},
  title = {Quantum error correction for long chains of trapped ions},
  author = {Ye, Min and Delfosse, Nicolas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1920},
  month = nov,
  year = {2025}
}

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[1] Isabelle Savill-Brown, Joseph J. Hope, Alexander K. Ratcliffe, Varun D. Vaidya, Haonan Liu, Simon A. Haine, C. Ricardo Viteri, and Zain Mehdi, "High-Speed and High-Connectivity Two-Qubit Gates in Long Chains of Trapped Ions", Physical Review Letters 136 19, 190802 (2026).

[2] Hengyun Zhou, Madelyn Cain, and Mikhail D. Lukin, "Opportunities in full-stack design of low-overhead fault-tolerant quantum computation", Nature Computational Science 5 12, 1110 (2025).

[3] Balint Pato, June Vanlerberghe, and Kenneth R. Brown, "Hyper-optimized Quantum Lego Contraction Schedules", Quantum 10, 2092 (2026).

[4] Min Ye, Dave Wecker, and Nicolas Delfosse, "Beam Search Decoder for Quantum Low-Density Parity-Check Codes", PRX Quantum 7 3, 033002 (2026).

[5] Sahil Khan, Suhas Vittal, Kenneth Brown, and Jonathan Baker, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 637 (2025) ISBN:979-8-3315-5736-2.

[6] Aleksandra Świerkowska, Jannik Pflieger, Emmanouil Giortamis, and Pramod Bhatotia, "ECCentric: An Empirical Analysis of Quantum Error Correction Codes", Proceedings of the ACM on Measurement and Analysis of Computing Systems 10 2, 1 (2026).

[7] Sahil Khan, Abhinav Anand, Kenneth R. Brown, and Jonathan M. Baker, 2026 IEEE International Symposium on High Performance Computer Architecture (HPCA) 1 (2026) ISBN:979-8-3315-9302-5.

[8] J. Pablo Bonilla Ataides, Hengyun Zhou, Qian Xu, Gefen Baranes, Bikun Li, Mikhail D. Lukin, and Liang Jiang, "Constant-Overhead Fault-Tolerant Bell-Pair Distillation Using High-Rate Codes", Physical Review Letters 135 13, 130804 (2025).

[9] Christian Kraglund Andersen and Eliška Greplová, "Small Quantum Low Density Parity Check Codes for Near-Term Experiments", arXiv:2507.09690, (2025).

[10] György P. Gehér, David Byfield, and Archibald Ruban, "Directional Codes: a new family of quantum LDPC codes on hexagonal- and square-grid connectivity hardware", arXiv:2507.19430, (2025).

[11] Abraham Jacob, Campbell McLauchlan, and Dan E. Browne, "Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes", arXiv:2508.08191, (2025).

[12] Arda Aydin, Nicolas Delfosse, and Edwin Tham, "Cyclic Hypergraph Product Code", arXiv:2511.09683, (2025).

[13] Isabelle Savill-Brown, Joseph J. Hope, Alexander K. Ratcliffe, Varun D. Vaidya, Haonan Liu, Simon A. Haine, C. Ricardo Viteri, and Zain Mehdi, "High-speed and high-connectivity two-qubit gates in long chains of trapped ions", arXiv:2506.11385, (2025).

[14] Edwin Tham, Min Ye, Ilia Khait, John Gamble, and Nicolas Delfosse, "Distributed fault-tolerant quantum memories over a 2xL array of qubit modules", arXiv:2508.01879, (2025).

[15] Aleksandra Świerkowska, Jannik Pflieger, Emmanouil Giortamis, and Pramod Bhatotia, "ECCentric: An Empirical Analysis of Quantum Error Correction Codes", arXiv:2511.01062, (2025).

[16] Rebecca Katharina Radebold, Stephen D. Bartlett, and Andrew C. Doherty, "Explicit Instances of Quantum Tanner Codes", arXiv:2508.05095, (2025).

[17] Shintaro Sato and Yasunari Suzuki, "Scheduling of syndrome measurements with a few ancillary qubits", arXiv:2508.07913, (2025).

[18] Tyler LeBlond, Peter Groszkowski, Justin G. Lietz, Christopher M. Seck, and Ryan S. Bennink, "Logical error rates for the surface code under a mixed coherent and stochastic circuit-level noise model inspired by trapped ions", Physical Review Research 7 4, 043184 (2025).

[19] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

[20] Edwin Tham and Nicolas Delfosse, "Optimized Clifford Noise Reduction: Theory, Simulations and Experiments", Quantum 9, 1829 (2025).

[21] Pranav S. Mundada, Aleksei Khindanov, Yulun Wang, Claire L. Edmunds, Paul Coote, Michael J. Biercuk, Yuval Baum, and Michael Hush, "Heterogeneous architectures enable a 138x reduction in physical qubit requirements for fault-tolerant quantum computing under detailed accounting", arXiv:2604.06319, (2026).

[22] Balint Pato, June Vanlerberghe, and Kenneth R. Brown, "Hyper-optimized Quantum Lego Contraction Schedules", arXiv:2510.08210, (2025).

[23] Nitish Kumar Chandra, Eneet Kaur, Reza Nejabati, and Kaushik P. Seshadreesan, "Distributed Quantum Error Correction with Bivariate Bicycle Codes in a Modular Architecture", arXiv:2605.04663, (2026).

[24] Nolan J. Coble, Min Ye, and Nicolas Delfosse, "Correction of chain losses in trapped ion quantum computers", arXiv:2511.16632, (2025).

[25] Tommaso Faorlin, Lorenz Panzl, Phoebe Grosser, Pablo Viñas, Alan Kahan, Walter Joseph Hörmann, Yannick Weiser, Giovanni Cerchiari, Thomas Feldker, Alexander Erhard, Georg Jacob, Juris Ulmanis, Rainer Blatt, Alejandro Bermudez, and Thomas Monz, "Entangling ions with engineered light gradients", arXiv:2603.07548, (2026).

[26] Cordell Mazzetti, Sayam Sethi, Rich Rines, Pranav Gokhale, and Jonathan Mark Baker, "Logical Compilation for Multi-Qubit Iceberg Patches", arXiv:2604.09956, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 17:31:13) and SAO/NASA ADS (last updated successfully 2026-08-13 17:31:14). The list may be incomplete as not all publishers provide suitable and complete citation data.

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