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  • Open Access

Adaptive Syndrome Extraction

Noah Berthusen*, Shi Jie Samuel Tan, Eric Huang, and Daniel Gottesman

  • *Contact author: nfbert@umd.edu
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PRX Quantum 6, 030307 – Published 16 July, 2025

DOI: https://doi.org/10.1103/ps3r-wf84

Abstract

Device error rates on current quantum computers have improved enough to where demonstrations of error correction below break even are now possible. Still, the circuits required for quantum error correction introduce significant overhead and sometimes inject more errors than they correct. In this work, we introduce adaptive syndrome extraction as a scheme to improve code performance and reduce the quantum error-correction cycle time by measuring only the stabilizer generators that are likely to provide useful syndrome information. We provide a concrete example of the scheme through the [[4,2,2]] code concatenated with a hypergraph product code and a syndrome-extraction cycle that uses quantum error detection to modify the syndrome-extraction circuits in real time. Compared to nonconcatenated codes and nonadaptive syndrome extraction, we find that the adaptive scheme achieves over an order of magnitude lower logical error rates while requiring fewer cnot gates and physical qubits. Furthermore, we show how to achieve fault-tolerant universal logical computation with [[4,2,2]]-concatenated hypergraph product codes.

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Popular Summary

Quantum error correction will be necessary to overcome the fragility of quantum information. To detect and correct errors, we must precisely probe certain properties of the system; unfortunately, interacting with the system can introduce noise, sometimes more than is removed. This phenomenon is apparent in near-term error-correction experiments, where unencoded qubits often have longer coherence times than error-corrected logical qubits. We introduce adaptive syndrome extraction as a protocol to mitigate this effect.

The protocol consists of two stages: in the first stage, we make measurements of a small set of “easy” stabilizer generators that will not introduce much noise. The results of the measurements from the first stage allow us to approximately determine where errors are located in the system. We then perform more “difficult” measurements to gather enough information to be able to correct errors. The effect of using this protocol is that less circuit noise is introduced, leading to lower logical error rates. Furthermore, this protocol enables a speedup in quantum error correction on certain quantum computing architectures, reducing the accumulation of idling errors. We corroborate these claims with circuit-level simulations. We also show that the overhead of fault-tolerant universal logical computation is not increased.

In general, our findings help motivate a move from statically compiled quantum error correction and algorithms to ones that make real-time decisions in order to reduce computational overheads. Doing so may hasten the arrival of large-scale quantum computation.

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References (94)

  1. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature 638, 920 (2025).
  2. B. W. Reichardt, D. Aasen, R. Chao, A. Chernoguzov, W. van Dam, J. P. Gaebler, D. Gresh, D. Lucchetti, M. Mills, and S. A. Moses et al., Demonstration of quantum computation and error correction with a tesseract code, arXiv:2409.04628.
  3. M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, A. Chernoguzov, J. M. Dreiling, C. Foltz, F. Frachon, J. P. Gaebler, T. M. Gatterman, L. Grans-Samuelsson, and D. Hayes et al., Demonstration of logical qubits and repeated error correction with better-than-physical error rates, arXiv:2404.02280.
  4. B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, W. van Dam, and M. B. Hastings et al., Logical computation demonstrated with a neutral atom quantum processor, arXiv:2411.11822.
  5. N. Lacroix, A. Bourassa, F. J. H. Heras, L. M. Zhang, J. Bausch, A. W. Senior, T. Edlich, N. Shutty, V. Sivak, and A. Bengtsson et al., Scaling and logic in the color code on a superconducting quantum processor, arXiv:2412.14256.
  6. X. Xu, N. d. Beaudrap, J. O’Gorman, and S. C. Benjamin, An integrity measure to benchmark quantum error correcting memories, New J. Phys. 20, 023009 (2018).
  7. N. Delfosse, M. E. Beverland, and M. A. Tremblay, Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes, arXiv:2109.14599.
  8. N. Baspin, O. Fawzi, and A. Shayeghi, A lower bound on the overhead of quantum error correction in low dimensions, arXiv:2302.04317.
  9. N. C. Brown, J. P. Campora III, C. Granade, B. Heim, S. Wernli, C. Ryan-Anderson, D. Lucchetti, A. Paetznick, M. Roetteler, K. Svore, and A. Chernoguzov, Advances in compilation for quantum hardware – a demonstration of magic state distillation and repeat-until-success protocols, arXiv:2310.12106.
  10. B. W. Reichardt, Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits, Quantum Sci. Technol. 6, 015007 (2020).
  11. C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, and N. C. Brown et al., Realization of real-time fault-tolerant quantum error correction, Phys. Rev. X 11, 041058 (2021).
  12. N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V. Gorshkov, and D. Gottesman, Toward a 2D local implementation of quantum low-density parity-check codes, PRX Quantum 6, 010306 (2025).
  13. M. B. Hastings, Weight reduction for quantum codes, arXiv:1611.03790.
  14. M. B. Hastings, On quantum weight reduction, arXiv:2102.10030.
  15. D. Poulin, Stabilizer formalism for operator quantum error correction, Phys. Rev. Lett. 95, 230504 (2005).
  16. D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A 73, 012340 (2006).
  17. H. Bombin, Topological subsystem codes, Phys. Rev. A 81, 032301 (2010).
  18. N. Baspin and D. Williamson, Wire codes, arXiv:2410.10194.
  19. D. Bacon, S. T. Flammia, A. W. Harrow, and J. Shi, Sparse quantum codes from quantum circuits, IEEE Trans. Inf. Theory 63, 2464 (2017).
  20. M. B. Hastings and J. Haah, Dynamically generated logical qubits, Quantum 5, 564 (2021).
  21. D. Gottesman, Opportunities and challenges in fault-tolerant quantum computation, arXiv:2210.15844.
  22. N. Delfosse and A. Paetznick, Spacetime codes of Clifford circuits, arXiv:2304.05943.
  23. J. C. M. de la Fuente, Dynamical weight reduction of Pauli measurements, arXiv:2410.12527.
  24. A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54, 4741 (1996).
  25. D. Gottesman, Stabilizer codes and quantum error correction, arXiv:quant-ph/9705052.
  26. C. N. Self, M. Benedetti, and D. Amaro, Protecting expressive circuits with a quantum error detection code, Nat. Phys. 20, 219 (2024).
  27. J.-P. Tillich and G. Zemor, Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Trans. Inf. Theory 60, 1193 (2014).
  28. Q. Xu, H. Zhou, G. Zheng, D. Bluvstein, J. Ataides, M. D. Lukin, and L. Jiang, Fast and parallelizable logical computation with homological product codes, arXiv:2407.18490.
  29. Y. Hong, Single-shot preparation of hypergraph product codes via dimension jump, arXiv:2410.05171.
  30. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  31. A. Steane, Multiple particle interference and quantum error correction, Proc. R. Soc. London. Ser. A: Math. Phys. Eng. Sci. 452, 2551 (1996).
  32. S. Bravyi and M. B. Hastings, in Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14 (Association for Computing Machinery, New York, NY, USA, 2014), p. 273.
  33. M. Sipser and D. Spielman, Expander codes, IEEE Trans. Inf. Theory 42, 1710 (1996).
  34. A. Leverrier, J.-P. Tillich, and G. Zemor, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (IEEE, Berkeley, CA, USA, 2015).
  35. A. O. Quintavalle and E. T. Campbell, Reshape: A decoder for hypergraph product codes, IEEE Trans. Inf. Theory 68, 6569 (2022).
  36. A. O. Quintavalle, P. Webster, and M. Vasmer, Partitioning qubits in hypergraph product codes to implement logical gates, Quantum 7, 1153 (2023).
  37. L. P. Pryadko, V. A. Shabashov, and V. K. Kozin, QDistRnd: A GAP package for computing the distance of quantum error-correcting codes, J. Open Source Software 7, 4120 (2022).
  38. E. Knill, Quantum computing with realistically noisy devices, Nature 434, 39 (2005).
  39. S. Yoshida, S. Tamiya, and H. Yamasaki, Concatenate codes, save qubits, npj Quantum Inf. 11, 88 (2025).
  40. H. Goto, High-performance fault-tolerant quantum computing with many-hypercube codes, Sci. Adv. 10, eadp6388 (2024).
  41. B. Criger and B. Terhal, Noise thresholds for the [[4, 2, 2]]-concatenated toric code, Quantum Information and Computation 16, 1261 (2016).
  42. E. Knill, Scalable quantum computing in the presence of large detected-error rates, Phys. Rev. A 71, 042322 (2005).
  43. P. Shor, in Proceedings of 37th Conference on Foundations of Computer Science (IEEE, Burlington, VT, USA, 1996), p. 56.
  44. H. Goto and H. Uchikawa, Fault-tolerant quantum computation with a soft-decision decoder for error correction and detection by teleportation, Sci. Rep. 3, 2044 (2013).
  45. D. Poulin, Optimal and efficient decoding of concatenated quantum block codes, Phys. Rev. A 74, 052333 (2006).
  46. N. Meister, C. A. Pattison, and J. Preskill, Efficient soft-output decoders for the surface code, arXiv:2405.07433.
  47. S. Huang and S. Puri, Increasing memory lifetime of quantum low-density parity check codes with sliding-window noisy syndrome decoding, Phys. Rev. A 110, 012453 (2024).
  48. D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Surface code quantum computing with error rates over 1%, Phys. Rev. A 83, 020302 (2011).
  49. Q. Xu, J. P. B. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou, Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays, Nat. Phys. 20, 1084 (2024).
  50. A. Gong, S. Cammerer, and J. M. Renes, Toward low-latency iterative decoding of QLDPC codes under circuit-level noise, arXiv:2403.18901.
  51. S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, and P. Blanchard et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  52. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, and D. Hangleiter et al., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024).
  53. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  54. A. G. Manes and J. Claes, Distance-preserving stabilizer measurements in hypergraph product codes, Quantum 9, 1618 (2025).
  55. S. J. S. Tan and L. Stambler, Effective distance of higher dimensional HGPS and weight-reduced quantum LDPC codes, arXiv:2409.02193.
  56. H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015).
  57. N. Berthusen and D. Gottesman, Partial syndrome measurement for hypergraph product codes, Quantum 8, 1345 (2024).
  58. D. J. C. MacKay and R. M. Neal, Near Shannon limit performance of low density parity check codes, Electron. Lett. 33, 457 (1997).
  59. J. Roffe, https://pypi.org/project/ldpc/ LDPC: Python tools for low density parity check codes (2022).
  60. T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes, arXiv:2406.18655.
  61. K. Fang, M. Zhang, R. Shi, and Y. Li, Dynamic quantum circuit compilation, arXiv:2310.11021.
  62. J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyenhuis, Demonstration of the trapped-ion quantum CCD computer architecture, Nature 592, 209 (2021).
  63. L. Pecorari, S. Jandura, G. K. Brennen, and G. Pupillo, High-rate quantum LDPC codes for long-range-connected neutral atom registers, Nat. Commun. 16, 1111 (2025).
  64. S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, arXiv:quant-ph/9811052.
  65. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (N. Y) 303, 2 (2003).
  66. H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).
  67. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  68. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2D systems, Phys. Rev. Lett. 104, 050503 (2010).
  69. N. Baspin and A. Krishna, Quantifying nonlocality: How outperforming local quantum codes is expensive, Phys. Rev. Lett. 129, 050505 (2022).
  70. C. A. Pattison, A. Krishna, and J. Preskill, Hierarchical memories: Simulating quantum LDPC codes with local gates, Quantum 9, 1728 (2025).
  71. S. H. Choe and R. Koenig, How to fault-tolerantly realize any quantum circuit with local operations, arXiv:2402.13863.
  72. A. A. Kovalev and L. P. Pryadko, Quantum Kronecker sum-product low-density parity-check codes with finite rate, Phys. Rev. A 88, 012311 (2013).
  73. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature 627, 778 (2024).
  74. Y. Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, Long-range-enhanced surface codes, Phys. Rev. A 110, 022607 (2024).
  75. C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A 89, 022317 (2014).
  76. A. Strikis and L. Berent, Quantum low-density parity-check codes for modular architectures, PRX Quantum 4, 020321 (2023).
  77. C. Gidney, M. Newman, P. Brooks, and C. Jones, Yoked surface codes, Nat. Commun. 16, 4498 (2025).
  78. G. Zhu, A topological theory for qLDPC: Non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the n1/3 distance barrier, arXiv:2501.19375.
  79. A. J. Malcolm, A. N. Glaudell, P. Fuentes, D. Chandra, A. Schotte, C. DeLisle, R. Haenel, A. Ebrahimi, J. Roffe, A. O. Quintavalle, S. J. Beale, N. R. Lee-Hone, and S. Simmons, Computing efficiently in QLDPC codes, arXiv:2502.07150.
  80. P. Panteleev and G. Kalachev, Degenerate quantum LDPC codes with good finite length performance, Quantum 5, 585 (2021).
  81. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2022).
  82. N. Berthusen, Adaptive Syndrome Extraction (2025), https://github.com/noahberthusen/adaptive_qec.
  83. A. Grospellier, L. Grouès, A. Krishna, and A. Leverrier, Combining hard and soft decoders for hypergraph product codes, Quantum 5, 432 (2021).
  84. A. A. Kovalev, S. Prabhakar, I. Dumer, and L. P. Pryadko, Numerical and analytical bounds on threshold error rates for hypergraph-product codes, Phys. Rev. A 97, 062320 (2018).
  85. M. A. Tremblay, N. Delfosse, and M. E. Beverland, Constant-overhead quantum error correction with thin planar connectivity, Phys. Rev. Lett. 129, 050504 (2022).
  86. C. Gidney, Stim: A fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  87. C. Pattison, https://github.com/qldpc/exp_ldpc QEC utilities for practical realizations of general qLDPC codes (2024).
  88. D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998).
  89. H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and D. E. Browne, Fault-tolerant logical Clifford gates from code automorphisms, arXiv:2409.18175.
  90. E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Sci. Technol. 4, 025006 (2019).
  91. A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, Single-shot error correction of three-dimensional homological product codes, PRX Quantum 2, 020340 (2021).
  92. A. Anshu, N. P. Breuckmann, and C. Nirkhe, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023 (Association for Computing Machinery, New York, NY, USA, 2023), p. 1090.
  93. N. P. Breuckmann and S. Burton, Fold-transversal Clifford gates for quantum codes, Quantum 8, 1372 (2024).
  94. R. Chao and B. W. Reichardt, Fault-tolerant quantum computation with few qubits, npj Quantum Inf. 4, 42 (2018).

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PRX Quantum 6, 030307– Published 16 July, 2025
  • Received 7 March 2025
  • Revised 22 May 2025
  • Accepted 28 May 2025
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