Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Fault-tolerant control of an error-corrected qubit

## Abstract

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3,4,5,6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon–Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.

This is a preview of subscription content, access via your institution

## Relevant articles

• ### Universal control of a six-qubit quantum processor in silicon

Nature Open Access 28 September 2022

• ### Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays

Nature Communications Open Access 09 August 2022

• ### Optical demonstration of quantum fault-tolerant threshold

Light: Science & Applications Open Access 05 July 2022

## Access options

\$32.00

All prices are NET prices.

## Data availability

The data that support the findings of this study are available from the corresponding author upon request. Source data are provided with this paper.

## Code availability

The code used for the analyses is available from the corresponding author upon request.

## References

1. Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 (1995).

2. Knill, E. & Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 55, 900–911 (1997).

3. Shor, P. W. Fault-tolerant quantum computation. In Proc. 37th Conference on Foundations of Computer Science (1996).

4. Knill, E., Laflamme, R. & Zurek, W. Threshold accuracy for quantum computation. Preprint at https://arxiv.org/abs/quant-ph/9610011 (1996).

5. Gottesman, D. Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127–137 (1998).

6. Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate. SIAM J. Comput. (2008).

7. Bravyi, S. & Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).

8. Feynman, R. P. Quantum mechanical computers. Found. Phys. 16, 507–531 (1986).

9. Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997).

10. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).

11. Reiher, M., Wiebe, N., Svore, K. M., Wecker, D. & Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl Acad. Sci. USA 114, 7555–7560 (2017).

12. Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999).

13. Von Burg, V. et al. Quantum computing enhanced computational catalysis. Preprint at https://arxiv.org/abs/2007.14460 (2020).

14. Gidney, C. & Ekerå, M. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Preprint at https://arxiv.org/abs/1905.09749 (2019).

15. Gottesman, D. E. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology (1997).

16. Córcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 6979 (2015).

17. Takita, M., Cross, A. W., Córcoles, A. D., Chow, J. M. & Gambetta, J. M. Experimental demonstration of fault-tolerant state preparation with superconducting qubits. Phys. Rev. Lett. 119, 180501 (2017).

18. Linke, N. M. et al. Fault-tolerant quantum error detection. Sci. Adv. 3, e1701074 (2017).

19. Harper, R. & Flammia, S. T. Fault-tolerant logical gates in the IBM quantum experience. Phys. Rev. Lett. 122, 080504 (2019).

20. Andersen, C. K. et al. Repeated quantum error detection in a surface code. Nat. Phys. 16, 875–880 (2020).

21. Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152–2155 (1998).

22. Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602–605 (2004).

23. Schindler, P. et al. Experimental repetitive quantum error correction. Science 332, 1059–1061 (2011).

24. Reed, M. D. et al. Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2012).

25. Riste, D. et al. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6, 6983 (2015).

26. Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015).

27. Gong, M. et al. Experimental verification of five-qubit quantum error correction with superconducting qubits. Preprint at https://arxiv.org/abs/1907.04507 (2019).

28. Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 345, 302–305 (2014).

29. Luo, Y. H. et al. Quantum teleportation of physical qubits into logical code-spaces. Preprint at https://arxiv.org/abs/2009.06242 (2020).

30. Heeres, R. W. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat. Commun. 8, 94 (2017).

31. Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).

32. Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016).

33. Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368–372 (2020).

34. de Neeve, B., Nguyen, T. L., Behrle, T. & Home, J. Error correction of a logical grid state qubit by dissipative pumping. Preprint at https://arxiv.org/abs/2010.09681 (2020).

35. Wilhelm Maunz, P. L. High Optical Access Trap 2.0 Report No. SAND2016-0796R (Sandia National Laboratories, 2016).

36. Debnath, S. et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 563, 63–66 (2016).

37. Wright, K. et al. Benchmarking an 11-qubit quantum computer. Nat. Commun. 10, 5464 (2019).

38. Bacon, D. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73, 012340 (2006).

39. Aliferis, P. & Cross, A. W. Subsystem fault tolerance with the Bacon–Shor code. Phys. Rev. Lett. 98, 220502 (2007).

40. Debroy, D. M., Li, M., Huang, S. & Brown, K. R. Logical performance of 9 qubit compass codes in ion traps with crosstalk errors. Quantum Sci. Technol. 5, 034002 (2020).

41. Li, M., Miller, D. & Brown, K. R. Direct measurement of Bacon–Shor code stabilizers. Phys. Rev. A 98, 050301 (2018).

42. Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015).

43. Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002).

44. Li, M., Miller, D., Newman, M., Wu, Y. & Brown, K. R. 2D compass codes. Phys. Rev. 9, 021041 (2019).

45. Reichardt, B. W. Quantum universality from magic states distillation applied to CSS codes. Quantum Inf. Process. 4, 251–264 (2005).

46. Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594–2597 (1998).

47. Kielpinski, D et al. A decoherence-free quantum memory using trapped ions. Science 291, 1013–1015 (2001).

48. Hu, J., Liang, Q., Rengaswamy, N. & Calderbank, R. Mitigating coherent noise by balancing weight-2z-stabilizers. Preprint at https://arxiv.org/abs/2011.00197 (2020).

49. Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102,110502 (2009).

50. Cetina, M. et al. Quantum gates on individually-addressed atomic qubits subject to noisy transverse motion. Preprint at https://arxiv.org/abs/2007.06768 (2020).

51. Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature 417, 709–711 (2002).

52. Home, J. P. et al. Complete methods set for scalable ion trap quantum information processing. Science 325, 1227–1230 (2009).

53. Pino, J. M. et al. Demonstration of the QCCD trapped-ion quantum computer architecture. Preprint at https://arxiv.org/abs/2003.01293 (2020).

54. Brown, K. R., Harrow, A. W. & Chuang, I. L. Arbitrarily accurate composite pulse sequences. Phys. Rev. A 70, 052318 (2004).

55. Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835–1838 (1999).

56. Maslov, D. Basic circuit compilation techniques for an ion-trap quantum machine. New J. Phys. 19, 023035 (2017).

## Acknowledgements

This work was performed at the University of Maryland with no material support from IonQ. We acknowledge discussions with N. M. Linke and the contributions of J. Mizrahi, K. Hudek, J. Amini, K. Beck and M. Goldman to the experimental setup. This work is supported by the ARO through the IARPA LogiQ programme, the NSF STAQ Program, the AFOSR MURIs on Dissipation Engineering in Open Quantum Systems and Quantum Interactive Protocols for Quantum Computation, and the ARO MURI on Modular Quantum Circuits. L.E. and D.M.D. are also funded by NSF award DMR-1747426.

## Author information

Authors

### Contributions

L.E. collected and analysed the data. L.E., D.M.D., C.N. and M.N. wrote the manuscript and designed figures. M.C. and C.M. led construction of the experimental apparatus with contributions from L.E., C.N., A.R., D.Z. and D.B. Theory support was provided by D.M.D., M.N., M.L. and K.R.B. C.M. and K.R.B. supervised the project. All authors discussed results and contributed to the manuscript.

### Corresponding author

Correspondence to Laird Egan.

## Ethics declarations

### Competing interests

K.R.B. is a scientific advisor for IonQ, Inc. and has a personal financial interest in the company.

Peer review information Nature thanks Daniel Gottesman, Jonathan Home, Philipp Schindler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Stabilizer measurement circuits.

a, b, Non-fault-tolerant (a, red, right) and fault-tolerant (b, blue, right) stabilizer measurement orderings, performed on a FT-encoded $$|0{\rangle }_{{\rm{L}}}$$ state (a, b, blue, left). In both cases, a variable error Z(θ) is introduced on the ancilla qubit in the middle of the stabilizer measurement operation. These circuits were used to generate the data in Fig. 2a. c, Direct measurement of the full error syndrome. Various single-qubit ‘errors’ are introduced on any one of the data qubits to generate different ancilla measurement outcomes. This circuit was used to generate the data in Fig. 2b.

## Supplementary information

### Supplementary Information

This file contains Supplementary Information, including Supplementary Figs. 1–16, Tables 1–5 and additional references.

## Rights and permissions

Reprints and Permissions

Egan, L., Debroy, D.M., Noel, C. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021). https://doi.org/10.1038/s41586-021-03928-y

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41586-021-03928-y

• ### Realizing repeated quantum error correction in a distance-three surface code

• Sebastian Krinner
• Nathan Lacroix
• Andreas Wallraff

Nature (2022)

• ### A quantum processor based on coherent transport of entangled atom arrays

• Dolev Bluvstein
• Harry Levine
• Mikhail D. Lukin

Nature (2022)

• ### Measurement-induced quantum phases realized in a trapped-ion quantum computer

• Crystal Noel
• Christopher Monroe

Nature Physics (2022)

• ### Silicon qubits move a step closer to achieving error correction

• Sophia E. Economou

Nature (2022)

• ### Fault-tolerant operation of a logical qubit in a diamond quantum processor

• M. H. Abobeih
• Y. Wang
• T. H. Taminiau

Nature (2022)

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

## Search

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing