Skip to main content

Thank you for visiting 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.

Demonstration of fault-tolerant universal quantum gate operations


Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error-correcting codes1,2. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design3,4,5, which, in general, increases the complexity of the implementation. Here we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In particular, we make use of the recently introduced paradigm of flag fault tolerance, where the absence or presence of dangerous errors is heralded by the use of auxiliary flag qubits6,7,8,9,10. We perform a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code11,12, and fault-tolerantly prepare a logical magic state8,13. We then realize a fault-tolerant logical T gate by injecting the magic state by teleportation from one logical qubit onto the other14. We observe the hallmark feature of fault tolerance—a superior performance compared with a non-fault-tolerant implementation. In combination with recently demonstrated repeated quantum error-correction cycles15,16, these results provide a route towards error-corrected universal quantum computation.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Quantum error-correction code, logical gates and experimental system.
Fig. 2: Fault-tolerant preparation of a logical basis state \({|0\rangle }_{{\rm{L}}}\) and logical Clifford operations.
Fig. 3: Fault-tolerant implementation of a logical entangling gate.
Fig. 4: Fault-tolerant generation of a logical magic state \({|{\boldsymbol{H}}\rangle }_{{\bf{L}}}\).
Fig. 5: Fault-tolerant T-gate injection.

Data availability

The data underlying the findings of this work and the quantum circuits are available at

Code availability

All codes used for data analysis are available from the corresponding author upon reasonable request.


  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge Univ. Press, 2010).

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

    MathSciNet  Article  ADS  Google Scholar 

  3. Shor, P. W. Fault-tolerant quantum computation. In Proc. 37th Conference on Foundations of Computer Science 56–65 (IEEE, 1996).

  4. Preskill, J. Reliable quantum computers. Proc. R. Soc. Lond. A 454, 385–410 (1998).

    Article  ADS  Google Scholar 

  5. Aliferis, P., Gottesman, D. & Preskill, J. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97–165 (2006).

    MathSciNet  MATH  Google Scholar 

  6. Chao, R. & Reichardt, B. W. Quantum error correction with only two extra qubits. Phys. Rev. Lett. 121, 050502 (2018).

    CAS  Article  ADS  Google Scholar 

  7. Chamberland, C. & Beverland, M. E. Flag fault-tolerant error correction with arbitrary distance codes. Quantum 2, 53 (2018).

    Article  Google Scholar 

  8. Chamberland, C. & Cross, A. W. Fault-tolerant magic state preparation with flag qubits. Quantum 3, 143 (2019).

    Article  Google Scholar 

  9. Chao, R. & Reichardt, B. W. Flag fault-tolerant error correction for any stabilizer code. PRX Quantum 1, 010302 (2020).

    Article  Google Scholar 

  10. Reichardt, B. W. Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits. Quantum Sci. Technol. 6, 015007 (2020).

    Article  ADS  Google Scholar 

  11. Bombin, H. & Martin-Delgado, M. A. Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006).

    CAS  Article  ADS  Google Scholar 

  12. Steane, A. Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2577 (1996).

    MathSciNet  Article  ADS  Google Scholar 

  13. Goto, H. Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code. Sci. Rep. 6, 19578 (2016).

    CAS  Article  ADS  Google Scholar 

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

    MathSciNet  Article  ADS  Google Scholar 

  15. Ryan-Anderson, C. et al. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11, 041058 (2021).

    CAS  Google Scholar 

  16. Abobeih, M. H. et al. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Preprint at (2021).

  17. Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997).

    MathSciNet  Article  Google Scholar 

  18. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).

    MathSciNet  Article  Google Scholar 

  19. Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate. SIAM J. Comput. 38, 1207–1282 (2008).

    MathSciNet  Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Paetznick, A. & Reichardt, B. W. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett. 111, 090505 (2013).

    Article  ADS  Google Scholar 

  22. Beverland, M. E., Kubica, A. & Svore, K. M. Cost of universality: a comparative study of the overhead of state distillation and code switching with color codes. PRX Quantum 2, 020341 (2021).

    Article  ADS  Google Scholar 

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

    MathSciNet  CAS  Article  ADS  Google Scholar 

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

    CAS  Article  ADS  Google Scholar 

  25. Erhard, A. et al. Entangling logical qubits with lattice surgery. Nature 589, 220–224 (2021).

    CAS  Article  ADS  Google Scholar 

  26. Satzinger, K. J. et al. Realizing topologically ordered states on a quantum processor. Science 374, 1237–1241 (2021).

    CAS  Article  ADS  Google Scholar 

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

    CAS  Article  Google Scholar 

  28. Marques, J. F. et al. Logical-qubit operations in an error-detecting surface code. Nat. Phys. 18, 80–86 (2021).

    Article  Google Scholar 

  29. Chen, Z. et al. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).

    Article  Google Scholar 

  30. Gottesman, D. Quantum fault tolerance in small experiments. Preprint at (2016).

  31. 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).

    Article  ADS  Google Scholar 

  32. Vuillot, C. Is error detection helpful on IBM 5Q chips? Quantum Inf. Comput. 18, 949–964 (2018).

    MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021).

    CAS  Article  ADS  Google Scholar 

  35. Hilder, J. et al. Fault-tolerant parity readout on a shuttling-based trapped-ion quantum computer. Phys. Rev. X 12, 011032 (2022).

    CAS  Google Scholar 

  36. Pogorelov, I. et al. Compact ion-trap quantum computing demonstrator. PRX Quantum 2, 020343 (2021).

    Article  ADS  Google Scholar 

  37. Sørensen, A. & Mølmer, K. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A 62, 022311 (2000).

    Article  ADS  Google Scholar 

  38. Nebendahl, V., Häffner, H. & Roos, C. F. Optimal control of entangling operations for trapped-ion quantum computing. Phys. Rev. A 79, 012312 (2009).

    Article  ADS  Google Scholar 

  39. Bermudez, A., Xu, X., Gutiérrez, M., Benjamin, S. C. & Müller, M. Fault-tolerant protection of near-term trapped-ion topological qubits under realistic noise sources. Phys. Rev. A 100, 062307 (2019).

    CAS  Article  ADS  Google Scholar 

  40. Riesebos, L., Fu, X., Varsamopoulos, S., Almudever, C. G. & Bertels, K. Pauli frames for quantum computer architectures. In DAC ’17: Proc. 54th Annual Design Automation Conference 2017 1–6 (Association for Computing Machinery, 2017).

  41. Knill, E. Quantum computing with realistically noisy devices. Nature 434, 39–44 (2005).

    CAS  Article  ADS  Google Scholar 

  42. Parrado-Rodríguez, P., Ryan-Anderson, C., Bermudez, A. & Müller, M. Crosstalk suppression for fault-tolerant quantum error correction with trapped ions. Quantum 5, 487 (2021).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  44. Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. Preprint at (2021)

  45. Ryan-Anderson, C. Quantum Algorithms, Architecture, and Error Correction. PhD thesis, The Univ. New Mexico (2018).

  46. Hradil, Z., Řeháček, J., Fiurášek, J. & Ježek, M. in Quantum State Estimation 59–112 (Springer, 2004).

Download references


We acknowledge support from the EU Quantum Technology Flagship grant AQTION under grant agreement number 820495, and by the US Army Research Office through grant number W911NF-21-1-0007; and funding by the Austrian Science Fund (FWF), through the SFB BeyondC (FWF project number F7109), by the Austrian Research Promotion Agency (FFG) contract 872766, and by the IQI GmbH. M. Ringbauer acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 840450. M. Müller acknowledges support by the ERC Starting Grant QNets grant number 804247. S.H. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy ‘Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1’ 390534769. The research is also based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the US Army Research Office grant number W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA or the US Government. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the US Army Research Office.

Author information

Authors and Affiliations



L.P., I.P. and T.F. carried out the experiments. L.P., I.P., T.F., M. Meth, C.D.M., R.S., M. Ringbauer, P.S. and T.M. contributed to the experimental setup. L.P. analysed the data. S.H. performed the numerical simulations. S.H., M. Rispler and M. Müller performed circuit analysis, characterization and theory modelling. L.P., S.H., I.P., M. Rispler, P.S. and M. Müller wrote the manuscript, with contributions from all authors. R.B., P.S., M. Müller and T.M. supervised the project.

Corresponding author

Correspondence to Philipp Schindler.

Ethics declarations

Competing interests

T.M., R.B. and T.F. are connected to Alpine Quantum Technologies, a commercially oriented quantum computing company.

Peer review

Peer review information

Nature thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

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 generators of a single logical qubit.

Expectation values of the stabilizer generators and the logical operators of the seven-qubit colour code for the six cardinal states of the Bloch sphere. Results for the non-fault-tolerant and fault-tolerant preparation scheme are depicted in orange and turquoise respectively, whereas results from numerical simulations are shown in lighter coloured bars. 2,500 and 106 runs were performed in the experiment and for simulations for each prepared state, respectively. For the calculation of the expectation values of the logical operators a round of perfect error correction is applied. For the measurements corresponding to the data presented in this figure but also in Fig. 2 the sign of the rotation angle of physical Y-rotations is flipped, effectively implementing an additional deterministic π phase flip on qubit 6 and a π bit flip on qubit 7 at the end of the circuit depicted in Fig. 2a. The effects of this redefinition do not amount to a change of measurement bases and can be readily accounted for in post-processing.

Extended Data Table 1 Acceptance rates of flag encoding circuits

Supplementary information

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Postler, L., Heuβen, S., Pogorelov, I. et al. Demonstration of fault-tolerant universal quantum gate operations. Nature 605, 675–680 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


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.


Quick links

Nature Briefing

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