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Low-overhead transversal fault tolerance for universal quantum computation

Nature volume 646, pages 303–308 (2025)Cite this article

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Abstract

Fast, reliable logical operations are essential for realizing useful quantum computers1,2,3. By redundantly encoding logical qubits into many physical qubits and using syndrome measurements to detect and correct errors, we can achieve low logical error rates. However, for many practical quantum error correction codes such as the surface code, owing to syndrome measurement errors, standard constructions require multiple extraction rounds—of the order of the code distance d—for fault-tolerant computation, particularly considering fault-tolerant state preparation4,5,6,7,8,9,10,11,12. Here we show that logical operations can be performed fault-tolerantly with only a constant number of extraction rounds for a broad class of quantum error correction codes, including the surface code with magic state inputs and feedforward, to achieve ‘transversal algorithmic fault tolerance’. Through the combination of transversal operations7 and new strategies for correlated decoding13, despite only having access to partial syndrome information, we prove that the deviation from the ideal logical measurement distribution can be made exponentially small in the distance, even if the instantaneous quantum state cannot be made close to a logical codeword because of measurement errors. We supplement this proof with circuit-level simulations in a range of relevant settings, demonstrating the fault tolerance and competitive performance of our approach. Our work sheds new light on the theory of quantum fault tolerance and has the potential to reduce the space–time cost of practical fault-tolerant quantum computation by over an order of magnitude.

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Fig. 1: Transversal algorithmic fault tolerance.
Fig. 2: Illustration of fault tolerance strategy.
Fig. 3: Numerical verification of fault tolerance.
Fig. 4: The \(| \overline{{\boldsymbol{S}}}\rangle \) state distillation factory.

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Data availability

The data that support the findings of this study are available at Zenodo103 (https://doi.org/10.5281/zenodo.16552626).

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Acknowledgements

We acknowledge the discussion with G. Baranes, P. Bonilla, E. Campbell, S. Evered, S. Geim, L. Jiang, M. Kalinowski, A. Krishna, S. Li, D. Litinski, T. Manovitz, Y. Wu and Q. Xu. We particularly thank C. Pattison for early discussions and for suggesting the simulation of the \(| \overline{S}\rangle \) state distillation factory, and J. Haah for stimulating discussions and insights. We also thank M. Beverland, K. Brown, U. Vazirani and the reviewers for suggestions on presentation. We acknowledge the financial support from IARPA and the Army Research Office, under the Entangled Logical Qubits program (cooperative agreement no. W911NF-23-2-0219), the DARPA ONISQ program (W911NF2010021), the DARPA IMPAQT program (HR0011-23-3-0012), the DARPA MeasQuIT program (HR0011-24-9-0359), the Center for Ultracold Atoms (a NSF Physics Frontiers Center, PHY-1734011), the National Science Foundation (grant nos. PHY-2012023 and CCF-2313084), the Army Research Office MURI (grant no. W911NF-20-1-0082), DOE/LBNL (grant no. DE-AC02-05CH11231), the Wellcome Leap Quantum for Bio program. M.C. acknowledges support from the Department of Energy Computational Science Graduate Fellowship (award no. DE-SC0020347). D.B. acknowledges support from the NSF Graduate Research Fellowship Program (grant no. DGE1745303) and the Fannie and John Hertz Foundation. This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the Department of Defense or the US government.

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Author notes

  1. These authors contributed equally: Hengyun Zhou, Chen Zhao

Authors and Affiliations

  1. QuEra Computing, Boston, MA, US

    Hengyun Zhou, Chen Zhao, Casey Duckering & Sheng-Tao Wang

  2. Department of Physics, Harvard University, Cambridge, MA, USA

    Hengyun Zhou, Madelyn Cain, Dolev Bluvstein, Nishad Maskara, Hong-Ye Hu & Mikhail D. Lukin

  3. AWS Center for Quantum Computing, Pasadena, CA, USA

    Aleksander Kubica

  4. California Institute of Technology, Pasadena, CA, USA

    Aleksander Kubica

  5. Department of Applied Physics, Yale University, New Haven, CT, USA

    Aleksander Kubica

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  1. Hengyun Zhou
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Contributions

H.Z. formulated the decoding strategy and developed an initial proof sketch through discussions with C.Z., M.C., D.B., C.D., S.-T.W., A.K. and M.D.L.; C.Z., M.C., H.Z. and H.-Y.H. performed numerical simulations; H.Z., A.K., C.Z., M.C., N.M. and C.D. proved the fault tolerance of the scheme. All authors contributed to the writing of the paper.

Corresponding authors

Correspondence to Hengyun Zhou or Mikhail D. Lukin.

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Competing interests

M.D.L. is a co-founder and shareholder, and H.Z., C.Z., C.D. and S.-T.W. are employees of QuEra Computing. US patent application PCT/US25/22108 (filed on 28 March 2025 and naming H.Z., M.C., C.Z., D.B., M.D.L. and C.D. as co-inventors) contains technical aspects of this paper.

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Extended data figures and tables

Extended Data Fig. 1 Surface code and transversal operations.

(a) Illustration of the surface code. White circles indicate data qubits. Orange (green) plaquettes are Z (X) stabilizers. The logical \(\overline{Z}\) (\(\overline{X}\)) operator runs vertically (horizontally), and we choose our convention for fixing Z (X) stabilizers to be performing a chain of X (Z) flips to the left (bottom) boundary, as illustrated by the red line. (b) Illustration of transversal \(\overline{H}\) gate, consisting of transversal H gates followed by a reflection along the diagonal. Note that this differs from the usual transversal \(\overline{H}\) gate, which applies a rotation in the second step. For the non-rotated surface code, both choices map X (Z) stabilizers to Z (X) stabilizers and hence are valid, but our choice leads to a smaller transversal partition size for the full circuit. (c) Illustration of transversal \(\overline{S}\) gate, consisting of S and S† gates along the diagonal, together with CZ gates between mirrored qubits.

Extended Data Fig. 2 Illustration of error recovery and frame repair procedures.

We illustrate the procedure for the surface code, where a cross-sectional view with one spatial axis and one time axis is shown. We only illustrate X errors and Z stabilizer measurement errors, which are relevant to interpreting the \(\overline{Z}\) measurement. X errors can terminate on orange boundaries, but cannot terminate on cyan boundaries. The transversal \(\overline{CNOT}\) copies X errors from the top to the bottom, resulting in a branching point (black cross) and an error cluster spanning both code blocks. (a) Error chains and frame flips. Chains of X-type errors (orange lines) lead to syndromes (end points) or terminate on appropriate boundaries. A line segment in the vertical direction is a data qubit X error, while a line segment in the horizontal direction is a measurement error. Note that the X-type error cannot terminate on the transversal Z measurement boundary. The random stabilizer initialization leads to a frame configuration on the logical \(| \,\overline{+}\,\rangle \) initialization, as illustrated by the blue line and the flipped Z stabilizer (blue point). This is similar to the frame stabilizer operator gs illustrated in Extended Data Fig. 1(a). (b) We first infer an error recovery operator, which has the same boundary as the error chain. Together, the error and recovery operator form the fault configuration, which triggers no detectors. We illustrate a few examples (orange lines) that do not lead to a logical error: (1) the fault configuration forms a closed loop and is equivalent to applying a stabilizer; (2) the fault configuration terminates on an initialization boundary; (3) the fault configuration terminates on an out-going, unmeasured logical qubit, but the forward-propagated errors onto the measured logical qubit are equivalent to a stabilizer. A logical error can only happen when the fault configuration spans across two opposing spatial boundaries (red line), which requires an error of weight Θ(d). (c,d) The frame repair operation returns the logical qubit to the code space with all stabilizers +1, corresponding to cancelling any residual flipped stabilizers on the initialization boundary. Note that the error recovery process may also lead to a change that needs to be accounted for by frame repair. An example choice of frame repair is shown in (c), which applies an overall X operator on the logical measurement result. Alternatively, a different choice of frame repair shown in (d), related to the previous one by a frame logical flip, results in identity operation on the logical measurement result.

Extended Data Fig. 3 Post-selection rates from the numerical simulation of \(| \overline{{\boldsymbol{S}}}\rangle \) state distillation factory at circuit noise p = 0.1%.

Here the ideal post-selection rates are defined in Eq. (12).

Extended Data Fig. 4 Illustration of a 15-to-1 \(| \overline{{\boldsymbol{T}}}\rangle \) magic state distillation factory, adapted from ref. 24.

The green lines illustrate the application of a logical stabilizer, which allows re-interpretation of measurement results and changes which feed-forwards should be performed.

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Zhou, H., Zhao, C., Cain, M. et al. Low-overhead transversal fault tolerance for universal quantum computation. Nature 646, 303–308 (2025). https://doi.org/10.1038/s41586-025-09543-5

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