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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.

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Fig. 1: The Bacon–Shor subsystem code implemented on a 15-ion chain.
Fig. 2: Fault-tolerant logical qubit-state preparation.
Fig. 3: Manipulating logical states.
Fig. 4: Detection of arbitrary single-qubit errors.

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.

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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

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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.

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Competing interests

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

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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.

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

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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

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