Abstract
Future fault-tolerant quantum computers will require storing and processing quantum data in logical qubits. Here we realize a suite of logical operations on a distance-2 surface code qubit built from seven physical qubits and stabilized using repeated error-detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.
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Data availability
The data supporting the plots and claims within this paper are available online at http://github.com/DiCarloLab-Delft/Logical_Qubit_Operations_Data.
References
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015).
Martinis, J. M. Qubit metrology for building a fault-tolerant quantum computer. npj Quantum Inf. 1, 15005 (2015).
Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015).
Chen, Z. et al. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).
Lescanne, R. et al. Exponential suppression of bit-flips in a qubit encoded in an oscillator. Nat. Phys. 16, 509–513 (2020).
Grimm, A. et al. Stabilization and operation of a Kerr-cat qubit. Nature 584, 205–209 (2020).
Ristè, D. et al. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6, 6983 (2015).
Cramer, J. et al. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nat. Commun. 5, 11526 (2016).
Ristè, D. et al. Real-time processing of stabilizer measurements in a bit-flip code. npj Quantum Inf. 6, 71 (2020).
Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 345, 302–305 (2014).
Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021).
Erhard, A. et al. Entangling logical qubits with lattice surgery. Nature 589, 220–224 (2021).
Negnevitsky, V. et al. Repeated multi-qubit readout and feedback with a mixed-species trapped-ion register. Nature 563, 527–531 (2018).
Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).
Andersen, C. K. et al. Entanglement stabilization using ancilla-based parity detection and real-time feedback in superconducting circuits. npj Quantum Inf. 5, 69 (2019).
Bultink, C. C. et al. Protecting quantum entanglement from leakage and qubit errors via repetitive parity measurements. Sci. Adv. 6, aay3050 (2020).
Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016).
Hu, L. et al. Quantum error correction and universal gate set operation on a binomial bosonic logical qubit. Nat. Phys. 15, 503–508 (2019).
Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368–372 (2020).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).
Andersen, C. K. et al. Repeated quantum error detection in a surface code. Nat. Phys. 16, 875–880 (2020).
Chow, J. M. et al. Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits. Phys. Rev. Lett. 109, 060501 (2012).
Versluis, R. et al. Scalable quantum circuit and control for a superconducting surface code. Phys. Rev. Appl. 8, 034021 (2017).
Tomita, Y. & Svore, K. M. Low-distance surface codes under realistic quantum noise. Phys. Rev. A 90, 062320 (2014).
Saira, O.-P. et al. Entanglement genesis by ancilla-based parity measurement in 2D circuit QED. Phys. Rev. Lett. 112, 070502 (2014).
Takita, M. et al. Demonstration of weight-four parity measurements in the surface code architecture. Phys. Rev. Lett. 117, 210505 (2016).
Aliferis, P., Gottesman, D. & Preskill, J. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97–165 (2005).
Paetznick, A. & Svore, K. M. Repeat-until-success: non-deterministic decomposition of single-qubit unitaries. Quantum Inf. Comput. 14, 1277–1301 (2014).
Varbanov, B. M. et al. Leakage detection for a transmon-based surface code. npj Quantum Inf. 6, 102 (2020).
McEwen, M. et al. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 12, 1761 (2021).
Battistel, F., Varbanov, B. M. & Terhal, B. M. Hardware-efficient leakage-reduction scheme for quantum error correction with superconducting transmon qubits. PRX Quantum 2, 030314 (2021).
Satzinger, K. J. et al. Realizing topologically ordered states on a quantum processor. Preprint at https://arxiv.org/abs/2104.01180 (2021).
Heinsoo, J. et al. Rapid high-fidelity multiplexed readout of superconducting qubits. Phys. Rev. Appl. 10, 034040 (2018).
Schreier, J. A. et al. Suppressing charge noise decoherence in superconducting charge qubits. Phys. Rev. B 77, 180502(R) (2008).
Negîrneac, V. et al. High-fidelity controlled-Z gate with maximal intermediate leakage operating at the speed limit in a superconducting quantum processor. Phys. Rev. Lett. 126, 220502 (2021).
IBM Quantum and Community Qiskit: an open-source framework for quantum computing. Zenodo https://zenodo.org/record/5670152 (2021).
de Jong, J. Implementation of a Fault-Tolerant SWAP Operation on the IBM 5-Qubit Device. MSc thesis, Delft University of Technology (2019).
Acknowledgements
We thank R. Sagastizabal, M. Sarsby and T. Stavenga for experimental assistance, and G. Calusine and W. Oliver (MIT Lincoln Laboratories) for providing the travelling-wave parametric amplifiers used in the readout amplification chain. This research is supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the US Army Research Office grant no. W911NF-16-1-0071 and by Intel Corporation. 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. B.M.V., F.B. and B.M.T. are supported by ERC grant EQEC no. 682726.
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J.F.M. performed the experiment and data analysis. M.B., N.H. and L.D. designed the device. N.M., C.Z. and A.B. fabricated the device. J.F.M. and H.A. calibrated the device. M.S.M. and W.V. designed the control electronics. B.M.V. performed the numerical simulations and F.B. implemented the MLE method. B.M.T. supervised the theory work. J.F.M. and L.D. wrote the manuscript with contributions from B.M.V., F.B. and B.M.T., and feedback from all co-authors. L.D. supervised the project.
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Marques, J.F., Varbanov, B.M., Moreira, M.S. et al. Logical-qubit operations in an error-detecting surface code. Nat. Phys. 18, 80–86 (2022). https://doi.org/10.1038/s41567-021-01423-9
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DOI: https://doi.org/10.1038/s41567-021-01423-9
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