Shor, P. W. Fault-tolerant quantum computation. In Proc. 37th Conference on Foundations of Computer Science 56–65 (IEEE, 1996); https://doi.org/10.1109/SFCS.1996.548464
Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002).
Fowler, A. G., Stephens, A. M. & Groszkowski, P. High-threshold universal quantum computation on the surface code. Phys. Rev. A 80, 052312 (2009).
Raussendorf, R., Harrington, J. & Goyal, K. Topological fault-tolerance in cluster state quantum computation. N. J. Phys. 9, 199 (2007).
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).
Krinner, S. et al. Realizing repeated quantum error correction in a distance-three surface code. Nature 605, 669–674 (2022).
Zhao, Y. et al. Realization of an error-correcting surface code with superconducting qubits. Phys. Rev. Lett. 129, 030501 (2022).
Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).
Ryan-Anderson, C. et al. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11, 041058 (2021).
Ryan-Anderson, C. et al. High-fidelity teleportation of a logical qubit using transversal gates and lattice surgery. Science 385, 1327–1331 (2024).
Lacroix, N. et al. Scaling and logic in the colour code on a superconducting quantum processor. Nature 645, 614–619 (2025).
Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2023).
Acharya, R. et al. Quantum error correction below the surface code threshold. Nature https://doi.org/10.1038/s41586-024-08449-y (2024).
Postler, L. et al. Demonstration of fault-tolerant universal quantum gate operations. Nature 605, 675–680 (2022).
Horsman, C., Fowler, A. G., Devitt, S. & Meter, R. V. Surface code quantum computing by lattice surgery. N. J. Phys. 14, 123011 (2012).
Bombin, H. & Martin-Delgado, M. A. Quantum measurements and gates by code deformation. J. Phys. A 42, 095302 (2009).
Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015).
Landahl, A. J. & Ryan-Anderson, C. Quantum computing by color-code lattice surgery. Preprint at https://arxiv.org/abs/1407.5103 (2014).
Litinski, D. Magic state distillation: not as costly as you think. Quantum 3, 205 (2019).
Erhard, A. et al. Entangling logical qubits with lattice surgery. Nature 589, 220–224 (2021).
Hetényi, B. & Wootton, J. R. Creating entangled logical qubits in the heavy-hex lattice with topological codes. PRX Quant. https://doi.org/10.1103/PRXQuantum.5.040334 (2024).
Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993).
Høyer, P. & Špalek, R. Quantum fan-out is powerful. Theory Comput. 1, 81–103 (2004).
Versluis, R. et al. Scalable quantum circuit and control for a superconducting surface code. Phys. Rev. Appl. 8, 034021 (2017).
Heinsoo, J. et al. Rapid high-fidelity multiplexed readout of superconducting qubits. Phys. Rev. Appl. 10, 034040 (2018).
Gehér, G. P., Jastrzebski, M., Campbell, E. T. & Crawford, O. To reset, or not to reset—that is the question. npj Quantum Inf. 11, 39 (2025).
Tomita, Y. & Svore, K. M. Low-distance surface codes under realistic quantum noise. Phys. Rev. A 90, 062320 (2014).
O’Brien, T. E., Tarasinski, B. & DiCarlo, L. Density-matrix simulation of small surface codes under current and projected experimental noise. npj Quantum Inf. 3, 39 (2017).
Edmonds, J. Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965).
Spitz, S. T., Tarasinski, B., Beenakker, C. W. J. & O’Brien, T. E. Adaptive weight estimator for quantum error correction in a time-dependent environment. Adv. Quant. Technol. 1, 1800012 (2018).
Remm, A. et al. Experimentally informed decoding of stabilizer codes based on syndrome correlations. Phys. Rev. Research 8, 013044 (2026).
Liang, Y.-C. et al. Quantum fidelity measures for mixed states. Rep. Prog. Phys. 82, 076001 (2019).
Lao, L. & Criger, B. Magic state injection on the rotated surface code. In Proc. 19th ACM International Conference on Computing Frontiers 113–120 (ACM, 2022); https://doi.org/10.1145/3528416.3530237
Ye, Y. et al. Logical magic state preparation with fidelity beyond the distillation threshold on a superconducting quantum processor. Phys. Rev. Lett. 131, 210603 (2023).
Bravyi, S. & Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).
Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024).
Schwartz, I. et al. Deterministic generation of a cluster state of entangled photons. Science 354, 434–437 (2016).
Motzoi, F., Gambetta, J. M., Rebentrost, P. & Wilhelm, F. K. Simple pulses for elimination of leakage in weakly nonlinear qubits. Phys. Rev. Lett. 103, 110501 (2009).
Lazăr, S. et al. Calibration of drive nonlinearity for arbitrary-angle single-qubit gates using error amplification. Phys. Rev. Appl. 20, 024036 (2023).
Walter, T. et al. Rapid high-fidelity single-shot dispersive readout of superconducting qubits. Phys. Rev. Appl. 7, 054020 (2017).
Gambetta, J. et al. Protocols for optimal readout of qubits using a continuous quantum nondemolition measurement. Phys. Rev. A 76, 012325 (2007).
Swiadek, F. et al. Enhancing dispersive readout of superconducting qubits through dynamic control of the dispersive shift: experiment and theory. PRX Quant. 5, 040326 (2024).
McEwen, M. et al. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 12, 1761 (2021).
Lacroix, N. et al. Fast flux-activated leakage reduction for superconducting quantum circuits. Phys. Rev. Lett. 134, 120601 (2025).
DiCarlo, L. et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature 460, 240–244 (2009).
Rol, M. A. et al. Time-domain characterization and correction of on-chip distortion of control pulses in a quantum processor. Appl. Phys. Lett. 116, 054001 (2020).
Rol, M. A. et al. Fast, high-fidelity conditional-phase gate exploiting leakage interference in weakly anharmonic superconducting qubits. Phys. Rev. Lett. 123, 120502 (2019).
Negirneac, 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).
Lisenfeld, J. et al. Observation of directly interacting coherent two-level systems in an amorphous material. Nat. Commun. 6, 6182 (2015).
Krinner, S. et al. Benchmarking coherent errors in controlled-phase gates due to spectator qubits. Phys. Rev. Appl. 14, 024042 (2020).