Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704 (2005).
Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484 (1997).
Google Quantum AI Hartree-fock on a superconducting qubit quantum computer. Science 369, 1084 (2020).
Bernstein, D. J. & Lange, T. Post-quantum cryptography. Nature 549, 188 (2017).
Google Quantum AI Phase transitions in random circuit sampling. Nature 634, 328 (2024).
Sun, W., Wei, F., Shao, Y. & Wei, Z. Sudden death of quantum advantage in correlation generations. Sci. Adv. 10, eadr5002 (2024).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).
Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307 (2015).
Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003).
Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452 (2002).
Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58 (2024).
Scholl, P. et al. Erasure conversion in a high-fidelity rydberg quantum simulator. Nature 622, 273 (2023).
Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. Science 339, 1169 (2013).
Lim, S., Vaganov, M. V., Liu, J. & Ardavan, A. Demonstrating experimentally the encoding and dynamics of an error-correctable logical qubit on a hyperfine-coupled nuclear spin qudit. Phys. Rev. Lett. 134, 070603 (2025).
Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281 (2021).
Erhard, A. et al. Entangling logical qubits with lattice surgery. Nature 589, 220 (2021).
Self, C. N., Benedetti, M. & Amaro, D. Protecting expressive circuits with a quantum error detection code. Nat. Phys. 20, 219 (2024).
Pogorelov, I. et al. Experimental fault-tolerant code switching. Nat. Phys. 21, 298 (2025).
Hong, Y., Durso-Sabina, E., Hayes, D. & Lucas, A. Entangling four logical qubits beyond break-even in a nonlocal code. Phys. Rev. Lett. 133, 180601 (2024).
Ryan-Anderson, C. et al. High-fidelity teleportation of a logical qubit using transversal gates and lattice surgery. Science 385, 1327 (2024).
Paetznick, A. et al. Improved quantum processor logical error rates via correction and detection. Nature 654, 349 (2026).
Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441 (2016).
Sivak, V. V. et al. Real-time quantum error correction beyond break-even. Nature 616, 50 (2023).
Ni, Z. et al. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature 616, 56 (2023).
Google Quantum AI Quantum error correction below the surface code threshold. Nature 638, 920 (2025).
Dasu, S. et al. Computing with many encoded logical qubits beyond break-even. Preprint at https://doi.org/10.48550/arXiv.2602.22211 (2026).
Wang, P. et al. Single ion qubit with estimated coherence time exceeding one hour. Nat. Commun. 12, 233 (2021).
Gaebler, J. et al. High-fidelity universal gate set for 9Be+ ion qubits. Phys. Rev. Lett. 117, 060505 (2016).
Ballance, C. J., Harty, T. P., Linke, N. M., Sepiol, M. A. & Lucas, D. M. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Phys. Rev. Lett. 117, 060504 (2016).
Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).
Chen, W., Gan, J., Zhang, J.-N., Matuskevich, D. & Kim, K. Quantum computation and simulation with vibrational modes of trapped ions. Chin. Phys. B 30, 060311 (2021).
Zhang, M. et al. Observation of spin-tensor induced topological phase transitions of triply degenerate points with a trapped ion. Phys. Rev. Lett. 129, 250501 (2022).
Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. Nat. Phys. 18, 1053 (2022).
Meth, M. et al. Simulating two-dimensional lattice gauge theories on a qudit quantum computer. Nat. Phys. 21, 570 (2025).
Calajó, G. et al. Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits. PRX Quantum 5, 040309 (2024).
Kubischta, E. & Teixeira, I. Family of quantum codes with exotic transversal gates. Phys. Rev. Lett. 131, 240601 (2023).
Kubischta, E. & Teixeira, I. Quantum codes from twisted unitary t-groups. Phys. Rev. Lett. 133, 030602 (2024).
Gross, J. A., Godfrin, C., Blais, A. & Dupont-Ferrier, E. Hardware-efficient error-correcting codes for large nuclear spins. Phys. Rev. Appl. 22, 014006 (2024).
Lim, S., Liu, J. & Ardavan, A. Fault-tolerant qubit encoding using a spin-7/2 qudit. Phys. Rev. A 108, 062403 (2023).
Gross, J. A. Designing codes around interactions: the case of a spin. Phys. Rev. Lett. 127, 010504 (2021).
Omanakuttan, S., Buchemmavari, V., Gross, J. A., Deutsch, I. H. & Marvian, M. Fault-tolerant quantum computation using large spin cat-codes. PRX Quantum 5, 020355 (2024).
Yu, X. et al. Schrödinger cat states of a nuclear spin qudit in silicon. Nat. Phys. 21, 362 (2025).
Jain, S. P., Hudson, E. R., Campbell, W. C. & Albert, V. V. Absorption–emission codes for atomic and molecular quantum information platforms. Phys. Rev. Lett. 133, 260601 (2024).
Kruckenhauser, A. et al. Dark spin-cat states as biased qubits. Phys. Rev. Lett. 135, 020601 (2025).
Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513 (2019).
Cai, W., Ma, Y., Wang, W., Zou, C.-L. & Sun, L. Bosonic quantum error correction codes in superconducting quantum circuits. Fund. Res. 1, 50 (2021).
Reiter, F., Sørensen, A. S., Zoller, P. & Muschik, C. A. Dissipative quantum error correction and application to quantum sensing with trapped ions. Nat. Commun. 8, 1822 (2017).
Ma, Y. et al. Error-transparent operations on a logical qubit protected by quantum error correction. Nat. Phys. 16, 827 (2020).
Gertler, J. M. et al. Protecting a bosonic qubit with autonomous quantum error correction. Nature 590, 243 (2021).
Li, Z. et al. Autonomous error correction of a single logical qubit using two transmons. Nat. Commun. 15, 1681 (2024).
Lachance-Quirion, D. et al. Autonomous quantum error correction of gottesman-kitaev-preskill states. Phys. Rev. Lett. 132, 150607 (2024).
Lihm, J.-M., Noh, K. & Fischer, U. R. Implementation-independent sufficient condition of the knill-laflamme type for the autonomous protection of logical qudits by strong engineered dissipation. Phys. Rev. A 98, 012317 (2018).
van Mourik, M. W. et al. Experimental realization of nonunitary multiqubit operations. Phys. Rev. Lett. 132, 040602 (2024).
Brock, B. L. et al. Quantum error correction of qudits beyond break-even. Nature 641, 612 (2025).
Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281 (2003).
Allcock, D. T. C. et al. omg blueprint for trapped ion quantum computing with metastable states. Appl. Phys. Lett. 119, 214002 (2021).
Yang, H.-X. et al. Realizing coherently convertible dual-type qubits with the same ion species. Nat. Phys. 18, 1058 (2022).
Schmidt, P. O. et al. Spectroscopy using quantum logic. Science 309, 749 (2005).
Ma, W. L. et al. Path-independent quantum gates with noisy ancilla. Phys. Rev. Lett. 125, 110503 (2020).
Xu, X., Beaudrap, N. D., O’Gorman, J. & Benjamin, S. C. An integrity measure to benchmark quantum error correcting memories. New J. Phys. 20, 023009 (2018).
Cai, W. et al. Protecting entanglement between logical qubits via quantum error correction. Nat. Phys. 20, 1022 (2024).
Putterman, H. et al. Hardware-efficient quantum error correction via concatenated bosonic qubits. Nature 638, 927 (2025).
Kwon, H., Fischer, U. R., Lee, S.-W. & Jiang, L. Restoring Heisenberg scaling in time via autonomous quantum error correction. npj Quantum Inf. 12, 103 (2026).
DeBry, K. et al. Error correction of a logical qubit encoded in a single atomic ion. Nat. Phys. 22, 1446–1452 (2026).
Li, Y. et al. Replication data for: Beating the break-even point with autonomous quantum error correction. Zenodo https://doi.org/10.5281/zenodo.20768851 (2026).