Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493(R)–R2496(R) (1995).
Steane, A. M. Overhead and noise threshold of fault-tolerant quantum error correction. Phys. Rev. A 68, 042322 (2003).
Google Quantum AI and Collaborators. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2025).
Google Quantum AI and Collaborators.Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).
Grassl, M., Beth, T. & Pellizzari, T. Codes for the quantum erasure channel. Phys. Rev. A 56, 33–38 (1997).
Kubica, A. et al. Erasure qubits: overcoming the T1 limit in superconducting circuits. Phys. Rev. X 13, 041022 (2023).
Teoh, J. D. et al. Dual-rail encoding with superconducting cavities. Proc. Natl Acad. Sci. USA 120, e2221736120 (2023).
Xu, Q. et al. Tailored XZZX codes for biased noise. Phys. Rev. Res. 5, 013035 (2023).
Sahay, K., Jin, J., Claes, J., Thompson, J. D. & Puri, S. High-threshold codes for neutral-atom qubits with biased erasure errors. Phys. Rev. X 13, 041013 (2023).
Wu, Y., Kolkowitz, S., Puri, S. & Thompson, J. D. Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays. Nat. Commun. 13, 4657 (2022).
Aliferis, P. & Preskill, J. Fault-tolerant quantum computation against biased noise. Phys. Rev. A 78, 052331 (2008).
Barrett, S. D. & Stace, T. M. Fault tolerant quantum computation with very high threshold for loss errors. Phys. Rev. Lett. 105, 200502 (2010).
Delfosse, N. & Zémor, G. Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel. Phys. Rev. Res. 2, 033042 (2020).
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Puri, S., Boutin, S. & Blais, A. Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving. npj Quantum Inf. 3, 18 (2017).
Puri, S. et al. Bias-preserving gates with stabilized cat qubits. Sci. Adv. 6, eaay5901 (2020).
Grimm, A. et al. Stabilization and operation of a Kerr-cat qubit. Nature 584, 205–209 (2020).
Putterman, H. et al. Hardware-efficient quantum error correction via concatenated bosonic qubits. Nature https://doi.org/10.1038/s41586-025-08642-7 (2025).
Mirrahimi, M. et al. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New J. Phys. 16, 045014 (2014).
Leghtas, Z. et al. Confining the state of light to a quantum manifold by engineered two-photon loss. Science 347, 853–857 (2015).
Tuckett, D. K. et al. Tailoring surface codes for highly biased noise. Phys. Rev. X 9, 041031 (2019).
Tuckett, D. K., Bartlett, S. D., Flammia, S. T. & Brown, B. J. Fault-tolerant thresholds for the surface code in excess of 5% under biased noise. Phys. Rev. Lett. 124, 130501 (2020).
Darmawan, A. S., Brown, B. J., Grimsmo, A. L., Tuckett, D. K. & Puri, S. Practical quantum error correction with the XZZX code and Kerr-Cat qubits. PRX Quantum 2, 030345 (2021).
Claes, J., Bourassa, J. E. & Puri, S. Tailored cluster states with high threshold under biased noise. npj Quantum Inf. 9, 9 (2023).
Chuang, I. L. & Yamamoto, Y. Simple quantum computer. Phys. Rev. A 52, 3489–3496 (1995).
Chuang, I. L. & Yamamoto, Y. Quantum bit regeneration. Phys. Rev. Lett. 76, 4281–4284 (1996).
Knill, E., Laflamme, R. & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001).
Slussarenko, S. & Pryde, G. J. Photonic quantum information processing: a concise review. Appl. Phys. Rev. 6, 041303 (2019).
Ma, S. et al. High-fidelity gates and mid-circuit erasure conversion in an atomic qubit. Nature 622, 279–284 (2023).
Kang, M., Campbell, W. C. & Brown, K. R. Quantum error correction with metastable states of trapped ions using erasure conversion. PRX Quantum 4, 020358 (2023).
Gu, S., Vaknin, Y., Retzker, A. & Kubica, A. Optimizing quantum error-correction protocols with erasure qubits. PRX Quantum https://doi.org/10.1103/985g-58gd (2025).
Shim, Y.-P. & Tahan, C. Semiconductor-inspired design principles for superconducting quantum computing. Nat. Commun. 7, 11059 (2016).
Campbell, D. L. et al. Universal nonadiabatic control of small-gap superconducting qubits. Phys. Rev. X 10, 041051 (2020).
Levine, H. et al. Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons. Phys. Rev. X 14, 011051 (2024).
Zakka-Bajjani, E. et al. Quantum superposition of a single microwave photon in two different ‘colour’ states Nat. Phys. 7, 599–603 (2011).
Chou, K. S. et al. A superconducting dual-rail cavity qubit with erasure-detected logical measurements. Nat. Phys. 20, 1454–1460 (2024).
Chapman, B. J. et al. High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits. PRX Quantum 4, 020355 (2023).
Lu, Y. et al. High-fidelity parametric beamsplitting with a parity-protected converter. Nat. Commun. 14, 5767 (2023).
Koottandavida, A. et al. Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity. Phys. Rev. Lett. 132, 180601 (2024).
de Graaf, S. J. et al. A mid-circuit erasure check on a dual-rail cavity qubit using the joint-photon number-splitting regime of circuit QED. npj Quantum Inf. 11, 1 (2025).
Rosenblum, S. et al. A CNOT gate between multiphoton qubits encoded in two cavities. Nat. Commun. 9, 652 (2018).
Rosenblum, S. et al. Fault-tolerant detection of a quantum error. Science 361, 266–270 (2018).
Chen, Y.-H. & Baldwin, C. H. Randomized benchmarking with leakage errors. Phys. Rev. Res. https://doi.org/10.1103/j8sz-6ws9 (2025).
Wood, C. J. & Gambetta, J. M. Quantification and characterization of leakage errors. Phys. Rev. A 97, 032306 (2018).
Huang, W. et al. Logical multi-qubit entanglement with dual-rail superconducting qubits. Nat. Phys. 22, 591–597 (2026).
Korotkov, A. N. Error matrices in quantum process tomography. Preprint at arxiv.org/abs/1309.6405 (2013).
Gidney, C. Stim: a fast stabilizer circuit simulator. Quantum 5, 497 (2021).
Baranes, G. et al. Leveraging qubit loss detection in fault-tolerant quantum algorithms. Phys. Rev. X 16, 011002 (2026).

