Gottesman, D. An introduction to quantum error correction. Proc. Symp. Appl. Math. 58, 221–236 (2002).
Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error. In Proc. Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC ’97 176–188 (Association for Computing Machinery, 1997).
Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate. SIAM J. Comput. 38, 1207 (2008).
Shor, P. Fault-tolerant quantum computation. In Proc. 37th Conference on Foundations of Computer Science 56–65 (IEEE, 1996).
Aliferis, P., Gottesman, D. & Preskill, J. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97–165 (2006).
Reichardt, B. W. in Automata, Languages and Programming (eds Bugliesi, M. et al.) 60–61 (Springer, 2006).
Yamasaki, H. & Koashi, M. Time-efficient constant-space-overhead fault-tolerant quantum computation. Nat. Phys. 20, 247–253 (2024).
Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).
Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024).
Acharya, R. et al. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2024).
Gottesman, D. Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology (1997).
Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102, 110502 (2009).
Bravyi, S. & Kitaev, A. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).
Knill, E. Fault-tolerant postselected quantum computation: schemes. Preprint at https://doi.org/10.48550/arXiv.quant-ph/0402171 (2004).
Gottesman, D. & Chuang, I. L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390–393 (1999).
Bombin, H. & Martin-Delgado, M. A. Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006).
Kubica, A. & Beverland, M. E. Universal transversal gates with color codes: a simplified approach. Phys. Rev. A 91, 032330 (2015).
Moussa, J. E. Transversal clifford gates on folded surface codes. Phys. Rev. A 94, 042316 (2016).
Łodyga, J., Mazurek, P., Grudka, A. & Horodecki, M. Simple scheme for encoding and decoding a qubit in unknown state for various topological codes. Sci. Rep. 5, 8975 (2015).
Li, Y. A magic state’s fidelity can be superior to the operations that created it. N. J. Phys. 17, 023037 (2015).
Litinski, D. Magic state distillation: not as costly as you think. Quantum 3, 205 (2019).
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).
Gidney, C. & Ekerå, M. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum 5, 433 (2021).
Bravyi, S. & Haah, J. Magic-state distillation with low overhead. Phys. Rev. A 86, 052329 (2012).
Haah, J. & Hastings, M. B. Codes and protocols for distilling t, controlled-s, and Toffoli gates. Quantum 2, 71 (2018).
Meier, A. M., Eastin, B. & Knill, E. Magic-state distillation with the four-qubit code. Quantum Inf. Comput. 13, 195–209 (2013).
Campbell, E. T., Anwar, H. & Browne, D. E. Magic-state distillation in all prime dimensions using quantum reed-muller codes. Phys. Rev. X 2, 041021 (2012).
Jones, C. Multilevel distillation of magic states for quantum computing. Phys. Rev. A 87, 042305 (2013).
Hastings, M. B. & Haah, J. Distillation with sublogarithmic overhead. Phys. Rev. Lett. 120, 050504 (2018).
Krishna, A. & Tillich, J.-P. Towards low overhead magic state distillation. Phys. Rev. Lett. 123, 070507 (2019).
Beverland, M., Campbell, E., Howard, M. & Kliuchnikov, V. Lower bounds on the non-clifford resources for quantum computations. Quantum Sci. Technol. 5, 035009 (2020).
Jones, C. Low-overhead constructions for the fault-tolerant toffoli gate. Phys. Rev. A 87, 022328 (2013).
Selinger, P. Quantum circuits of t-depth one. Phys. Rev. A 87, 042302 (2013).
Gidney, C. & Fowler, A. G. Efficient magic state factories with a catalyzed \(\left\vert CCZ\right\rangle\) to \(\left\vert CCZ\right\rangle\) transformation. Quantum 3, 135 (2019).
Goppa, V. D. Algebraico-geometric codes. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 46, 762 (1982).
Gottesman, D. Surviving as a Quantum Computer in a Classical World (Self-Published, 2024).
Vasmer, M. & Kubica, A. Morphing quantum codes. PRX Quantum 3, 030319 (2022).
Golowich, L. & Guruswami, V. Asymptotically good quantum codes with transversal non-Clifford gates. Preprint at https://doi.org/10.48550/arXiv.2408.09254 (2024).
Nguyen, Q. T., Good binary quantum codes with transversal ccz gate. Preprint at https://doi.org/10.48550/arXiv.2408.10140 (2024).
Calderbank, A. R. & Shor, P. W. Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098 (1996).
Steane, A. Multiple-particle interference and quantum error correction. Proc. R. Soc. London A 452, 2551 (1996).
MacWilliams, F. J. & Sloane, N. J. A. The Theory of Error-Correcting Codes vol. 16 (Elsevier, 1977).
Stichtenoth, H. Algebraic Function Fields and Codes vol. 254 (Springer, 2009).
Houshmand, M., Zamani, M. S., Sedighi, M. & Arabzadeh, M. Decomposition of diagonal hermitian quantum gates using multiple-controlled pauli z gates. ACM J. Emerg. Technolog. Comput. Syst 11, 1 (2014).
Tsfasman, M. A., Vlădut, S. G., & Nogin, D. Algebraic Geometric Codes: Basic Notions vol. 139 (American Mathematical Society, 2007).
Panteleev, P. & Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 375–388 (Association for Computing Machinery, 2022).
Leverrier, A. and Zemor, G. Quantum tanner codes. In Proc. 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science 872–883 (IEEE Computer Society, 2022).
Dinur, I., Hsieh, M.-H., Lin, T.-C., and Vidick, T. Good quantum LDPC codes with linear time decoders. In Proc. 55th Annual ACM Symposium on Theory of Computing, STOC 2023 905–918 (Association for Computing Machinery, 2023).
Devetak, I. & Winter, A. Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207 (2005).
Bennett, C. H., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996).
Veitch, V., Ferrie, C., Gross, D. & Emerson, J. Negative quasi-probability as a resource for quantum computation. N. J. Phys. 14, 113011 (2012).
Veitch, V., Mousavian, S. A. H., Gottesman, D. & Emerson, J. The resource theory of stabilizer quantum computation. N. J. Phys. 16, 013009 (2014).
Howard, M. & Campbell, E. Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett. 118, 090501 (2017).
Hayashi, M. & Yamasaki, H. Generalized quantum Stein’s lemma and second law of quantum resource theories. Preprint at https://doi.org/10.48550/arXiv.2408.02722 (2024).