• Gottesman, D. An introduction to quantum error correction. Proc. Symp. Appl. Math. 58, 221–236 (2002).

    Article 
    MathSciNet 

    Google Scholar
     

  • 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).

    Article 
    MathSciNet 

    Google Scholar
     

  • 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).

    MathSciNet 

    Google Scholar
     

  • 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).

    Article 

    Google Scholar
     

  • Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).

    Article 

    Google Scholar
     

  • Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024).

    Article 

    Google Scholar
     

  • Acharya, R. et al. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2024).


    Google Scholar
     

  • 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).

    Article 
    ADS 

    Google Scholar
     

  • Bravyi, S. & Kitaev, A. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • 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).

    Article 
    ADS 

    Google Scholar
     

  • Bombin, H. & Martin-Delgado, M. A. Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006).

    Article 
    ADS 

    Google Scholar
     

  • Kubica, A. & Beverland, M. E. Universal transversal gates with color codes: a simplified approach. Phys. Rev. A 91, 032330 (2015).

    Article 
    ADS 

    Google Scholar
     

  • Moussa, J. E. Transversal clifford gates on folded surface codes. Phys. Rev. A 94, 042316 (2016).

    Article 
    ADS 

    Google Scholar
     

  • 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).

    Article 

    Google Scholar
     

  • Li, Y. A magic state’s fidelity can be superior to the operations that created it. N. J. Phys. 17, 023037 (2015).

    Article 

    Google Scholar
     

  • Litinski, D. Magic state distillation: not as costly as you think. Quantum 3, 205 (2019).

    Article 

    Google Scholar
     

  • 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).

    Article 
    ADS 

    Google Scholar
     

  • Gidney, C. & Ekerå, M. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum 5, 433 (2021).

    Article 

    Google Scholar
     

  • Bravyi, S. & Haah, J. Magic-state distillation with low overhead. Phys. Rev. A 86, 052329 (2012).

    Article 
    ADS 

    Google Scholar
     

  • Haah, J. & Hastings, M. B. Codes and protocols for distilling t, controlled-s, and Toffoli gates. Quantum 2, 71 (2018).

    Article 

    Google Scholar
     

  • Meier, A. M., Eastin, B. & Knill, E. Magic-state distillation with the four-qubit code. Quantum Inf. Comput. 13, 195–209 (2013).

    MathSciNet 

    Google Scholar
     

  • 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).


    Google Scholar
     

  • Jones, C. Multilevel distillation of magic states for quantum computing. Phys. Rev. A 87, 042305 (2013).

    Article 
    ADS 

    Google Scholar
     

  • Hastings, M. B. & Haah, J. Distillation with sublogarithmic overhead. Phys. Rev. Lett. 120, 050504 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Krishna, A. & Tillich, J.-P. Towards low overhead magic state distillation. Phys. Rev. Lett. 123, 070507 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Beverland, M., Campbell, E., Howard, M. & Kliuchnikov, V. Lower bounds on the non-clifford resources for quantum computations. Quantum Sci. Technol. 5, 035009 (2020).

    Article 
    ADS 

    Google Scholar
     

  • Jones, C. Low-overhead constructions for the fault-tolerant toffoli gate. Phys. Rev. A 87, 022328 (2013).

    Article 
    ADS 

    Google Scholar
     

  • Selinger, P. Quantum circuits of t-depth one. Phys. Rev. A 87, 042302 (2013).

    Article 
    ADS 

    Google Scholar
     

  • 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).

    Article 

    Google Scholar
     

  • Goppa, V. D. Algebraico-geometric codes. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 46, 762 (1982).

    MathSciNet 

    Google Scholar
     

  • 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).

    Article 
    ADS 

    Google Scholar
     

  • 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).

    Article 
    ADS 

    Google Scholar
     

  • Steane, A. Multiple-particle interference and quantum error correction. Proc. R. Soc. London A 452, 2551 (1996).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • 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).

    Article 

    Google Scholar
     

  • 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).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Bennett, C. H., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996).

    Article 
    ADS 

    Google Scholar
     

  • Veitch, V., Ferrie, C., Gross, D. & Emerson, J. Negative quasi-probability as a resource for quantum computation. N. J. Phys. 14, 113011 (2012).

    Article 

    Google Scholar
     

  • Veitch, V., Mousavian, S. A. H., Gottesman, D. & Emerson, J. The resource theory of stabilizer quantum computation. N. J. Phys. 16, 013009 (2014).

    Article 
    MathSciNet 

    Google Scholar
     

  • Howard, M. & Campbell, E. Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett. 118, 090501 (2017).

    Article 
    ADS 

    Google Scholar
     

  • 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).