We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of
. Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes.
 J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, ``Magic state distillation with low space overhead and optimal asymptotic input count,'' Quantum 1, 31 (2017), 1703.07847v1.
 S. Bravyi and A. Kitaev, ``Universal quantum computation with ideal Clifford gates and noisy ancillas,'' Phys. Rev. A 71, 022316 (2005), quant-ph/0403025.
 D. Gottesman, ``A class of quantum error-correcting codes saturating the quantum hamming bound,'' Phys. Rev. A 54, 1862 (1996), quant-ph/9604038.
 A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, ``Quantum error correction and orthogonal geometry,'' Phys. Rev. Lett. 78, 405-408 (1997), quant-ph/9605005.
 T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, et al., ``Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majorana zero modes,'' Physical Review B 95, 235305 (2017), 1610.05289.
 E. T. Campbell and M. Howard, ``Unifying gate-synthesis and magic state distillation,'' Phys. Rev. Lett. 118, 060501 (2017a), 1606.01906v2.
 E. T. Campbell and M. Howard, ``Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost,'' Physical Review A 95, 022316 (2017b), 1606.01904v3.
 A. Paetznick and B. W. Reichardt, ``Universal fault-tolerant quantum computation with only transversal gates and error correction,'' Phys. Rev. Lett. 111, 090505 (2013), 1304.3709v2.
 A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, ``Surface codes: Towards practical large-scale quantum computation,'' Phys. Rev. A 86, 032324 (2012), 1208.0928.
 N. Alon and J. H. Spencer, The probabilistic method (John Wiley & Sons, 2004).
 G. Song and A. Klappenecker, ``Optimal realizations of simplified toffoli gates,'' Quantum Information & Computation 4, 361-372 (2004).
 E. Arikan, ``Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,'' IEEE Transactions on Information Theory 55, 3051-3073 (2009), 0807.3917.
 J. M. Renes, F. Dupuis, and R. Renner, ``Efficient polar coding of quantum information,'' Physical Review Letters 109, 050504 (2012), 1109.3195.
 M. Grassl and T. Beth, ``Quantum bch codes,'' in Proceedings X. International Symposium on Theoretical Electrical Engineering, Magdeburg (1999) pp. 207-212, quant-ph/9910060.
 A. R. Calderbank and P. W. Shor, ``Good quantum error-correcting codes exist,'' Phys. Rev. A 54, 1098-1105 (1996), quant-ph/9512032.
Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.