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We systematically construct and classify fault-tolerant logical gates implemented by constant-depth circuits for quantum codes using cohomology operations and symmetry.
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A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, in Proc. ACM SODA (2023) pp. 1216–1244
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G. Zhu, M. Hafezi, and M. Barkeshli, Quantum origami: Transversal gates for quantum computation and measurement of topological order, Phys. Rev. Research 2
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M. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: breaking the n 1 / 2 polylog ( n ) n^{1/2}\text{polylog}(n) barrier for quantum ldpc codes, in Proc. ACM STOC (Association for Computing Machinery, New York, NY, USA, 2021) pp. 1276–1288
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S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature 627
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