Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates
<|reference_start|>Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates: We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension $q$ (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$ (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent $\gamma=\log(n/k)/\log(d)\rightarrow 0$ as the block length $n\rightarrow\infty$, where $k$ and $d$ denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size $q$.   We construct our codes by combining two modular components, namely,   (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and   (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield.   We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters ($k,d=n/2^{O(\log^*n)}$) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).<|reference_end|>
@article{golowich2024asymptotically,
    title={Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates},
    author={Louis Golowich and Venkatesan Guruswami},
    journal={arXiv preprint arXiv:2408.09254},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.09254},
    primaryClass={quant-ph cs.IT math.IT}
}