As techniques for fault-tolerant quantum computation keep improving, it is natural to ask: what is the fundamental lower bound on redundancy? In this paper, we obtain a lower bound on the redundancy required for $\epsilon$-accurate implementation of a large class of operations that includes unitary operators. For the practically relevant case of sub-exponential depth and sub-linear gate size, our bound on redundancy is tighter than the known lower bounds. We obtain this bound by connecting fault-tolerant computation with a set of finite blocklength quantum communication problems whose accuracy requirements satisfy a joint constraint. The lower bound on redundancy obtained here leads to a strictly smaller upper bound on the noise threshold for non-degradable noise. Our bound directly extends to the case where noise at the outputs of a gate are non-i.i.d. but noise across gates are i.i.d.

翻译：随着容错量子计算技术的持续改进，一个自然的问题是：冗余度的基本下界是什么？在本文中，我们针对包含酉算子在内的一类广泛操作，获得了实现$\epsilon$-精度所需冗余度的下界。对于指数级深度和亚线性门规模的实际相关情形，我们的冗余度下界比已知下界更严格。我们通过将容错计算与一组有限块长量子通信问题联系起来获得该下界，这些问题的精度要求满足联合约束。此处得到的冗余度下界导致非退化噪声的噪声阈值上界严格更小。该下界可直接推广至门的输出噪声非独立同分布但跨门噪声独立同分布的情形。