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不断进步，一个自然的问题是：冗余度的基本下界是什么？本文针对包括酉算子在內的一类广泛操作，给出了实现$\epsilon$精度所需的冗余度下界。对于次指数深度和次线性门规模的实用相关情况，我们的冗余度下界比已知下界更紧。通过将容错计算与一组满足联合精度约束的有限块长量子通信问题联系起来，我们得到了这一下界。此处获得的冗余度下界导致非退化噪声的噪声阈值上界严格变小。我们的下界可直接推广至门输出噪声非独立同分布但门间噪声独立同分布的情形。