A Converse for Fault-tolerant Quantum Computation

With improvements in achievable redundancy for fault-tolerant quantum computing, it is natural to ask: what is the minimum required redundancy? In this paper, we obtain a lower bound on the minimum redundancy required for $\epsilon$-accurate implementation of a large class of operations, which includes unitary operators. For the practically relevant case of sub-exponential (in input size) depth and sub-linear gate size, our bound on redundancy is tighter than the best known lower bound in \cite{FawziMS2022}. 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. This bound gives a strictly lower noise threshold for non-degradable noise and captures its dependence on gate size. This bound directly extends to the case where noise at the outputs of a gate are correlated but noise across gates are independent.