Magic state distillation at intermediate size

Recently [1] we proposed a family of magic state distillation protocols that obtains asymptotic performance that is conjectured to be optimal. This family depends upon several codes, called “inner codes” and “outer codes.” In Ref. [1], some small examples of these codes were given as well as an analysis of codes in the asymptotic limit. Here, we analyze such protocols in an intermediate size regime, using hundreds to thousands of qubits. We use BCH inner codes [2], combined with various outer codes. We extend the protocols of Ref. [1] by adding error correction in some cases. We present a variety of protocols in various input error regimes; in many cases these protocols require significantly fewer input magic states to obtain a given output error than previous protocols.