No-go theorem and a universal decomposition strategy for quantum channel compilation

Abstract

We rigorously prove a no-go theorem that, in sharp contrast to the case of compiling unitary gates, it is impossible to compile an arbitrary channel to arbitrary precision with any given finite elementary channel set, regardless of the length of the decomposition sequence. However, for a fixed error bound $\ensuremath{\epsilon}$, we find a general and systematic strategy to compile arbitrary quantum channels. We construct a universal set with a constant number of $\ensuremath{\epsilon}$-dependent elementary channels, such that an arbitrary quantum channel can be decomposed into a sequence of these elementary channels followed by a unitary gate, with the sequence length bounded by $O(\frac{1}{\ensuremath{\epsilon}}log\frac{1}{\ensuremath{\epsilon}})$ in the worst case. We further optimize this approach by exploiting proximal policy optimization---a powerful deep reinforcement learning algorithm for the gate compilation. We numerically evaluate the performance of our algorithm concerning topological compiling of Majorana fermions, and we show that our algorithm can conveniently and effectively reduce the use of expensive elementary operations.