Efficient Topological Compilation for Weakly-Integral Anyon Model

In a recent series of two research papers Cui, Wang and Hong proposed a class of anyonic models for universal quantum computation based on weakly-integral anyons. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. From the mathematical standpoint the underlying unitary bases arising in various versions of the weakly-integral anyonic models are defined over a certain ring of Eisenstein rationals, that has useful number-theoretic properties. In this paper we develop a compilation algorithm to approximate arbitrary n-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model, the most recent instance of the weakly-integral anyonic class. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in O(32 n log1/ε) and entanglement cost that is exponential in n. Another flavor of the algorithm produces efficient circuits with upper complexity bound in O(n32 n log1/ε) and no additional entanglement cost.