Higher order quantum operations of unitaries

A supermap is a transformation from a map to a map representing a higher order function.  The general properties required in quantum mechanics for performing a supermap on an input given by a black box that implements an unknown quantum operation were formulated with the framework of quantum networks and quantum combs by Chiribella et al.  We regard such direct implementations of supermaps in quantum mechanics as higher order quantum operations.  There are several known no-go theorems for higher order quantum operations with a single use of the black box.  However, it is not yet well known which supermaps are achievable with finite uses of the black box.  In this talk, we present go-results for higher order quantum operations of unitaries with finite uses of the black box.  We show quantum algorithms for performing a complex conjugate unitary, an inverse of unitary, and controllization of the d-th power of a d-dimensional unitary.