Quantum Hypothesis Testing for Non-Abelian Representations

How can prior knowledge of the query set be leveraged to construct efficient protocols for hypothesis testing among quantum channels? Specifically this work addresses the problem of distinguishing multiple unitary quantum channels in the serial adaptive query model under the assumption of known, non-trivial algebraic relations between them. This work shows, by explicit construction, that when the query set faithfully represents a finite subgroup of SU(2) the recently developed technique of quantum signal processing can be applied to build efficient algorithms for quantum multiple hypothesis testing. These algorithms intuitively encode information about the query set and are exponentially more efficient in the instance size than naive reductions to binary hypothesis testing. Moreover, this method illustrates a novel technique for transforming questions in quantum inference to those in the well-understood field of functional approximation. We provide generalizations to larger groups, shows robustness to noise, and give indication that, for a rich subset of quantum hypothesis testing, knowledge of the algebraic structure of the query set can be employed to improve algorithmic performance.