  The problem of discriminating between many quantum channels with certainty is
analyzed under the assumption of prior knowledge of algebraic relations among
possible channels. It is shown, by explicit construction of a novel family of
quantum algorithms, that when the set of possible channels faithfully
represents a finite subgroup of SU(2) (e.g., $C_n, D_{2n}, A_4, S_4, A_5$) the
recently-developed techniques of quantum signal processing can be modified to
constitute subroutines for quantum hypothesis testing. These algorithms, for
group quantum hypothesis testing (G-QHT), intuitively encode discrete
properties of the channel set in SU(2) and improve query complexity at least
quadratically in $n$, the size of the channel set and group, compared to
na\"ive repetition of binary hypothesis testing. Intriguingly, performance is
completely defined by explicit group homomorphisms; these in turn inform simple
constraints on polynomials embedded in unitary matrices. These constructions
demonstrate a flexible technique for mapping questions in quantum inference to
the well-understood subfields of functional approximation and discrete algebra.
Extensions to larger groups and noisy settings are discussed, as well as paths
by which improved protocols for quantum hypothesis testing against structured
channel sets have application in the transmission of reference frames, proofs
of security in quantum cryptography, and algorithms for property testing.
