Classification and implementation of unitary-equivariant and permutation-invariant quantum channels

Many quantum information tasks use inputs of the form ρ^{otimes m}, which naturally induce permutation and unitary symmetries. We classify all quantum channels that respect both symmetries - i.e. unitary-equivariant and permutation-invariant quantum channels from \(mathbb{C}^d\)^{otimes m} to \(mathbb{C}^d\)^{otimes n} - via their extremal points. Operationally, each extremal quantum channel factors as unitary Schur sampling → an irrep-level unitary-equivariant quantum channel → the adjoint unitary Schur sampling. We give a streaming implementation ansatz that uses an efficient streaming implementation of unitary Schur sampling together with a resource-state primitive, and we apply it to state symmetrization, symmetric cloning, and purity amplification. In these applications we obtain polynomial-time algorithms with exponential memory improvements in m,n. Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.