Unitary operator bases as reduced size universal averaging sets

We construct finite averaging sets for twirling finite-dimensional quantum states over a broad class of symmetry transformations represented by reductive Lie groups. The averaging is defined through the Cartan decomposition of the group, with Haar-uniform integration over the compact components and an arbitrary normalized measure over the non-compact Abelian component. We show that the resulting twirling map can always be represented as a finite probabilistic mixture of unitary 1-design channels acting on irreducible sectors of the representation restricted to the maximal compact subgroup. Thus, the finite construction is determined by the compact-sector decomposition, while the non-compact Cartan component contributes only scalar sector weights. This distinction is important for implementation: no Schur transform for the full reductive group is required. Compared with standard t -design-based finite averaging, our construction generally uses fewer averaging operators, at the price of replacing local tensor-power sampling by sector-wise unitary averaging. This could be especially beneficial for simulations and quantum algorithms on higher-dimensional systems as reduction of the size of the averaging set relatively increases with dimension.