Improve the local-dimension dependence of the SWAP-test implementation

Determine whether the dependence on the local dimension can be improved for the repeated SWAP-test implementation of unitary Schur sampling for qudits.

Background

The paper constructs a unitary Schur sampling protocol for permutation-invariant states of n qudits using repeated pairwise SWAP tests. After replacing ideal antisymmetric measurements with SWAP-test subroutines, the protocol uses $O\!\left(n\min\{n,d\}^{3}\log^2(n/\epsilon)\right)$ SWAP tests to achieve diamond-distance error at most $\epsilon$, where $d$ is the local qudit dimension. The authors emphasize that the resulting gate complexity is $\widetilde O(nd^3)$ for the relevant regime, and explicitly leave open whether this dependence on the local dimension can be reduced.