High-dimensional quantum Schur transforms

The quantum Schur transform has become a foundational quantum algorithm, yet even after two decades since the seminal 2005 paper by Bacon, Chuang, and Harrow (BCH), some aspects of the transform remain insufficiently understood. Moreover, an alternative approach proposed by Krovi in 2018 was recently found to contain a crucial error. In this paper, we present a corrected version of Krovi's algorithm along with a detailed treatment of the high-dimensional version of the BCH Schur transform. This high-dimensional focus makes the two versions of the transform practical for regimes where the number of qudits $n$ is smaller than the local dimension $d$, with Krovi's algorithm scaling as $\widetilde{O}(n^4)$ and BCH as $\widetilde{O}(\min(n^5,nd^4))$. Our work addresses a key gap in the literature, strengthening the algorithmic foundations of a wide range of results that rely on Schur--Weyl duality in quantum information theory and quantum computation.