An efficient high dimensional quantum Schur transform

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an n fold tensor product V ⊗n of a vector space V of dimension d. Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n, d and log ǫ, where ǫ is the precision. Following this, it had been an open question whether one can obtain an algorithm that is polynomial in log d. In a footnote in Harrow’s thesis [14], a brief description of how to make the algorithm of [5] polynomial in log d is given using the unitary group representation theory (however, this has not been explained in detail anywhere). In this article, we present a quantum algorithm for the Schur transform that is polynomial in n, log d and log ǫ using a different approach. We build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a “dual” algorithm to [5]. A novel feature of our algorithm is that we construct the quantum Fourier transform over permutation modules that could have other applications.