An efficient high dimensional quantum Schur transform
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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 BCH07 gave a quantum algorithm for this transform that is polynomial in n, d and ϵ^-1, where ϵ is the precision. Following this, it had been an open question whether one can obtain an algorithm that is polynomial in d. In a footnote in Harrow's thesis H05, a brief description of how to make the algorithm of BCH07 polynomial in 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, d and ϵ^-1 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 BCH07. A novel feature of our algorithm is that we construct the quantum Fourier transform over permutation modules that could have other applications.
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