Abstract
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 logϵ−1, where ϵ is the precision. In a footnote in Harrow's thesis [18], a brief description of how to make the algorithm of [5] polynomial in logd 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, logd and logϵ−1 using a different approach. Specifically, 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 the so called permutation modules, which could have other applications.
Highlights
Schur-Weyl duality is a remarkable correspondence between the irreducible representations of the symmetric group and those of the unitary group acting on an n fold tensor product of a vector space V
We present a circuit that can be used for strong Schur sampling using the representation theory of the symmetric group that runs in time polynomial in log d, n and log(1/ )
The dimension of the multiplicity of λ in the permutation module of the partition μ is Kλμ. This space has a basis in terms of semi-standard Young tableau (SSYT) of shape λ and content μ i.e., a Young diagram of shape λ filled with μ1 ones, μ2 twos etc., such that the numbers are strictly increasing in the columns and weakly increasing in the rows
Summary
Schur-Weyl duality is a remarkable correspondence between the irreducible representations of the symmetric group and those of the unitary group acting on an n fold tensor product of a vector space V. A circuit for Schur sampling block diagonalizes the unitary group (or equivalently the symmetric group) representation on the n fold tensor power of a d dimensional space. We present a circuit that can be used for strong Schur sampling using the representation theory of the symmetric group that runs in time polynomial in log d, n and log(1/ ). BCH developed a quantum circuit for the Clebsch-Gordan (CG) decomposition problem for the unitary group, which entails block diagonalizing a tensor product of two irreps of the unitary group. They use this circuit to construct the Schur transform by applying it iteratively.
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