An efficient high dimensional quantum Schur transform

Hari Krovi

doi:10.22331/q-2019-02-14-122

An efficient high dimensional quantum Schur transform

Quantum Engineering and Computing, Physical Sciences and Systems, Raytheon BBN Technologies, Cambridge, MA

Published:	2019-02-14, volume 3, page 122
Eprint:	arXiv:1804.00055v2
Doi:	https://doi.org/10.22331/q-2019-02-14-122
Citation:	Quantum 3, 122 (2019).

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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 $V^{\otimes n}$ 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 $\log\epsilon^{-1}$, where $\epsilon$ is the precision. In a footnote in Harrow's thesis [18], 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 $\log\epsilon^{-1}$ 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 $\textit{permutation modules}$, which could have other applications.

► BibTeX data

@article{Krovi2019efficienthigh,
  doi = {10.22331/q-2019-02-14-122},
  url = {https://doi.org/10.22331/q-2019-02-14-122},
  title = {An efficient high dimensional quantum {S}chur transform},
  author = {Krovi, Hari},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {122},
  month = feb,
  year = {2019}
}

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