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Title: Quantum tensor singular value decomposition
Authors: Wang, X 
Gu, L 
Lee, HW 
Zhang, G 
Issue Date: Jul-2021
Source: Journal of physics communications, July 2021, v. 5, no. 7, 75001
Abstract: Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time O(Npolylog(N)) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.
Keywords: Quantum fourier transform
Quantum singular value estimation
t-product
t-svd
Tensor singular value decomposition
Publisher: Institute of Physics Publishing
Journal: Journal of physics communications 
EISSN: 2399-6528
DOI: 10.1088/2399-6528/AC0D5F
Rights: © 2021 The Author(s). Published by IOP Publishing Ltd
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence (https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
The following publication Wang, X., Gu, L., Lee, H. W., & Zhang, G. (2021). Quantum tensor singular value decomposition. Journal of Physics Communications is available at https://doi.org/10.1088/2399-6528/AC0D5F
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