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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.35
URN: urn:nbn:de:0030-drops-158458
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15845/
Gribling, Sander ;
Nieuwboer, Harold
Improved Quantum Lower and Upper Bounds for Matrix Scaling
Abstract
Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting.
We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n × n matrices with at most m non-zero entries and with ?₂-error ε = Θ~(1/m) must make Ω(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for ε ∈ [1/n,1/2], any quantum algorithm capable of producing ε/100-?₁-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε) queries, and that there exists a constant ε₀ > 0 for which this problem takes Ω(n^{1.5}) queries.
To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε-?₁-scaling in time O~(n^{1.5}/ε²), whereas the best previously known bounds were O~(n²polylog(1/ε)) (classical) and O~(n^{1.5}/ε³) (quantum).
BibTeX - Entry
@InProceedings{gribling_et_al:LIPIcs.STACS.2022.35,
author = {Gribling, Sander and Nieuwboer, Harold},
title = {{Improved Quantum Lower and Upper Bounds for Matrix Scaling}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {35:1--35:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15845},
URN = {urn:nbn:de:0030-drops-158458},
doi = {10.4230/LIPIcs.STACS.2022.35},
annote = {Keywords: Matrix scaling, quantum algorithms, lower bounds}
}
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Keywords: |
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Matrix scaling, quantum algorithms, lower bounds |
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Collection: |
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39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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09.03.2022 |
