eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-03-09
35:1
35:23
10.4230/LIPIcs.STACS.2022.35
article
Improved Quantum Lower and Upper Bounds for Matrix Scaling
Gribling, Sander
1
Nieuwboer, Harold
2
IRIF, Université de Paris, CNRS, France
Korteweg-de Vries Institute for Mathematics and QuSoft, University of Amsterdam, The Netherlands
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 𝓁₂-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-𝓁₁-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-𝓁₁-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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol219-stacs2022/LIPIcs.STACS.2022.35/LIPIcs.STACS.2022.35.pdf
Matrix scaling
quantum algorithms
lower bounds