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Documents authored by van Apeldoorn, Joran

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.110
Quantum Algorithms for Matrix Scaling and Matrix Balancing

Authors: Joran van Apeldoorn, Sander Gribling, Yinan Li, Harold Nieuwboer, Michael Walter, and Ronald de Wolf

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.
Cite as

Joran van Apeldoorn, Sander Gribling, Yinan Li, Harold Nieuwboer, Michael Walter, and Ronald de Wolf. Quantum Algorithms for Matrix Scaling and Matrix Balancing. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 110:1-110:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vanapeldoorn_et_al:LIPIcs.ICALP.2021.110,
  author =	{van Apeldoorn, Joran and Gribling, Sander and Li, Yinan and Nieuwboer, Harold and Walter, Michael and de Wolf, Ronald},
  title =	{{Quantum Algorithms for Matrix Scaling and Matrix Balancing}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{110:1--110:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.110},
  URN =		{urn:nbn:de:0030-drops-141793},
  doi =		{10.4230/LIPIcs.ICALP.2021.110},
  annote =	{Keywords: Matrix scaling, matrix balancing, quantum algorithms}
}
Document
DOI: 10.4230/LIPIcs.TQC.2021.9
Quantum Probability Oracles & Multidimensional Amplitude Estimation

Authors: Joran van Apeldoorn

Published in: LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Abstract
We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for 𝓁₁-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for 𝓁₂-norm. We also show that these results are optimal up to logarithmic factors.
Cite as

Joran van Apeldoorn. Quantum Probability Oracles & Multidimensional Amplitude Estimation. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vanapeldoorn:LIPIcs.TQC.2021.9,
  author =	{van Apeldoorn, Joran},
  title =	{{Quantum Probability Oracles \& Multidimensional Amplitude Estimation}},
  booktitle =	{16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)},
  pages =	{9:1--9:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-198-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{197},
  editor =	{Hsieh, Min-Hsiu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.9},
  URN =		{urn:nbn:de:0030-drops-140046},
  doi =		{10.4230/LIPIcs.TQC.2021.9},
  annote =	{Keywords: quantum algorithms, amplitude estimation, monte carlo}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.99
Improvements in Quantum SDP-Solving with Applications

Authors: Joran van Apeldoorn and András Gilyén

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case when the input matrices are proportional to mixed quantum states that one can query. For this model Brandão et al. [Fernando G. S. L. Brandão et al., 2018] showed that the dependence on n can be replaced by a polynomial dependence on both the rank and the trace of the input matrices. We remove the dependence on the rank and hence require only a dependence on the trace of the input matrices. After we obtain these results we apply them to a few different problems. The most notable of which is the problem of shadow tomography, recently introduced by Aaronson [Aaronson, 2018]. Here we simultaneously improve both the sample and computational complexity of the previous best results. Finally we prove a new Omega~(sqrt{m}alpha gamma) lower bound for solving LPs and SDPs in the quantum operator model, which also implies a lower bound for the model of Brandão et al. [Fernando G. S. L. Brandão et al., 2018].
Cite as

Joran van Apeldoorn and András Gilyén. Improvements in Quantum SDP-Solving with Applications. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 99:1-99:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{vanapeldoorn_et_al:LIPIcs.ICALP.2019.99,
  author =	{van Apeldoorn, Joran and Gily\'{e}n, Andr\'{a}s},
  title =	{{Improvements in Quantum SDP-Solving with Applications}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{99:1--99:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.99},
  URN =		{urn:nbn:de:0030-drops-106750},
  doi =		{10.4230/LIPIcs.ICALP.2019.99},
  annote =	{Keywords: quantum algorithms, semidefinite programming, shadow tomography}
}
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