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Documents authored by Roland, Jérémie

Document
DOI: 10.4230/LIPIcs.TQC.2024.7
Eigenpath Traversal by Poisson-Distributed Phase Randomisation

Authors: Joseph Cunningham and Jérémie Roland

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
We present a framework for quantum computation, similar to Adiabatic Quantum Computation (AQC), that is based on the quantum Zeno effect. By performing randomised dephasing operations at intervals determined by a Poisson process, we are able to track the eigenspace associated to a particular eigenvalue. We derive a simple differential equation for the fidelity, leading to general theorems bounding the time complexity of a whole class of algorithms. We also use eigenstate filtering to optimise the scaling of the complexity in the error tolerance ε. In many cases the bounds given by our general theorems are optimal, giving a time complexity of O(1/Δ_m) with Δ_m the minimum of the gap. This allows us to prove optimal results using very general features of problems, minimising the problem-specific insight necessary. As two applications of our framework, we obtain optimal scaling for the Grover problem (i.e. O(√N) where N is the database size) and the Quantum Linear System Problem (i.e. O(κlog(1/ε)) where κ is the condition number and ε the error tolerance) by direct applications of our theorems.
Cite as

Joseph Cunningham and Jérémie Roland. Eigenpath Traversal by Poisson-Distributed Phase Randomisation. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cunningham_et_al:LIPIcs.TQC.2024.7,
  author =	{Cunningham, Joseph and Roland, J\'{e}r\'{e}mie},
  title =	{{Eigenpath Traversal by Poisson-Distributed Phase Randomisation}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.7},
  URN =		{urn:nbn:de:0030-drops-206779},
  doi =		{10.4230/LIPIcs.TQC.2024.7},
  annote =	{Keywords: Randomisation method, Non-unitary adiabatic theorems, Grover problem, Quantum linear systems problem}
}
Document
DOI: 10.4230/LIPIcs.TQC.2016.5
Robust Bell Inequalities from Communication Complexity

Authors: Sophie Laplante, Mathieu Laurière, Alexandre Nolin, Jérémie Roland, and Gabriel Senno

Published in: LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)

Abstract
The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bounds. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.
Cite as

Sophie Laplante, Mathieu Laurière, Alexandre Nolin, Jérémie Roland, and Gabriel Senno. Robust Bell Inequalities from Communication Complexity. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{laplante_et_al:LIPIcs.TQC.2016.5,
  author =	{Laplante, Sophie and Lauri\`{e}re, Mathieu and Nolin, Alexandre and Roland, J\'{e}r\'{e}mie and Senno, Gabriel},
  title =	{{Robust Bell Inequalities from Communication Complexity}},
  booktitle =	{11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-019-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{61},
  editor =	{Broadbent, Anne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.5},
  URN =		{urn:nbn:de:0030-drops-66867},
  doi =		{10.4230/LIPIcs.TQC.2016.5},
  annote =	{Keywords: Communication complexity, Bell inequalities, nonlocality, detector efficiency}
}
Document
DOI: 10.4230/LIPIcs.TQC.2015.163
A Universal Adiabatic Quantum Query Algorithm

Authors: Mathieu Brandeho and Jérémie Roland

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

Abstract
Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.
Cite as

Mathieu Brandeho and Jérémie Roland. A Universal Adiabatic Quantum Query Algorithm. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 163-179, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brandeho_et_al:LIPIcs.TQC.2015.163,
  author =	{Brandeho, Mathieu and Roland, J\'{e}r\'{e}mie},
  title =	{{A Universal Adiabatic Quantum Query Algorithm}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{163--179},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.163},
  URN =		{urn:nbn:de:0030-drops-55556},
  doi =		{10.4230/LIPIcs.TQC.2015.163},
  annote =	{Keywords: Quantum Algorithms, Query Complexity, Adiabatic Quantum Computation, Adversary Method}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.434
Explicit relation between all lower bound techniques for quantum query complexity

Authors: Loïck Magnin and Jérémie Roland

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
Cite as

Loïck Magnin and Jérémie Roland. Explicit relation between all lower bound techniques for quantum query complexity. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 434-445, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{magnin_et_al:LIPIcs.STACS.2013.434,
  author =	{Magnin, Lo\"{i}ck and Roland, J\'{e}r\'{e}mie},
  title =	{{Explicit relation between all lower bound techniques for quantum query complexity}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{434--445},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.434},
  URN =		{urn:nbn:de:0030-drops-39548},
  doi =		{10.4230/LIPIcs.STACS.2013.434},
  annote =	{Keywords: Quantum computation, lower bound, adversary method, polynomial method}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2009.2322
Non-Local Box Complexity and Secure Function Evaluation

Authors: Marc Kaplan, Iordanis Kerenidis, Sophie Laplante, and Jérémie Roland

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
A non-local box is an abstract device into which Alice and Bob input bits $x$ and $y$ respectively and receive outputs $a$ and $b$ respectively, where $a,b$ are uniformly distributed and $a \oplus b = x \wedge y$. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function $f$. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We show that non-local box complexity has interesting applications to classical cryptography, in particular to secure function evaluation, and study the question posed by Beimel and Malkin \cite{BM} of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function $f$. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 \nlbs, while no finite bound was previously known.
Cite as

Marc Kaplan, Iordanis Kerenidis, Sophie Laplante, and Jérémie Roland. Non-Local Box Complexity and Secure Function Evaluation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 239-250, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kaplan_et_al:LIPIcs.FSTTCS.2009.2322,
  author =	{Kaplan, Marc and Kerenidis, Iordanis and Laplante, Sophie and Roland, J\'{e}r\'{e}mie},
  title =	{{Non-Local Box Complexity and Secure Function Evaluation}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{239--250},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2322},
  URN =		{urn:nbn:de:0030-drops-23226},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2322},
  annote =	{Keywords: Communication complexity, non-locality, non-local boxes, secure function evaluation}
}
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