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DOI: 10.4230/LIPIcs.ITCS.2017.47

The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

Authors: Mathieu Laurière and Dave Touchette

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract

In two-party interactive quantum communication protocols, we study a recently defined notion of quantum information cost (QIC), which has most of the important properties of its classical analogue (IC). Notably, its link with amortized quantum communication complexity has been used to prove an (almost) tight lower bound on the bounded round quantum complexity of Disjointness. However, QIC was defined through a purification of the input state. This is valid for fully quantum inputs and tasks but difficult to interpret even for classical tasks. Also, its link with other notions of information cost that had appeared in the literature was not clear. We settle both these issues: for quantum communication with classical inputs, we characterize QIC in terms of information about the input registers, avoiding any reference to the notion of a purification of the classical input state. We provide an operational interpretation of this new characterization as the sum of the costs of revealing and of forgetting information about the inputs. To obtain this result, we prove a general Information Flow Lemma assessing the transfer of information in general interactive quantum processes. Specializing this lemma to interactive quantum protocols accomplishing classical tasks, we are able to demistify the link between QIC and other previous notions of information cost in quantum protocols. Furthermore, we clarify the link between QIC and IC by simulating quantumly classical protocols. Finally, we apply these concepts to argue that any quantum protocol that does not forget information solves Disjointness on n-bits in Omega(n) communication, completely losing the quadratic quantum speedup. Hence forgetting information is here a necessary feature in order to obtain any significant improvement over classical protocols. We also prove that QIC at 0-error is exactly n for Inner Product, and n (1 - o(1)) for a random Boolean function on n+n bits.

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Mathieu Laurière and Dave Touchette. The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, p. 47:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lauriere_et_al:LIPIcs.ITCS.2017.47,
  author =	{Lauri\`{e}re, Mathieu and Touchette, Dave},
  title =	{{The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{47:1--47:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.47},
  URN =		{urn:nbn:de:0030-drops-81898},
  doi =		{10.4230/LIPIcs.ITCS.2017.47},
  annote =	{Keywords: Communication Complexity, Information Complexity, Quantum Computation and Information}
}

Document

DOI: 10.4230/LIPIcs.CCC.2017.23

Augmented Index and Quantum Streaming Algorithms for DYCK(2)

Authors: Ashwin Nayak and Dave Touchette

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract

We show how two recently developed quantum information theoretic tools can be applied to obtain lower bounds on quantum information complexity. We also develop new tools with potential for broader applicability, and use them to establish a lower bound on the quantum information complexity for the Augmented Index function on an easy distribution. This approach allows us to handle superpositions rather than distributions over inputs, the main technical challenge faced previously. By providing a quantum generalization of the argument of Jain and Nayak [IEEE TIT'14], we leverage this to obtain a lower bound on the space complexity of multi-pass, unidirectional quantum streaming algorithms for the DYCK(2) language.

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Ashwin Nayak and Dave Touchette. Augmented Index and Quantum Streaming Algorithms for DYCK(2). In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{nayak_et_al:LIPIcs.CCC.2017.23,
  author =	{Nayak, Ashwin and Touchette, Dave},
  title =	{{Augmented Index and  Quantum Streaming Algorithms for DYCK(2)}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{23:1--23:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.23},
  URN =		{urn:nbn:de:0030-drops-75437},
  doi =		{10.4230/LIPIcs.CCC.2017.23},
  annote =	{Keywords: Quantum streaming algorithms, Space complexity, Quantum communication complexity, Quantum information cost, DYCK(2), Augmented Index}
}

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The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

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Augmented Index and Quantum Streaming Algorithms for DYCK(2)

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