When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1801
URN: urn:nbn:de:0030-drops-18014
URL: http://drops.dagstuhl.de/opus/volltexte/2009/1801/
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### Quantum Query Complexity of Multilinear Identity Testing

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### Abstract

Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.

### BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2009:1801,
author =	{Vikraman Arvind and Partha Mukhopadhyay},
title =	{{Quantum Query Complexity of Multilinear Identity Testing}},
booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
pages =	{87--98},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-09-5},
ISSN =	{1868-8969},
year =	{2009},
volume =	{3},
editor =	{Susanne Albers and Jean-Yves Marion},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},