文章基本信息

标题：Quantum Lower Bounds for Approximate Counting via Laurent Polynomials
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作者：Scott Aaronson ; Robin Kothari ; William Kretschmer 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：169
页码：7:1-7:47
DOI：10.4230/LIPIcs.CCC.2020.7
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S âS [N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = Î©( S ) or T = Î©(â^S{N/ S }). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA. In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state SâY© = 1/â^S S â^'_{i â^^ S} iâY© - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either Î~(â^S{N/ S }) queries or else Î~(min{ S ^{1/3},â^S{N/ S }}) QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.
关键词：Approximate counting; Laurent polynomials; QSampling; query complexity