文章基本信息

标题：Lower Bounds for Function Inversion with Quantum Advice
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作者：Kai-Min Chung ; Tai-Ning Liao ; Luowen Qian 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：163
页码：8:1-8:15
DOI：10.4230/LIPIcs.ITC.2020.8
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Function inversion is the problem that given a random function f: [M] â' [N], we want to find pre-image of any image f^{-1}(y) in time T. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size S that only depends on f but not on y. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Groverâs algorithm [Grover, 1996], which does not leverage the power of preprocessing. Nayebi et al. [Nayebi et al., 2015] proved a lower bound STÂ² â¥ ÌfÎ©(N) for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. [Minki Hhan et al., 2019] subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where M = O(N). In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al. [Ambainis et al., 1999], to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.
关键词：Cryptanalysis; Data Structures; Quantum Query Complexity