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Lower Bounds for Function Inversion with Quantum Advice

Authors Kai-Min Chung, Tai-Ning Liao, Luowen Qian

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Kai-Min Chung
  • Academia Sinica, Taipei, Taiwan
Tai-Ning Liao
  • National Taiwan University, Taipei, Taiwan
Luowen Qian
  • Boston University, MA, USA


The authors would like to thank Nai-Hui Chia, Luca Trevisan, Xiaodi Wu, and Penghui Yao for their helpful insights during the discussions. We also thank the anonymous reviewers at QIP and ITC for pointing out various issues in the paper.

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Kai-Min Chung, Tai-Ning Liao, and Luowen Qian. Lower Bounds for Function Inversion with Quantum Advice. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 8:1-8:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


Function inversion is the problem that given a random function f: [M] → [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 Grover’s algorithm [Grover, 1996], which does not leverage the power of preprocessing. Nayebi et al. [Nayebi et al., 2015] proved a lower bound ST² ≥ ̃Ω(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Quantum query complexity
  • Cryptanalysis
  • Data Structures
  • Quantum Query Complexity


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