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Impossibility of Succinct Quantum Proofs for Collision-Freeness

Author(s)
Aaronson, Scott
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Abstract
We show that any quantum algorithm to decide whether a function f:\left[n\right] \rightarrow\left[ n\right] is a permutation or far from a permutation\ must make \Omega\left( n^{1/3}/w\right) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that \mathsfSZKA\mathsfQMAA , answering an eight-year-old open question of the author. Indeed, we show that relative to some oracle, \mathsfSZK is not in the counting class \mathsfA\mathsf0\mathsfPP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem..
Date issued
2011-01
URI
http://hdl.handle.net/1721.1/72073
Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Journal
Electronic Colloquium on Computational Complexity
Publisher
Hasso-Plattner-Institut für Softwaresystemtechnik GmbH
Citation
Aaronson, Scott. "Impossibility of Succinct Quantum Proofs for Collision-Freeness." Electronic Colloquium on Computational Complexity (2011): TR11-001.
Version: Author's final manuscript
ISSN
1433-8092

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