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Quantum Pseudorandomness and Classical Complexity

Author William Kretschmer

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William Kretschmer
  • University of Texas at Austin, TX, USA


Thanks to Scott Aaronson for suggestions on the writing, Adam Bouland for insightful discussions, and Qipeng Liu for clarifying some questions about [Kai-Min Chung et al., 2020].

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William Kretschmer. Quantum Pseudorandomness and Classical Complexity. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 2:1-2:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We construct a quantum oracle relative to which BQP = QMA but cryptographic pseudorandom quantum states and pseudorandom unitary transformations exist, a counterintuitive result in light of the fact that pseudorandom states can be "broken" by quantum Merlin-Arthur adversaries. We explain how this nuance arises as the result of a distinction between algorithms that operate on quantum and classical inputs. On the other hand, we show that some computational complexity assumption is needed to construct pseudorandom states, by proving that pseudorandom states do not exist if BQP = PP. We discuss implications of these results for cryptography, complexity theory, and quantum tomography.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • pseudorandom quantum states
  • quantum Merlin-Arthur


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