Classical Simulation of One-Query Quantum Distinguishers

Authors Andrej Bogdanov, Tsun Ming Cheung, Krishnamoorthy Dinesh, John C. S. Lui

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Author Details

Andrej Bogdanov
  • School of EECS, University of Ottawa, Canada
Tsun Ming Cheung
  • School of Computer Science, McGill University, Montreal, Canada
Krishnamoorthy Dinesh
  • Dept. of Computer Science and Engineering, Indian Institute of Technology, Palakkad, India
John C. S. Lui
  • Dept. of Computer Science and Engineering, Chinese University of Hong Kong, China

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Andrej Bogdanov, Tsun Ming Cheung, Krishnamoorthy Dinesh, and John C. S. Lui. Classical Simulation of One-Query Quantum Distinguishers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the relative advantage of classical and quantum distinguishers of bounded query complexity over n-bit strings, focusing on the case of a single quantum query. A construction of Aaronson and Ambainis (STOC 2015) yields a pair of distributions that is ε-distinguishable by a one-query quantum algorithm, but O(ε k/√n)-indistinguishable by any non-adaptive k-query classical algorithm. We show that every pair of distributions that is ε-distinguishable by a one-query quantum algorithm is distinguishable with k classical queries and (1) advantage min{Ω(ε√{k/n})), Ω(ε²k²/n)} non-adaptively (i.e., in one round), and (2) advantage Ω(ε²k/√{n log n}) in two rounds. As part of our analysis, we introduce a general method for converting unbiased estimators into distinguishers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Query complexity
  • quantum algorithms
  • hypothesis testing
  • Grothendieck’s inequality


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