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.
@InProceedings{bogdanov_et_al:LIPIcs.APPROX/RANDOM.2023.43, author = {Bogdanov, Andrej and Cheung, Tsun Ming and Dinesh, Krishnamoorthy and Lui, John C. S.}, title = {{Classical Simulation of One-Query Quantum Distinguishers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {43:1--43:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.43}, URN = {urn:nbn:de:0030-drops-188684}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.43}, annote = {Keywords: Query complexity, quantum algorithms, hypothesis testing, Grothendieck’s inequality} }
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