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Testing Classical Properties from Quantum Data

Authors Matthias C. Caro , Preksha Naik , Joseph Slote

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum Property Testing
  • Quantum Data
  • Boolean Functions

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Abstract

Many properties of Boolean functions can be tested far more efficiently than the function itself can be learned. However, this dramatic advantage often disappears when testers are limited to random samples of f instead of adaptively chosen queries to f. In this work we investigate the quantum version of this restriction: quantum algorithms that test properties of a Boolean function f solely from copies of either the function state |f⟩∝ ∑_x|x,f(x)⟩ or the phase state |(-1)^f⟩∝ ∑_x (-1)^{f(x)}|x⟩.
Quantum advantage in testing from data. For monotonicity, symmetry, and triangle-freeness, we show passive quantum testers are unboundedly or super-polynomially better than their classical passive testing counterparts. They are competitive with classic query-based testers in each case.
Inadequacy of Fourier sampling. Our new testers use techniques beyond quantum Fourier sampling, and it turns out this is necessary: we show a certain class of bent functions can be tested from 𝒪(1) function states but has a sample complexity lower bound of 2^{Ω(n)} for any tester relying exclusively on Fourier and classical samples.
Classical queries vs. quantum data. Our passive quantum testers are competitive with classical query-based testers, but this isn't universal: we exhibit a testing problem that can be solved from 𝒪(1) classical queries but requires Ω(2^{n/2}) function state copies. The Forrelation problem provides a separation of the same magnitude in the opposite direction, so we conclude that quantum data and classical queries are "maximally incomparable" resources for testing.
Towards lower bounds. We also begin the study of lower bounds for testing from quantum data. For quantum monotonicity testing, we prove that the ensembles of [Goldreich et al., 2000; Black, 2024], which give exponential lower bounds for classical sample-based testing, do not yield any nontrivial lower bounds for testing from quantum data. New insights specific to quantum data will be required for proving copy complexity lower bounds for testing in this model.

Cite As Get BibTex

Matthias C. Caro, Preksha Naik, and Joseph Slote. Testing Classical Properties from Quantum Data. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 34:1-34:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.34

Author Details

Matthias C. Caro
  • University of Warwick, Coventry, UK
  • Freie Universität Berlin, Germany
Preksha Naik
  • California Institute of Technology, Pasadena, CA, USA
Joseph Slote
  • California Institute of Technology, Pasadena, CA, USA

Funding

Part of this work was completed while JS was visiting the Simons Institute for the Theory of Computing, supported by DOE QSA grant #FP00010905.
  • Caro, Matthias C.: MCC was partially supported by a DAAD PRIME fellowship and by the BMBF (QPIC-1).
  • Slote, Joseph: JS is funded by Chris Umans' Simons Foundation Investigator Grant.

Acknowledgements

We thank Srinivasan Arunachalam, Fernando Jeronimo, Kyle Gulshen, Jiaqing Jiang, John Bostanci, Yeongwoo Hwang, Shivam Nadimpali, John Preskill, Akshar Ramkumar, Abdulrahman Sahmoud, Mehdi Soleimanifar, and Thomas Vidick for discussions. Moreover, we thank Nathan Harms for suggesting we look at monotonicity and discussions on classical bounds for related problems. We thank Fermi Ma for helpful discussions that provided inspiration for Theorem 12. We are grateful to Francisco Escudero Gutiérrez for pointing us to the Fourier-analytic characterization of monotonicity which allowed us to improve a previous passive quantum monotonicity tester. Finally, we thank the anonymous STOC 2025 reviewers for helpful feedback.

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