Fractional Certificates for Bounded Functions

Authors Shachar Lovett , Jiapeng Zhang

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Shachar Lovett
  • Department of Computer Science and Engineering, University of California San Diego, CA, USA
Jiapeng Zhang
  • Department of Computer Science, University of Southern California, Los Angeles, CA, USA

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Shachar Lovett and Jiapeng Zhang. Fractional Certificates for Bounded Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold more generally for any bounded function computed by a low degree polynomial. In this work we prove two new results towards establishing this conjecture: first, that any such polynomial has a small fractional certificate complexity; and second, that many inputs have a small sensitive block. We show that these would imply the Aaronson and Ambainis conjecture, assuming a conjectured extension of Talagrand’s concentration inequality. On the technical side, many classical techniques used in the analysis of Boolean functions seem to fail when applied to bounded functions. Here, we develop a new technique, based on a mix of combinatorics, analysis and geometry, and which in part extends a recent technique of Knop et al. (STOC 2021) to bounded functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
  • Aaronson-Ambainis conjecture
  • fractional block sensitivity
  • Talagrand inequality


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