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  1. Free, publicly-accessible full text available June 10, 2025
  2. Free, publicly-accessible full text available June 10, 2025
  3. Tauman_Kalai, Yael (Ed.)
    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. 
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  4. Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Our proof uses elementary counting together with a novel connection to the sunflower lemma. In addition to a simplified proof, our approach opens up a new avenue of attack towards proving lifting theorems with improved gadget size - one of the main challenges in the area. Focusing on one of the most widely used gadgets - the index gadget - existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas. 
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