Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms

Authors Nikhil Bansal, Makrand Sinha, Ronald de Wolf

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Author Details

Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Makrand Sinha
  • Simons Institute, Berkeley, CA, USA
  • University of California Berkeley, CA, USA
Ronald de Wolf
  • QuSoft, CWI, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands


We thank Scott Aaronson, Srinivasan Arunachalam, Jop Briët, Shachar Lovett and Ryan O'Donnell for helpful comments and pointers to the literature.

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Nikhil Bansal, Makrand Sinha, and Ronald de Wolf. Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every d-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a poly(d)-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-d block-multilinear form with constant variance, there always exists a variable with influence at least 1/poly(d). In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Briët and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance k-fold Forrelation. Our main technical result relies on connections to free probability theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Oracles and decision trees
  • Aaronson-Ambainis conjecture
  • Quantum query complexity
  • Classical query complexity
  • Free probability
  • Completely bounded norm
  • Analysis of Boolean functions
  • Influence


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