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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

Funding

  • Bansal, Nikhil: Supported in part by the NWO VICI grant 639.023.812.
  • Sinha, Makrand: Supported by a Simons-Berkeley postdoctoral fellowship.
  • de Wolf, Ronald: Partially supported by the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037), and through QuantERA ERA-NET Cofund project QuantAlgo (680-91-034).

Acknowledgements

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

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.CCC.2022.28

Abstract

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
Keywords
  • Aaronson-Ambainis conjecture
  • Quantum query complexity
  • Classical query complexity
  • Free probability
  • Completely bounded norm
  • Analysis of Boolean functions
  • Influence

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