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Quantum Query Algorithms are Completely Bounded Forms

Authors Srinivasan Arunachalam, Jop Briët, Carlos Palazuelos

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Srinivasan Arunachalam
Jop Briët
Carlos Palazuelos

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Srinivasan Arunachalam, Jop Briët, and Carlos Palazuelos. Quantum Query Algorithms are Completely Bounded Forms. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ITCS.2018.3

Abstract

We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Using this characterization, we show that many  polynomials of degree at least 4 are far from those coming from quantum query algorithms.
Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms.
We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.

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Keywords
  • Quantum query algorithms
  • operator space theory
  • polynomial method
  • approximate degree.

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