paper

Quantum Query Algorithms are Completely Bounded Forms

arXiv:1711.07285 · doi:10.1137/18M117563X

Abstract

We prove a characterization of -query quantum algorithms in terms of the unit ball of a space of degree- polynomials. 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). 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. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. 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. Revision note: A mistake was found in the proof of the second result on degree-4 polynomials far from 2-query quantum algorithms. An explanation of the issue, a corrected proof and stronger examples are presented in work of Escudero Gutiérrez and the second author.

24 pages, 3 figures. v2: 27 pages, minor changes in response to referee comments. v3: addresses an error in a proof and gives a reference for a corrected proof

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