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.
@InProceedings{arunachalam_et_al:LIPIcs.ITCS.2018.3, author = {Arunachalam, Srinivasan and Bri\"{e}t, Jop and Palazuelos, Carlos}, title = {{Quantum Query Algorithms are Completely Bounded Forms}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {3:1--3:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.3}, URN = {urn:nbn:de:0030-drops-83383}, doi = {10.4230/LIPIcs.ITCS.2018.3}, annote = {Keywords: Quantum query algorithms, operator space theory, polynomial method, approximate degree.} }
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