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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

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.

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)

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@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.} }

Document

**Published in:** LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

We study how generic is the property of nonlocality among the set of quantum correlations for bipartite dichotomic measurements. To do so, we consider the characterization of these quantum correlations as those of the form gamma = ( < u_i , v_j > )_{i,j=1}^n , where the vectors u_i and v_j are in the unit sphere of a real Hilbert space. The important parameters in this description are the number of vectors n and the dimension of the Hilbert space m. Thus, it is natural to study the probability of a quantum correlation being nonlocal as a function of alpha = m/n , where the previous vectors are independent and uniformly distributed in the unit sphere of R^m. In this situation, our main result shows the existence of two completely different regimes: There exists an alpha_0 > 0 such that if alpha leq alpha_0, then gamma is nonlocal with probability tending to 1 as n rightarrow infty. On the other hand, if alpha geq 2 then gamma is local with probability tending to 1 as n rightarrow infty.

Carlos E. González-Guillén, C. Hugo Jiménez, Carlos Palazuelos, and Ignacio Villanueva. How Many Quantum Correlations Are Not Local?. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 39-47, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{gonzalezguillen_et_al:LIPIcs.TQC.2015.39, author = {Gonz\'{a}lez-Guill\'{e}n, Carlos E. and Jim\'{e}nez, C. Hugo and Palazuelos, Carlos and Villanueva, Ignacio}, title = {{How Many Quantum Correlations Are Not Local?}}, booktitle = {10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)}, pages = {39--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-96-5}, ISSN = {1868-8969}, year = {2015}, volume = {44}, editor = {Beigi, Salman and K\"{o}nig, Robert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.39}, URN = {urn:nbn:de:0030-drops-55475}, doi = {10.4230/LIPIcs.TQC.2015.39}, annote = {Keywords: nonlocality, quantum correlations, Bell inequalities, random matrices} }

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