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.

Approximate degree, Operator space theory, Polynomial method, Quantum query algorithms
Innovations in Theoretical Computer Science
Quantum and classical data transmission
This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/639.071.409 - Quantum and classical data transmission
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Arunachalam, S, Briët, J, & Palazuelos, C. (2018). Quantum query algorithms are completely bounded forms. In Leibniz International Proceedings in Informatics, LIPIcs (pp. 3:1–3:21). doi:10.4230/LIPIcs.ITCS.2018.3