Srinivasan ArunachalamJop BriëtPalazuelos Cabezón, Carlos2024-01-292024-01-292019-01Arunachalam, S.; Briët, J.; Palazuelos, C. Quantum Query Algorithms Are Completely Bounded Forms. SIAM Journal on Computing 2019, 48(3), 903–925. doi:10.1137/18M117563X.0097-53971095-711110.1137/18m117563xhttps://hdl.handle.net/20.500.14352/95983We 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. [``Polynomials, Quantum Query Complexity, and Grothendieck's Inequality,” in Proceedings of the 31st Conference on Computational Complexity, CCC 2016, Schloss Dagstuh, 2016, pp. 25:1--25:19]. Our proof is based on a fundamental result of Christensen and Sinclair [J. Funct. Anal., 72 (1987), pp. 151-181] 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.engjournal articlehttps//doi.org/10.1137/18m117563xopen accessQuantum algorithmsQuery complexityPolynomial methodMultilinear formsOperator space theoryAnálisis matemático12 Matemáticas