We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-(2t) 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.

Additional Metadata
Keywords Quantum algorithms, Query complexity, Multilinear forms, C* -algebras, Operator space theory, Polynomial method
Persistent URL dx.doi.org/10.1137/18M117563X
Journal SIAM Journal on Computing
Project Quantum and classical data transmission
Grant This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/639.071.409 - Quantum and classical data transmission
Citation
Arunachalam, S, Briët, J, & Palazuelos, C. (2019). Quantum query algorithms are completely bounded forms. SIAM Journal on Computing, 48(3), 903–925. doi:10.1137/18M117563X