Quantum Query Algorithms are Completely Bounded Forms

TL;DR

This paper characterizes quantum query algorithms using degree-2t polynomials and introduces a refined polynomial degree measure that matches quantum query complexity, providing new insights into the structure of quantum algorithms and their polynomial representations.

Contribution

It offers a polynomial characterization of t-query quantum algorithms and refines the approximate polynomial degree concept to exactly match quantum query complexity.

Findings

Many degree-4 polynomials are far from those generated by 2-query algorithms.

One-query quantum algorithms are equivalent to bounded quadratic polynomials.

A new polynomial characterization simplifies understanding quantum query complexity.

Abstract

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. (CCC'16). 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. 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. Revision note: A mistake was found in the proof of the second result on degree-4 polynomials far from 2-query quantum algorithms. An explanation of the issue, a corrected proof and stronger examples are presented in work of Escudero Guti\'errez and the second author.