Optimal inequalities for completely bounded polynomials and the limitations of quantum query algorithms

2026-09-04

AI Summary

本文通过完全有界多项式方法，证明了几个最优函数不等式，以限制量子查询算法的能力，超越了先前的研究成果。

Abstract

We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works. 1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials $p$ of degree $t$ satisfy a root-influence bound, $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t^2$, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t$. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as $t$-fold forrelation- are block-multilinear polynomials with $\|p\|_{\text{cb}}\leq 1,$ our inequality shows that they satisfy $t\geq \sum_i\sqrt{\mathrm{Inf}_i[p]}$. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive. 2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial $p$ defined on $\{-1,1\}^n$ of degree $2t$, the Fourier Growth at the level $2t,$ namely $\|\widehat p_{2t}\|_{\ell_1}$, satisfies $\|\widehat p_{2t}\|_{\ell_1}\leq (en/(2t-1))^{\frac{2t-1}{2}}\|p\|_{\text{cb}}$. This is optimal up to the factor $e$, as witnessed by $2t$-fold forrelation. As quantum query algorithms that make $t$ queries (to the whole input) satisfy $\|p\|_{\text{cb}}\leq 1$, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).

Problem

• quantum query algorithms
• completely bounded polynomials
• functional inequalities
• limitations
• Fourier Growth

Innovation

• completely bounded polynomials
• root-influence bound
• quantum query algorithms
• Fourier growth
• nonadaptive classical queries