完全有界多项式的最优不等式及量子查询算法的局限性
Optimal inequalities for completely bounded polynomials and the limitations of quantum query algorithms
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- Quriosity, Inria, France（库里奥西蒂，法国国家信息与自动化研究所）
- Centre de Recerca Matemàtica, Spain（加泰罗尼亚数学研究中心）
- Universidad Complutense de Madrid, Spain（马德里康普顿斯大学）
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中文总结 AI 辅助
该研究通过完全有界多项式方法证明了块多重线性多项式和量子查询算法最高层傅里叶增长的最优不等式，导出了更优的量子查询算法极限定理，部分解决了Girish的问题。
中文摘要 AI 辅助
我们通过完全有界多项式方法研究量子查询算法能力的局限性问题，特别证明了涉及不同完全有界多项式概念的若干最优函数不等式，这些不等式导出了优于先前研究的量子查询算法能力的极限定理。1. 块多重线性多项式的最优根影响界：先前研究表明，t次块多重线性多项式p满足根影响界∥p∥_cb≥∑_i√Inf_i[p]/t²，该界强于Aaronson-Ambainis猜想中的界；我们找到了该不等式的最优常数：∥p∥_cb≥∑_i√Inf_i[p]/t。由于查询输入不相交块的量子算法（如t重forrelation）的振幅是满足∥p∥_cb≤1的块多重线性多项式，我们的不等式表明这些算法满足t≥∑_i√Inf_i[p]。我们证明该不等式既产生了比基于Aaronson-Ambainis论证的先前结果更高效的经典模拟，又实现了定性改进：所有经典查询均为非自适应。2. 量子查询算法最高层的最优傅里叶增长：我们证明，对{-1,1}^n上每个2t次多项式p，其2t层的傅里叶增长∥p̂_{2t}∥_{ℓ₁}满足∥p̂_{2t}∥_{ℓ₁}≤(en/(2t-1))^((2t-1)/2)∥p∥_cb，该界在因子e范围内是最优的，2t重forrelation可作为例证。由于进行t次（对整个输入）查询的量子算法满足∥p∥_cb≤1，这为这些算法提供了傅里叶增长界，部分解决了Girish（STOC,2026）提出的问题。
英文摘要
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).
