发表机构
Université Paris-Cité, CNRS, IRIF; University of Latvia, Faculty of Science and Technology（巴黎西岱大学，法国国家科学研究中心，信息研究与研究所; 拉脱维亚大学，理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出操作性证书概念，用于刻画布尔函数在零误差和精确量子查询模型下的证书复杂度，并给出新下界，展示了一个函数使证书复杂度对Q0紧而下界优于有理度。
AI 中文摘要
证书复杂度 $(C(f))$ 是衡量布尔函数 $f$ 复杂性的一个基本指标，它统计为了确定函数值而需要知道的输入位数。证书可以看作是一个部分赋值，或者是一个函数为常数的布尔子立方体。证书复杂度在确定性查询（或决策树）复杂度 $(D)$ 以及其他查询模型（如有界误差随机和量子复杂度 $(R, Q)$）中已被充分理解，但在量子零误差 $(Q_0)$ 和精确查询复杂度 $(Q_E)$ 中则不然，因为对于这些模型（甚至对于 $Q$）没有公认的证书“对象”。相反，我们研究了一种操作性的证书概念，并将其应用于各种基于查询的模型，重点关注零误差和精确量子查询复杂度，同时也关注多项式度度量。我们给出了 $C, RC$（随机证书复杂度）和 $QC$（量子证书复杂度）的新刻画，这些刻画基于各种度量，如经典和量子破坏复杂度、无歧义证书复杂度以及多项式度的变体。证书复杂度还为 $Q_E$ 和 $Q_0$ 提供了新的下界，$Q_E$ 和 $Q_0$ 是 $D$ 和 $R_0$ 的量子类比，对于这些复杂度度量，已知的下界技术很少，而这些技术并非已经是双侧误差量子查询复杂度的下界。我们展示了一个全布尔函数，对于该函数，我们的证书复杂度度量给出了 $Q_0$ 的紧下界，但有理度数和 $Q$ 渐近更小。
英文摘要
Certificate complexity $(C(f ))$ is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity $(D)$ and other query models such as bounded-error randomized and quantum complexity $(R, Q)$, but not as well for quantum zero-error $(Q_0)$ and exact query complexity $(Q_E)$, where there is no agreed-upon certificate 'object' (even for $Q$). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of $C, RC$ (randomized certificate complexity) and $QC$ (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on $Q_E$ and $Q_0$, the quantum analogues of $D$ and $R_0$, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for $Q_0$, but rational degree and $Q$ are asymptotically smaller.
