What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires $\Theta(n)$ queries to an NP oracle to compute a witness to a given SAT formula, quantumly $\Theta(\log n)$ queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires $\Omega(\log n)$ queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires $\Omega(\log n)$ queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making $o(log n)$ NP queries, then $\text{BPP}^{\text{NP}[o(n)]}$ contains a $\text{P}^{\text{NP}}$-complete problem if the algorithm is classical and $\text{FBQP}^{\text{NP}[o(n)]}$ contains an $\text{FP}^{\text{NP}}$-complete problem if the algorithm is quantum.

翻译：多项式时间量子计算在拥有NP预言机访问权限时具有怎样的能力？本文聚焦于布尔可满足性问题研究中的两个基本任务：搜索到决策的转化以及近似计数。我们首先证明，与经典环境下多项式时间图灵机需要向NP预言机进行Θ(n)次查询才能求解给定SAT公式的证据不同，量子环境下仅需Θ(log n)次查询即可完成。随后我们在黑盒模型中证明该结果是紧的——任何具有"类NP"查询访问权限的量子算法，若要以恒定概率提取解，至少需要Ω(log n)次查询。进一步关注SAT解的近似计数问题，通过利用搜索到决策转化与近似计数之间的量子关联，我们发现现有经典近似计数算法可能已达最优。首先，我们在"类NP"黑盒查询场景中给出下界：即使在量子计算机上，近似计数也需要Ω(log n)次查询。随后我们给出白盒下界（即输入公式无需隐藏于预言机中）：若存在多项式时间随机化经典或量子算法能以o(log n)次NP查询完成近似计数，那么当算法为经典时，BPP^NP[o(n)]包含一个P^NP-完全问题；当算法为量子时，FBQP^NP[o(n)]包含一个FP^NP-完全问题。