A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP $\subseteq$ QIP(2).

中文翻译：

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量子约简下基于NP难问题的单向排列

复杂性理论的一个基本追求是将最坏情况的问题简化为平均情况的问题。存在诸如 PSPACE 之类的复杂性类别，它们承认最坏情况到平均情况的减少。然而，对于诸如 NP 之类的许多其他类，到目前为止的证据通常是否定的，因为这种减少的存在会导致多项式层次结构 (PH) 的崩溃。将密码原语（例如，逆向单向排列的平均情况硬度）基于 NP 完整性是一个特别有趣的例子。由于有证据表明从 NP 难问题到破坏这些原语的经典归约会导致 PH 崩溃，因此似乎不太可能将密码原语建立在 NP 难问题上。尽管如此，这些结果并不排除存在量子还原的可能性。在这项工作中，我们开始研究这些问题的量子类似物。除了形式化量子归约的基本概念和通过分离的例子证明量子归约的能力之外，我们的主要结果表明，如果 NP 完全问题归结为使用某些类型的量子归约反转单向排列，那么 coNP $\subseteq$ QIP (2).