While seemingly undesirable, it is not a surprising fact that there are certain problems for which quantum computers offer no computational advantage over their respective classical counterparts. Moreover, there are problems for which there is no `useful' computational advantage possible with the current quantum hardware. This situation however can be beneficial if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. In such a situation, we would like to have evidence that it is difficult to solve those problems on quantum computers; but what is their exact complexity? To do so one has to prove lower bounds, but proving unconditional time lower bounds has never been easy. As a result, resorting to conditional lower bounds has been quite popular in the classical community and is gaining momentum in the quantum community. In this paper, by the use of the QSETH framework [Buhrman-Patro-Speelman 2021], we are able to understand the quantum complexity of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT, and also are able to comment on the non-trivial complexity of approximate-#CNFSAT; both of these have interesting implications about the complexity of (variations of) lattice problems, strong simulation and hitting set problem, and more. In the process, we explore the QSETH framework in greater detail than was (required and) discussed in the original paper, thus also serving as a useful guide on how to effectively use the QSETH framework.

翻译：尽管看似不尽人意，但存在某些问题使得量子计算机相较于经典计算机无法提供计算优势，这一事实并不令人意外。此外，对于当前量子硬件而言，有些问题甚至不可能实现“有用的”计算优势。然而，若我们不希望量子计算机快速解决某些问题——例如与后量子密码学相关的问题——这种情形反而可能是有益的。在此类情境下，我们希望获得证据表明这些问题在量子计算机上难以求解；但它们的精确复杂性究竟如何？为此，必须证明下界，然而证明无条件时间下界绝非易事。因此，在经典计算领域中，转向条件性下界的方法颇为流行，并且正在量子计算领域中获得势头。本文通过使用QSETH框架[Buhrman-Patro-Speelman 2021]，我们得以理解CNFSAT的几个自然变体（如奇偶性-CNFSAT或计数-CNFSAT）的量子复杂性，并能就近似-#CNFSAT的非平凡复杂性做出评述；这两者都对（各类变体）格问题、强模拟、击中集问题等的复杂性具有有趣的含义。在此过程中，我们比原始论文中（所需且已讨论的）更详尽地探索了QSETH框架，从而也作为如何有效使用QSETH框架的有用指南。