QSETH Strikes Again Finer Quantum Lower Bounds for Lattice Problem, Strong Simulation, Hitting Set Problem, and More

Despite the wide range of problems for which quantum computers offer a computational advantage over their classical counterparts, there are also many problems for which the best known quantum algorithm provides a speedup that is only quadratic, or even subquadratic. Such a situation could also be desirable if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. When searching for algorithms and when analyzing the security of cryptographic schemes, we would like to have evidence that these problems are difficult to solve on quantum computers; but how do we assess the exact complexity of these problems? For most problems, there are no known ways to directly prove time lower bounds, however it can still be possible to relate the hardness of disparate problems to show conditional lower bounds. This approach has been popular in the classical community, and is being actively developed for the quantum case [1, 15, 14, 7]. In this paper, by the use of the QSETH framework [15] 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 versions of counting-CNFSAT. Without considering such variants, the best quantum lower bounds will always be quadratically lower than the equivalent classical bounds, because of Grover's algorithm; however, we are able to show that quantum algorithms will likely not attain even a quadratic speedup for many problems. These results have implications for the complexity of (variations of) lattice problems, the strong simulation and hitting set problems, and more. In the process, we explore the QSETH framework in greater detail and present a useful guide on how to effectively use the QSETH framework.