We prove that the generic quantum speedups for brute-force search and counting only hold when the process we apply them to can be efficiently inverted. The algorithms speeding up these problems, amplitude amplification and amplitude estimation, assume the ability to apply a state preparation unitary $U$ and its inverse $U^\dagger$; we give problem instances based on trace estimation where no algorithm which uses only $U$ beats the naive, quadratically slower approach. Our proof of this is simple and goes through the compressed oracle method introduced by Zhandry. Since these two subroutines are responsible for the ubiquity of the quadratic "Grover" speedup in quantum algorithms, our result explains why such speedups are far harder to come by in the settings of quantum learning, metrology, and sensing. In these settings, $U$ models the evolution of an experimental system, so implementing $U^\dagger$ can be much harder -- tantamount to reversing time within the system. Our result suggests a dichotomy: without inverse access, quantum speedups are scarce; with it, quantum speedups abound.

翻译：我们证明，通用量子加速在暴力搜索和计数问题中有效的前提是，所应用的流程能够被高效地逆向操作。加速这些问题的核心算法——振幅放大与振幅估计——假设我们能够执行量子态制备酉算子$U$及其逆算子$U^\dagger$。我们基于迹估计构造了问题实例，其中仅使用$U$的算法无法超越经典朴素方法的二次加速上限。该证明简洁明了，利用了Zhandry提出的压缩预言方法。由于这两个子程序是量子算法中二次"Grover加速"广泛存在的根源，我们的结果解释了为何在量子学习、计量学和传感等场景中这类加速难以实现——在这些场景中，$U$模拟实验系统的演化过程，而实现$U^\dagger$相当于逆转系统时间，其难度远高于正向操作。本研究揭示了一个二分性结论：缺乏逆算子访问能力时，量子加速难得一见；而具备该能力时，量子加速将无处不在。