Amplitude amplification and estimation require inverses

Understand

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.