Amplitude amplification and estimation require inverses

TL;DR

This paper demonstrates that quantum speedups for search and counting are only achievable when the process can be efficiently inverted, explaining limitations in quantum algorithms where inversion is hard.

Contribution

It proves that quantum speedups require the ability to invert the process, using trace estimation problems and the compressed oracle method to establish this necessity.

Findings

Quantum speedups depend on invertibility of the process.

Without inverse access, quantum algorithms do not outperform classical ones.

Inversion of the process is crucial for amplitude amplification and estimation.

Abstract

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.