Cost analysis of quantum phase estimation: quantum Fourier transform versus Hadamard test approaches using Fisher information

Quantum phase estimation (QPE) is indispensable to many quantum algorithms, and numerous implementations have been proposed. However, a systematic understanding of which QPE method minimizes resource costs for a given problem setting is still lacking. Estimation accuracy varies with both the overlap between the input and eigenstates and the separation of the eigenphases, making conventional numerical comparisons inadequate.

In this study, we investigate quantum-Fourier-transform-based QPE (QFT-QPE) and Hadamard-test-based QPE (HT-QPE). By introducing the Finsher information and the Cramér–Rao lower bound (CRLB), we quantify the ultimate precision limit with the eigenphase and overlap as parameters, enabling a fair comparison of resource costs. Our analysis reveals a crossover region: HT-QPE incurs lower cost when the overlap is large, whereas QFT-QPE becomes more economical when the overlap is small. These findings offer guidance for selecting QPE algorithms according to problem characteristics and will facilitate further advancements in QPE techniques.