Some Error Analysis for the Quantum Phase Estimation Algorithms

This paper is concerned with the phase estimation algorithm in quantum computing algorithms, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is not exactly implemented; (3) random approximations are used for the unitary operator, e.g., the QDRIFT method. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with {error less or equal to $2^{-n}$ } and probability at least $1-\epsilon$, the required number of qubits is $t \geq n + \log \big(2 + \frac{\delta^2 }{2 \epsilon \Delta\!E^2 } \big).$ The parameter $\delta$ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and $\Delta\! E$ characterizes the spectral gap, i.e., the separation from the rest of the phase values. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.