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.

翻译：本文涉及量子计算算法中的阶段估计算法, 特别是以下两种假设情况:(1) 输入矢量不是电子元;(2) 单体操作员没有完全执行;(3) 单体操作员使用随机近似法, 例如QDRIFT 方法。 我们用一致性错误来描述计算阶段值的概率, 包括残余错误、 Trotter 分裂错误或统计平均方差差错误。 在前两种情况中, 我们显示, 为了以 {eror 小于或等于$2 ⁇ - n}}} 和概率至少1\ epslon$获得阶段值, 单体操作员需要的 qubits数量是 $ t\ geq n +\ log\ big (2 +\ frac kdel%2\\\\\ epsilon\\\!\ Delta\!\! 在前两种情况中, 我们的参数 $delta$ 量化了与exact eigenctor 和 i/ or slon 等值相关的错误, 。