Some error analysis for the quantum phase estimation algorithms
Some error analysis for the quantum phase estimation algorithms
复制标题
DOI：
10.1088/1751-8121/ac7f6c
复制
发表时间：
2021-11
期刊：
影响因子：
--
通讯作者：
Xiantao Li
中科院分区：
文献类型：
--
作者：
Xiantao Li
本文研究量子计算中的相位估计算法，特别是（1）输入向量不是本征向量;（2）幺正算符用Trotter或Taylor展开方法近似;（3）幺正算符用随机近似的情形。我们的特点的概率计算相位值的一致性误差，包括残差，特罗特分裂误差，或统计均方误差。在前两种情况下，我们证明，为了获得误差小于或等于2−n且概率至少为1 − 1的相位值，所需的量子位数为t <$n+ log 2 +δ22 <$ΔE2。参数δ量化了与不精确特征向量和/或酉算子相关联的误差，并且ΔE表征了谱间隙，即，与其余相位值的分离。该分析通过包括这些效应来概括标准结果（Cleve等1998 Phys. Rev X 11 011020; Nielsen和Chuang 2002 Quantum Computation and Quantum Information）。更重要的是，它表明当δ < ΔE时，复杂度保持不变。对于第三种情况，我们发现了类似的估计，但随机步骤的数量必须足够大。
This paper is concerned with the phase estimation algorithm in quantum computing, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is approximated by Trotter or Taylor expansion methods; (3) random approximations are used for the unitary operator. 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 − ϵ, the required number of qubits is t⩾n+log2+δ22ϵΔE2 . The parameter δ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and ΔE characterizes the spectral gap, i.e., the separation from the rest of the phase values. This analysis generalizes the standard result (Cleve et al 1998 Phys. Rev X 11 011020; Nielsen and Chuang 2002 Quantum Computation and Quantum Information) by including these effects. More importantly, it shows that when δ < ΔE, the complexity remains the same. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.