相位估计的傅里叶分析法
Fourier Analytic Approach to Phase Estimation
摘要：
为了对相位估计进行统一的分析，我们着重讨论了极限分布。证明了极限分布可由支集属于$[-1,1]$的$L^2$函数的傅里叶变换的绝对平方给出。利用这个关系，我们研究了极限分布的方差与其尾概率之间的关系。作为我们的结果，我们证明了最小化渐近方差的协议不最小化尾概率。根据区间的宽度，我们推导出最小化尾概率的估计协议。这种最优方案由常出现在小波或时间限制傅立叶分析中的长椭球波函数给出。在区间估计的框架下，得到了保证给定置信系数的最小置信区间。
英文摘要：
For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.
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