Considering Grover's Search Algorithm (GSA) with the standard diffuser stage applied, we revisit the $3$-qubit unique String Detection Problem (SDP) and extend the algorithm to $4$-qubit SDP with multiple winners. We then investigate unstructured search problems with non-uniform distributions and define the Orthogonal Vector Problem (OVP) under quantum settings. Although no numerically stable results is reached under the original GSA framework, we provide intuition behind our implementation and further observations on OVP. We further perform a special case analysis under the modified GSA framework which aims to stabilize the final measurement under arbitrary initial distribution. Based on the result of the analysis, we generalize the initial condition under which neither the original framework nor the modification works. Instead of utilizing GSA, we also propose a short-depth circuit that can calculate the orthogonal pair for a given vector represented as a binary string with constant runtime.
翻译:考虑到格罗佛的Search Algorithm(GSA)与标准扩散阶段的应用,我们重新审视了3美元平方位独特的字符串探测问题(SDP),并将算法扩大到4美元平方位SDP与多个赢家。然后我们调查非统一分布的无结构搜索问题,并在量子设置下定义Orthogoal矢量问题(OVP)。虽然在原GSA框架下没有达到数字稳定的结果,但我们提供了执行的直觉和对OVP的进一步观察。我们还根据经过修改的GSA框架进行了特别分析,目的是在任意的初步分布下稳定最终测量。根据分析结果,我们概括了原始框架和修改工作都没有起作用的初始条件。我们不使用GSA,还提议了一个短深度的电路,可以计算作为连续运行时间的二联线代表的某一矢量的矢量的矩形对。