The quadratic reduction of query complexity of Grover’s search algorithm (GA), while significant, would not be enough to enjoy exponentially fast data searching in large-scale quantum computation. One of the ways to enhance the speedup in the framework of Grover’s algorithm is to employ a novel quantum operation, i.e., inversion against an unknown state; however, this is not possible at least in quantum theory. We thus extend the Grover algorithm assisted by closed timelike curves (CTCs), in which the unknown-state inversion is achievable by combining the superposition of two unknown states with cloning. We dubbed this refined algorithm CTC-assisted Grover algorithm (CTC-GA). We show that the CTC-GA can vastly reduce the query complexity compared to the original algorithm; remarkably, from polynomial to poly-logarithmic. These results will provide a broader intuition for exponential quantum speedup of data searching problems.

中文翻译：

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Grover搜索算法和闭合的类似时间的曲线的加速

虽然格罗弗搜索算法（GA）的查询复杂度二次降低，但意义不大，不足以在大规模量子计算中享受指数级快速数据搜索。在格罗弗算法的框架内提高速度的一种方法是采用一种新颖的量子运算，即对未知状态的反演。但是，这至少在量子理论中是不可能的。因此，我们借助封闭的类似时间的曲线（CTC）扩展了Grover算法，其中可以通过将两个未知状态的叠加与克隆相结合来实现未知状态的反转。我们称这种改进的算法为CTC辅助的Grover算法（CTC-GA）。我们证明，与原始算法相比，CTC-GA可以大大降低查询复杂度；从多项式到多元对数。