In this paper, we describe an alternative circuit implementation for the Grover search algorithm by replacing the amplitude amplification part with a non-unitary gate which can be implemented by using an additional ancilla register. We show that the final quantum state in the Grover search algorithm is the normalized marked quantum state in the Gram-Schmidt process. Therefore, one can try to generate this vector by using a non-unitary gate or an approximation of this non-unitary gate. Since we still use the marking part of the original algorithm, $U_{mark}$, the complexity of the algorithm is bounded by the complexity of this operator. We discuss how the implementation of the non-unitary may not be easy task and show the approximations to this operator e.g. through linear combination of unitary matrices or similar methods. Finally we discuss, how these approximations may change the complexity.

翻译：本文提出了一种Grover搜索算法的替代电路实现方案，其核心思想是将振幅放大部分替换为可通过辅助寄存器实现的非酉门。我们证明Grover搜索算法中的最终量子态即为Gram-Schmidt过程中归一化的标记量子态。因此，可以尝试通过非酉门或该非酉门的近似实现来生成该向量。由于算法仍采用原始算法的标记部分$U_{mark}$，其复杂度受该算子复杂度限制。我们讨论了非酉门实现可能存在的困难，并展示了通过酉矩阵线性组合等方法的近似实现方案。最后探讨了这些近似处理对算法复杂度的影响。