Grover's search algorithm is renowned for its dramatic speedup in solving many important scientific problems. The recently proposed Variational Quantum Search (VQS) algorithm has shown an exponential advantage over Grover's algorithm for up to 26 qubits. However, its advantage for larger numbers of qubits has not yet been proven. Here we show that the exponentially deep circuit required by Grover's algorithm can be replaced by a multi-controlled NOT gate together with either a single layer of Ry gates or two layers of circuits consisting of Hadamard and NOT gates, which is valid for any number of qubits greater than five. We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near-perfect reachability in finding the good element of an arbitrarily large unstructured data set, and its reachability exponentially improves with the number of qubits, where the reachability is defined to quantify the ability of a given Ansatz to generate an optimal quantum state. Numerical studies further validate the excellent reachability of the VQS. Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of qubits completes an essential step in proving its exponential advantage over Grover's algorithm for any number of qubits, and the latter proving is significant as it means that the VQS can efficiently solve NP-complete problems.

翻译：Grover搜索算法因能显著加速解决许多重要科学问题而闻名。最近提出的变分量子搜索（VQS）算法在最多26个量子比特的问题中展现出比Grover算法更优的指数级优势。然而，该算法在更多量子比特情况下的优势尚未得到证实。本文证明，Grover算法所需的指数深度电路可替换为一个多控非门加上单层Ry门或由Hadamard门和NOT门构成的双层电路，且该结论对多于五个量子比特的任何系统均成立。我们证实，采用单层Ry门作为拟设的VQS算法在任意规模的无序数据集中寻找目标元素时具有近乎完美的可达性，且该可达性随量子比特数呈指数级提升。此处，可达性被定义为衡量给定拟设生成最优量子态能力的量化指标。数值研究进一步验证了VQS算法的卓越可达性。证明在任意量子比特数下采用深度-1拟设的VQS算法具有近乎完美的可达性，为证明其对任意量子比特数均优于Grover算法奠定了关键基础。后者若得以证明，则意味着VQS算法能有效解决NP完全问题，具有重大意义。