使用量子元的精确和定点格罗弗搜索
Exact and Fixed-Point Grover Search with Qudits
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中文总结 AI 辅助
针对利用量子元的量子平台，提出基于量子元的格罗弗搜索统一框架，构造预言机和扩散算子，推广搜索变体，分析相位匹配技术并提供电路分解，比较布洛赫球轨迹，为量子计算和传感应用提供实用工具包。
中文摘要 AI 辅助
格罗弗算法为搜索非结构化数据库提供了二次加速，传统上是在维度为2的幂的希尔伯特空间中用量子比特实现的。随着利用量子元（具有两个以上能级的量子系统）的量子平台的出现，需要将格罗弗搜索推广到这些架构，包括具有不同维度量子元的异构系统。在此，我们提出了一个基于量子元的格罗弗搜索统一框架，详细说明了有无辅助量子比特的预言机和扩散算子的构造，并推广了确定性和定点搜索变体，以确保精确或有界的成功概率。我们分析了相位匹配技术，并提供了适用于不同硬件平台的显式电路分解。我们还比较了布洛赫球上的相应轨迹，以直观地展示不同相位选择如何放大目标状态。这些结果有助于在量子元处理器上实现格罗弗搜索的灵活、面向硬件的协议，可能减少电路深度并提高成功概率，从而为利用多能级量子系统的量子计算和传感应用提供实用工具包。
英文摘要
Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.
