AI 中文总结
该研究提出基于d维晶格离散时间量子行走的量子空间搜索显式电路，通过特定编码和操作实现移位，验证其在周期晶格上的动力学，扩展到非周期边界等系统，分析资源并验证噪声鲁棒性，为量子空间搜索提供实用框架。
AI 中文摘要
我们提出了一种基于d维晶格上离散时间量子行走的量子空间搜索显式量子电路。在该算法中，触发器移位算子将行走者沿选定空间方向移动到相邻位置，并在移动后反转相应方向标签。通过对每对相反方向进行编码，使其仅在硬币寄存器的最低有效量子比特上不同，我们使用对位置寄存器的硬币控制模增量和减量操作以及一个反转方向标签的单X门来实现移位。我们验证了所提出的电路在二维和三维周期晶格上再现了理论动力学。我们进一步将电路构造扩展到具有位置依赖移位规则的系统，如非周期边界，并在二维晶格上验证了这种扩展。资源分析表明电路宽度呈对数增长。对于二维晶格，合成深度与\(O(\sqrt{N}(\log N)^{3/2})\)一致，而三维深度在研究范围内经验上遵循\(O(\sqrt{N})\)依赖关系。在CX门去极化噪声模型下，CX优化电路表现出更好的噪声鲁棒性。这些结果为在规则晶格上实现量子空间搜索并将其扩展到有缺陷和其他不规则晶格结构提供了一个实用框架。
英文摘要
We propose an explicit quantum circuit for quantum spatial search based on discrete-time quantum walks on $d$-dimensional lattices. In this algorithm, the flip-flop shift operator moves the walker to a neighboring site along the selected spatial direction and reverses the corresponding direction label after the move. By encoding each pair of opposite directions so that they differ only in the least significant qubit of the coin register, we implement the shift using coin-controlled modular increment and decrement operations on the position registers, together with a single $X$ gate that reverses the direction label. We verify that the proposed circuits reproduce the theoretical dynamics on two- and three-dimensional periodic lattices. We further extend the circuit construction to systems with position-dependent shift rules, such as non-periodic boundaries, and validate this extension on a two-dimensional lattice. The resource analysis shows logarithmic growth of the circuit width. For the two-dimensional lattice, the synthesized depth is consistent with $O(\sqrt{N}(\log N)^{3/2})$, while the three-dimensional depth empirically follows an $O(\sqrt{N})$ dependence over the investigated range. Under a CX-gate depolarizing noise model, CX-optimized circuits exhibit improved noise robustness. These results provide a practical framework for implementing quantum spatial search on regular lattices and extending it to defective and other irregular lattice structures.
Comments9 pages, 7 figures
