AI 中文总结
该研究提出了广义高效的量子电路框架，实现任意维度凯莱图上的离散时间量子行走，通过移位算子分解降低CNOT成本，为近期量子设备的实用DTQW提供硬件友好的可扩展途径。
AI 中文摘要
我们提出了一种广义且高效的量子电路框架，用于实现任意维度凯莱图上的离散时间量子行走（DTQWs）。基于Razzoli等人的边界量子傅里叶变换（Boundary QFT）方案，我们针对一维凯莱图的三类生成集（无对合的逆闭生成集、含对合的逆闭生成集、非逆闭生成集），引入了移位算子的系统性多阶段分解方法。该分解将经QFT对角化的移位算子分层分解为结构化块组件，逐步降低所需旋转门的控制位数，并将高阶多量子比特受控操作替换为低阶等效操作的集合。我们将该构造扩展至d维环面图，并以8-凯莱图和ℤ₁₆×ℤ₈环面图为例提供了明确的电路实现。利用Rosa等人的线性CNOT缩放方法进行的门复杂度分析表明，在逆闭生成集的生成集度k≤64、非逆闭生成集的k≤16的范围内，分解后的实现相较于 naive 实现的上界CNOT成本大幅降低。基准测试进一步显示，该效率增益对系统规模N基本不敏感，生成集度k是移位算子的主导资源参数。这些结果为近期量子设备上实用DTQW的实现提供了可扩展且硬件友好的途径。
英文摘要
We present a generalized and efficient quantum circuit framework for implementing discrete-time quantum walks (DTQWs) on Cayley graphs of arbitrary dimension. Building on the Boundary QFT scheme of Razzoli et al., we introduce a systematic multi-stage decomposition of the shift operator for 1D Cayley graphs across three classes of generating sets: inverse-closed without involutions, inverse-closed with an involution, and non-inverse-closed. The decomposition hierarchically factorizes the QFT-diagonalized shift operator into structured block components, progressively reducing the control degree of the required rotation gates and replacing high-degree multi-qubit controlled operations with collections of lower-degree equivalents. We extend this construction to $d$-dimensional torus graphs and provide explicit circuit implementations for an 8-Cayley graph and a $\mathbb{Z}_{16} \times \mathbb{Z}_8$ torus graph as concrete illustrations. Gate complexity analysis using the linear CNOT scaling of Rosa et al. demonstrates that the decomposed implementation achieves a substantial reduction in upper-bound CNOT cost relative to the naive implementation within the regime $k \leq 64$ for inverse-closed graphs and $k \leq 16$ for non-inverse-closed graphs, where $k$ denotes the degree of the generating set. Benchmarking further reveals that this efficiency gain is largely insensitive to the system size $N$, identifying $k$ as the dominant resource parameter for the shift operator. These results provide a scalable and hardware-conscious pathway toward practical DTQW implementations on near-term quantum devices.
Comments40 pages
